Gravitation and Cosmology: From the Hubble Radius to the Planck Scale (Fundamental Theories of Physics)

Gravitation and Cosmology: From the Hubble Radius to the Planck Scale Fundamental Theories of Physics An Internationa...

Author: Richard L. Amoroso | Geoffrey Hunter | Menas Kafatos and Jean-Pierre Vigier

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Gravitation and Cosmology: From the Hubble Radius to the Planck Scale

Fundamental Theories of Physics An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application

Editor: ALWYN VAN DER MERWE, University of Denver, U.S.A.

Editorial Advisory Board: JAMES T. CUSHING, University of Notre Dame, U.S.A. GIANCARLO GHIRARDI, University of Trieste, Italy LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, U.K. CLIVE KILMISTER, University of London, U.K. PEKKA J. LAHTI, University of Turku, Finland ASHER PERES, Israel Institute of Technology, Israel EDUARD PRUGOVECKI, University of Toronto, Canada TONY SUDBURY, University of York, U.K. HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der Wissenschaften, Germany

Volume 126

Gravitation and Cosmology: From the Hubble Radius to the Planck Scale Proceedings of a Symposium in Honour of the 80th Birthday of Jean-Pierre Vigier Edited by

Richard L. Amoroso Noetic Advanced Studies Institute, Orinda, CA, U.S.A.

Geoffrey Hunter York University, Toronto, Canada

Menas Kafatos George Mason University, Fairfax, VA, U.S.A. and

Jean-Pierre Vigier Pierre et Marie Curie Université, Paris, France

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-48052-2 1-4020-0885-6

©2003 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2002 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

http://kluweronline.com http://ebooks.kluweronline.com

Dedicated to Les heretiques de la science The Jean-Pierre Vigier resistant potential struggling within each of us

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TABLE OF CONTENTS

Dedication Foreword Preface Group Photos

v xi xiii xvii

Part I – Astrophysics & Cosmology

1. From the Cosmological Term to the Planck Constant Jose G. Vargas and D. G. Torr 2. Creation of Matter and Anomalous Redshifts Jayant V. Narlikar 3. The Origin of CMBR as Intrinsic Blackbody Cavity-QED Resonance Inherent in the Dynamics of the Continuous State Topology of the Dirac Vacuum Richard L. Amoroso and Jean-Pierre Vigier 4. Some New Results in Theoretical Cosmology Wolfgang Rindler 5. Whitehead Meets Feynman and the Big Bang Geoffrey Chew 6. Developing the Cosmology of a Continuous State Universe Richard L. Amoroso 7. The Problem of Observation in Cosmology and the Big Bang Menas Kafatos 8. Absorber Theory of Radiation in Expanding Universes Jayant V. Narlikar 9. Bohm & Vigier Ideas as a Basis for a Fractal Universe Corneliu Ciubotariu, Viorel Stancu and Ciprian Ciubotariu 10. A Random Walk in a Flat Universe Fotini Pallikari 11. Multiple Scattering Theory in Wolf’s Mechanism and Implications in QSO Redshift Sisir Roy and S. Datta 12. Connections Between Thermodynamics, Statistical Mechanics, Quantum Mechanics, and Special Astrophysical Processes Daniel C. Cole

1 11

27 39 51 59 65 81 85 95

103

111

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TABLE OF CONTENTS Part II - Extended Electromagnetic Theory

13. New Developments in Electromagnetic Field Theory Bo Lehnert 14. Comparison of near and Far Field Double-slit Interferometry for Dispersion in Propagation of the Photon Wave-packet Richard L. Amoroso, Jean-Pierre Vigier, Menas Kafatos & Geoffrey Hunter 15. Photon Diameter Measurements G. Hunter, M. Kowalski, R.Mani, L.P. Wadlinger, F. Engler & T. Richardson 16. What Is the Evans-vigier Field? Valeri V. Dvoeglazov 17. Non-Abelian Gauge Groups for Real and Complex Amended Maxwell’s Equations Elizabeth Rauscher 18. Experimental Evidence of Near-Field Superluminally Propagating Electromagnetic Fields William D. Walker 19. The Photon Spin and Other Topological Features of Classical Electromagnetism Robert M. Kiehn 20. The Process of Photon Emission from Atomic Hydrogen Marian Kowalski 21. Holographic Mind - Overview: The Integration of Seer, Seeing, and Seen Edmond Chouinard 22. Photons from the Future Ralph G. Beil

125

147 157 167

183

189

197 207 223 233

Part III – Gravitation Theory

23. Can One Unify Gravity and Electromagnetic Fields? Jean-Pierre Vigier & Richard L. Amoroso 24. The Dipolar Zero-Modes of Einstein Action Giovanni Modanese 25. Theoretical and Experimental Progress on the GEM (Gravity-Electro-Magnetism) Theory of Field Unification John Brandenburg, J. F. Kline and Vincent Di Pietro 26. Can Gravity Be Included in Grand Unification ? Peter Rowlands and John P. Cullerne 27. Gravitational Energy-Momentum in the Tetrad and Quadratic Spinor Representation of General Relativity Roh S. Tung and James M. Nester

241 259

267 279

287

GRAVITATION AND COSMOLOGY 28. Spinors in Affine Theory of Gravity Horst V. Borzeszkowski and Hans-J. Treder 29. A New Approach to Quantum Gravity, An Overview Sarah B. Bell, John P. Cullerne and Bernard M. Diaz 30. Multidimensional Gravity and Cosmology and Problems of G M.A. Grebeniuk and Vitaly N. Melnikov 31. Quantum Gravity Operators and Nascent Cosmologies Lawrence B. Crowell 32. Gravitational Magnetism: An Update Saul-Paul Sirag

ix

295 303 313 321 331

Part IV - Quantum Theory

33. Quantum Hall Enigmas Malcolm H. Macgreggor 34. On the Possible Existence of Tight Bound States in Quantum Mechanics A. Dragic, Z. Marie & J-P Vigier 35.A Chaotic-stochastic Model of An Atom Corneliu Ciubotariu, Viorel Stancu & Ciprian Ciubotariu 36. Syncronization Versus Simultaneity Relations, with Implications for Interpretations of Quantum Measurements Jose G. Vargas and Douglas G. Torr 37. Can Non-local Interferometry Experiments Reveal a Local Model of Matter? Joao Marto and J. R. Croca 38. Beyond Heisenberg’s Uncertainty Limits Josee R. Croca 39. Towards a Classical Re-interpretation of the Schrodinger Equation According to Stochastic Electrodynamics K. Dechoum, Humberto Franca and C. P. Malta 40.The Philosophy of the Trajectory Representation of Quantum Mechanics Edward R. Floyd 41. Some Physical and Philosophical Problems of Causality in the Interpretation of Quantum Mechanics Bogdan Lange

the Power and the Basic Equations 42. The Force of Quantum Mechanics Ludwik Kostro 43. Progress in Post-Quantum Physics and Unified Field Theory Jack Sarfatti

337 349 357

367 377 385

393 401

409

413 419

x

TABLE OF CONTENTS

Part V –Vacuum Dynamics & Spacetime 44. Polarizable-vacuum Approach to General Relativity Harold E. Puthoff 45. The Inertia Reaction Force and its Vacuum Origin Alfonso Rueda and Bernard Haisch 46. Engineering the Vacuum Trevor Marshall 47. The Photon as a Charge-Neutral and Mass-Neutral Composite Particle Hector A. Munera 48 . Pregeometry Via Uniform Spaces Mark Stuckey and Wyeth Raws 49 . A ZPF-Mediated Cosmological Origin of Electron Inertia Michael Ibison 50 . Vacuum Radiation, Entropy and the Arrow of Time Jean Burns 51. Quaternions, Torsion and the Physical Vacuum: Theories of M. Sachs and G. Shipov Compared David Cyganski and William S. Page 5 2 . Homoloidal Webs, Space Cremona Transformations and the Dimensionality and Signature of Macro-spacetime Metod Saniga 53. Pulse Interaction in Nonlinear Vacuum Electrodynamics A. M. Ignatov & Vladimir Poponin 54. Proposal for Teleportation by Help of Vacuum Holes Constantin Leshan, S. Octeabriscoe and R.L. Singerei 55. Cosmology, the Quantum Universe, and Electron Spin Milo Wolff 56. On Some Implications of the Local Theory Th(G)and of Popper’s Experiment Thomas D. Angelidis

Index

431 447 459 469 477

483 491

499

507 511 515 517

525 537

FOREWORD

Jean-Pierre Vigier continually labeled one of les heretiques de la science, l’eternel resistant et le patriarche is yet a pillar of modern physics and mathematics, with one leg firmly planted in theory and the other in empiricism spanning a career of nearly 60 years with a publication vitae quickly approaching 400! He wrote of his mentor Louis de Broglie “Great physicists fight great battles”, which perhaps applies even more so to Jean-Pierre Vigier himself1. If fortune allows a visit to Paris, reported to be the city of love, and certainly one of the most beautiful and interesting cities in the world; one has been treated to a visual and cultural feast. For example a leisurely stroll from the Musee du Louvre along the Champs-Elysees to the Arc de Triomphe would instill even the least creative soul with the entelechies of a poets muse. It is perhaps open to theoretical interpretation, but if causal conditions have allowed one to be a physicist, visiting Paris, one may have taken opportunity to visit the portion of the old Latin quarter in place Jussieu where Pierre et Marie Curie Universite, reported to be ‘the best university in France’, is stationed. While there at Paris - VI you might have been even more fortunate still to visit professor Jean-Pierre Vigier and meet with him in his office near the department of Gravitation et Cosmologie Relativistes (GCR) where he has an emeritus position. The probability in the relativistic approximation has now approached unity that your de Broglie wave has entered superposition with a certain chair in Vigier’s office; and now causal conditions are such that it is immanent that you will be fortunate enough to settle into de Broglie’s revered chair saved from the years at the Institut Henri Poincare for your conversation with Jean-Pierre Vigier, currently the only living scientist who was a student of de Broglie. Now if all this isn’t too distracting to the heart and soul of the physicist whose daydream vision clears sufficiently during a pensive gaze out the window of this office at the top of the university, one beholds Notre Dame, the Eiffel tower, the Sorbonne and numerous other awe inspiring Parisian landmarks. Habituation to such a panorama would take a concerted effort even for a regular visitor as scrutiny often reveals something unnoticed before. This December we were working a little late in the excitement of having just perused a videotape of cold fusion research arriving that afternoon from a laboratory Belgrade. While we were taking turns making phone calls to Belgium, San Francisco, Belgrade and Texas; I stood gazing out that window again. The Eiffel tower was dressed as a Christmas tree with a flashing light display which will continue every evening for 10 minutes on the hour until the start of the new millennium- Also a fitting tribute to the year of Jean-Pierre’s 80th birthday. 1 A brief biography of Vigier by Stan Jeffers is found in S. Jeffers, B. Lehnert, N. Abramson & L.Chebotarev (eds.) Jean-Pierre Vigier and the Stochastic Interpretation of Quantum Mechanics, 2000, Montreal: Apeiron.

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You may have noticed I have said little so far about J-P the man himself, but only a few things about ‘the emperor’s shimmering clothes’. Oh that I had a thousand pages of ultafine print! If you are interested in similar areas of physical research Professor Vigier’s office is a treasure trove, a tumble of piles of papers and books like Einstein’s office was reported to be. He is at the point in his noted career where everyone sends him ‘stuff, for PLA, for gifts, for his information, for his critique, to support his life’s work etc. etc... Strolling along with him during his daily trip to his mail box in GCR is also an adventure because every day papers, books, videos arrive – lots of them, ‘things’ on the forefront of physics and cosmology. This recursive loop having justifiably achieved an ‘ideal state’ of self-organized superposition is one of the factors helping to maintain Jean-Pierre as an advanced guard soldiering in the pursuit of knowledge of the physical world. At 80+ he is still immeasurably prolific, probably more prolific than most men half his years. I hope his full biography is written as there are a thousand thousand stories that should be told and preserved from his personal, political and scientific life; but he says he wont stand for it, wont use the extensive time required for such a project as long as he can still work effectively on Physics. For example, the well-known incident when Vigier was a young student of de Broglie around age 25. Vigier arriving to meet with de Broglie found the prime minister of France already waiting for some time to discuss the possibility of his membership in the French academy. De Broglie called for Vigier to come in for his usual discussions and proclaimed loudly “as for the Prime Minister, tell him to come back next week!” Finally if one has been most fortunate of all to have had opportunity to work and collaborate with this great man of science, as perhaps most of the authors in this volume have to one degree or another, it’s possible to learn more physics in an hour or a day than gleaned from many months or years past in ambles of ones career. As an octogenarian Vigier is still as sharp and focused as the Einstein Nadelstrahlung that emanate from his penetrating eyes. He must be a genius, not only as evidenced from the quality of the quantity of his published lore; but pose a question and he prattles off authors, dates, books with little pause... All of us thank you Jean-Pierre Vigier for the opportunity to hold this symposium in honor of your 80th birthday and 60 years of physics with 50 years of that time devoted to work on the nature of the photon and aspects of quantum theory. Finally now after such a duration, your work begins to find acceptance in the general physics community. It is a grand inspiration to us all! If [all physicists] follow the same current fashion in expressing and thinking about electrodynamics or field theory, then the variety of hypotheses being generated... is limited. Perhaps rightly so, for possibly the chance is high that the truth lies in the fashionable direction. But, on the off chance that it is in another direction – a direction obvious from an unfashionable view of field theory – who will find it? Only someone who sacrifices himself... from a peculiar and unusual point of view, one may have to invent for himself– Richard Feynman, from the Nobel Prize lecture.

R.L.Amoroso, Paris December, 2000

PREFACE

The Physics of the twentieth century has been dominated by two ideas: the relativity of space and time and the quantization of physical interactions. A key aspect of Relativity is the Principle of Causality, which says that one event (a point in the 4-dimensions of space-time) can only possibly cause another event if their separation in space is not greater than their separation in time (measured as -ict, with c the velocity of light), or in other words that physical interactions cannot travel faster than the speed of light. Relativity is a classical theory in the sense that physical interactions are thought to be essentially localized at points in space and time; i.e. that there is no action at a distance. This theory thus embodies the idea of locality as an essential condition for physical interactions to take place. Yet quantum theory is diametrically opposed to this concept of local realism because of the Heisenberg Uncertainty Principle (HUP) by which the precision with which we can simultaneously determine the position and velocity (momentum) of a particle is limited by the finite value of Planck’s constant of action h. Whether this Principle is merely a limitation on experiments involving interactions, or whether it is intrinsic to Nature has been the subject of an ongoing controversy, with Niels Bohr as the founding proponent of the Copenhagen Interpretation, and such notables as Einstein, de Broglie, Schrodinger, Bohm and Vigier as its opponents. Experiments involving observation of particles that have not interacted with the experimental apparatus, may resolve this question. The theory of relativity grew out of the integration of the theory of electricity and magnetism into Maxwell's equations, culminating in the latter half of the 19th century with the notable inference (now widely applied in modern technology) that light is simply electromagnetic radiation, which travels at a universal speed, the velocity of light. The concept of particle (as energy concentrated at or in the vicinity of a point in space-time) is ingrained in the thinking of almost all physicists perhaps because it is implicit in that foundation of elementary physics, classical mechanics (Newton's equations of motion and their ramifications in Hamiltonian and Langrangian mechanics). The non-classical aspect of Quantum theory is its formulation in terms of wave-like amplitudes for physical processes which can interfere with each other to produce a net intensity, intensities being measurable whereas amplitudes are not. Since quantum theory is essentially wave-like (characterized by interference phenomena), when it is applied to the mechanics of particles it leads to absurdities, the most notable examples being 2-path experiments, whose results are consistent with the idea that the particle travels along both paths simultaneously with the quantum mechanical amplitudes for these alternative routes interfering with each other to produce the observed intensity pattern like that observed in

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PREFACE

Young’s seminal double slit experiment. Such experiments with Photons (Aspect in the 1970s) are not too surprising, because photons are not classical particles (you cannot bring them to rest in the laboratory to measure their intrinsic properties such as mass), and indeed some physicists subscribe to the idea that the very concept of photons is fictitious, and that all photon-like phenomena can be explained by a statistical (stochastic) model of electromagnetic interactions, the quantization being a result of the quantum mechanics of electrons in atoms and molecules, rather than of the light itself. This resistance to the photon concept was present from its inception; it was 18 years after Einstein wrote his 1905 paper proposing" light quanta" as the explanation for the photoelectric effect before the photon concept became accepted by "respectable" physicists, and it took Arthur Compton's discovery of the effect that now bears his name to effect this acceptance. However, 2-path experiments with real particles (i.e. having a rest mass) are not so easily explained away; Zeilinger and his associates working in Vienna have observed singleparticle interference phenomena with neutrons (circa 1985)and most recently (1999) with C60 (Buckyball) molecules. The latter especially are almost macroscopic (the 60 individual atoms can been seen in high-power (electron and scanning-tunneling) microscopes. The idea that a Buckyball molecule can go through 2 slits in a screen simultaneously, and then interfere with itself to produce the observed intensity pattern, is so bizarre as to be ridiculed by any chemist who works with molecular beams (gas-phase chemistry). Yet it is the conventional interpretation of the Zeilinger C60 experiments. This is the great mystery, puzzle and paradox of the quantum mechanics of particles. The majority of professional physicists simply accept the phenomena using quantum theory to predict the results of experiments without being concerned with the logical inconsistency of the concept of a point-particle with the quantum interference phenomena. The discreteness of physical interactions is quantified by the value of Planck's constant, and the physical origin and nature of this discreteness, and what in nature determines the value of Planck's constant, remain elusive questions. The question of whether the fundamental constants (Planck's h, the electric charge on the electron e, the velocity of light c, and the gravitational constant G) may differ in different regions of space and time, must await an answer in terms of our future understanding of their physical origin. Specifically, whether the electron charge is truly fundamental has been brought into question by the quark theory of elementary particle structure and Wheeler’s spacetime wormholes. One physical phenomenon is apparently inconsistent with both the Relativity and Quantum theories; Gravity (regarded as a physical interaction) appears to travel at infinite speed (corroborated by classical mechanical calculations of the motions of celestial bodies assuming that the gravitational interaction is instantaneous over astronomical distances), and while attempts have been made to develop a quantized theory of gravitation, no quantization of gravity has yet been observed. One way out of this dilemma has been to regard gravity as simply the creation of the physical framework (curved space-time) within which all (other) physical phenomena take

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place. This viewpoint that gravity is not a physical interaction allows what are classically regarded as gravitational forces to be consistent with the relativistic principle of causality (that no interaction can travel faster than the speed of light), and it leads to the possibility of an infinite universe, our own observable univererse being limited to the part that is receding from us at ‘observed’ speeds less than the velocity of light. However, the observation of red-shifts not attributable to the receding velocity of the source (non-velocity red-shifts) in recent years, calls into question the cosmological model of an expanding universe, and it may eventually throw light on the nature of gravity. The Physicists who have attended the three Vigier Symposia (at York University in 1995 and 1997, and at the Berkeley Campus of the University of California in 2000) (Planned for Paris in 2003) are, like Jean-Pierre Vigier himself and his mentor Louis de Broglie, committed to the intrinsic logicality of Nature in terms of determinism; as Einstein remarked: "God does not play dice". The lectures presented at the Symposia (and at similar conferences over a period of several decades) are a variety of attempts to resolve the intrinsic paradox of the quantum mechanics of particles and to reconcile the locality intrinsic to Relativity theory with the manifest non-local realism of quantum interference phenomena. This rigorous investigation continues to proceed from the Hubble radius to the Planck scale; and if there ever should be a demise of the bigbang or a handle on the Dirac’s polarizable vacuum- perhaps these investigations will lead us infinitely beyond... Geoffrey Hunter York University, Toronto, Canada Richard L. Amoroso Noetic Advanced Studies Institute, Orinda, USA March, 2001

The organizers gratefully acknowledge generous financial support from: The California Institute of Physics and Astrophysics (CIPA) International Space Sciences Organization (ISSO) The Noetic Advanced Studies Institute

and extend special thanks to: Edmond Chouinard of Measurements Research, Inc. Roh Tung of CIPA for help in preparing the final manuscript!

PREFACE

xvi

The International Organizing Committee 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

R.L. Amoroso, The Noetic Advanced Studies Institute Physics Lab, USA Chairman. B. Haisch,Solar & Astrophysics Laboratory,Lockheed Martin,Palo Alto,USA B. Hiley,Theoretical Physics Research Unit, Birkbeck College, London, UK. G. Hunter, Chemistry Department, York University, Canada - Coorganizer. S. Jeffers, Department of Physics & Astronomy, York University, Canada Coorganizer. M. Kafatos, Center for Earth Observation & Space Research,George Mason Univ.,USA - Coorganizer. C. Levit, NASA Ames Research Center, Molecular Nanotechnology Group, Moffet Field, CA USA. M. Moles, Instituto de Mathematicas y Fisica, Spain . J. Narlikar, Inter-University Center for Astronomy and Astrophysics, India. S. Roy, Indian Statistical Institute, Calculta, India. A. Rueda, California State University, Dept. of Electrical Engineering, USA. A. van der Merwe, Physics Department, University of Denver, USA. J-P. Vigier, Gravitation et Cosmologie Relativistes, Pierre et Marie Curie Universite, Paris VI, France

Program Committee F. Pallikari, University of Athens, Physics Department, Greece. A. Rueda, California State University, Dept. of Electrical Engineering, USA. R. Amoroso, The Noetic Advanced Studies Institute - Physics Lab, USA. B. Haisch, CIPA, USA. G. Hunter, Department of Chemistry, York Univ, Canada. Keynote Speakers 1. Jean-Pierre Vigier - France 2. Jayant Narlikar - India 3. Wolfgang Rindler - USA 4. Bo Lehnert - Sweden

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FROM THE COSMOLOGICAL TERM TO THE PLANCK CONSTANT J. G. VARGAS1,2 AND D. G. TORR2 Center for Science Education 2 Department of Physics, University of South Carolina, Columbia, SC 29210 1

1.

Introduction

In this paper, we show the potential of classical differential geometry to unify gravity with the other interactions and, specially, quantum mechanics. Consider Cartan’s comments[1]: “...a Riemannian space... in the immediate neighborhood of a given point, it can be assimilated to a Euclidean space.” And also: “A general space with a Euclidean connection may be viewed as made of an infinite amount of infinitesimally small pieces of Euclidean space, endowed with a connecting law permitting to integrate two contiguous pieces into one and the same Euclidean space.” And finally: “...collections of small pieces of Euclidean spaces, oriented relative to the neighboring pieces.” Hence, had general relativity not been born by 1925, i.e. at a time when the present form of Quantum Mechanics was not yet known, one might have expected that Riemannian geometry (RG) would have to do, if anything, with the realm of the very small, not of the very large. Why are the mathematics of the quantum world and of the very small (as defined above) so different? Enough mathematical knowledge exists to solve the puzzle. The key to the solution of the problems that we have posed lies in the affine connection of spacetime. At the end of section 2, we summarize the topics to be considered, other than the connection itself. 2.

The Affine Connection Of Spacetime

The field equations of General Relativity (GR) concern the Einstein tensor, which is a piece of the curvature. A modern lecturer in GR would likely explain curvature through the transport of vectors around closed paths. But Einstein himself could not, at the birth of GR in 1915, have explained curvature in this way; this concept of curvature through transport of vectors was foreign to RG until the introduction in 1917 by Levi-Civita[2] (LC) of a rule for parallel transport, i.e. an affine connection, in a Riemannian space. The 1 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 1-10. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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J. G. VARGAS & D. G. TORR

quantities known as Christoffel symbols then became also the quantities for the LC affine connection. The set of quantities known as Riemann’s curvature then started to represent both the original or metric curvature and the new or affine curvature, a property of affine connections. In 1917, RG ceased to be a purely metric theory and became an affine cum metric theory. As Cartan put it[3]: “With the introduction of his definition of parallelism, Levi-Civita was the first one to make the false metric spaces of Riemann become (not true Euclidean spaces, which is impossible, but) at least spaces with Euclidean connection ... (emphasis in the original). To our knowledge, general relativists never questioned the physical correctness of attaching an affme meaning to the geometric objects of 1915-GR, i.e. those of 1915-RG. By the time that general affine connections were created and understood, the LC connection (younger than GR!) had become part of the foundations of Einstein’s theory of gravitation. This happened by default, as physicists did not know about other possibilities. The obvious alternative to a LC connection is a teleparallel connection. They are the canonical connections of the pair constituted by a metric and a preferred frame field, namely connections which are zero in that frame field. The affine curvature is then zero in any frame field. All geometric quantities (torsion, affine curvature, etc.) again derive from the metric only, as in RG, though there is now a frame field that plays a special role in the derivation. The earth, punctured at the poles and endowed with the connection where the rhumb lines rather than the maximum circles are the lines of constant direction, has zero affine curvature. This connection is so natural that Christopher Columbus and his sailors entrusted their lives to it: they maintained “a constant direction” by staying on the same parallel when they sailed to the New World. Since Columbus’ earth also is round, it has non-zero metric curvature. The terms teleparallelism (TP) and parallelism at a distance refer to these connections. Einstein [3] tried to replace RG (cum LC connection) with TP. His motivation was “Given points A and B separated by a finite distance, the lengths of two linear elements placed at A and B can be compared, but one cannot do the same with their directions; in RG there is no parallelism at a distance.” (Emphasis in original). Einstein then proceeded to develop TP. Notice from the quote that it was self-evident for him that physics must be based on TP: he objected to RG (endowed with the LC connection) for no other reason than for not being teleparallel! We now show that replacing the 1917 LC extension of GR with a TP affine extension one obtains a physical theory with the same metric relations as GR but which enriches Einstein’s theory by producing additional relations with profound implications. Einstein tried to develop physical TP and failed. The mathematics of the time was not ripe. Whereas in Riemannian geometry the only independent differential invariants are the (defined by we now have as independent differential invariants and But even this is not rich enough for a unified theory of the interactions. When one

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3

knows the mathematics, it is also a self-evident truth that the Lorentzian signature is the canonical, preferred signature of Finsler geometry. In the modern view of differential geometry, Finsler geometry must be viewed as pertaining to Finsler bundles, not to Finsler metrics, the metric origins of this geometry not withstanding. The view that Finslerian connections are the connections determined by Finslerian metrics is as incorrect as the view that the theory of affine connections is about connections determined by Riemannian metrics. In Finsler geometry, the differential invariants may be split into the and The which are the components of the following pieces: vector-valued 1-forms and span the Finslerian base space or phase-spacetime. In Finslerian TP there are cross sections where (but not are zero. The torsion, which will represent the non-gravitational interactions in Finslerian TP, is simply the exterior covariant derivative of the differential invariant dP. TP thus is Aharonov-Bohm compliant, and there is a new philosophy. The geometric expectations that we have raised might prompt readers to expect that, in the same way as the metric represents the gravitational interaction, some other differential invariant might represent the other interactions and, still some other, quantum mechanics. The new philosophy, however, is that different members of the set of differential invariants combine in different ways to give rise to different physical concepts, interactions, representative equations, etc. Thus the Finsler-invariant quantity ( modulo ) contains all the metric relations of the manifold. It satisfies mod The Riemannian case corresponds to when mod is a quadratic form on the velocity coordinates. The set contains all the information about the classical description of motion for all the interactions, to the extent that each individual interaction admits such a description. It also constitutes the input for Dirac equations that exclude the weak and “a combined strong-weak” interactions, etc. The understanding of how Dirac equations enter the geometric picture is the main issue. The beginnings of the solution were provided by Kähler [4] through the construction of a calculus that, by combining the exterior and interior derivatives, generalizes Cartan’s calculus and gives rise to a theory of “Kähler-Dirac” equations, the standard Dirac equation being a particular case. Although Kähler confined himself to spaces endowed with the LC connection, his work is easily reformulated to apply to TP [5]. The reformulation does not, however, eliminate some peculiar features of his calculus. These disappear, however, when, in order to remove from the total set of differential invariants the invariants that embody the macroscopic rotations, Finslerian TP is further reformulated as a Kaluza-Klein (KK) theory based on ( [6]. A final feature of the resulting picture is that the connection has to be stochastic: without it, there is no gravitation of neutral matter. This is known as the Sakharov-Puthoff conjecture, which is a consequence here. To conclude, the LC affine connection (and curvature) entered physics surreptitiously. In case it is the wrong affine connection of spacetime, one should try TP.

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J. G. VARGAS & D. G. TORR

3. The Gravitational Sector Of Teleparallelism

Given the structural richness of TP, there is no need for ad hoc introductions of additional structure, at least not until one has developed the consequences of TP as a postulate. Let be the affine connection and let be LC object (no longer the affine connection). We have:

where represents a 2-tensor-valued 1-forrn. Its components are linear combinations of the components of the torsion and have three indices, the form index being hidden in Users of the tensor calculus should think of as representing the and of as representing the Christoffel symbols. The metric curvature, or Riemannian curvature, is

The known from the tensor calculus, are components of the 2-tensor-valued 2form the form indices being hidden. The affine curvature of the space, or curvature of the rule that compares vectors at different points A and B, is

Substituting (1) in (2):

Expanding (4), or its tensor calculus equivalent, and setting

equal to zero:

The contents of the parenthesis is a tensor, and so is The term “Einstein contraction” will refer to the process that takes us from the curvature 2-form to the so-called Einstein tensor: get the components, contract them to obtain Ricci, etc. One obtains a completely geometric Einstein equation by applying this process to both sides of Eq. (5). If we deal with point masses and symmetry arguments (binary pulsar), one cannot distinguish between the Einstein contraction of (5) and Einstein’s equations, since the torsion (and its derivatives) for these masses will be non zero only at the point masses positions. Let us now show that the equivalence principle is part of the rich contents of Eq. (5). For a homogeneous and time independent configuration of all fields, would be zero. Since we can choose to be zero, the whole contents of the parenthesis (and, with it, its Einstein contraction) can be made to disappear. One then interprets this term as gravitational energy. For confirmation observe that the equivalence of an accelerated frame and a gravitational field is only as large as homogeneity allows (Einstein elevator). Hence it is only as large as the differentials remain negligible. The gravitational energy contained in the Einstein contraction of the contents of the parenthesis thus exhibits the equivalence principle and that it is valid only locally (physical sense of

COSMOLOGICAL TERM TO PLANCK CONSTANT

5

“locally”). The contracted quadratic term, then represents non-gravitational energy. The term is the key to controlling gravity through inhomogeneous and/or time dependent electromagnetic fields. It causes a variation in the weight of a body by an inhomogeneous electric field. The experimental detection of this effect at the University of South Carolina by M. Yin and T. Datta will soon be submitted for publication. Previous work was reported by Dimofte [7]. Our present interest, however, is in establishing the connection between the very large and the very small. Since the same differential invariants generate the macro and microscopic sectors, implications of one sector for the other will propagate through these differential invariants.

4. Completion Of The Classical Sector Of Teleparallelism The torsion is the formal exterior derivative of (notice the Finslerian generalization). The “d” in does not mean the differential of anything. is not an exact form but notation for the vector-valued 1-form that gives the translation vector assigned to a path by integration on it of One differentiates using the definition of which relates bases at x+dx and x. The first Bianchi identity for TP reads i.e. that the exterior covariant derivative of is 0. The “Kähler complement” to this system is constituted by the specification of the interior covariant derivative of the torsion, where is a vector-valued 3-form and is the interior covariant derivative defined in the TP Kähler calculus [5]. The system contains Maxwell equations in vacuum (charges but no material media) up to a constant. The zero-current equations are not self dual, in general, since is not the usual of the zero torsion, except as an approximation [8]. The structure of Finsler geometry allows one to perform an identification (within the limits imposed by the state of the theory) between pieces of the torsion and the different interactions [9]. Equations of motion in the form of autoparallels (up to particledependent “dressing constants”) have been obtained [9]. It has been found that some pieces of the Finslerian torsion do not contribute to the classical motion (meaning that they have zero classical range), except through the gravitational effects produced by their energy-momentum tensors [9]. This fact, together with the O(3) symmetry they enjoy (which becomes SU(2) in the Dirac sector) suggests their association with the weak and weak-cum-strong interactions of this theory. It has further been argued that the corresponding particles constitute “a second cosmological fluid” of dark matter whose pressure term is “the” cosmological term [9].

5. The Quantum Sector of Teleparallelism The Kähler generalization of the Dirac equation refers to any equation of the type [4]:

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J. G. VARGAS & D. G. TORR

where and constitute the input and output differential forms. is the sum of the interior and exterior derivatives, defined as in Kähler, but with generalized connection instead of the LC connection. The free torsion equations can be written as:

and thus are Kähler-Dirac equations where

Kähler showed that, if

then

i.e. another solution for the input A conjugate Dirac equation can be defined for the same form From the solutions for the direct and conjugate equations for given a conserved current follows, at least for the LC connection (we have not yet dealt with all these issues in TP). With the form Kähler solved the hydrogen atom [10]. Note the unusual location of Also, the existence of different Clifford algebra representations of Dirac’s equation speaks of the fact that the geometric meaning of the equation has not yet been understood. The form is scalar-valued. Assume we had a vector-valued Since has zero-valuedness, must be of inhomogeneous valuedness, going, in principle, up to infinity. The non-ad-hoc way to control this explosion of valuedness is to assume a tangent Clifford algebra, not a tensor algebra. This provides the Kähler equation with additional structural richness and solves structural problems of present day Finsler geometry [6,11]. One defines a vector-valued 1-form in a canonical KK space, namely The interpretation of (ds in previous publications) is proper time. is a unit vector spanning the “fifth” dimension of the (4+l)-KK space and corresponds to the velocity. The differentiation of involves both but not the In this structure, one encounters U(1) as a spacetime symmetry [11]. We have not yet developed the algebraic details of this theory, so as to give the opportunity to SU(3) to show up as an external symmetry. We expect that its appearance is just a matter of time, since Schmeikal has already shown that the SU(3) symmetry is contained in basic spin representations of the octahedral space group thus linking this inner symmetry with spatial geometry [12]. One can then think of

as the aforementioned, extremely rich “canonical” equation where has Cliffordvaluedness and not just scalar-valuedness, like the electromagnetic field, or even vectorvaluedness.

6.

Relation Between Gravitation And Quantum Physics In Teleparallelism

We now show deep connections between the very large and the very small in TP. To start with, we emphasize the significance of Kähler’s theory of Dirac equations (the Kähler who generalized Cartan’s theory of exterior differential systems). With the humility of

COSMOLOGICAL TERM TO PLANCK CONSTANT

7

great mathematicians when they touch on matters of physics, Kähler stated: “So that the interior calculus may have its confirmation test in the quantum and relativity theories,...” [13] (he used the term interior calculus for what is nowadays called exterior-interior calculus). When the humble dressing of this subordinate statement is removed, Kähler has claimed the creation of a language suitable for two sectors of physics which are advertised as seemingly irreconcilable. This language should be studied by those prominent and, therefore, influential physicists who dare to pass negative judgment on these issues from a position of ignorance of the mathematics involved, or future generations will taint their memories for such serious oversight. Indeed, unless the God that Einstein rhetorically invoked in a now famous statement is not utterly malicious, one wonders why nature would ignore such a simple and rich option to create a most sophisticated and elegant world in the main or tangent bundle, rather than in the auxiliary bundles of gauge geometry.. The key to understanding the relation between TP gravitation and quantum physics is the realization that, whereas Einstein’s equations constitute 10 equations for 20 unknowns (the 20 independent components of the curvature tensor), the gravitational equations are now 20 for the same 20 unknowns. Of course, the torsion components also appear in the same equation, but the torsion has to be considered as a given in this part of the argument, like the energy-momentum tensor. Because there are now 20 field equations, the Cauchy problem involves the specification of the initial condition just at a point of space (not on a hypersurface) at an instant of time. To make the argument clear, imagine the sudden switching-on of sources of torsion all over the universe. The initial conditions that those sources create will start to arrive at A from within the past light cone of any given spacetime point A. Of course, the geometric fields that the infinity of sources of torsion produce do not match at A. What this implies is that, the “exact” solution for the basic differential invariants (or connection, of our closed system of differential equations does not even make sense except as a stochastic solution, though allowing for a dominant non-stochastic component of the stochastic solution. We would have a background of torsion in the universe as we now have the electromagnetic background. In fact, it will look electrodynamic away from matter, but, close to it, it will have the dynamical richness of a vacuum whose governing differential equations are non-linear, with all the concomitants of such non-linearity. For instance, we cannot just say that the background torsion field (or, better, the background and/or field) is very small. This may be the case here (i.e. at some point) and now (at some instant of time). But the derivatives may be very large and virtually cancel among themselves. Large fields may resurrect further down the line (in space and or time) in the form of solitons, as for Muraskin’s equations [14]:

These equations are being considered here because different types of computer-generated solutions of them exist [14] and because their quadratic terms are highly similar to those that appear in the equations for the electromagnetic vacuum of our own system [8] (We have also provided these equations with an interpretation within TP [15]). Notice that,

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J. G. VARGAS & D. G. TORR

when the quadratic terms are eliminated, these equations reduce to whose solution obviously is where the are integration constants. It is worth noting the randomization of the spacetime structure (i.e. of the basic differential invariants of the geometry) that would take place in a radiation-dominated spacetime where the background could not be treated linearly. Imagine integrating the field equations and finding all the geometric quantities at a spacetime point under the assumption, for simplicity, that there were just two sources in the universe, namely at points B and C. The field equations being quadratic, the signals coming from B and C become badly scrambled. Imagine now that this launching of initial conditions happens for the signals coming from any source in the universe (fermionic fields), not to mention the fact that even the torsion of the background field (and of any field for that matter) is itself a source of the metric structure of spacetime, as exhibited by equation 5. This is not unlike the fact that any tiny piece of energy contributes to the Einstein equations. The difference between this stochastic picture and the classical picture of GR arises with their differing Cauchy problems! By increasing the sought precision of the solution, one eventually reaches the stochastic background. The latter may be endowed with little energy and yet have huge effects, the fluctuations arising when the derivatives of the field do not cancel among themselves. In our geometric picture, these fluctuations are to be associated with the stochasticity of the solutions for the field equations satisfied by the differential invariants of the geometry. The preceding argument referred to the bosonic fields (in the absence of similar studies for our own equations, we argued by analogy with the closely related Muraskin equations). Through the sharing of common differential invariants, whose basic equations constitute a closed system of equations when a Kähler-Dirac equation is included, the stochasticity of the connection has to be associated with the value of h, since the quantum effects, and the vacuum fluctuations with it, vanish as h goes to zero (Notice that, although one needs to include the canonical Kähler-Dirac equation in the geometrically closed system of field equations, there is no need to include among the basic differential invariants, since a knowledge of and allows one to obtain Without such stochasticity, there are no vacuum fluctuations and no quantum physics as we know it. It is the nature of the gravitational sector of the system of field equations for TP that causes the necessarily stochastic nature of the vacuum of TP.

7.

The Sakharov-Puthoff Conjecture As An Integral Part Of Teleparallel Physics

Let us finally deal with the Sakharov-Puthoff conjecture [16, 17]. In a nutshell, it states that the gravitational interaction is wiped out when one switches the other interactions off. Specifically, gravitation is an effect caused by the vacuum fields of the other interactions. This is a consequence of TP in the following way. Without stochasticity, the vacuum would have to be considered as empty space in a true sense. The torsion field would then be zero. Since, by postulate, the affine curvature also is zero, the spacetime is flat. In other words, it becomes affine space [Only the flat metrics (Euclidean, Minkowskian, etc.) are consistent with affine space, or else the affine connection would not be metric-

COSMOLOGICAL TERM TO PLANCK CONSTANT

9

compatible. Equation (5) indeed shows that making the torsion (and, therefore, equal to zero, annuls the metric curvature] One is not claiming that the energy of the vacuum (bosons) causes the metric curvature of spacetime. This may be the relativistic way of thinking, but is not what TP dictates. The gravitational energy to be associated with the gravitational force in TP is the Einstein contraction of (See Eq. 5). This term becomes significant next to matter, where the torsion field becomes associated with fermions ( of type Eq. 6). As for the nature of the background field, the solutions of the Muraskin equations [14] suggest that the magnitude of the fields which are solutions of the geometric equations of this theory may be very small at some point and very large at some other point. It is conceivable in principle, that the zero point field will become, far away from matter, a linear-looking non-linear version of the cosmic background field. The problem, to the different extents that it is perceived to be one, of the infinite energy density of the zero point field of QED and stochastic electrodynamics (and concomitant infinite curvature of spacetime) does not even arise in this theory. Very close to particles, phenomena like vacuum polarization would show up. For matter in bulk, regular gravitation would then occur, as intimated by the work of Haish-Rueda-Puthoff [18]. If we want to alter gravitation, i.e. how the background field appears to matter in bulk, we would have to put very large gradients where matter and background fields meet, i.e. very large inhomogeneous fields. Most of the statements in the previous paragraph are not theory but speculation, as the mathematical proof is not there yet. It took many of the very best physicists of the beginning of the twentieth century to develop the paradigm of Quantum Mechanics. Each of them provided a piece to the puzzle, piece which eluded everybody else in such distinguished group. Hence, the solution that TP may provide to the grave problems that afflict present day theoretical physics will remain largely speculative until a significant amount of theorists join this effort.

8. Acknowledgments

One of us (J.G.V.) deeply acknowledges generous funding from the Offices of the ViceProvost for Research and of the Dean of the School of Science and Mathematics of the University of South Carolina at Columbia.

References 1. Cartan, É.: Oeuvres Complètes, Editions du C.N.R.S., Paris, 1983. 2. Levi-Civita, T.: Nozione di parallelismo in una varietà qualunque e consequente specificazione geometrica della curvatura Riemanniana, Rendiconti di Palermo 42 (1917), 173-205. 3. Einstein, A.: Théorie Unitaire du Champ Physique”, Ann. Inst. Henri Poincaré 1 (1930), 1-24. 4. Kähler, E.: Innerer und äusserer Differentialkalkül, Abh.Dtsch. Akad. Wiss. Berlin, Kl. Math., Phy. Tech., 4 (I960), 1-32. 5. Vargas, J. G. and Torr, D. G.: Teleparallel Kähler calculus for spacetime, Found. Phys. 28 (1998), 931 -958. 6. Vargas, J. G. and Torr, D. G.: Clifford-valued clifforms: a geometric language for Dirac equations, in R. Ablamowicz and B. Fauser (eds.), Clifford Algebras and their Applications in Mathematical Physics,

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Birkhäuser Boston, 2000, pp. 135-154. 7. Dimofte, A.: An experiment concerning electrically induced gravitation, Master’s Thesis, University of South Carolina, Columbia, 1999. 8. Vargas, J. G. and Torr, D. G.: The Cartan-Einstein unification with teleparallelism and the discrepant measurements of Newton’s constant G, Found. Phys. 29 (1999), 145-200. 9. Vargas, J. G. and Torr, D. G.: The theory of acceleration within its context of differential invariants: the roots of the problem with the cosmological term, Found. Physics 29 (1999), 1543-1580. 10. Kähler, E.: Die Dirac-Gleichung, Abh.Dtsch. Akad. Wiss. Berlin, Kl. Math., Phy. Tech, 1 (1961), 1-38. 11. Vargas, J. G. and Torr, D. G.: Marriage of Clifford algebra and Finsler geometry: a lineage for unification? Int. J. Theor. Phys. 39 (2000, July). In press. 12. Schmeikal, B.: The generative process of space-time and strong interaction quantum numbers of orientation. In R. Ablamowicz, P. Lounesto and J. M. Parra (Eds.), Clifford algebras with numeric and symbolic computations, Birkhäuser Boston, 1996, pp. 83-100. 13. Kähler, E.: Der innere Differentialkalkül, Rendiconti di Matematica 21 (1962), 425-523. 14. Muraskin, M.: Mathematical Aesthetic Principles/ Nonintegrable Systems, World Scientific, Singapore, 1995. 15. Vargas, J. G. and Torr, D. G.: The construction of teleparallel Finsler connections and the emergence of an alternative concept of metric compatibility, Found. Phys. 27 (1997), 825-843. 16. Sakharov, A. D.: Spectral density of eigen values of the wave equation and vacuum polarization”, Theor. Math. Phys., 23 (1975), 435-444. 17. Puthoff, H, E.: Gravity as a zero-point fluctuation force, Phys. Rev. A 39 (1989), 2333-2342. 18. Haish, B., Rueda, A. and Puthoff, H.: Inertia as a zero-point field force, Phys. Rev. A 49 (1994), 678-694.

CREATION OF MATTER AND ANOMALOUS REDSHIFTS

JAYANT V. NARLIKAR Inter-University Centre for Astronomy and Astrophysics Post Bag 4, Ganeshkhind, Pune 411 007

Abstract. This presentation discusses the role of creation of matter in cosmology. While the phenomenon is considered a singular event in the big bang model, a more physical description is given in the quasi-steady state cosmology. Some highlights of this model are presented. Finally, the observations of anomalous redshifts are briefly described and viewed as a consequence of newly created matter ejected from older matter.

1. Introduction Modern cosmology began in 1917, with Einstein's model of the universe, in which the universe was homogeneous and isotropic and also static (Einstein 1917). The general belief in a static universe in which the galaxies etc., are at rest was so strong that when in 1922 Aleksandr Friedmann (1922, 1924) proposed expanding models of the cosmos, they were largely ignored by everybody, including Einstein. However, the first significant observational result in cosmology came in 1929 when Edwin Hubble announced the velocity-distance relation for galaxies, based on the redshifts observed in their spectra (Hubble 1929). This led people to the interpretation that the universe is not static but expanding. And the Friedmann models, which had also been independently found by Abbe Lemaitre (1927), became the recognized models for the universe. As Lemaitre had observed, these models appeared to start from the state of infinite density, which he interpreted to mean a dense primeval ‘atom'. In modern jargon this is called the state of ‘big bang'. For a decade or so after World War II, George Gamow, Ralph Alpher, Robert Herman and others explored this supposed dense primordial state. They concluded that it was dominated by high temperature radiation and other subatomic particles moving at near-light speeds (Alpher, et al 1948). They felt that this was ideally suited for nuclear fusion making all the chemical elements from protons and neutrons. However, they soon learned that this could not be done, because of the absence of stable nuclei at mass numbers 5 and 8. But they also realized that, if there had been such an early ultradense stage, the universe might well 11 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 11-26. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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contain an expanding cloud of primordial radiation that would preserve its blackbody form as the universe evolved (Alpher and Herman 1948). In the 1940s, however, another new idea challenging the hot big bang evolved, and in 1948 three British astrophysicists, Hermann Bondi, Tommy Gold and Fred Hoyle, proposed the steady state model (Bondi and Gold 1948, Hoyle 1948). It not only assumed the universe to be homogeneous and isotropic in space, but also unchanging in time. Thus there was no big bang, no hot phase; in fact the universe was essentially without a beginning and would be without an end. It, however, steadily expanded, thus creating new volumes of space which got filled up with new matter that was continually created. Hoyle in fact proposed a slight modification of Einstein's general relativity to account for matter creation out of a negative energy reservoir of energy. As more and more matter got created, energy conservation required the reservoir to become more and more negative; but taking into consideration the fact that space was expanding, the energy density ofthis reservoir remained steady. Thus in the steady state theory there was no mystical event like the ‘big bang' and no sudden appearance of all the matter into the universe (in violation of the energy conservation law). Instead there was a steady expansion supported by a continuous creation of matter. In 1948, the estimates of the age of the big bang universe showed it to be very small (of the order of years), smaller than the geological age of the Earth years). In the 1950s and the 1960s the debate between the big bang and steady state theories continued unabated. However, two events in the mid-1960s swung the argument in favour of the big bang cosmology. One was the realization that the observed abundances of light nuclei in the universe required their manufacture in a very hot dense stage (Hoyle and Tayler 1964). The other was the observation of the microwave background radiation (Penzias and Wilson 1965) which was quickly interpreted as the relic of the early hot era. Thus the big bang model acquired the status of the ‘standard model' of the universe. However, as we will now discuss, this reasoning may have been too simplistic.

2. The Standard Cosmological Model: Some Critical Issues The issue related to matter creation is perhaps the most ticklish conceptual issue in standard cosmology. This can be shown by the following line of reasoning. The Hilbert action which leads to the equations of general relativity and which in turn provide the dynamical basis of standard cosmology, is given in standard notation by

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13

Here V is the 4-volume of the spacetime region in which the action is defined. The variational principle which leads to the Einstein tensor of relativity, requires the integrand to be well defined with continuous second derivatives with respect to the spacetime coordinates. However, when we solve the cosmological equations based on this action, we hit the big bang event at the cosmic epoch t = 0, where (1) breaks down. Thus there is a mathematical inconsistency in the entire procedure which is usually ascribed to the presence of a spacetime singularity. Normally the existence of an inconsistency is taken to rule out the solution via the logic of reductio ad absurdum. Here, however, the singularity is dignified to the level of a metaphysical event beyond the scope of physics and mathematics. The physical limitations are shown by the fact that at the breakdown of the action principle, the law of conservation of matter and energy also breaks down. Which is why, the sudden appearance of all matter and energy in the universe remains unexplained. Instead the cosmologist tries to work within the zone t > 0, going as close to the singular epoch as the equations permit, but keeping away from it. The attempt by Gamow and his colleagues to understand the origin of nuclei in terms of nucleosynthesis at the early epochs can be seen in this context. As mentioned earlier, the attempt had a very limited success. In 1967, Robert Wagoner, William Fowler and Fred Hoyle (1967) repeated a calculation originally reported by Gamow, Alpher and Herman. They calculated that a synthesis of the light elements in the early hot universe yielded abundances of deuterium,

and

that were satisfactorily in agreement with

astrophysical observations if the average cosmological density

of baryonic

matter was related to the radiation temperature T (in Kelvin) by Cosmological theory requires this relationship between density and temperature to be maintained throughout the expansion of the universe from its early hot state. So, putting in the measured value of the present background temperature, T = 2.73 K, yields about for the present-day average density of the cosmos. For a comparison, the standard model predicted the present density of the universe to be close to This density is sometimes referred to as the closure density: models more dense than this are closed in the topological sense, while those with less density are open. Although this density was almost two orders of magnitude less than the standard model's closure density, it agreed with galactic astronomer Jan Oort's estimate for the average cosmic density of observable material. The higher “closure” value of about

given by standard cosmological theory, is explained in terms of

nonbaryonic matter that has changed its identity over the years from neutrinos to esoteric “cold-dark matter" particles, perhaps with some remaining admixture of neutrinos. For some of us, it is not reassuring that this line of reasoning from the 1960s is still the best available

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in favor of Big Bang cosmology, despite the continuing failure of attempts to identify the required nonbaryonic matter. This standard-cosmology argument on relic radiation can be countered by a still more precise calculation with a very different implication.

We know that

is

synthesized from hydrogen in stars with an energy yield of about ergs for each gram of helium, the energy being radiated by the stars to produce a radiation background. If all ofthe about one

in the universe has been produced in this way (the observed abundance is for every 12 hydrogen atoms), then the accompanying radiation background

should have an energy density of

That is quite close to the

observed energy density of the microwave background, namely Either this agreement is coincidental, or we must conclude that the was created, not by Big Bang nucleosynthesis, but rather by hydrogen burning inside stars (a process that we know to exist), and that the radiation background from stars has become subsequently thermalized into the far infrared (as discussed here at a later stage). We turn now to further problems associated with the so-called standard model. If negative values of the energy density are prohibited, one can argue that the observed expansion of the universe requires not only that the universe was more compressed in the past, but additionally that it was also expanding in the past. If we denote the time dependence of the linear scale factor of the universe S(t), general relativity tells us that the scale factor has always been increasing in the past and, as we look back in time, we see the universe become more and more compressed at earlier and earlier times. Ultimately to what? In attempts to answer this question, it is accepted that particle energies increase up to values in the TeV range, and then, by speculation, all the way to the Big Bang. Up to symmetry arguments are invoked and the theory departs increasingly from known physics, until ultimately the energy source of the universe is put in as an initial condition, as are other physical conditions like the fluctuations of matter density that became enhanced later to form galaxies in an otherwise homogeneous universe. Because the intitial conditions are beyond the present observer's ability to observe and verify, and because the particle physics has remained untested at energies of the order we are completely at the mercy of speculations! More so, as the primordial conditions are never repeated at any later stage, and so we are in fact violating the repeated testability criterion of a physical theory. Ironically, the existence of the microwave background which is claimed as the best evidence for the big bang cosmology, itself prevents direct observations of these very early epochs. For as we try to probe the universe prior to the 'surface of last scattering' the scattering of photons prevents any coherent observations from being made. Thus one has to rely on a string of consistency arguments only in lieu of direct proof for very profound statements about the early universe. These and other unsatisfactory features led Fred Hoyle, Geoffrey Burbidge and myself to take a fresh look at cosmology and try a new approach.

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3. Quasi Steady-State Cosmology Since 1993, we have been developing an alternative cosmology, beginning from an action principle by which we seek to explain how matter and radiation appeared in the universe (Hoyle, et al 1993). That is to say, the action principle includes the possibility that the world-line of a typical particle can have a beginning. The details involve a scalar field analogous to that which appears in popular inflationary models favored by standard cosmology, and also very similar to the scalar field used to describe creation of matter in the steady state cosmology (Hoyle and Narlikar 1964). As it does in the inflationary models, the scalar field exerts a negative pressure that explains the universal expansion. In our theory, the field also acts negatively in the creation process, balancing the positive energy of matter production. That permits new matter to appear in an already existing universe, instead of requiring the creation of the entire universe de novo, in a Big Bang. We regard the creation as being triggered locally in what we call minicreation events or minibangs, with the negative field component subsequently escaping from the region of creation, which has experienced an accumulation of positive energy. It is in this way, we argue, that black holes are formed - not through the infall of matter. The popular black hole paradigm at present assumes that the high energy activity in the nuclei of certain 'active' galaxies is triggered by a spinning massive black hole with several billion solar masses. However, this interpretaion runs into problems like the following. Matter moving at velocity transverse to the radius vector from the center of a spherical black hole of mass

solar masses

at the critical distance

has angular momentum of order per gram. But matter rotating about a galactic centre typically has ten thousand times more angular momentum than that. Therefore it is difficult for us to see how a large quantity of matter in a galaxy could come to be packed into the small scale of a black hole, even when the black hole has a mass as large as

solar masses. The conventional interpretation has as yet, found no satisfactory way around this difficulty. But if, at the centres of galaxies, there are black holes that act as minicreation events, the escape of the negative energy field generated in the creation process provides a ready explanation for the accumulation of the positive material component, leading to an easily understood development of the central black hole. While there are several interesting applications of this idea to high energy astrophysics, we will discuss next how the combined effect of such minicreation events drives the dynamics of the universe. It turns out that while the long term result of this interaction is the steady state model, there are significant short term effects which make the universe oscillate around the steady state solution. Which is why the cosmology is called the ‘quasi-steady state cosmology'. We outline the important features of this model. For details, see Hoyle, et al (1994, a, b, 1995, 2000)

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4. Cosmological Solutions The spacetime geometry of the quasi-steady state cosmology (QSSC) is described, just as in standard cosmology, by the Robertson-Walkar line element, with the expansion of the universe determined by the scale factor S(t). The difference in this theory is that the equation for the square of the time derivative of S now carries a negative term that decreases like Thus, in a time-reversed picture, in which the scale factor S grows smaller, a stage will eventually be reached in which this new term will dominate over the positive term, due to the material content of the universe, that goes like The effect, as one goes backward in time, is to produce an oscillation of the scale factor:

In the time-dependent scale factor, the parameter is the temporal period of the periodic function which turns out to be 5-10 times longer than the “age of the universe" arrived at in the Big Bang scenario. The other characteristic-time parameter, describes an exponential growth that is very slow on the time scale of the periodic function. P is determined by the rate of matter creation averaged over a large number of minicreation events. For details of these solutions see Sachs, et al (1996). The quasi steady-state model also has two dimensionless parameters: the ratio between the amplitudes of at its maxima and minima, and the ratio of the present scale factor to its periodic minimum. Typical values of these four parameters that best fit the observational data are

Notice that the oscillations of the scale factor are always within finite limits: in short there is no spacetime singularity.

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Among the broad observational data that these parameters must reproduce are (1) the relationship between the redshifts of galaxies and their visual magnitudes, (2) the angular sizes of quasars at different redshifts, (3) the population counts of galaxies and radio sources, (4) the largest observed redshifts, (5) the microwave background and (6) the cosmic abundances of the lightest nuclear isotopes. We begin with the discussion of the microwave background first, as it is widely regarded as tour de force for the standard model.

5. The Microwave Background As seen in the cosmological model described above, the stars shining in the previous cycles would leave a relic radiation background. This can be estimated with the help of starlight distribution in the present cycle, since all cycles are ideally identical. It turns out that the total energy density of this starlight at the present epoch is adequate to give a radiation background of ~2.7 K, in good agreement with the observations. The question is, would this relic radiation be thermalized to a near-perfect blackbody spectrum and distributed with a remarkable degree of homogeneity? The answer to the first question is ‘yes’. The thermalizers are metallic whiskers which work most efficiently for this process, much more so than the typical spherical grains. These are formed when supernovae make and eject metals in vapour form. Experimental work on the cooling of carbon and metallic vapors has shown that there is a strong tendency for condensates to appear as long thread-like particles, often called whiskers. Carbon and metal whiskers are particularly effective at converting optical radiation into the far infrared. Calculations show that a present-day intergalactic density of for such whiskers would suffice to thermalize the accumulated starlight at an oscillatory minimum. Such a whisker density could readily be accounted for by the ejecta of supernovae, which can easily leave the confines of their parent galaxies. For details of this process see Narlikar et al (1997) and Hoyle, et al (2000). But near an oscillatory maximum, the universe is sufficiently diffuse that such intergalactic particulates have a negligible effect on starlight. Light propagation is then essentially free and, because of the long time scale of the maximum phase of each cycle, there is a general mixing of starlight from widely separated galaxies. Because of this mixing and the large-scale cosmic homogenity and isotropy, the energy density of the radiation also acquires a high degree of homogeneity. That homogeneity persists, because the absorption and reemission of the starlight at the next minimum does not change the energy density. Thus we have an explanation of the remarkable uniformity of the cosmic microwave background. Small deviations from this uniformity, on the order of a part in are expected for regions near rich clusters of distant galaxies. This implies that the microwave background should exhibit temperature fluctuations on the sky of a few tens of microkelvin on an angular scale determined by the clustering of distant galaxies. For a distant cluster of diameter 10 megaparsecs observed at a redshift of 5 (about the highest redshift that's been

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seen), that angular scale is about 0.7°, in good agreement with the largest observed fluctuations in the microwave background. The ease with which the complexities of the microwave background can be understood in the quasi steady-state cosmology is, in our opinion, a strong indication that the theory is on the right track. Rather than being put in by parametric choices, the observed fluctuations of the microwave background arise naturally from the clustering of galaxies.

6. Origin Of The Light Nuclei

There are more than 320 known isotopes of the elements. In 1957 Geoffrey and Margaret Burbidge, Willy Fowler and Fred Hoyle (1957) showed that, with the possible exceptions of deuterium, and all the isotopes were synthesized by nuclear processes in stellar interiors. Burbidge and Hoyle (1998) have recently reviewed the situation as follows. The list of eight problematic cases was soon reduced to five, as and were found to be produced in the spallation reactions of cosmic rays. More recently, it has been found that the depletion of Fe in old stars correlates closely with the abundance of strongly suggesting that was produced in association with the iron in supernovae. Thus the original list of eight light nuclear species that at one time were candidates for association with a hot Big Bang cosmology is now reduced to four. Of these, lithium can possibly be made under stellar circumstances, in view of the finding of lithium rich supergiant stars. Then we restress the striking fact that the energy density of the microwave background is very close to what we calculate for the production of the observed abundance solely by hydrogen burning in stars. When all this is put together, we are left with only two of the eight special cases, as they seemed in 1957 - namely deuterium and What is the likehood, we now ask, that even these last two will turn out to have purely astrophysical origins?

is accumulated in large quantities in dwarf stars whose

masses are too small for the isotope to be destroyed by the radiation There is also a class of earlier-type, more massive stars (including 3 Cen A), in which most of the helium is

On the

year time scale of the quasi steady-state

cosmology, it seems likely that the cosmic abundance of

(Big Bang nucleosynthesis

predicts about one for every ten thousand nuclei) is to be explained by an escape from stars of these types in stellar winds. Deuterium, the last survivor from our original list of problematic light nuclei, is a particularly difficult case. It is both produced and destroyed by astrophysical processes. Deuterons are made in high-energy processes, such as solar flares, that generate free neturons, and destroyed by burning in stellar interiors. Arguments over whether astrophysical production suffices, with no need to invoke cosmological deuteron production,

MATTER CREATION AND ANOMALOUS REDSHIFTS

19

therefore turn on measurements of the cosmic D/H abundance ratio, which are difficult to accomplish with precision. In these circumstances, I think that the deuterium case can reasonably be regarded as uncertain. With all other nuclides (except, of course, ) produced in adequate abundance by astrophysical means, it would seem best to extend this generalization to the deuteron and presume that any nucleus heavier than the proton has been synthesized by processes associated with stars.

7. Observations Of Discrete Sources One of the most interesting developments in recent times in extragalactic astronomy is the use of Type IA supernovae to determine distances of galaxies and using the results to test the redshift-distance relations predicted by different cosmological models. Standard cosmology, after years of discounting the cosmological constant introduced by Einstein in his static model of 1917 as unnecessary, suddenly found it a very attractive parameter! There are two reasons for the same, (i) that the age of the universe comes out too low for comfort when compared to the estimated ages of stars and galaxies and (ii) the extension of Hubble's law to distant galaxies does not fit the standard models. By introducing the cosmological constant these difficulties can be overcome. On the other hand, as has been shown by Banerjee, et al (2000), there is excellent agreement between the Hubble relation based on the measured distances of galaxies, using Type IA supernovae and the predictions of the QSSC. This happens precisely because of the absorption caused by intergalactic dust postulated by the QSSC to thermalize the microwave background. Indeed, for best fit to the data,the optimum value of the adjustible parameter, viz. the whisker density, lies very well in the range required to generate thermalized microwave background. So far as stellar and galactic ages are concerned, the long time scales of the QSSC model ensure that there are no such problems. There are also excellent agreements on two other cosmological tests. In one we look at the angular sizes of the tiny cores of distant radio sources whose redshifts are known. The angular size redshift relation predicted by the QSSC provides a very good fit to the data (see Banerjee and Narlikar 1999). The other test relates to counts of radio sources up to varying levels of flux density. This number-flux density relation can also be closely simulated by the QSSC.

8. The Minicration Events, Dark Matter And Active Nuclei In the period 1958-74, Ambartsumian first developed the idea that many groups and clusters of galaxies are systems of positive total energy - that is to say, expanding systems not gravitationally bound - and that many small galaxies were formed in and ejected from the nuclei of larger systems. He also accepted the evidence of explosive events in radio sources and Seyfert galaxies. In the 1960s, when quasi-stellar objects with large redshifts were being

20

J. V. NARLIKAR

identified in increasing numbers, it was realized that they are also highly energetic objects closely related to explosive events in galaxies. How are we to understand such great outpourings of matter and energy? As far as the associations and clusters of galaxies are concerned, most theorists, unlike Ambartsumian, have simply not been prepared to accept the observations at face value. For many years, they have clung to the belief that the protogalaxies and galaxies were formed early in the history of the universe. From that point of view, it is impossible to believe that many galaxies are less than a billion years old, which must be the case if galaxies are, even now, being formed and ejected in expanding associations. It is generally agreed that, in such groups and clusters, the kinetic energy of the visible objects is much greater than their potential energy. The conventional way out nowadays is to assume that such groupings are indeed gravitationally bound - by large quantities of unseen “dark matter or energy". This conejcture was already put forward for some of the great clusters of galaxies by Fritz Zwicky in the 1930s. In the 1970s, the view that the masses of systems of galaxies on all scales are proportional to their sizes became widely believed, but it was not stressed that this result is only obtained by assuming that they are bound and therefore obey the virial condition for which there is no other evidence. The QSSC suggests that these open systems are in fact the remnants of mini-creation events and that their excess kinetic energy is the result of their explosive origin. As we shall see later in this article, the minicreation events play a key role in forming the large scale structure observed in the universe today. At the same time, we do have considerable evidence - from the flat rotation curves of spiral galaxies - for the existence of dark matter in them. This dark matter could very well be stars of previous cycles which are burnt out and devoid of any radiation. They could also be white dwarfs of very large ages of the kind not possible in the relatively limited lifespan of the standard model. What about radio sources, active galactic nuclei and quasi-stellar objects? It is generally accepted that they all release very large amounts of energy from dense regions with dimensions no larger than our solar system. It has been clear since the early 1960s that there are only two possibilities: This energy is either gravitational in origin, or it is released in creation processes. Conservatively, the total energy release in powerful sources is at least where M is the mass of the Sun. In the radio sources, much of this energy resides in highly relativistic particles. To get such enormous energy releases in gravitational collapse it is necessary to consider processes very close to the Schwarzschild radius, where it would be very difficult to get the energy out. Even if the efficiency of the initial process is as high as a few percent, the efficiency with which the gravitational energy is then converted through several stages into relativistic particles and magnetic flux would be very small. Despite these difficulties, the standard model explaining active galacitc nuclei asserts that, in all such situations, there is a massive black hole at the centre of the galaxy, surrounded by an accretion disk, and that all of the observed energy, emitted in whatever form, is gravitational in origin. All of it, we are told, arises from matter falling into the disk and then into the black hole. But this type of model cannot convincingly explain the many observed phenomena, largely because the

MATTER CREATION AND ANOMALOUS REDSHIFTS

21

efficiency with which gravitational energy can be transformed into relativistic particles and photons is so small. It is much more likely that, in active galactic nuclei, we are seeing the creation of mass and energy as proposed in the QSSC. Massive near-black holes are undoubtedly present in the centres of most galaxies. But when they are detected, the galaxy is typically not active. The important feature is probably the quasi steady-state creation process, which can take place in the presence of a large mass concentration.

9. Large Scale Structure

The minicreation centres act as nuclei for large scale structure. Imagine that most of the matter creation goes on near the oscillatory minima. Thus to restore `steady state' from one cycle to next, the decline in density by the factor must be made by the creation process. The creation of new galaxies to take the place of old ones must therefore be by a factor

multiplying the density of the population at the minima. One may therefore assume that in typical creation centres new coherent objects are created and ejected, some of which may act as nuclei of fresh creation later. Taking these features of the QSSC into consideration, Ali Nayeri, Sunu Engineer, J. V. Narlikar and Fred Hoyle (1999) simulated a toy model on a computer to see how the real process may work. The steps in this simulation are as follows. (a) Produce

points randomly in a unit cube.

(b) Around a fraction neighbor within a distance

of these points produce a randomly oriented where is a fraction less than 1.

(c) Expand the cube and all scales within it homologously in the ratio directions.

in all

(d) From the expanded cube retain the central cubical portion of unit size, deleting the rest.

These operations describe the creation process during one QSSC cycle. We repeat this exercise many times to see how the distribution of points evolves. From the work of Nayeri et al (1999), it is clear that the distribution of points soon develops clusters and voids,

22

J. V. NARLIKAR

typically like that in the real universe. A two-point correlation analysis confirms this visual impression quantitatively. The relative ease with which this type of distribution can be generated is in sharp contrast to the not inconsiderable efforts spent in standard cosmology in arriving at a cluster+void distribution through gravitational clustering.

10. Distinguishing Tests What are the specific tests that may distinguish the QSSC from standard cosmology? A few are as follows: (1) If a few light sources like galaxies or clusters are found with modest (~ 0.1) blueshifts, they can be identified with those from the previous cycle, lying close to the peak of the scale factor. In standard cosmology there should be no blueshifts. (2) If low mass stars, say with half a solar mass, are found in red giant stage, they will have to be very old, ~ 40-50 Gyr old, and as such they cannot be accommodated in the standard model, but will naturally belong to the generation born in the previous cycle of the QSSC. (3) If the dark matter in the galaxies is proved largely to be baryonic, or if other locations like clusters of galaxies turn out to have large quantities of baryonic matter, then the standard cosmology would be in trouble. For, beyond a limit the standard models do not allow for baryonic matter as it drastically cuts down the predicted primordial deuterium and also spoils the scenario for structure formation. These observations lie just beyond the present frontiers of astronomical observations. So we hope that the cosmological debate will spur observers to scale greater heights and push their observing technology past the present frontiers, as happened fifty years ago with the debate about the original steady state cosmology.

11. Anomalous Resdshifts I will end with a very fundamental question: Is Hubble's law obeyed by all extragalactic objects? Throughout this discussion we have taken it for granted that the redshift of an extragalactic object is cosmological in origin, i.e., that it is due to the expansion of the universe. Indeed, we may call this assumption the Cosmological Hypothesis (CH). The Hubble diagram on which the CH is based shows a fairly tight m - z relationship for first ranked galaxies in clusters, thus justifying our belief in the CH. However, a corresponding plot for quasars has enormous scatter (Hewitt and Burbidge 1993). Although people discuss the cosmological tests on the basis of CH for quasars as well as galaxies, it is found that in

MATTER CREATION AND ANOMALOUS REDSHIFTS

23

some cases special efforts are needed to make the CH consistent with data on quasars. These include, apart from the Hubble diagram, the superluminal motion in quasars, rapid variability, the absence of absorption trough, etc., which I will not have time to go into here (see, however, Narlikar 1989 for a discussion) To what extent is the CH valid for quasars? Let us begin with the type of data Stockton (1978) had collected in which quasars and galaxies were found in pairs or groups of close neighbours on the sky. The argument was that if a quasar and a galaxy are found to be within a small angular separation of one another, then very likely they are physical neighbours and according to the CH their redshifts must be nearly equal. This argument is based on the fact that the quasar population is not a dense one and if we consider an arbitrary galaxy, the probability of finding a quasar projected by chance within a small angular separation from it is very small. If the probability is < 0.01, say, then the null hypothesis of projection by chance is to be rejected. In that case the quasar must be physically close to the galaxy. This was the argument Stockton used. While Stockton found evidence that in such cases the redshifts of the galaxy and the quasars, and say, were nearly the same, there have been data of the other kind also. In two books H.C. Arp (1987, 1998) has described numerous examples where the chance projection hypothesis is rejected but Over the years four types of such discrepant redshift cases have emerged: 1. There is growing evidence that large redshift quasars are preferentially distributed closer to low redshift bright galaxies . 2. There are alignments and redshift similarities in quasars distributed across bright galaxies. 3. Close pairs or groups of quasars of discrepant redshifts are found more frequently than due to chance projection. 4. There are filaments connecting pairs of galaxies with discrepant redshifts. It is worth recording that there are continuing additions to the list of anomalous cases. They are not limited to optical and radio sources only, but are also found in X-ray sources, as shown recently by Arp (1998). The reader may find it interesting to go through the controversies surrounding these examples. The supporters of CH like to dismiss all such cases either as observational artefacts or selection effects. Or, they like to argue that the excess number density of quasars near bright galaxies could be due to gravitational lensing. While this criticism or resolution of discrepant data may be valid in some cases, it is hard to see why this should hold in all cases.

Another curious effect which was first noticed by G. Burbidge (1968) in the late 1960s concerns the apparent periodicity in the redshifts distribution of quasars. The

24

J. V. NARLIKAR

periodicity of first found by Burbidge for about seventy QSOs is still present with the population multiplied thirtyfold. What is the cause of this structure in the z-distribution? Various statistical analyses have confirmed that the effect is significant. Another claim, first made by Karlsson (1977), is that log (1 + z) is periodic with a period of 0.206. This also is very puzzling and does not fit into the simple picture of the expanding universe, that we have been working with in this book. On a much finer scale W. Tifft (1996) has been discovering a redshift periodicity (and also at half this value) for differential redshifts for double galaxies and for galaxies in groups. The data have been refined over the years with accurate 21 cm redshift measurements. If the effect were spurious, it would have disappeared. Instead it has grown stronger and has withstood fairly rigorous statistical analyses. For a universe regulated by Hubble's law, it is hard to fit in these results. The tendency on the part of the conventional cosmologist is to discount them with the hope that with more complete data they may disappear. At the time of presenting this account the data show no such tendency! It is probable that the effects are genuine and our reluctance to ignore them also stems from the lack of availability of any reasonable explanation. The explanation may bring in a significant non-cosmological component in the observed redshift z. Thus we should write:

The cosmological component

obeys Hubble's law while the noncosmological part

exhibits the anomalous behavior. What could

be due to? There are a few possibilities,

none of which is thoroughly tested for full satisfaction: 1. Doppler effect arises from peculiar motions relative to the cosmological rest frame. It is a well known phenomenon in physics. 2. Gravitational redshift arises from compact massive objects,as discussed in general relativity. 3. Spectral coherence discussed by E. Wolf (1986) causes a frequency shift in propagation when light fluctuations in the source are correlated. 4. In the tired light theory a photon of nonzero rest mass loses energy while propagating through space. 5. In the variable mass hypothesis arising from the Machian theory of F. Hoyle and the author (see Narlikar 1977), particles may be created in small and large explosions and those created

MATTER CREATION AND ANOMALOUS REDSHIFTS

25

more recently will have smaller mass and hence larger redshift. I am partial to the last alternative as it seems to fit most of the important features of anomalous redshifts. It explains why there are no anomalous blueshifts seen. It reproduces the observed quasar-galaxy configurations and also provides an explanation for the anomalous redshifts of companion galaxies as older stages of quasars in an evolutionary sequence. For details see (Narlikar and Das 1980, Narlikar and Arp 1993). To what extent can these alternatives provide explanations for the discrepant data? Does matter creation hold the clue to these so-called anomalies? Or, would the discrepancies dwindle away as observations improve? On the other hand, how will theorists explain them if they grow in significance? Clearly these issues have enormous implications for Hubble's law in particular and for cosmology in general.

Acknowledgement

I thank the organizers of this meeting for giving me the opportunity to convey my personal greetings to Jean-Pierre Vigier on this occasion through this particular presentation. He has always looked for original interpretations, and shown us how one needs courage to progress away from the beaten track. My good wishes to him on his eightieth birthday and for a long active life ahead.

References Alpher, R.A. and Herman, R.C. (1948), Nature, 162 774 Alpher, R.A., Bethe, H. and Gammow, G. (1948), Phys. Rev., 73, 80 Arp, H.C. (1987), Quasars, Redshifts and Controversies, Berkeley, Interstellar Media Arp, H.C. (1998), Seeing Red, Apeiron, Montreal Banerjee, S.K. and Narlikar, J.V. (1999), M.N.R.A.S., 307, 73 Banerjee, S.K., Narlikar, J.V., Wickramasinghe, N.C., Hoyle, F. and Burbidge, G. (2000), A.J., 119, 2583 Bondi, H. and Gold, T. (1948), M. N. R. A. S., 108, 252 Burbidge, G. (1968), Ap.J. Lett., 154, L41 Burbidge, G. and Hoyle, F. (1998), Ap.J, 509, L1 Burbidge, E.M., Burbidge, G., Fowler, W.A. and Hoyle, F. (1957), Rev. Mod. Phys., 29, 547 Einstein, A. (1917), Preuss. Akad. Wiss. Berlin, Sitzber, 778, 799, 844 Friedmann, A. (1922), Z. Phys., 10, 377 Friedmann, A. (1924), Z. Phys., 21, 326 Hewitt, A. and Burbidge, G. (1993), Ap.J. Supp., 87, 451 Hoyle, F. (1948), M. N. R. A. S., 108, 372 Hoyle, F. and Narlikar, J.V. (1964), Proc. R. Soc. A., 278, 465 Hoyle, F. and Tayler, R.J. (1964), Nature, 203, 1108 Hoyle, F., Burbidge, G. and Narlikar, J.V. (1993), Ap.J, 410, 437 Hoyle, F., Burbidge, G. and Narlikar, J.V. (1994a), M.N.R.A.S., 267,1007 Hoyle, F., Burbidge, G and Narlikar, J.V. (1994b), A \& A, 289, 729 Hoyle, F., Burbidge, G and Narlikar, J.V. (1995), Proc. R. Soc. A., 448, 191 Hoyle, F., Burbidge, G and Narlikar, J.V. (2000), A Different Approach to Cosmology, Cambridge Hubble, E. (1929), Proc. N. Acad. Sci., 15, 168

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Karlsson, K.G. (1977), A\& A, 58, 237 Lemaitre, G (1927), Ann. Soc. Sci. Bruxelles, 47, 49 Narlikar, J.V. (1977), Ann. Phys. (N. Y.), 107, 325 Narlikar, J.V. and Das, P.K. (1980), Ap.J., 240, 401 Narlikar, J.V. and Arp, H.C. (1993). Ap.J, 405, 51 Narlikar, J.V., Wickramasinghe, N.C., Sachs, R. and Hoyle, F. (1997), Int. J. Mod. Phys. D., 6,125 Nayeri, A., Engineer, S., Narlikar, J.V. and Hoyle, F. (1999), Ap.J, 525,10 Penzias, A. A. and Wilson, R.W. (1965), Ap.J, 142, 419 Sachs, R., Narlikar, J.V. and Hoyle, F. (1996), A\& A, 313, 703 Stockton, A. (1978), Ap.J, 223, 747 Tifft, W. (1996), Ap.J, 468,491 Wagoner, R.V., Fowler, W.A. and Hoyle, F. (1967), Ap.J, 148, 3 Wolf, E. (1986), Phys. Rev. Lett., 56, 1370

THE ORIGIN OF CMBR AS INTRINSIC BLACKBODY CAVITY-QED RESONANCE INHERENT IN THE DYNAMICS OF THE CONTINUOUS STATE TOPOLOGY OF THE DIRAC VACUUM Aplications of Quantum Gravity Part II 1

RICHARD L. AMOROSO The Noetic Advanced Studies Institute - Physics Lab 120 Village Square #49 Orinda, Ca 94563-2502 USA [email protected]

JEAN-PIERRE VIGIER Pierre et Marie Curie Université Gravitation et Cosmologie Relativistes Tour 22 – Boite 142, 4 place Jussieu, 75005 Paris, France

Abstract. The isotropic Cosmic Microwave Background Radiation (CMBR) is reinterpreted as emission from the geometric structure of spacetime. This is postulated to occur in the context of the Wheeler/Feynman transactional radiation law, extended to include the dynamics of spacetime topology in a framework of continuous state Spacetime Cavity-QED (STCQED) where the Planck Blackbody spectrum is described as an equilibrium condition of cosmic redshift as absorption and CMBR as emission. The continuous state spin-exchange compactification dynamics of the Dirac vacuum hyperstructure is shown to gives rise naturally to a 2.735° K Hawking type radiation from the topology of Planck scale micro-black hole hypersurfaces. This process arises from the richer open Kaluza-Klein dimensional structure of a post Bigbang continuous state cosmology.

1. Introduction A putative model of CMBR/Redshift as blackbody emission/absorption equilibrium is predicted to occur in the Cavity-QED (STCQED) Spacetime topology of the polarized Dirac vacuum is presented in terms of a continuous state, periodic dimensional reduction, topological spin-exchange

1. For part I see Amoroos, R.L, Kafatos, M. & Ecimovic, P. , The origin of cosmological redshift in spin exchange vacuum compactification and nonzero rest mass photon anisotropy, in G. Hunter, S. Jeffers & J-P Vigier (eds.) Causality and Locality in Modern Physics, 1998, Dorderecht: Kluwer Academic. 27 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 27-38. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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R. L. AMOROSO AND J-P VIGIER

compactification process. The dynamics of this new CMBR model take place in the context of a post Bigbang continuous state universe (CSU) model where the present is a standing wave of the future-past, (Amoroso, 2002a). The well known Wheeler-Feynman absorber radiation law (1945)was extended by Cramer (1986) and independently by Chu (1993)to include quantum theory; (WFCC) for short. In this preliminary paper, WFCC theory is further extended to include continuous state transformations, futurepast topological dynamics, of a 12D complex

Minkowski spacetime

(Amoroso,

2000, 2002). The 3-torus singularity structure of f-p standing wave dynamics are a foundational principle of the Continuous State Universe (CSU). Thus in WFCC-CSU theory any present state or event is a result of a complex tier of future-past influences as illustrated in unexpanded form by

It is common knowledge that photon emission results from electromagnetic dipole oscillations in boundary transitions of atomic Bohr orbitals. Bohr’s quantization of atomic energy levels is applied to the topology of Spacetime CQED boundary conditions (STCQED) in accordance with equation (1) where spacetime QED cavities of energy to a higher state emission) according to

undergo continuous harmonic transition

(redshift-absorption mode) or to a lower state

(CMBR-

Thus we postulate that boundary conditions

inherent in continuous standing wave spacetime spin exchange cavity compactification dynamics of vacuum topology also satisfy the requirements for photon emission. In metaphorical terms, periodic phases or modes in the continuous spacetime transformation occur where future-past exiplex2 states act as torque moments of CMBR/Redshift BB emission/absorption equilibrium. Compactification appears as localized scalar potentials to standard quantum measurement, but nonlocally, in the WFCC-CSU model, are a continuous transformation of QED or SED hyperdimensional cavities in black body equilibrium. Delocalized compactification dynamics produce a periodic mass equivalency by oscillations of the gravitational potential (GP) providing the action principle for absorption and emission (see section7). Theoretical feasibility of Planck scale Black Holes (BH) has long been demonstrated (Markov, 1966). Thus the CMBR can be considered a form of Hawking radiation (Hawking, 1976) from the hypertiling of the Dirac sea. The CSU is modeled as a type of hyperdimensional Klein bottle, topologically representative of Kant's antinomy of an open/closed spacetime. The hypergeometry of which translates in a metric of comoving Birkhoff spheres (Birkhoff, 1923) where is preserved through all levels of scale (Amoroso, Kafatos & Ecimovic, 1998, Kafatos, Roy & Amoroso, 2000). Taking the Hubble sphere as the arbitrary radius of the observable universe, the GP is opposed within the sphere, not by inflation but by a nonlocal equivalence to the GP, i.e dark energy of the megaverse (Amoroso, 2002). Both CMBR-emission and Redshift-absorption arise from an 'electromotive torque' (17) in the GP equivalent acceleration of the translation of the co-moving topology of higher and lower spacetime dimensions fundamentally equivalent to a Planck scale black body hypersurface.

2 An exiplex (a form of eximer), usually chemistry nomenclature, used to describe an excited, transient, combined state, of two different atomic species (like XeCl) that dissociate back into the constituent atoms rather than reversion to some ground state after photon emission.

THE ORIGIN OF CMBR

29

2. General Properties Of Black BodyRadiation A BB cavity radiates at every possible frequency dependent on the temperature of the walls of the cavity. In thermodynamic equilibrium the amount of energy U(v) depends only on temperature and is independent of the material of the walls or shape of the container. The radiation field behaves like a collection of simple harmonic oscillators that can arbitrarily be chosen to have a set of boundary conditions of dimension L which is repeated periodically through spacetime in all directions. These boundary conditions will yield the same equilibrium radiation as any other boundary conditions, and with this result no walls are actually required because the walls thermodynamically only serve in the conservation of energy (Bohm, 1951); allowing the putative feasability of a STCQED origin for CMBR to be compatible with natural law.

2.1 BLACKBODY CAVITY - COSMOLOGICAL CONSTRAINTS Defining the observable universe as an Einstein 3-sphere, any spherical distribution of matter of arbitrary size (according to the general theorem proven by Birkhoff) (1923) maintains a uniform contribution of the GP with any particle in the volume. Metaphorically the WFCC-CSU model defines the radius of the universe R in terms of a comoving Hubble sphere with the topology of a hyper-Klein bottle. This relation maintains itself through all levels of scale. Therefore Birkhoff’s theorem can apply hyperdimensionally to all matter in the megaverse. This can explain the origin of the cosmological constant, why space appears universally flat and why 3-sphere dark matter is not required to explain galactic rotation since in CSU cosmology (Amoroso,2002a), it is instead a magaversal dark energy. This arbitrary cavity putatively modeling the structure of the universe, as drawn from current astrophysical data, is generally accepted to be a perfect BB radiator of 2.75° K . Einstein introduced the cosmological constant to balance the GP in a static universe. Which he then retracted when Hubble discovered what was erroneously thought to be a Doppler recessional redshift, apparently obviating the need for a cosmological constant. Further Einstein postulated the existence of singularities derived from the field equations of general relativity; from which Friedman suggested that the universe itself originated in a temporal singularity giving rise to the Bigbang model of recent history. It has been shown in Part I (Amoroso, Kafatos & Ecimovic, 1997) that redshift is intrinsic to photon mass anisotropy; suggesting that recession is an observational rather than a Doppler Bigbang effect. When the CMBR was discovered it was interpreted as definitive proof that the Bigbang was the correct model of creation. However, the same observational data may be also interpreted in the manner here. CSU Gravity, which models compactification as a rich dynamic hyperstructure provides an inherent mechanism to balance the GP in a static universe where the CMBR is not a remnant of adiabatic inflation but intrinsic to the equilibrium conditions of Planck scale spacetime CQED or CSED.

2.2 BLACKBODY MICROCAVITY CONSTRAINTS Dirac vacuum CQED boundary conditions are taken to represent the walls of Birkhoff BB-BH microcavities comprised of a tiled stochastic hyperstructure of Planck scale

phase cells with the

lower limit of dimensional size determined by the Heisenberg uncertainty principle with the cavity volume defined by

and the energy for each coordinate defined by

30

R. L. AMOROSO AND J-P VIGIER (Amoroso, 2002). During the continuous cycles of dimensional reduction the

energy

is parallel transported by an energyless Topological Switching3 of higher to lower

dimensionality

without distorting the smoothness of perceived macroscopic realism

because of the standing wave spin exchange process. Although in CSU reality the Planck backcloth is a 11(12)D hypertiling of topologically comoving hyperstructures, not a rigid tiling of 3D cubes with primal fixed compactification as in Bigbang theory.

2.2.1 CMBR Energy Damping by Vacuum Conductivity Planck’s radiation law for a harmonic oscillator is energy per unit time per unit volume. An order of magnitude calculation for the energy of a single transverse CMBR cavity wave mode for the energy density is

According to Lehnert & Roy (1998) energy where R is radius of the universe and r is direction of propagation.

This implies that the energy density has an e-folding decay length

where

conductivity of the vacuum because the conductivity is extremely small. The corresponding energy decay time (damping time for to decay from original value) would be absorption time of the “tired light” redshift absorption effect. This applies to all waves where

is radius of universe.

3. Black Holes Any number of bosons may cohere in a phase cell while Fermions must have energy to occupy the same domain because of the Pauli exclusion principle and therefore must be degenerate in black holes. These Planck volumes considered as the boundary conditions of the cavity ground state, cohere stochastically to embody any required energy configuration. The general expression for BB radiation derived by Planck takes the form:

where

is spectral emittance, and k is the Boltzmann constant. Hawking found a similar

relationship for the hypersurface of a black hole (Hawking, 1974a, 1976). The topology of the Planck backcloth has been considered to be a latticework of micro black holes. The thermodynamic relationship between black hole area and entropy (Beckenstein, 1973) and emittivity (Hawking, 1974a,b; Berezin, 1997) found to occur at the hyperstructure surface of a black hole is putatively developed here for similar emittivity for CMBR black body emission intrinsic to the CQED features of spacetime topology.

3 Topological Switching refers to the optical illusion occurring when fixating on a panel of Necker cubes where a background vertex switches to a foreground vertex; here utilized as a metaphor of how parameters of a higher dimensional topology may interplay harmonically by parallel transport into lower dimensional structures.

THE ORIGIN OF CMBR

31

3.1 SIZE TEMPERATURE RELATIONSHIP OF KERR BLACK HOLES Bekenstein, (1973) suggested a relationship between the thermodynamics of heat flow and the surface temperature of a BH, which led Hawking, 1974a to the finding that all BH's can radiate energy in BB equilibrium because the entropy of a black hole

is related to the surface area A of its event

horizon. Where k is Boltzmann's constant,

(Sung, 1993). This

leads to the expression for the surface temperature of a black hole:

Q = charge, and L= momentum (Sung, 1993). This

where

shows that the BB temperature of a BH is the inverse of its mass, which for a typical Kerr BH represents a temperature of one ° K for a BH a little larger than the moon or for each gm.. Accordingly the Beckenstein - Hawking relationship, while a stellar mass BH has the expected fractional degree temperature, the predicted temperature for microcavity Planck scale BH would be about . Therefore the additional physics of WFCC-CSU spin exchange dynamics must be added to account for the difference in the geometry of a black hole having a fixed internal singularity structure with a lifetime of billions of years and a Planck scale black hole with an open singularity (Amoroso, 2002) rotating at the speed of light c with a Planck time lifetime of While a micro-BH might be considered to have a temperature of billions of degrees Kelvin if the nature of its internal singularity and total entropy is derived through the predictions of GR and bigbang cosmology; because according to GR a singularity occupies no volume and has infinite energy density. But GR breaks down and is known to be incomplete at the quantum level; requiring new physics to describe spacetime quantization. Further, although Einstein said 'spacetime is the ether' (Einstein, 1922) radiation was still considered to be independent of the vacuum, which is now known not to be the case (Amoroso, Kafatos, & Ecimovic, 1997).

3.2 TEMPERATURE RELATIONSHIP OF DIRAC CAVITY 'BLACK HOLES' In the transition from the Newtonian continuum to quantum theory, what still remains to be properly addressed is the ultimate nature of a discrete point. The infinite density Einstein singularity is still too classically rooted. In terms of WFCC-CSU the energy density is delocalized in terms of the equivalent GP of compactification dynamics. Planck scale black body cavities are topologically open nonlocally and spin exchange entropy through a continuous flux of energy; and are not scalar compactified singularities originating in a Bigbang, but constantly accelerate toward an open propagating ground that is never reached nonlocally. The inertia inherent in this dynamic results in the intrinsic

4.

2.75° K CMBR

Spin Exchange

Starting with the Hawking radiation modification of the Planck BB relationship as applied to BH surface dynamics, the requirement for application to a quantum BB QED cavity generally defined as

32 the phase space of

R. L. AMOROSO AND J-P VIGIER in (5) is the addition of spin exchange parameters. Where

N is the complex sum of Planck hyperunits comprising one BB QED microcavity. Spin dynamics can be readily described using the density matrix formalism. Spin states are represented as linear combinations of

and

states corresponding to the spin eigenvalues; and

can be used in terms of the wave function to determine the value of spin characteristics

The density matrix

is made up of the spin coupling coefficients

and

The diagonal

elements correspond to real local spin orientations, and the nondiagonal elements correspond to complex quantities representing spin projection on planes perpendicular to axes of quantization. For the purposes of discussion any arbitrary axis may be chosen as an axis of quantization; but in the spin exchange process the geometry of the complex topology of the Argand plane transforms from real to complex in the retiling of compactification dynamics. The variance in the diagonal elements effects the longitudinal spin polarization along the axis of quantization; and the nondiagonal variances effect transverse spin polarizations. It is the phase of the elements that determine the angle of spin coupling with each dimensional axis. This relates CMBR emission/absorption to the cycle of torque moments. The mass equivalent inertial properties comprising the linear and angular momentum components of spin exchanged in the nonlocal compactification structure allow the Dirac vacuum to maintain perfect BB equilibrium inside the scale invariant Hubble Birkhoff sphere.

5. Spontaneous Emission Of CMBR By Spacetime Cavity QED This preliminary model for continuous spontaneous emission of STCMBR directly from CQED dynamics of the stochastic properties of the Dirac sea, obviates CMBR origin as the relic of an initial state Bigbang cosmology as the standard model has predicted. In this model we make one speculative new assumption that is not based on the published body of empirical data for CQED. Spontaneous emission by atomic coupling to vacuum zero-point fluctuations of the Dirac sea is already an integral part of CQED both in the laboratory and theory; here we postulate that a similar process can occur in free space. In classical electrodynamics the vacuum has no fluctuation; by contrast quantum radiation can be viewed as partly due to emission stimulated by vacuum zero-point fluctuations. The literature on CQED is rich in descriptions of the nature of spontaneous emission of radiation by atoms in a cavity (Berman, 1994) We begin development by choosing, for historical reasons, the upper limit of the number of atoms in the vacuum of space to the figure of one atom per cubic centimeter as derived by Eddington, (1930). This figure could be considered arbitrary, but for

THE ORIGIN OF CMBR

33

our purposes it is sufficient to note that there are sufficient free atomic panicles moving in space for spontaneous CSU-STCQED emission of WFCC-CMBR. Charged particles are coupled to the electromagnetic radiation field at a fundamental level. Even in a vacuum, an atom is purturbed by the zero-point field, and this coupling is responsible for some basic phenomena such as the Lamb shift and spontaneous radiative decay. (E.A. Hinds, 1993)

Recent developments in CQED have included descriptions of emission by Rydberg atoms in microwave cavities that include optical frequencies. (Carmichael et al, 1993 ; Jhe et al, 1987; Heinzen et al, 1987; Raizezn et al, 1989; Zhu et al, 1990; Thompson et al, 1992 and Rempe et al, 1991. The Rydberg formula for atomic spectra is related to the binding energy of an electron by:

where

is the magnetic permeability which is the ratio of the magnetic flux density B of an atom to

an external field strength H.

which is also related to the permeability of free space

the Coulomb constant k and the magnetic constant

by

where is the vacuum permittivity of free space; m and e are mass and charge of an electron respectively, c is the speed of light and h is Planck’s constant. In the nonperturbative regime strength of the dipole coupling is larger than the dissipation rate and quantum mechanical effects have been shown to include multi-photon resonance, frequency shifts and atomic two state behavior at vacuum Rabi resonance, the latter of which will be of most interest in our discussion (Carmichael, et al,1993).

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R. L. AMOROSO AND J-P VIGIER

Spontaneous emission requires only a single quantum so the internal state of the atom-vacuum coupled cavity system may be described by the simple quantum basis

where

and

are the Fock photon states and

and

are two states of the Rabi/Rydberg

atom. Momentum operators x(p) and y(p) relate center of mass and atom ground state dynamics where a master equation can describe the two state atom interacting with the mode of the vacuum cavity momentum distribution after spontaneous emission and the emission spectra (Ren et al, 1992; Carmichael et al, 1993).

where the a’s are the boson creation and annihilation operators and the sigma’s the raising and lowering operators for the atom (Carmichael, 1993). We assume that the atom acts classically as a free wave-packet where the internal state of the system which can be described by

describes

With where

And

In addition to the atoms classical motions as a free wave-packet, the vacuum coupled system when excited, has two harmonic potentials related to the atoms motion and spontaneous emission process as in the following from Carmichael, 1993.

Vacuum Rabi atomic orbital splitting is the normal mode splitting of the coupled harmonic oscillators; one mode describing the atomic dipole and the other the cavity field mode. This system of coupled harmonic oscillation is extremely versatile and can be applied to describe Dirac vacuum cavity QED

THE ORIGIN OF CMBR

35

emission of the CMBR when driven by the vacuum quantum mechanical stochastic field. Our application to the CMBR is based on the work of Agarwal, 1991 and Carmichael, 1993 on the nature of stochastic driving fields in CQED. Starting with the hamiltonian for a coupled harmonic oscillator

where are the coordinates and momenta of the one dimensional oscillator; with the subscripts A and C referring to atomic dipole and cavity modes respectively of the Rabi/Rydberg atom in free space. The oscillator coupling is modulated by the Doppler frequency with phase modulating the dipole coupling constant for atomic motion; the equations of which take the form of equations (12) (Carmichael, 1993). This has been a non-perturbative formalism much simpler to interpret than a QED perturbative expansion that we deem sufficient for this stage of development of the Vigier-Amoroso CQED CMBR Dirac spacetime emission theory.

6. Possibility Of Blackbody Emission From Continuous Spacetime Compactification It is also suggested that further development of the CQED model of CMBR emission could be extended to include spontaneous emission from the continuous dimensional reduction process of Compactification. This would follow from modeling spacetime cavity dynamics in a manner similar to that in atomic theory for Bohr orbitals. In reviewing atomic theory Bohm, (1967 states : Inside an atom, in a state of definite energy, the wave function is large only in a toroidal region surrounding the radius predicted by the Bohr orbit for that energy level. Of course the toroid is not sharply bounded, but reaches maximum in this region and rapidly becomes negligable outside it. The next Bohr orbit would appear the same but would have a larger radius confining with wave vector

and propagated

with the probability of finding a particle at a given region proportional

to Since f is uniform in value over the toroid it is highly probable to find the particle where the Bohr orbit says it should be (Bohm, 1967)

A torus is generated by rotating a circle about an extended line in its plane where the circles become a continuous ring. According to the equation for a torus. where r is the radius of the rotating circle and R is the distance between the center of the circle and the axis of rotation. The volume of the torus is and the surface area is in the above Cartesian formula the z axis is the axis of rotation. Electron charged particle spherical domains fill the toroidal volume of the atomic orbit by their wave motion. If a photon of specific quanta is emitted while an electron is resident in an upper more excited Bohr orbit, the radius of the orbit drops back down to the next lower energy level decreasing the volume of the torus in the emission process.

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R. L. AMOROSO AND J-P VIGIER

We suggest that these toroidal orbital domains have properties similar to QED cavities and apply this structure to topological switching during dimensional reduction in the continuous state universe (CSU) model (Amoroso, 2000, 2002). To summarize pertinent aspects of CSU cosmology: 1. Compactification did not occur immediately after a big bang singularity, but is a continuous process of dimensional reduction by topological switching in view of the WheelerFeynman absorber model where the present is continuously recreated out of the future-past. Singularities in the CSU are not point like, but dynamic wormhole like objects able to translate extension, time and energy. 2. The higher or compactified dimensions are not a subspace of our Minkowski 3(4)D reality, but our reality is a subspace of a higher 12D megaverse of three 3(4)D Minkowski spacetime packages. During the spin-exchange process of dimensional reduction by topological switching two things pertinent to the discussion at hand : 1. There is a transmutation of dimensional form from extension to time to energy ; in a sense like squeezing out a sponge as the current Minkowski spacetime package recedes into the past down to the Planck scale ; or like an accordian in terms of the future-past recreating the present. 2. There is a tension in this process that could be like string tension in superstring theory that allows only specific loci or pathways to the dimensional reduction process during creation of the transient Planck scale domain . Even though there are diccrete aspects to this process it appears continuous FAPP from the macroscopic level (like the film of a movie); the dynamics of which are like a harmonic oscillator. With the brief outline of CSU parameters in mind, the theory proposes that at specific modes in the periodicity of the Planck scale pinch effect, cavities of specific volume reminiscent of Bohr toroidal atomic orbits occur. It is proposed rather speculatively at present that these cavities, when energized by stochastically driven modes in the Dirac ether or during the torque moment of excess energy during the continuous compactification process, or a combination of the two as in standard CQED theory of Rabi/Rydberg spontaneous emission, microwave photons of the CMBR type could be emitted spontaneously from the vacuum during exiplex torque moments. This obviously suggests that Bohr atomic orbital state reduction is not the only process of photon emission; (or spacetime modes are more fundamental) but that the process is also possible within toroidal boundary conditions in spacetime itself when in a phase mode acting like an atomic volume. A conceptualization of a Planck scale cavity during photon emission is represented in figure 1c with nine dimensions suppressed.

7. Deriving The Topological Action Principle For Dirac Cavity CMBR Emission Well known forms of the Schrodinger equation central to quantum theory have correspondence to Newton’s second law of motion

which is also chosen as the formal basis for CSU

CMBR emission theory. A more rigorous defense of the logic for this choice will be given elsewhere. Here only the postulate that CMBR emission is governed by a unified electo-gravitation action principle is stated.

Neither Newtonian

(although it was derived from f = ma) nor

Einsteinian gravitation is utilized for deriving the advanced/retarded description of CMBR emission because the related structural-phenomenological boundary conditions of the cavities topology has no relation to classical dynamics which both of these theories do. Newton’s gravitation law also

THE ORIGIN OF CMBR

37

contains a constant of undesired dimensionality; whereas f= ma is without dimensionality. For similar reasons Einstein’s gravity is also not chosen. Since relativistic energy momentum and not mass is required, first we substitute Einstein’s mass energy relation

into Newton’s second law and obtain:

Where

will become the unitary emission/absorption force and E arises from the complex self-organized electro-gravitational Geon energy related to

of the CSU complex Minkowski metric

as defined in the basic premise of CSU theory (Amoroso, 2002) where

and

E is scale invariant through all levels of the CSU beginning at the highest level in the supralocal Megaverse as a hyperdimensional Wheeler Geon (Wheeler, 1955). A Geon is a ball of photons of sufficient size that it will self cohere through gravitational action. At the micro level the Geon becomes synonymous with the E term and quantized as a unit of Einstein’s, the fundamental physical quantity defined as a ‘mole or Avogadro number of photons’. Next the equation is generalized for the CSU as derived from the work of Kafatos, Roy & Amoroso, 2000. Taking an axiomatic approach to cosmological scaling, such that all lengths in the universe are scale invariant, we begin with the heuristic relation that or where represents the rate of change of scale in the universe. This corresponds to the Hubble relation for perceived expansion of the universe where continuing for final substitution we have

and

or substituting Since

So The

terms cancel and we are left with:

Which is the formalism for the fundamental unitary action equilibrium conditions of the GP. It should be noted that R is a complex rotational length and could also be derived in terms of angular momentum spacetime spinors or Penrose twistors at higher levels closer to domains described by conventional theory. But the derivation above is more fundamental to CSU CMBR. The Hubble Einstein 3-sphere, a subspace in CSU cosmology, is covered by the scale invariant hyper-geon (unified) field. The spin exchange mechanism of continuous dimensional reduction-compactification dissipates the putative heat predicted by gauge theory for the Planck scale BH backcloth (Markov, 1966; Sung, 1993). The free energy for CMBR emission during the periodic exiplex moment arises by parallel transport during continuous dimensional reduction. Spacial dimensions, by the boundary of a boundary = 0 condition, first transport to temporal dimensionality (Ramon & Rauscher, 1980) and then to energy (Cardone et al, 1999) This key concept will be clarified in an ensuing paper.

8. Summary A preliminary formalism for CMBR emission and tired light redshift absorption as BB equilibrium from the continuous state topological dynamics of the Dirac vacuum in a CSU has been presented. This has taken two possible forms: 1. A stochastically driven CQED effect on Eddington free space Rabi/Rydberg atoms coupled to vacuum zero-point field fluctuations. 2. A composite exiplex of

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R. L. AMOROSO AND J-P VIGIER

advanced - retarded spacetime topological cavity modes which may act as an atom-cavity formed on the basis of gravito-quantum coherence effects by unitary action of

molecule

Both postulated

by only two new theoretical concepts, from already observed CQED effects in the laboratory: 1. A Dirac type vacuum coupling between the atom and vacuum cavities of the structure of spacetime itself, and 2. CMBR photon emission can also occur from the Bohr-type boundary conditions of spacetime topology without the presence of an atom with E transport by topological switching in D-reduction of BH's have been demonstrated to emit BB radiation in the quasiclassical limit, and the lower limit has been shown to be the Plank mass providing a firm theoretical foundation for intrinsic vacuum emmitivity. A non inflationary origin of CMBR obviates the Bigbang requiring reinterpretation of the standard cosmological model with profound implications for the future of cosmological theory.

Acknowledgment Appreciation to Bo Lehnert for helpful discussions in preparation of section 2.2.1

References Agarwal, G.S., 1991, Additional vacuum-field Rabi splittings in cavity QED, Phys Rev A, 43:5, 2595-2598. Amoroso, R.L., Kafatos, M. & Ecimovic, P. 1998. The Origin of Cosmological Redshift in Spin Exchange Vacuum Compactification and Nonzero Rest Mass Photon Anisotrophy. In G. Hunter, S. Jeffers and J-P Vigier Eds. Causality and Locality in Modern Physics, Dordrecht: Kluwer. Amoroso, R.L., 2002, Developing the cosmology of a continuous state universe, this volume, pp.59-64. Bekenstein, J.D. 1973. Black holes and entropy, Physical Review D, V.7 N8, 2333-2346. Berezin, V. 1997. Quantum black hole model and Hawking's radiation, Phys. Rev. D, V.55, N4, 2139-2151. Berman, P.R. Ed. 1994. Cavity Quantum Electrodynamics, New York, Academic Press. Birkhoff,G.D. 1923, Relativity and Modern Physics, Cambridge: Harvard Univ. Press. Bohm, D. (1951) Quantum Theory. Englewood Cliffs, Prentice-Hall. Cardone, F., Francaviglia, M. and Mignani, R. 1999, Energy as a fifth dimension, Found. Phys. L. 12:4, 347-69. Carmichael, H.J., 1993, Phys. Rev.Let. 70:15, 2273-2276. Chu, S-Y, 1993, Physical Review Letters 71, 2847. Cramer, J, 1976, The transactional interpretation of quantum mechanics, Reviews of Modern Physics, 58, 647. Eddington, A.S., 1930, Internal Constitution of the Stars, Cambridge: University Press. Einstein, A. 1922. Sidelights on Relativity, London, Methuen & Co. Feynman, R.P. (1961) Quantum Electrodynamics. New York: Benjamin. Haroche, S. & Raimond, J-M. 1993. Cavity Quantum Electrodynamics. Scientific American. Hawking, S.W. 1976, Black holes and thermodynamics, Physical Review D, V 13, No.2, 191-197. Hawking, S.W. 1974a, Black hole explosions? Nature, v 248. 30-31. Hawking, S.W.1974b, The anisotropy of the universe at large times. In: IAU Symposium No. 63 on Confrontation of Cosmological Theories with Observational Data, Ed: M.S. Longair, Dordrecht, Netherlands. Heinzen, D.J., Feld, M.S., 1987, Phys. Rev. Let. 59:23, 2623-2626. Jhe, W., Anderson, A. Hinds, E.A., Meschede, D., Haroche.S, 1987, Phys. Rev. Let. 58:7, 666-669. Markov, M.A. 1966, Zh. Eksp. Theor. Fiz. v51, p. 878. Milonni, P. 1994. The Quantum Vacuum, San Diego, Academic Press. Raizezn, M.G., Thompson, R.J., Brecha, R.J., Kimble, H.J. & Carmichael, H.J.,1989, Phys. Rev. L. 63:3, 240-3. Ramon, C. and Rauscher, E., 1980, Superluminal transformations in complex Minkowski spaces, Foundations of Physics 10:7/8, 661-669. Rempe, G., 1993, Contemp. Phys., 34:3, 119-129 Ren, W., Cresser, J.D., and Carmichael, J.H., 1992, Phys. Rev. A, 46, 7162 Sung, J.C. 1993. Pixels of Space-Time, Woburn, Scientific Publications. Thompson, R.J., Rempe, G. & Kimble, H.J., 1992, Phys. Rev. Let. 68:8, 1132-1135. Wheeler, J.A., 1955, Geons, Physical Review, 97:2, pp. 511-536. Zhu, Y, 1990, Phys. Rev. Lett. 64 , 2499.

SOME NEW RESULTS IN THEORETICAL COSMOLOGY

WOLFGANG RINDLER The University of Texas at Dallas – Physics Dept. Richardson TX 75083-0688 [email protected]

Part 1 Abstract

After a general introduction to the standard Friedman models we discuss the topology of the big bang and the horizon structure of inflationary universes. 1. Introduction

Some fifty years ago, when I first began to study cosmology, observations were few and theoretical cosmology was not greatly restricted by specific data. Cosmology was more speculation than science. Fanciful ideas could flourish freely. One of the most attractive of these ideas was the steady state theory ("SST") invented by Bondi and Gold in 1948 and later elaborated by Hoyle and Narlikar and others. In retrospect it is a little embarrassing to remember how readily and fervently one believed in that theory simply on the strength of its philosophical appeal. You will recall that it took account of the expansion of the universe by postulating that matter was being created continuously so as to keep the average density and thus the average appearance of the universe constant for all eternity. The SST cured one specific problem: the Hubble expansion of the universe was then thought to be about ten times bigger than it actually is, and so the universe -- if it started with a big bang -- seemed very young, younger, in fact, than the age of the earth as determined by radiocarbon dating. Obviously an eternal universe solves this "age problem". But it had already been solved along quite different lines by Lemaitre with a lambda-term in Einstein's field equations. When in the fifties and sixties the observations of the radio astronomers began to accumulate and the data spoke against the SST, most people eventually abandoned it. But more emotion than meets the eye often goes into our apparently rational science. Dennis Sciama, well known for his many fine contributions to cosmology and the teacher of people like Stephen Hawking, George Ellis, Martin Rees and others, wrote in his memoirs that the day on which he finally had to give up the SST under the weight of the evidence was one of the saddest in all his life. And he was not alone in this sentiment. Today inflationary cosmology is all the rage. But I cannot help drawing parallels to the SST of the fifties. Both theories are based on philosophically attractive 39 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 39-50. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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hypotheses outside the standard physics of the day, and both purport to solve certain outstanding problems. In the case of inflation, one of these problems is the homogeneity of the universe. But as Schücking has said, the homogeneity of the inflationary universe is bought at an exorbitant cost in credibility: apart from yet untested physics, we are getting some 1056 other now independent universes that nobody needed... Since the late fifties and the coming of age of radio astronomy, cosmological data have been coming in at an ever-accelerating rate. Clearly cosmology has become a hard science. So much so that today we are on the verge of having answers to at least two of the most basic questions: is the universe finite or infinite, and will it expand indefinitely or will it eventually halt and re-collapse to a big crunch. Already this last alternative seems to be ruled out by the latest data from supernova spectra, which indicate that the expansion is accelerating: the death of our universe will be cold rather than hot. 2. The Friedman Models

What I would like to do in this lecture is to address a few of my favorite topics, some quite new like the structure of the big bang and the explanation of the enhanced horizon size in inflationary universes, and some not so new, like Milne's "toy" universe and the phase diagram, for which I merely want to make propaganda. So let us begin at the beginning. One of the most striking things about the universe, apart from its immensity, is its regularity. On the largest scale it seems to be both homogeneous and isotropic. And this is a piece of great good luck for cosmologists trying to construct models for it. As pointed out by A.G.Walker, cosmological homogeneity means that the logbooks that observers on any galaxy ("fundamental observers") can keep are identical except for an arbitrary choice of zero point of time. Matching all the logbooks synchronizes all the fundamental observers' clocks. In this way a homogeneous universe defines a unique time, called "cosmic time", t. Its progress is determined by the standard clocks of all the fundamental observers, and its simultaneities (or moments) are determined by equal local states (e.g. of the density, the expansion rate, space curvature, time elapsed since the big bang, etc.) But testing for homogeneity is tricky, because we see distant regions, as they were perhaps billions of years ago, not as they are today. On the other hand, we are fortunate: we see almost perfect isotropy around us. The modern modesty principle suggests that we are not so special as to be at the center of the universe. So if we see isotropy around us, every other fundamental observer will see isotropy too. But then homogeneity follows. For if region A evolved differently from region B, this would be perceived as anisotropy from region C equidistant from A and B. To construct a mathematical model of such a universe we need general relativity. Luckily, the facts of homogeneity and isotropy make the task fairly straightforward. Let me give you a totally unrigorous argument. Special relativity, where gravity is switched off, plays out in flat Minkowski spacetime with metric

NEW RESULTS IN COSMOLOGY

41

As soon as there is gravity, spacetime becomes curved. Now, at each "cosmic moment" t=const, the universe must be a 3-space of constant curvature, because of homogeneity and isotropy. So we replace the Euclidean space-metric by the metric of a 3-space of constant curvature where k is the "curvature index" and takes only the values 1, -1, or 0, while R is the radius of curvature (when ) and usually depends on time, So we have

Even when k = 0 we can write this in the form of (think of sphere!)

where R(t) is now called the "expansion factor" and the coordinates of are attached to the galaxies permanently ("comoving coordinates"). This is the so-called Friedman-Robertson-Walker ("FRW") metric, discovered by Friedman in 1922 and elaborated by Robertson and Walker in the thirties. Friedman is really the father of modern cosmology. To him belongs the enormous distinction of being the first ever, in the history of mankind, to envision a dynamic universe, a universe that moves under its own gravity. To learn how it moves, we must apply Einstein's field equations to its metric (1). Recall that the field equations relate the geometry to the sources. These sources, i.e. the contents of the universe, are forced by the assumed isotropy of the FRW model to be a "perfect fluid" characterized only by its density and its isotropic pressure p. In the idealized model, the actual contents of the universe are ground up and smoothly redistributed with the same overall motion pattern. It is usually assumed that near the big bang the universe was radiation-dominated, whereas after it was matterdominated with negligible pressure: one speaks of a "dust" universe. For this dust phase the field equations reduce to one equation of continuity:

where G is the constant of gravity, and one equation of motion ("Friedman's differential equation"):

where the over dot denotes d/dt and

is the so-called cosmological constant.

A large variety of models are compatible with these equations, depending on the value of k and of the parameters C and There are models that contract from infinity and re-expand, and others with a big bang in the infinite past. Those are of no practical

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WOLFGANG RINDLER

interest. We here consider only models having a big-bang origin (R = 0) and a finite present age These models can be positively or negatively curved (k = 1 or k = –1) or flat (k = 0). They can either re-collapse ("oscillating" models) or expand indefinitely. The "phase diagram", Figure 1, contains a most useful summary of these solutions, in terms of the present values (suffix zero) of (the Hubble expansion parameter), (the present age), (the dimensionless density parameter) and (the dimensionless deceleration parameter). Each point in the diagram represents the present state of one of the Friedman universes. If we knew the product and ( or alternatively ), we would know precisely which universe we live in. The diagram exhibits the important boundaries between k > 0 and k < 0 universes; between and universes: and between oscillatory and nonoscillatory universes; the latter boundary consists of the line for and the dotted line for Present observations suggest and Augmented by realistic estimates for and these numbers strongly favor a non-oscillating universe: but it is too early to decide between k > 0 and k < 0. The inflationary hypothesis favors k = 0. 3. Milne's Model And The Point-Like Big Bang

The total volume of a positively curved universe (a 3-sphere) is finite and the big bang presents no topological problems. It is a singular point-event, before which neither space nor time existed. If, on the other hand, the universe is negatively curved (and we assume no "funny" artificial topology), its volume as well as its matter content is infinite. And this is true at every cosmic moment, no matter how close to the big bang. So can the big bang be point-like in this case, or is it infinitely extended? To understand this problem, I like to make propaganda for an old "toy" model-universe discussed by Milne in 1934, where the topology is quite transparent. There is a solution of Friedman's differential equation (2) that has and C = O (which corresponds to G = 0 and thus to gravity being "switched off"). Under these conditions the model must live in special-relativistic Minkowski space! Its Friedman description, as we see at once from eq. (2), must have k = -1 and R = ct; regarded at successive cosmic moments it is thus infinitely extended and expands at a constant rate. Milne found that by slicing differently through the fundamental-observer worldlines, he could describe the model as an expanding finite ball in an ordinary Euclidean inertial frame. Any one of the fundamental observers can be the center of the ball, and the others move radially away from that one at all speeds short of the speed of light. The time T used in this view is the "ordinary" inertial time of the central observer. Cosmic tints t is indicated on the standard clocks carried by all the other fundamental observers, starting from zero at the big bang. At a given T-instant these clocks read more and more behind T (by time dilation) as we go towards the outer unattained edge of the ball, which moves at the speed of light. The infinite cosmic-time sections consist of infinitely many identical origin-neighborhoods of Milne's ball.

NEW RESULTS IN COSMOLOGY

43

I have recently demonstrated (in a paper to be published elsewhere) that even in the general k = -1 case, the universe can be sliced into finite sections and thus be regarded as an expanding ball -- though not in flat 3-space -- starting with a point-like big bang. Here the question arises how an infinite amount of matter can be contained in a finite volume. (This question has no answer in Milne's model, where the "matter" can only consist of geometric points.) In the general case the answer hinges on the phenomenon of gravitational collapse. Just as in the case of a black hole, there comes a stage in the collapse of a collapsing universe where the collapse cannot be halted and the concentration of matter must increase without limit. Playing such a collapse in reverse, we get a big bang near which the matter concentration becomes infinite. In the picture of the universe as an expanding ball, this infinite concentration occurs towards the edge, and integrating the matter over the volume yields infinity.

4. Particle Horizons

The last topic I want to examine in some detail is horizons. As we look with ever better telescopes into the night sky, we see ever farther and thus fainter galaxies. And since light travels at finite speed, we see the distant galaxies as they were at ever-earlier times. In principle, there are two barriers to this looking backwards in time. The first we reach when the age of the matter we see has dropped about 500,000 years after the big bang, the so-called recombination time. Earlier than that, the universe was not transparent to light: ionized matter interacting with radiation formed and opaque fluid. (Later on the ions and electrons combined into atoms and no longer interacted strongly with photons.) However, if we could "see" with neutrinos instead of photons, we could see through this barrier -- all the way to the big bang: in principle we could receive neutrinos emitted at the big bang itself. So as we look in any direction (say with neutrinos -- these are purely theoretical considerations) as far as the big bang, we may well believe that we have surveyed all the matter of the universe. But this is far from being the case. We have merely reached the second barrier, our "particle horizon", which we shall now discuss. A very good model of a closed universe (k = 1) is provided by an ordinary balloon that is being blown up. Galaxies are ink dots uniformly distributed over the balloon. Photons are little bugs crawling over the balloon along great circles, always at the same speed. Such "balloons" of radius R (t) are actually sub universes of the full 3-D universe. (In the cases k = 0 and k = -1 the balloon becomes a plane rubber sheet or a rubber saddle, respectively.) Now cover the balloon with silver dollars. Their rims momentarily are the light fronts (or neutrino fronts) emitted at their center at the big bang. (At the big bang space itself expanded much faster around each particle than the first light front emitted at that particle!) As my creation light front (read: my particle horizon) passes you, you see me for the very first time, at my creation. By symmetry, at that same cosmic instant I see you at creation. So my particle horizon at any instant is as far back into the past as I can possibly see at that instant. But that is only a very small fraction of the entire universe. And thereby hangs a puzzle. How can we explain the observed homogeneity of the universe -- since the influence from each particle cannot have spread further than its

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particle horizon? So the universe cannot have homogenized itself. Of course, in the description we gave (the nice spherical balloon, the nice circular horizons), the big bang itself is assumed a priori to be so delicately designed as to produce homogeneity and isotropy from the start. If there were some flaw in the execution, the universe could not correct itself. Thus if among all the natural miracles you find such delicately designed initial conditions unacceptable, you have the "horizon problem" or "homogeneity problem". Inflationists claim to have cured the homogeneity problem by producing vastly larger horizons -- though it is still not clear how this would lead to isotropy. The argument usually given is that during the inflationary period, when the universe is stretched by a factor of (in seconds!!), the horizons are also stretched by that factor, and that is what makes them so big. Let us look at the situation in some detail. Even inflationists agree that today the observable universe is of FRW type. So in principle we can determine the best-fitting FRW dust model. We can follow this model back in time. Somewhere around recombination time, for increased accuracy, we can replace the dust-dominated model by a radiation-dominated model -- and inflationists would have no quarrel with that. Only after we come down to a radius do the continuations backwards in time diverge. (We are now within seconds of the big bang!) Whereas the FRW model just keeps on going according to general-relativistic dynamics, inflation has a sudden exponential drop. In the FRW model all the matter of the universe is created at the big bang. In the inflationary model, by contrast, the big bang is a mere "big whimper", where only a minute fraction of the matter is created -- the bulk being created during the inflation, when the density remains constant! ("The ultimate free lunch" according to Alan Guth, who invented inflation.) Now is it really the stretching that makes the inflationary horizons big? Evidently not, since the standard model must stretch equally to reach only not quite so fast. The real reason why inflation produces larger horizons is that a whimper produces larger horizons than a bang. The advantage of the inflationary horizon over the standard horizon was present even before inflation! For it can be shown that in radiative FRW models (and all models are radiative near the big bang) the radius of the particle horizon near the big bang is given by

and is thus proportional to Since after inflationary and noninflationary FRW models have the same density, the density of the standard model at is the same as that of the inflationary model at the onset of inflation, i.e. at Let us assume that the whimper and its development until the onset of inflation is strictly FRW. Then the horizon size in the whimper universe at is identical to that in the standard universe at What inflation then does to the whimper horizon is pure gain: it expands it by more than a factor of By that factor, owing to the space expansion, and by more, since the horizon light front does not

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stand still during inflation. So at the end of inflation the ratio of the horizon radii is better than in favor of the inflationary model. Thereafter that ratio decreases owing to the equal motion of the horizon light fronts through space, but the advantage is permanent. Even if the whimper is chaotic (and inflationists clearly do not posit perfectly Friedmannian initial conditions), the Friedmannian horizon-density relation may well be indicative of a similar relation holding locally when the density is not homogeneous. The same horizon enhancing argument would then still apply to inflation.

Part 2

In this second part, added only for the printed version of this lecture. I provide some of the mathematics that I omitted before. There is little point in repeating the rigorous derivation of the Friedman-Robertson-Walker metric (1) for homogeneous-isotropic universes, since that can be found in all the textbooks [1,2,3,4]. Equally standard is the application of Einstein's field equations to this metric, which yields, first, the conservation equation for the later, matter-dominated universe (zero pressure,

and

for the early, radiation-dominated universe

It also yields the evolution equation (2), often called Friedman's differential equation, when matter dominates, and a similar equation with the first term on the right side replaced by where when radiation dominates. The very interesting phase diagram, Figure 1, is obtained by integrating the equation for the age of the universe -- for which the first ~500,000 years can be safely fudged and the entire universe treated as matter-dominated. One defines the Hubble parameter the density parameter and the deceleration parameter as we did after eq. (2) in Part 1, whereupon one finds, directly from the Friedman eq. (2) and its time-derivative, the following important identities:

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With these one goes back into eq. (2), rewriting it in the form

This can at once be integrated. Since we are interested in the present age models, we assume R(0) = 0 and perform a definite integration:

of big-bang

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the empty brace denotes the braced expression in (6), and is defined by the last equation. This function can be machine integrated, whereupon we can tabulate and graph corresponding values of and with

The line by (4) corresponds to and separates universes from those with The line k = 0 by (5) corresponds to and separates positively from negatively curved universes. Solutions of

Friedman's equation for which has a zero at a finite time are necessarily "oscillatory". For this to happen, we need the right side of (2), regarded as a function of R, to have a zero, which is certainly the case when but it also happens when that right side has a minimum less than zero. The condition for that is easily seen to be which by (3) - (5), corresponds to the region below the locus

shown as a dotted line in Figure 1. Now for the mathematics of the Milne model. Looking at eq. (2), we see that when C = 0 (no gravity) and there is only one non-static solution: k = -1, R = ct. This is a uniformly expanding, negatively curved, infinite universe, with FRW metric (1) specialized to

where is the metric of the unit sphere. Milne realized that because of the absence of gravity and of this model must live in Minkowski space. Indeed, if we set

the metric (9) becomes Minkowskian:

The worldlines of the substratum --

-- now become the bundle of straight lines

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filling the forward light cone

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with vertex at the big bang

(R = 0).

Every time-cut through this bundle, yields a 3-sphere of radius Milne's "infinite" universe now appears as a finite ball of matter expanding uniformly from zero radius at It is my contention that even the most general "infinite" FRW universes can be similarly regarded as expanding finite balls -- though this time not with flat interiors -i.e., they all permit, like the Milne universe, a space like foliation into finite ball-like sections. For this purpose I consider a typical k = -1, nonempty FRW universe (2), choosing the radiative equation of state, since our main interest lies in the vicinity of the big bang:

in units making c = 1. This universe I then Embed in five dimensional Minkowski space essentially following a method originally developed by Robertson [5]. The usual embedding procedure is to introduce a redundant fifth variable (here denoted by ) into the metric, in hopes of making it five-dimensionally flat The relation of that fifth variable to the others determines a hyper surface, which represents the original space as a subspace of Here our FRW universe has the metric

with R given by (13). A useful trick (due to Schücking) is to replace t by R as coordinate, using (13), whereupon (14) becomes

This suggests defining

which has the effect of replacing the last term in (15) by procedure (10), and define and r by

Finally we follow Milne's

thus converting (15) into

which evidently represents conoidal hyyersurface

Our FRW universe is now embedded therein as the

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which osculates tile big-bang light cone just as does the Milne universe. However, we have here mapped the FRW universe twice: once into the "front" half (u > 0) of the conoid, once into its "back" half (u < 0). The null lines (u= 0) do not belong to the map. And now it is seen at once that successive sections t = const represent successively bigger balls, of surface area and that the big bang was a point-event. As for the last topic of my lecture, the horizon enhancement in inflationary universes, the only mathematics I omitted earlier was the proof that the particle horizon in FRW universes near the big bang is given by the formula

now again working in full units. Consider the FRW metric in the form (14), where the sin h term -- characterizing k = -1 universes -- is irrelevant for our present purposes ( is replaced by if k = 1, and by if k = 0 ). Radial light signals in any case satisfy A "creation" light signal, emitted forward at the big bang (i.e., at t = 0) will therefore satisfy

If this integral converges, i.e., if it yields a finite value for that light signal is still within the substratum at time t and determines, in fact, the particle horizon. For its proper distance from the origin at an early time we have, utilizing (13),

Dropping the index 1 and substituting for

we then obtain (19).

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Acknowledgements I dedicate this paper to Jean-Pierre Vigier on the occasion of his eightieth birthday. And special thanks to Edmond Chouinard for typesetting my manuscript.

References 1. C.W.Misner, K.S.Thorne, J.A.Wheeler, Gravitation, Freeman, San Francisco, 1973 2. H.C.Ohanian, R.Ruffini, Gravitation and Spacetime, 2nd. ed., Norton, New York, 1994 (This reference also deals with inflation.) 3. W.Rindler, Essential Relativity, 2nd.ed., Springer, New York, 1977 (This reference describes the Milne model in detail. Also the phase diagram.) 4. S.Weinberg, Gravitation and Cosmology, Wiley, New York, 1972 5. H.P.Robertson, Proc. Nat. Acad. Sci. 15(1929) 822.

WHITEHEAD MEETS FEYNMAN AND THE BIG BANG

GEOFFREY F. CHEW Theoretical Physics Group, Physics Division Lawrence Berkeley National Laboratory Berkeley, California 94720, U.S.A.

Abstract.. Historical quantum cosmology (HQC) is based not on matter but on a chain of local history—a chain lengthened by many local steps in each global step that expands a double-cone spacetime. The universe’s forward-lightcone lower bound corresponds to the big bang while its backward-lightcone upper bound corresponds to the present. (All history occurs after the big bang and before the present.) HQC adapts continuous string-theoretical and Feynman-graphical notions to discrete Whiteheadian process. While standard physics-cosmology posits a spatially-unbounded universe of matter that (continuously) carries conserved energy-momentum, angular momentum and electric charge, in HQC only a tiny “rigid” component (“enduring process” in Whitehead’s terminology) of a discrete and finite history corresponds to matter. The huge majority of history is “nonrigidly meandering” in time as well as in space and carries none of the above conserved quantities. Dense “vacuonic” history, unobservable by the scientific method, nevertheless carries conserved magnetic charge, contacts material history and participates in magnetodynamic action at a distance. One outcome is zitterbewegung for most standard-model elementary particles, leading to rest mass and collapse of material wave function. Another is probabilistic nature of predictions based solely on past material history. There is prospect of understanding gravity as outcome of interplay between magnetically polarized vacuum and matter. Origin of the standard model’s 3 colors, 3 generations and (approximately) 30-degree Weinberg angle will be sketched and related to the internal structure of the photon and other “elementary” particles.

1. Introduction Alfred North Whitehead is celebrated among philosophers for representing reality not in terms of matter but through process [1]. Whitehead saw matter not as fundamental but rather as a very-special “enduring” type of process. His approach has seemed useless to the practice of a science founded on the “reproducible measurement” notion that posits matter as a priori. However science within the last century has come (reluctantly) to recognize that no measurements are exactly reproducible—that none of its matter-based descriptions of the universe can be more than approximations appropriate to some selected scale. The huge ratio between different scales displayed by our universe accounts for the high accuracy of certain scientific descriptions. Once unavoidability of scale-based approximation in science is acknowledged, it becomes tempting to use process as basis for a mathematical model of reality more general than any scientific model. A process basis can represent non-reproducible phenomena while explaining the high accuracy exhibited by the reproducibility principle known to physicists as “Poincaré invariance”. If process 51

R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 51-58. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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patterns interpretable as “material” and others interpretable as “measurement” can be found, and if these patterns conform to the (approximate) scale-based scientific “knowledge” accumulated by mankind, then the mathematical model enjoys not only philosophical value but also scientific. I shall here describe a process model that is illuminating mysteriously arbitrary features of the standard particle-physics model. Application to wider mysteries is looming. I call the model “historical quantum cosmology” [2]. HQC employs lattice coherent states (a Von Neumann invention [3]) within the direct product of 16 simple Fock spaces. [4] This basis, which exploits isomorphism between the complex-conformal group for 3+1 spacetime and the group GL(4,c) minus center, was serendipitously suggested 16 years ago by topological twistor-related remarks of V. Poenaru that dovetailed with Hilbert-space considerations known to my Berkeley colleague Eyvind Wichmann and generously passed along. A “history coherent state” is labeled by the discrete (global) age spacing between the big bang and the present together with a long but finite chain of (local) “pre-events” each of which carries a 16-valued label comprising a complex 4-vector of inverse-time dimension and a dual (right-handed and left-handed) pair of real time-dimension 4-vectors. A pre-event’s complex 4-vector label not only prescribes an impulse but locates it within a spacetime doublecone whose forward-lightcone lower bound corresponds to big bang and whose backward-lightcone upper bound corresponds to the present. The age of a pre-event is its Minkowski distance from the big-bang vertex. The Minkowski distance between big-bang and present vertices sets doublecone “size”, both spatial and temporal. This size exceeds Hubble time (or length), by a huge although finite factor that reflects our distance from doublecone center. Existence of a center violates the homogeneity principle underlying standard cosmology, but mankind locates so far from the center that within our observation-accessible neighborhood -- a redshift interval of order -- homogeneity is an excellent approximation. The impulse means that at each pre-event “something happens”, even though not to an “object”. Impulse together with location attaches to each pre-event a phase—i.e., an “action”. The impulse at a pre-event is determined through electro-magnetodynamic “action at a distance” by the intersections of the history chain with the backward lightcone of this pre-event [5]. Any history chain comprises 3 successive closed loops, each with a distinct electric-charge label (+, 0, -). A loop begins at the big-bang vertex and meanders, except for rare rigid segments interpretable as matter, throughout the doublecone interior, contacting the present boundary in “ongoing” material segments but not contacting the big-bang boundary, before returning to origin. Chain lengthening in doublecone expansion occurs at the present boundary. The dual time-dimension 4-vector label pair assigns to each pre-event electric and magnetic “pre-currents” constrained by charge labels. The magnetic-charge label maintains a single value (+) throughout all 3 loops. (The magnetic-electric asymmetry generates parity asymmetry.) By Feynman’s rule, which ensures charge conservation, physical charges reverse sign during those history-chain segments that retreat in age. Feynman, through his graphical representation of quantum electrodynamics, recognized that a single electron line, meandering in spacetime, can represent all electrons and positrons throughout the history of the universe. A discrete generalization of Feynman’s continuous idea is a cornerstone of historical quantum cosmology. Our generalization represents by a single history chain not only all matter but also a dynamic vacuum that profoundly influences the behavior of matter. Only a tiny fraction of the history chain exhibits the rigid (straight-line) structure interpretable as matter. (Whitehead’s concept of “matter” is equivalent to “inertia”.) The vast majority of history, meandering in age (as well as in space) almost randomly from one pre-event to the next, is interpreted as “vacuum”. Each step along the history chain is light-like and the magnitude of localage step has a fixed (scale-setting) value believed somewhat below Planck scale, but the sign of localage step may be either positive or negative. One consequence of meandering is a (spatial) density of pre-events at the scale of local step and, at much larger scales, approximate homogeneity for a

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“dynamical vacuum”. Homogeneity of dense HQC vacuum underpins approximate Poincaré invariance for dilute rigid material history. The density of material history may not exceed a limit corresponding to a spatial “parton scale” (hugely larger than the local step) that characterizes the very special rigid patterns of local history corresponding to “creation, propagation and annihilation of an elementary particle”. Already at parton scale the HQC vacuum is extremely homogeneous.

2. Elementary-Particle Propagation The special “tower” history pattern corresponding to propagation of a standard-model zero-rest-mass elementary particle (lepton, quark, gluon or electroweak boson) comprises 4 distinct parallel segments of the history chain, two segments advancing in age and two retreating so net magnetic charge vanishes [6]. The tower may be described as a “4-beaded closed string” moving in discrete steps. “String tension” is provided by magnetic coulomb “attraction” between adjacent beads, of opposite-sign magnetic charge, around the closed string. Two of the 4 pre-events building a “string quartet” share the same age and same magnetic charge and are “retarded” with respect to the other pre-event pair, which carries the opposite magnetic charge and a larger age. Each of the two retarded pre-events within a string quartet lies on the backward lightcones of the two advanced pre-events. The age difference between advanced and retarded quartet members defines parton scale. Transverse tower extension (“string radius”) is determined by the value of elementary magnetic charge together with local-step magnitude and is believed to be in the neighborhood of Planck scale. A tower thus displays 3 distinct scales: (1) The longitudinal displacement between successive quartets—the universal local-age step-- is believed to be below Planck scale. (2) The longitudinal displacement between communicating advanced and retarded “halves” of the tower—defines a parton scale that sets an upper limit to elementary-particle rest masses. (3) The transverse (spatial) extension—“radius”—is believed to be near Planck scale. The ratio between parton scale and local step has been tentatively located in the logarithmic neighborhood of while (using Dirac’s relation between elementary electric and magnetic charges [7]) the ratio between radius and local step is estimated to be near Standard-model chiral-fermion propagation is represented by a “pinched” tower—pre-events along the two retarded chain segments sharing spacetime locations along tower central axis, whereas vector-boson towers exhibit (“unpinched”) advanced-retarded symmetry. Within a “tower half” (advanced or retarded) the wave function is either symmetric or antisymmetric under interchange of the two constituent chain segments [8]. The half wave functions of a vector-boson tower are both antisymmetric while for chiral-fermion towers the retarded (pinched) half is symmetric. Fermion advanced-half wave functions match the antisymmetric half wave functions of vector bosons. An advanced quark half matches a gluon half while an advanced lepton half matches an electroweak vector-boson half. A pre-event label related to the complex 4-vector, on the chain segments building a retarded symmetric fermion half, represents the 3-valued standard-model attribute that has been called “generation”. This same label within an antisymmetric half represents color. How does a complex 4vector manage to represent both color and generation for elementary particles, as well as spacetime location and impulse for general pre-events?

3. Cosmospin A complex 4-vector of inverse-time dimension is equivalent to a complex 2x2 matrix, which may be written as the product of a general (dimensionless) unitary matrix and a (dimension-carrying) hermitian

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impulse. The former may be factored into a (“special”) unitary matrix of unit determinant, an element of the group SU(2), multiplied by a unit-modulus complex number - a “phase factor”, an element of the group U(1). The 4-dimensional parameter space for a general 2X2 unitary-matrix would be the product of a 3-sphere (unit sphere in 4 dimensions) and a circle, except that “doubly-antipodal” points (antipodal on both 3-sphere and circle) are equivalent—corresponding to the same unitary matrix. The entire circle may be used but only half of the 3-sphere [4]. The compact 4-space spanned by a general unitary matrix has been called “gauge space”. Requiring pre-event action to be given by a Lorentz inner product of impulse 4-vector with spacetime-location 4-vector, establishes a unique 4 to 1 mapping of gauge space onto the interior of a spacetime doublecone [4]. Four distinct gauge-space sectors map separately onto this spacetime. Each closed loop of the history chain passes once around the circle, beginning and ending at a special point in gauge space that maps onto the vertex of the big-bang (forward) lightcone. All four gauge sectors are encountered in each loop. History-chain direction distinguishes “first half” of circle, which we label “cosmospin up”, from “second half” which we label “cosmospin down”. (Two of the four sectors occupy the first circle-half, the remaining two the second.) Mapping between gauge space and spacetime is singular at circle midpoint; a 3-dimensional gauge subspace projects there onto a single spatial location that has been called “center of universe”. History-chain passage between cosmospin up and down is “passage through universe center”. The special locally-enduring character of material history is incompatible with such passage, but that portion of history described by human physics locates within a doublecone region far from universe center (from which universe center is invisible). In our region the 2-valued cosmospin label on a history-chain segment, like the 3-valued (+, 0, –) electric-charge label, does not change. Elementary-matter in our neighborhood, built from patterns of 4 rigidly-correlated history-chain segments close to each other in spacetime and each carrying a 6-valued label, reflects distinction between gauge-space sectors that connect through universe center. Tower-half wave functions (in a 36-dimensional space) are products of cosmospin wave functions and electric-charge wave functions. Their symmetry is correspondingly the product of cosmospin symmetry and electric-charge symmetry. Gluon-half wave functions are symmetric in cosmospin and antisymmetric in electric charge while the converse is true for electroweak-boson half wave functions. The antisymmetric cosmospin wave function carries zero cosmospin and the antisymmetric electric-charge wave function carries zero electric charge. A triplet of symmetric cosmospin wave functions is responsible in antisymmetric tower halves for the particle feature called “color” and, in symmetric halves, for “generation”. Symmetric electric-charge wave functions represent chiral isospin (chiral asymmetry stemming from magnetic-electric asymmetry). Irrelevant to particle physics, even though important, is the second gauge-space doubling (with respect to spacetime)—a doubling that relates to meaning for “the present”. The boundary separating the two cosmospin-up gauge-space sectors maps onto the present doublecone boundary, as also does the boundary between the two cosmospin-down sectors. Subtle issues ignored by science (except at the 1927 Solvay Conference) surround the concept of “present”. With the excuse that we are today meeting as scientists, nothing will here be reported in this regard. If HQC survives, other meetings will hear surprising ideas about the meaning of time.

4. Structure of the HQC Photon Although a photon propagator tower has the general characteristics of any electroweak vector boson (each half odd in cosmospin symmetry and even in electric- charge symmetry), the photon electriccharge wave function has a special character allowing photon coupling to the net electric charge carried by a tower regardless of color, generation and chirality [8]. Orthogonality, between photon

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carried by a tower regardless of color, generation and chirality [8]. Orthogonality, between photon and the other weakly-coupled (massless) elementary neutral boson that overlaps with the vector-boson isotriplet coupled to left-handed fermions, requires unambiguously a 30-degree Weinberg angle. (Acquisition of rest mass by weak bosons may change this angle.) Within each photon-tower half, one history-chain segment carries electric charge while the other is neutral. (In photon creation or annihilation, the 2 electric-charge –carrying segments transfer to other towers, while the 2 electrically-neutral segments connect to each other.) Because preevent labels on the two halves of a photon are the same, net photon charge vanishes, but the presence of electric charge as well as magnetic charge “inside a photon” is essential. Its internal magnetic charge “stabilizes” a propagating photon, allowing it to “endure” (Whitehead’s terminology), while its internal electric charge couples the photon to any matter that carries (net) electric charge. The material component of the universe thereby becomes “self observable”. Direction difference between magnetic and electric pre-currents precludes electric-charge contribution to the endurance of elementary matter. Magnetic charge is responsible for the “existence” of matter while electric charge renders matter “observable”. Although history patterns describable as “observations” remain to be worked out in detail, massless-photon emission, propagation and absorption are necessary ingredients, together with “classical electromagnetic fields” from electriccharge sources. Ideas developed two decades ago by Henry Stapp, [13] on the basis of Feynman’s formulation of quantum electrodynamics, together with the magnetoelectrodynamics of Reference [5], show how electric charges on the material segments of history associate to each history chain a classical electromagnetic field. It is presumed that, in historical quantum cosmology as well as in standard physics, zero photon rest mass will be maintained by the photon’s coupling to conserved electric charge.

5. Vacuons I now turn attention to a nonrigid local-history pattern very different from a tower or even from a tower half—a pattern that builds the great bulk of nonmaterial history. The pattern, called “vacuon”, is a single pair of history-chain segments, carrying “opposite” cosmospin and electric-charge indices, along which successive pre-event pairs occupy the same spacetime locations (as in the pinched retarded half of a fermion propagator)[9]. Along a vacuon (in contrast to any propagator half), age advance and age retreat, together with change of spatial-step direction, occur incoherently. “Opposite” cosmospin and electric charge indices, furthermore, mean vanishing of net cosmospin and electric charge at each pre-event pair along a vacuon. Magnetic charge need not vanish locally, although age meandering yields zero average magnetic charge at scales large compared to local step. In an “elementary material event”– a parton-scale pattern of ~ pre-events that represents a standard-model Feynman-graph vertex-- a vacuon may be “absorbed” or “emitted” by a material history-chain segment, “transmuting” that segment to another of opposite sense but nevertheless carrying the same observable material quantum numbers. Rigid material history provides sources and sinks of meandering vacuum history. Although, as the doublecone universe expands, the rigidity of material-history ageing requires diminishing matter density, vacuum history by meandering in age can maintain constant spatial density. Beyond direct contact with material history, vacuum history “at a distance” magnetodynamically affects material action (as well as vacuum action) through those vacuon pre-event pairs at which magnetic charge is nonzero. It is believed that rest mass and material wave-function collapse result from direct contact between vacuum history and matter, while gravity results from vacuum action at a distance.

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6. Rest Mass from Vacuum-Induced Zitterbewegung The simplest material event within a history is “trivial”—involving a single (zero-rest-mass) elementary particle whose spatial-direction of propagation is reversed through a direct vacuum-history contact that endows the material wave function with a phase decrement (MED action) of order unity [9]. Accumulation of negative phase through a succession of many velocity reversals is presumed, following Dirac,[10] to be phenomenologically equivalent (at scales far above parton scale) to rest mass. Dirac, through his celebrated equation for spin-1/2 propagation (where the velocity operator is

interpreted chirality reversal

as (maximum-velocity) spatial-direction

reversal at fixed momentum and spin that with repetition amounts (at “large scale”) through quantum superposition to propagation with nonzero rest mass. Historical quantum cosmology accepts, not only for chiral fermions but also for vector bosons, such “zitterbewegung” meaning for “elementary” rest mass, attributing the “source” of zitterbewegung to direct contact between matter and vacuum history. Rest-mass magnitude (which must lie below the inverse of parton scale) is expected to depend on particle “structure” partly through trivial-event phase and partly through interplay between particle structure and vacuum structure. The phenomenological wave-function collapse model publicized by John Bell indicates correlation between rest mass and rate of collapse [11]. It is anticipated that quantum superposition of different history chains--a superposition providing meaning for material energy-momentum and angular momentum--will not only confirm Dirac’s meaning for rest mass but show that contact between vacuum history and matter is a source of material-wave-function collapse. Dense dynamicalvacuum history is a promising “environment” to induce material decoherence.

7. Gravity Although the nonrigidity of vacuum history precludes it carrying energy-momentum in a material sense, there is vacuum-vacuum and vacuum-matter magnetic-coulomb action at a distance. (Material source of magnetic coulomb action is screened outside tower radius.) Vacuum history that is homogeneous in absence of matter (in the sense that magnetic charge fluctuates randomly at the scale of local step) may be “polarized” by presence of matter. Because large-scale inhomogeneity of vacuum would affect the large-scale behavior of matter, speculation is irresistible that gravity is a manifestation of large-scale magnetic-charge interaction between matter and polarized vacuum. Before verification of such an origin for gravity, precise meaning for material energymomentum and angular momentum must be given through (“quantum”) superposition of different rigid history patterns according to unitary infinite-dimensional Lorentz-group representations [12]. Although meaning is apparent in outline, precision remains to be achieved. One obstacle is the coordination of local step with global step. The global doublecone expands in discrete age steps—“global occasions”-- that, while huge on parton scale, are small on the scale of human consciousness and relate to “measurement” in Copenhagen quantum mechanics, accommodating the “observable” discrete-process notion of nuclear or atomic “transition”. (The scale of such transitions, although large on parton scale, is smaller than the global step.) A sample unresolved question, relevant to superposition of different histories, is whether a parton-scale “elementary material event” (comprising pre-events) may overlap the boundary between successive global occasions or must be complete within a single such occasion.

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8. Conclusions Still not understood is the Copenhagen rule probabilistically predicting material history in a global occasion from that in the preceding occasion. (That such a rule can be no more than probabilistic is unavoidable from its disregard of vacuum history.) Grasping the HQC basis for standard quantum mechanics will require understanding those special history patterns that qualify as “measurements through classical electromagnetism plus photons”. Although usually unacknowledged, the interpretability of Copenhagen quantum mechanics rests squarely on electromagnetism.

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Historical quantum cosmology is a collaborative work in progress, even though some scientist collaborators may hesitate to acknowledge involvement in an enterprise with philosophical implications. I have been functioning as cheer leader and coordinator. Foundational mathematical input, as already mentioned, came from V. Poenaru and Eyvind Wichmann. Conceptual contributions have regularly been made by Jerry Finkelstein and Henry Stapp. (Henry introduced Whitehead, coherent states and the concept of “present”.) Paul Masson, Leewah Yeh and Peter Pebler have contributed group-theoretically. Mahiko Suzuki has aided contact with the standard model, while Dave Jackson has influenced the model’s magneto-electrodynamics. (Schwinger’s ideas about magnetic charge [14] have been influential.) Two philosophers, Philip Clayton and Ralph Pred, have attended to the quixotic enterprise and provided encouragement. The cited unpublished LBNL reports are draft material for a book whose publication date and authors at this stage remain uncertain. (If insurmountable inconsistencies are encountered with respect to quantum superposition of different histories, there may never be publication.)

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

A. N. Whitehead, Process and Reality, MacMillan, New York (1929). G. F. Chew, Historical Quantum Cosmology, Berkeley Lab Preprint LBNL-42946 J. von Neumann, Math.. Foundations of Quantum Mechanics, Princeton Univ. Press, Princeton (1955). G. F. Chew, Coherent-State Dyonic History, Berkeley Lab Preprint LBNL-42648. G. F. Chew, Pre-Event Magneto-Electrodynamics, Berkeley Lab Preprint LBNL-42647. G. F. Chew, Elementary-Particle Propagation Via 3-Scale “Towers of Quartet Rings” Within a Dyonic History Lattice, Berkeley Lab Preprint LBNL-42649. P. A. M. Dirac, Phys. Rev. 74, 817 (1948). G. F. Chew, Cosmological Origin of Color and Generation, Berkeley Lab Preprint LBNL-44253. G. F. Chew, Vacuum History and Rest Mass, Berkeley Lab Preprint LBNL 44285. P. A. M. Dirac, Quantum Mechanics, third edition, p. 260, Oxford University Press, New York (1947). P. Pearle and E. Squires, Phys. Rev. Letters 73, 1 (1994). M. A. Naimark, Linear Representations of the Lorentz Group, Pergamon Press, New York (1964). H.P. Stapp, Phys. Rev. D 28, 1386 (1983). J. Schwinger, Phys. Rev. 173, 1536 (1968); Science 165, 757 (1969).

DEVELOPING THE COSMOLOGY OF A CONTINUOUS STATE UNIVERSE

RICHARD L. AMOROSO Noetic Advanced Studies Institute - Physics Lab 120 Village Square MS 49, Orinda, CA 94563-2502 USA “It is sensible and prudent... to think about alternatives to the standard model, because the evidence is not all that abundant... and we do know that the standard cosmological model is pointing to another surprise... because (it) traces back to a singularity. ” P.J.E Peebles (1993)

Abstract. Although popular, Bigbang cosmology still contains untested assumptions and unresolved problems. Recent observational and theoretical work suggest it has become feasible to consider introducing a new standard model of cosmology. Parameters for developing a Continuous State Universe (CSU) are introduced in a primitive initial form.

1. Introduction We have recently entered one of the periodic transitional phases in the evolution of fundamental theories of physics, giving sufficient pause to reinterpret the general body of empirical data. Recent refinements in observation of cosmic blackbody radiation [1] and various programs of theoretical modeling [2,3] suggest it might be reasonable to explore replacing the naturalistic Bigbang cosmology (BBC). A Continuous State Universe (CSU) based on alternative interpretations of the observational data is introduced in preliminary form. We begin reexamining pillars of BBC, briefly review alternate interpretations, then introduce general parameters for a continuous state universe (CSU). Reviewing the historical development of physical theory [4] illustrates the fact that two general models, one unitary and the other dualistic, have evolved simultaneously in the scientific literature: Unitary Model. Naturalistic, Darwinian, Newtonian; a classically oriented model aligned with current interpretations of the standard models - i.e. Bigbang Cosmology, Bohr’s phenomenological interpretation of Quantum Theory, standard Maxwellian electromagnetism and Einstein’s General theory of Relativity. Many unanswered questions like the breakdown of Maxwell’s equations at singularities remain. Dualistic Model. Includes all conventional wisdom plus extended theory; Bohm, de Broglie, Vigier, & Proca implying a polarizable Dirac vacuum with additional parameters and interactions. Best evidence is the Casimir effect. Offers plausible explanation for many unanswered questions, for example the Proca equations satisfy problems in electromagnetic theory. Also allows room for teleological causalities.

Only in the context of dualistic parallels of extended theory can a CSU cosmology be viably presented. The concept of a polarizable Dirac vacuum introduces an additional 59 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck scale, 59-64. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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causal order not deemed acceptable in physical theory because it was considered unreasonable that spacetime could contain an ordered periodicity or significant additional symmetry. As discussed below a dual causality and additional vacuum symmetry invites extension of the Wheeler/Feynman [5] radiation law to dynamics of spacetime topology itself where the present state is comprised of a continuous future-past standing wave [6]. The CSU is intended as the next evolutionary step in the progression of modern cosmological modeling stemming from Einstein's 1917 proposal of a Static Universe (ESU) and the banner 1948 development of both the Steady-State Universe (SSU) of Bondi, Gold & Hoyle and the BBC by Gamow, Alpher and Bethe. Although the CSU could be considered a form of ESU or SSU modeling, it is sufficiently different to require a proliferation of nomenclature. For example the CSU has neither inflation or expansion; and the CSU is not confined to the limits of the Einstein/Minkowski/ Riemann/Hubble sphere of the current standard BBC and SSU models. The CSU introduces a revolutionary structural change in the universe. The Hubble sphere represents only an observational limit. Fundamental CSU space is an absolute holographic-like space projecting a megaverse of a potentially infinite number of nested relational Hubble-type domains, each with different laws of physics and complete causal separation from our realm [7]. The additional subspace dimensions hypothesized as compactified in the initial BBC event are not a subspace in the CSU; instead ‘our’ whole relational Hubble sphere is a subspace of an absolute hyperspace without dimensionality as now defined. Additional dimensions are not compact, but ‘open’, undergoing a process of continuous compactification and dimensional reduction as the ‘standing wave’ of the present is continuously created and recreated [8]. 2. Parallel Interpretations Of Cosmological Data

3. Philosophy Of Space In CSU Cosmology - Origin Of Structure

Although the concept of Absolute Space (AS) as defined by Newton is discarded in contemporary physics, a deeper more fundamental form of AS nevertheless seems to

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exist and is a required foundation for CSU Cosmology. The CSU reintroduces a complementary AS that is non Newtonian because Newtonian AS, once considered the basis of ‘our space’, first of all is only a form of Euclidian space without sufficient degrees of freedom to incorporate Quantum or Relativity theory. CSU AS is different, but similar enough that Newton deserves credit for realizing the importance of AS. Secondly the relational space of the Einstein universe contains insufficient symmetry parameters to describe the additional causal properties of a supralocal megaverse. The AS proposed by the CSU) (defined in postulate 1) represents the ground of all existence and ‘resides’ beyond the observed Hubble universe or even the infinite number of other possible supralocal nested Hubble-type spheres (with varied laws of physics) [7]. The ultimate nature of CSU AS remains ineffable at the moment, but empirical tests are being prepared [14, 19]. In the meantime we can deduce some AS properties to steer empirical investigations to higher order properties these deductions suggest. Postulate 1: Space is the most fundamental ‘form or substance’ of existence; and the origin of all structure. The demarcation and translation of which constitutes the basis of all energy or phenomenology. Space takes two forms in CSU cosmology, Absolute Space and the temporal relational subspaces that arise from it. A basis for energy (space geometry) is a fundamental form of information which signifies the cosmological foundation of causality. This postulate also connotes the most rudimentary basis of structural-phenomenology.

The complementarity between the new concept of AS in CSU Cosmology and the contemporary relational space suggested by Einstein’s theories of relativity can be simplistically represented as a ‘virtual reality’ by interpreting CSU AS as a fundamental background space of the related space fields referred to by Einstein’s quote below. Time is a complex process only just beginning to be addressed by physicists [9]. One can say that all forms of time [6, 9] represent various types of motion and in that sense time can be discounted as a concept (i.e. - not absolutely fundamental). Then geometric translation or field propagation becomes more fundamental. Thus space (whatever it is) is the most fundamental concept of the universe. Space with boundary conditions or energy is fundamental to all forms of matter. Difficulty in defining the fundamental nature of a spacetime stems from the incomplete unification of quantum and gravitational theories with electromagnetism [3]. The conceptual disparity arises in terms of correspondence between the Newtonian worldview of a continuous AS in opposition to current Einsteinian view of discreteness. This debate about the nature of space has continued at least since Aristotle. Einstein in his last published statement regarding the nature of space and time said: “The victory over the concept of absolute space or over that of the inertial system became possible only because the concept of the material object was gradually replaced as the fundamental concept of physics by that of the field...The whole of physical reality could perhaps be represented as a field whose components depend on four space-time parameters. If the laws of this field are in general covariant, then the introduction of an independent (absolute) space is no longer necessary. That which constitutes the spatial character of reality is then simply the four-dimensionality of the field. There is then no ‘empty space’, that is, there is no space without a field.” [10].

Einstein’s view is a form of the relational theory of space developed by Leibniz and Huygens [12,13]. The relational model is limited to the Hubble sphere of human observation. The HD supralocal megaverse retains an absolute character of which Einstein’s relational domain is a corresponding subspace. Relationalism is in opposition to ‘substantivalism’ which gives space the ontological status of an independent reality as

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a kind of substance[12]; the Newtonian concept of absolute space being the prime example. As stated above the CSU redefines the nature of absolute space. 3.1 THE WHEELER GEON CONCEPT Wheeler [17] postulates a photonic mass of sufficient size to self cohere spherically. In Wheeler’s notation the Geon is described by three equations. The first (1) is the wave equation, followed by two field equations the first (2) of which gives a mass distance relationship and the second (3) variation of the factor

with circular frequency such that

related to the dimensionless radial coordinate

is the abbreviation for

Wheeler states that this system of equations permits change of distance scale without change of form [17]. 3.2 THE HYPER-GEON DOMAIN OF CSU FIELD THEORY Wheeler originally defined the Geon as a classical spacetime construct. A more complex Hyper-Geon postulated to reside beyond 3(4)D relational spacetime is utilized in the CSU[9]; and acts as an HD cover engulfing the Einstein/Hubble Universe. It forms the lower bound energy of a projected 12D space and action principle of the unified field. Postulate 2: The Supralocal Hyper-Geon is the most fundamental energy or phenomenology of existence. This Energy arises from the ordering and translation of AS ‘space’ (i.e. information or change of entropy). This fundamental Geon energy, is the unified field, the primary quantum of action of all temporal existence; filling the immensity of space (nonlocally) controls the evolution of the large scale structure of the universe, the origin of life (‘elan vital’) of classical philosophy and finally is the root and ‘light of consciousness’.

4. Introduction To The CSU Spacetime Formalism Extending work by Rauscher [8], and Cole [20] on 8D complex Minkowski space the CSU is instead formalized utilizing a 12D complex Minkowski metric developed from the standard four real dimensions plus 8 imaginary D representing a retarded and advanced complex hyperspace topology Cramer [15] expanded the Wheeler/Feynamn absorber radiation law [5] to include quantum theory. The complex hyperspace representation further extends Wheeler/Feynman law to the continuous topological transformation of spacetime itself. For symmetry reasons the

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standard Minkowski line element metric

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is expanded into periodic

retarded and advanced topological elements fundamental to relational space ‘extension’:

which adapts the complex

Minkowski metric from a standard form

to a periodic form for application to 11(12)D CSU spacetime where 3(4)D ‘standing wave’ Minkowski ‘present’ spacetime; and

the new and

for complex correspondence to the standard 4 real dimensions utilizing 8 imaginary dimensions. The 8 imaginary dimensions, while not manifest generally on the Euclidean real line, are nevertheless ‘physical’ in the CSU [6]; and can be represented by coordinates designating correspondence to real and retarded/advanced continuous spacetime transformation. The complex 12 dimensional CSU space, can be constructed so that and likewise for 1, -1) [8].

where the indices and run 1 to 4 yielding (1, 1,

Hence, we now have a new complex twelve space metric We can further develop this space in terms of the Penrose twistor

algebra, asymptotic twister space and spinor calculus since twister algebra as already developed by Penrose falls naturally out of complex spaces and the twistor is derived from the imaginary part of the spinor field [8]. In CSU singularities take a 3-torus form. The Penrose twistor SU(2,2) or is constructed from four spacetime, where

is the real part of the space and

twister

is usually a pair of spinors

is the imaginary part of the space. The and

which Penrose uses to represent a

twistor as in the case of the null infinity condition a zero spin field is

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5. Conclusions

Scientific theory, whether popular or unpopular at any point in history, must ultimately be based on description of natural law, not creative fantasies of a scientist’s imagination. Only by adequate determination of natural law can a theory successfully model reality. “There is good reason for the taboo against the postulate of new physics to solve new problems, for in the silly limit one invents new physics for every new phenomena [15]”. Cosmology is

becoming a mature science; mature enough that there is no room for surprises? References [1] de Bernardis, P. et al., 2000, Nature, 404, 955-959. [2] Amoroso, R.L., Kafatos, M. & Ecimovic, P. 1998, in Hunter, G., S. Jeffers & J-P Vigier (eds.), Causality & Locality in Modern Physics, Dordrecht: Kluwer [3] Vigier, J-P & Amoroso, R.L. 2002, Can gravity & electromagnetism be unified? (this volume) [4] Vigier, J-P, Amoroso, R.L. & Lehnert, B., 2002, Physics, or not two Physics, submitted. [5] Wheeler, J.A., & Feynman, R., 1945, Rev. Mod. Physics, 17, 157. [6] Amoroso, R.L. 2002, NATO ARW,R. Buccheri, & M. Saniga, (eds.) Dordrecht, Kluwer, in press. [7] Kafatos, M. Roy, S. & Amoroso, R. 2000, in Buccheri, di Gesu & Saniga, (eds.) Studies on Time, Kluwer [8] Rauscher, E., 2002, Non-Abelian guage groups for real and complex Maxwell’s equations (this volume) [9] Amoroso, R.L, 2000, 2000, in Buccheri, di Gesu & Saniga, (eds.) Studies on Time, Dordrecht: Kluwer [10] Misner, C.W.,Thorne, K. & Wheeler, J.A. 1973, Gravitation, San Francisco: Freeman. [11] Amoroso, R.L, 2002, Noetic Field Theory: The Cosmology of Mind, Book in progress. [12] Sklar, L. 1995, Philosophy and Spacetime Physics, Berkeley: Univ. of California Press. [13] Reichenbach, H. 1957, Philosophy of Space and Time, New York: Dover. [14] Amoroso, R.L, et al, 2002, Dirac Vacuum Interferometry, in progress. [15] Peebles, P.J.E. 1993, Principles of Physical Cosmology, Princeton: Princeton University Press. [16] Cramer, J.G., 1986, Reviews of Mod. Physics, 58:3, 647-687. [17] Wheeler, J.A. 1955, Geons, Physical Review, 97:2, 511-536. [18] Amoroso, R.L, 2002 The origins of CMBR (this volume) [19] Arkani-Hamed, N, Dimopoulos, S. & Dvali, G. 1999,. Phys. Rev. D 59, 086004 [20] Cole, E.A.B., 1977, Il Nuovo Cimento, 40:2, 171-180.

THE PROBLEM OF OBSERVATION IN COSMOLOGY AND THE BIG BANG MENAS KAFATOS Center for Earth Observing and Space Research, School of Computational Sciences, and Department of Physics, George Mason University, Fairfax, VA 22030-4444, USA

Abstract. The understanding of universe has to utilize experimental data from the present to deduce the state of the universe in distant regions which implies in the distant past. Also, theories have to account for certain peculiarities or “coincidences” observed, first discovered by Eddington and Dirac. The prevalent view today in cosmology is the big bang, inflationary evolutionary model. This theory has to be finely tuned to account for these coincidences such as the flatness, the horizon and more recently the cosmological constant problems. Certain nagging problems have remained, e.g. the need to postulate cold, dark matter in amounts much larger than all the observable matter put together and more recently the need to postulate that the universe is even accelerating, i.e. the totally unknown and postulated cosmological constant prevails over the large structure of the universe. Big bang cosmology has been extrapolated to realms beyond its observational applicability and despite its impressive achievements, this methodology may go counter to the usual requirement of verification on which all science rests. We will present fundamentally different approaches that may be more in accord with quantum epistemology. Consequences of these approaches will be explored. 1. Cosmological Realm Issues Evolutionary models of the universe The most accepted theory of the large-scale structure of the universe is big bang cosmology which has achieved impressive results (Silk 1989). Evidence from 2.73 K black body radiation The evidence from the existence of the 3°K (or more exactly 2.735°K) black body radiation observed with such hig accuracy from COBE is attributed to a primordial radiation field associated with the big bang itself and now redshifted to very low temperatures, peaking in the microwave region (Smoot, 1996). Formation of the elements Although the evidence here is not as compelling as the previous one, supporters of the big bang point to the fact that the light elements can be accounted for as forming in an expanding universe. The problem though here is that the required baryon density is more than a magnitude smaller than the closure density (see below), (Peebles, 1993; Kafatos and Nadeau, 2000). Distant sources 65 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 65-80. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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M. KAFATOS The relationship between redshift and apparent visual magnitude (or velocity of recession vs. distance if redshifts are cosmological) is shown in Figure 1 for quasars. It is immediately obvious that either quasars have discordant redshifts (a view which I favor) or that the most distant objects in the universe cannot be used for distance estimates. Is universe open, closed or exactly flat? Closure density Whether the universe is open, closed or (four-dimensionally) geometrically flat is determined by how far the density of matter/energy deviates from the closure density. The observed density of the universe is within one or two orders of magnitude (depending on the size of volume/matter surveyed) equal to the closure density (see below). This is close enough to speculate that the universe may be exactly flat (because in an expanding universe, the present approximate relationship becomes an identity for early-on densities). Inflationary models Exactly flat Require cold dark matter—so far unseen From the previous argument, one is tempted to speculate that the universe may be exactly flat as suggested by Kazanas, Guth and others (Guth, 1981). In these inflationary scenarios, the univers is much, much larger than the observed volume 20 billion light years or so across. Because the inflationary model is so elegant theoretically, it has gained acceptance. The problem is though that the exact flatness requires 10–100 times more mass/energy than all the observable matter in galaxies put together. This so-called “dark matter” (most favorably being cold) has so far not been detected. Recent evidence indicates universe may be accelerating cosmological constant (Einstein’s “biggest blunder”) Too make matters worse, if one assumes that distant supernovae of Type I behave the same way as nearby supernovae, evidence indicates that the universe may not even be dominated but instead by vacuum physics, in the sense that a “negative” kind of gravity exists, its magnitude described by the cosmological constant. Einstein himself, who introduced this hypothetical constant to keep a static, spherical universe, later called it his biggest blunder.

The above brief exposition points to the fact that although elegant and in many ways supported by evidence, big bang, of the inflationary kind, cosmology has progressively become ensnarled by current evidence and by its own strong predictions. Any general relativistic Friedmann- Robertson-Walker big bang model, as well as any other non-big bang cosmological model, such as the steady state model, etc., cannot be considered outside the process of cosmological observations, and its predictions are ultimately intricately interwoven with the process of observation itself and the limits imposed by (Kafatos 1989, 1996, 1998). Any theoretical construct is subject horizons of knowledge at some ultimate, faint observational limit. For example, for the big bang theory, light cannot be used to observe further back in time or for very large redshifts (redshift being the relative difference of observed wavelength and the emitted wavelength of light, which in the big bang cosmology is a measure of the distance to the source) to test the big bang theory close to the beginning. As such the whole cannot be studied from the parts, the beginning is forever hidden from the present. Ultimately, observational limitations prohibit verifying cosmological theories to any degree of accuracy for any given observational test. For example, for all practical purposes, the big bang galaxy formation theory runs into verification problems at redshifts, z ~ 4 -10, close to distances discerned by the Hubble Space Telescope and future space telescopes. The reason is that the type and evolutionary history of the

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“standard candles” (such as galaxies) used to measure the Hubble expansion rate and overall structure of the universe cannot be unequivocally determined independently of the cosmology itself (Kafatos 1989).

1.1

COSMOLOGICAL CONSTRAINTS

We now turn to a more detailed discussion of the constraints imposed on any, not just big bang, cosmology. In cosmology, there are a number of facts about the large scale structure which must be considered. These in turn provide constraints for physical theory (see also Kafatos, 1999). The universe is essentially flat, this observed property is known as the flatness problem The density of the present-day universe is close to the closure or critical density, the limit between forever expansion and future re-collapse, i.e.

where is the present-day value of the Hubble constant defined as and R is the scale of the universe. The Hubble constant provides an estimate of the current expansion rate of the universe (measurements by the Hubble Space Telescope indicate that its value is close to The argument goes that if the universe is close to flat today, it was exactly flat close to the time of the big bang itself, to one part in This is known as the flatness problem. The usual interpretation proposed in the early 80’s (Guth, 1981; and others) is that early on, the universe was in an inflationary state, washing out any departures from flatness on time scales of sec. In more general terms, it would appear that the universe somehow followed the simplest possible theoretical construct (flatness) in its large-scale 4-D geometry. The universe is remarkably homogeneous at large scales as revealed by the 2.73 K black body background radiation — T is constant to 1 part in , known as the horizon problem The universe is remarkably homogeneous at large scales as related to the radiation that fills all space. This is known as the horizon problem. The inflationary model proposed by Guth and others (cf. Guth, 1981) was developed in various forms to account for the flatness of the universe and also was proposed to solve the horizon problem. This problem is manifesting in terms the apparent

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homogeneity of the 2.73 K black body radiation seen as by COBE (Smoot 1996). Although the 2.73 K radiation was years after the beginning, opposite sides of the sky at that time were out of causal contact, separated light years. Other correlations in the large-scale structure of the universe exist such as very large structures in the distribution of matter itself (Geller and Huchra 1989). These structures may or may not be manifesting at all scales all the way to the scale of the universe itself R~ Hubble radius.

Cosmological constant “coincidence” Recent observations indicate a cosmological constant might be needed in a flat universe framework, known as the Cosmological constant problem Where observations of distant supernovae indicated that Although the universe appears to be close to a flat, Euclidean (Peebles, 1993), Einstein-de Sitter state as indicated from the fact that the density is close to closure, it is still not clear what the geometry of the universe is today; exactly flat (as many theoretical constructs require); open (yielding a forever-expanding, negatively curved space-time); closed (yielding a maximum expansion and a positively curved space-time); or maybe even open and accelerating (requiring a non-zero cosmological constant as recent observations seem to indicate). The cosmological constant was first introduced by Einstein to counter gravity and produce a closed, static universe stable - it essentially acts as negative gravity. It was later abandoned when observations by Hubble and others were interpreted as favoring an expanding universe. It has recently been reintroduced by cosmologists as the present observations seem to be indicating at face value that the universe not only is expanding but it is also accelerating in its expansion. Observations indicate that baryons as well as the observed luminous matter contribute 0.1 or even less of the value of the closure density at the present era. As such, if one insists on exact flatness, one needs to introduce unknown forms of “dark matter” for the other 90% or more of what is required. To make matters worse, unknown physics is required by the existence of a nonzero cosmological constant. In other words, the mathematical model is simple in its assumptions but the underlying physics required to maintain it is complex and even unknown. This reminds us the historical analogy of the Ptolemaic Universe: To keep the orbits of the planets circular in a geocentric universe (which was also a “simple” universe), required an increasing amount of complexity, more and more epicycles. The universe appears to be extremely fined tuned (cf. Kafatos 1998). Eddington (1931, 1939) and Dirac (1937, 1938) noticed that certain “coincidences” in dimensionless ratios can be constructed and these ratios link microscopic with macroscopic quantities (cf. Kafatos 1998). For example, the ratio of the electric force to gravitational force (both presumably a constant), is a large number

while the ratio of the observable size of the universe (presumably changing) to the size of an elementary particle is also a large number, surprisingly close to the first number, or

It is hard to imagine that two very large and unrelated numbers, one from microphysics, the other from macrophysics, would turn out to be so close to each other, Dirac argued. The two, Dirac argued, must be related. The problem though is that in (3) the numerator is changing as the universe expands while (2) is presumably constant. Why should two such large numbers, one variable and the other not, be so close to each other? Dirac’s (1937) Large Number Hypothesis states that the fact that the two ratios in (2) and (3) are equal is not a mere coincidence. He and

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others (cf. Dyson 1972) have attempted to account for the apparent equality between (2) and (3) by assuming that constants such as the gravitational constant may be varying. Other ratios such as the ratio of an elementary particle to the Planck length,

large numbers such as “Eddington’s number”, etc. exist and “harmonic” numbers can be constructed from them (Harrison 1981). For example, according to Harrington (1981), Eddington’s number is approximately equal to the square root of (2) or (3). These “coincidences” may be indicating the existence of some deep, underlying unity involving the fundamental constants and linking the microcosm to the macrocosm. Other, less traditional ways, such as the Anthropic Principle (Barrow and Tipler 1986) have been proposed to account for the above fine tuning properties of the universe. It might though be possible to invoke quantum non-locality as the underlying principle. To recapitulate, The Universe is Extremely Fined Tuned In the following sense, we have 1. Flatness Problem or Age Problem At Big Bang to 1 part in A correlate is: Why is the Universe so old? 2. Horizon Problem 3 K blackbody radiation emitted years after big bang. Temperature constant to 1 part in But opposite side of the sky were out of causal contact at time of emission (separated by light years). 3. Isotropy Problem Or, Why is the Universe expanding at such a regular fashion today? 4. Homogeneity Problem Is the Universe truly homogeneous? Large superclusters (e.g. Pisces-Cetus) are abserved that extend up to 10% of the radius of the Universe. 5. Cosmological Constant Problem Smaller by at least 46 orders of magnitude from what standard particle theory might predict. But what is its value? Cosmological Theoretical Constructs Inflation/Big Bang Is based on the Doppler interpretation of redshifts Requires CDM, maybe mixed CDM and HDM (hot dark matter) and probably , all unknown theoretically and unseen in the laboratory Attributes the universe to quantum fluctuations of the vacuum Model has lost its original appeal-continuous refinements with more and more parameter fitting are required Spectrum of primordial fluctuations, background inhomogeneities, etc. require more and more observations with, probably, an ever-increasing introduction of new ideas and parameters to “save the day” Anthropic Principle Anthropic Principle (Barrow and Tipler 1986) has been proposed to account for the above fine tuning properties of the universe

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M. KAFATOS Superstring Theory or future theoretical developments? Possibly but it does add nothing to current knowledge Large Number Hypothesis Possibly but not theoretically clear Quantum non-locality may be invoked as the underlying principle As result of the wholeness inherent in the measurement process and quantum processes themselves? Ratio of the electric force to gravitational force

Ratio of the observable size of the universe (presumably changing) to the size of an elementary particle is also a large number, surprisingly close to the first number, or

Dirac’s Large Number Hypothesis Ratio of an elementary particle to the Planck length,

large numbers such as “Eddington’s number, etc. exist. “Harmonic” numbers can be constructed from them, e.g. Eddington’s number is approximately equal to the square root of Dirac’s relations, etc.

2. Quantum And Cosmology Some general considerations apply: Quantum processes are fundamental in the universe Basic nature of matter and energy described in terms of quantum theories (QED, QCD, Supersymmetry & String Theories) Universe was in quantum state early-on At Planck time and at high energies/densities in an evolving universe Quantum gravity is the ultimate frontier Will help us understand high gravity astrophysical phenomena as well as early state of the universe It has become clear that quantum non-locality as revealed by the Aspect and Gisin experiments (Aspect, Grangier & Roger, 1982; Tittel, Brendel, Zbinden & Gisin, 1998; Kafatos & Nadeau 1990, 2000; Nadeau and Kafatos, 1999) has demonstrated the inadequacy of classical, local realistic theories to account for quantum-like correlations and the nature of underlying reality. The epistemological and ontological consequences are far-reaching (Kafatos and Nadeau 2000) and imply a non-local, undivided reality. Moreover, Drãgãnescu and Kafatos (2000) explore the possibility that foundational principles operate at all levels in the physical as well as beyond the physical aspects of the cosmos. These foundational principles are meta-mathematical or pre-mathematical in the sense that

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mathematical constructs of the physical universe emerge from them. If truly universal, these principles should apply at all scales. Non-locality also appears to be prevalent at different scales. Quantum theory has shown that the whole is not just the sum of its constituent parts. For example, the quantum vacuum is much richer and complex than any system of particles interacting among themselves. Studying particle interactions, no matter how complex, will not tell us much about the vacuum as the latter is unaffected by such interactions. These developments are indicative of the need to develop a new way to approach problems that have so far eluded ordinary physical science.

2.1.THE NON-LOCAL UNIVERSE In the generalized complementarity framework (Kafatos and Nadeau 1990, 2000), complementary constructs need to be considered to formulate a complete picture of a scientific field under examination (e.g. the large-scale structure of the universe) as a horizon of knowledge is approached. This means that as a horizon is approached, ambiguity as to a unique view of the universe sets in. It was precisely these circumstances that apply at the quantum level, which prompted Bohr to affirm that complementary constructs should be employed (Bohr 1961). Moreover, the remarkable correlations exhibited at cosmological scales are reminiscent of Bell-type quantum correlations (Bell 1964) that were so abhorrent to Einstein (Einstein, Podolsky and Rosen 1935) and yet confirmed by the Aspect and Gisin experiments. Kafatos (1989) and Roy and Kafatos (1999) proposed that Bell-type correlations would be pervasive in the early universe arising from the common electron-positron annihilations: Binary processes involving Compton scattering of the resultant gamma-ray photons with electrons would produce N-type correlations (Figure 2). In these conditions, the outcome of the cascade of processes (even in the absence of observers) would produce space-like correlations among the original entangled photons. Kafatos and Nadeau (1990, 2000) and Kafatos (1998) have in turn proposed three types of

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non-localities: Spatial or Type I non-locality occurs when 2 quanta (such as photons) remain entangled at all scales across space-like separated regions, even over cosmological scales (Figure 3).

Temporal or Type II non-locality (or Wheeler’s Delayed Choice Experiment) occurs in situations where the path that a photon follows is not determined until a delayed choice is made (Figure 4).

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In some strange sense, the past is brought together (in the sense that the path is not determined) by the experimental choice. This non-locality confirmed in the laboratory could also occur over cosmological distances (Wheeler 1981). Type III non-locality (Kafatos and Nadeau 1990, 1999) represents the unified whole of space-time revealed in its complementary aspects as the unity of space (Type I) and the unity of time (Type II non-locality). It exists outside the framework of space and time and cannot, therefore, be discerned by the scientific method although its existence is implied. To recapitulate, EPR thought experiment and Bell’s Theorem 1. Nonlocality 2. Measurements Aspect and Gisin experiments and quantum non-locality 1. Cosmological correlations Delayed-choice experiment 2. From laboratory scales to cosmological scales (Wheeler, 1981) 3. Alternative Ideas in the Cosmological Real The starting point here is: Large structures in the universe are difficult to produce in an expanding universe 3. Traditional approach is to attribute these to primordial inhomogeneities in an inflationary model Bell-type correlations in the early universe 4. Correlations may be related to the nature of quantum processes in the early universe This would tie present-day observations to quantum processes early on

3.1 HORIZONS OF KNOWLEDGE IN COSMOLOGY Observational Horizons of knowledge in cosmology have been discussed before (Kafatos and Nadeau, 2000). Basically, one cannot observe events beyond certain finite z-values, which for galaxies are in the approximate range ~5-30; photons pick up from where galaxies leave up to z ~ 1,000 for the 3°K radiation; while for neutrinos (if they ever get observed for cosmological import) one can in principle get as far back as The big bang itself is, however, unobservable. Moreover, as Kafatos (1989) showed, ultimately source position ans spectra from sources would become confused due to the existence of very few photons from distant sources and the wave-particle duality which forces experiment choices (see Figure 5). There are a number of theoretical constants as well (discussed in detail in Kafatos and Nadeau, 2000). To recapitulate, z ~ 5-30 (?) galaxies; ~30-1,000 photons; (Big Bang) is unobservable Source and spectra confusion

neutrinos

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Theoretical If inflation is correct, most of universe is unobservable Big Bang extrapolates to unobservable realms what may be local conditions Red shift controversy remains Distance ladder assumes we know astrophysics of very distant regions in space time (“standard candles”)

3.2

THE ARROW OF TIME – AN ALTERNATE VIEW

Recently, Kafatos, Roy and Amoroso (2000) have shown that these coincidences could be reinterpreted in terms of relationships linking the masses of elementary particles as well as the total number of nucleons in the universe (or Eddington’s number) to other fundamental “constants” such as the gravitational constant, G, the charge of the electron, e, Planck’s constant, h, and the speed of light, c. They conclude that scale-invariant relationships result, e.g. all lengths are then proportional to the scale of the universe R, etc. The arrow of time is introduced in an observer-dependent universe as these fundamental “constants” change (e.g. Eddington’s number varies from at the time of big bang today, etc). Time does not exist independently of conscious observers. Specifically, one may adopt Weinberg’s relationship which in one of its forms is

where is the electron mass, is the (present) Hubble constant and the other parameters in (5) are the usual physical constants. Weinberg’s relation can be shown to be equivalent to Dirac’s relationships (2) and (3) when the latter are equated to each other (Kafatos, Roy and Amoroso, 2000). We can then obtain a relationship linking the speed of light c to the rate of change of the scale of the

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universe. In fact, the proportionality factor is ~ 1, whnone substitutes for values of fundamental quantities like the present number of particles in the universe, etc. The next step assumes that the relationship linking c and R is an identity, (for example, at the Planck time, one observes that this relationship still holds if the ratios of all masses and the number of particles also As such, in this picture all the fundamental constants are changing and not just one of them as was assumed in past works. It is interesting that, recently, the possibility that the cosmological constant itself might be changing (Glanz 1998) has been suggested. As such, what is suggested as a framework for the universe is a natural extension of previous ideas. Therefore, as changes from an initial value of 1 to the present value of the universe would be appearing to be evolving to an observer inside it or the arrow of time is introduced. Finally, the outcomes of this prescription are not just that an arrow of time is introduced and the mysterious coincidences of Dirac and Eddington now can be understood as scale-invariant relationships linking the microcosm to the macrocosm; but in addition, all scales are linked to each other and what one calls, e.g. fundamental length, etc. is purely a convention. The existence of horizons of knowledge in cosmology, indicate that as a horizon is approached, ambiguity as to a unique view of the universe sets in. It was precisely these circumstances that apply at the quantum level, requiring that complementary constructs be employed (Bohr 1961). At the initial time, if we set the conditions like as proposed by Kafatos, Roy and Amoroso (2000), we can axiomatize the numerical relations connecting the microcosm and the macrocosm. In other words, after setting at the initial time of Big Bang, this relationship remains invariant even at the present universe. This relation is a type of scaling law at the cosmological scale and connects the microcosm and the macrocosm. In a sense, Light connects everything in the universe. Now if there is expansion of the Universe, R itself is changing and more specifically, then the fundamental constants like G, and c may also all vary with time. Due to the variation of these fundamental constants, will also be changing from the initial value 1. This implies that more and more particles will be created due to expansion of the universe. So an observer, who is inside the universe will instead see an arrow of time and evolutionary universe. As the present number of the nucleons in the universe, the fundamental constants achieve their present values. In a sense, if one considers that the universe is undergoing evolutionary processes, one would conclude in this view that of the fundamental constants themselves are changing. The other aspect of this view is that if one considers the fundamental constants as changing, the observer will observe an arrow of time in the Universe. So, the arrow of time can be related to a kind of complementarity between two constructs, i.e., the universal constants are constant, on the one hand, and constants are changing, on the other hand.

4. The Universal Diagrams – Visualization The Wholeness Of The Universe A series of Universal Diagrams (UD) have been constructed (Kafatos, 1986; Kafatos and Nadeau, 2000; Kafatos and Kafatou, 1991) and reveal deep underlying wholeness. These can be constructed by placing various physical quantities of many different objects in the universe on common, multidimensional plots. 2-D diagrams have been constructed involving the mass, size, luminous output, surface temperature and entropy radiated away of different objects in the universe. These diagrams originally constructed for astronomical objects (Kafatos 1986) have been revised and extended to all scales including biological entities, industrial and man-made objects, etc. Two of these 2-D diagrams are shown here.

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Figure 6, entropy radiated versus mass; Figure 7, luminosity versus mass). The diagrams show continuity among different classes of objects and can even be used to find likely regions where to-date undiscovered objects could be located are (such as super-superclusters, large planets, etc.). The overall appearance of the UDs does not change as more objects are introduced, rather the specifics of smaller regions become more refined. Over smaller regions, different power laws can be found to fit the data, while more global relationships can be found that approximately fit many different classes of objects (such as an approximately linear relationship between entropy radiated away and mass). It is found that black holes provide boundaries in the UDs and often cut across the main relationships in these diagrams. The values of the constants (and their ratios) and the laws of physics are determining the overall relationships and as such the diagrams must be related to the ratios (2) and (3), although it is

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not totally clear at present if additional principles may or may not be required. There are large scale correlations revealed in these diagrams among different dimensions (other than space and time examined above) or parameters which extend beyond the quantum or cosmological realms, to realms such as living organisms, etc. It follows that non-locality in the sense of global multidimensional correlations, is revealed by the UDs to be a foundational principle of the structure of the cosmos along with complementarity (Kafatos and Nadeau 1990, 2000).

5. Foundational Principles A new approach of starting with foundational principles is proposed (Drãgãnescu and Kafatos, 2000a, and Struppa, Kafatos, Roy, Kato, and Amoroso, 2000). There are good reasons to believe that the present-day science (which concerns itself with explanations of structural realities and as such can be considered to be a structural science) is limited in its approach, in the sense that it cannot completely explain life, mind and consciousness, as well as the nature of matter and reality. The proposed approach is to explore foundational principles as the underlying structures themselves similar to the Ideas of Plato (rather than relying on the physical structures to account for the underlying nonstructural or phenomenological levels). Although one cannot neglect the impressive accomplishments and impact of science, including cosmology, as it has been developed over the last few centuries, it is also clear that a new, fundamentally different approach is needed to avoid an alienation between

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science and other human endeavors. Accepting that few fundamental principles are the source of all scientific and philosophical human endeavors, it may then follow that reductionism (one of the main operating principles of modern science) can be reinstalled in new philosophical and scientific approaches. A foundational approach has to be developed to assure that there is no danger of absolute and complete reductionism. In fact, in exploring foundational principles one can re- examine whether reductionism itself is a consequence of a generalized principle of Simplicity: A whole is composed of simpler parts yielding discreteness. Reductionism is then the methodology of exploring the discreteness and relationships arising from it. It may be supposed that all existence, consisting of the physical, mental and psychological worlds, consists of complementary principles in the deepest sense. It may be supposed that from the depths of existence a single universe (or world) manifests (or many universes as in the Many Worlds Interpretation of quantum theory) which maintains a direct connection with the original foundational principles and underlying levels. It may also be supposed that a variety of other possibilities in the sense of different levels of existence or universes are possible as well. As such, ontological model of the entire nature of reality is needed, a new model that extends present science, which should be able to respond to such onto logical problems. It follows that, perhaps, foundational principles are more fundamental than physical theories (Kafatos, 1998). Still, the foundational principles have to rely on a general model of existence and need to be developed in a systematic way (Drãgãnescu and Kafatos, 2000b). The epistemological and ontological consequences are far-reaching (Kafatos and Nadeau 1990, 2000; Nadeau and Kafatos, 1999) and imply a non-local, undivided reality which reveals itself in the physical universe through non-local correlations and which can be studied through complementary constructs or views of the universe. Quantum theory and its implications open, therefore, the door for the thesis that the universe itself may be conscious (although this statement cannot be proven by the usual scientific method which separates object from subject or the observed from the observer), Kafatos and Nadeau (2000), Nadeau and Kafatos (1999). To recapitulate, Drãgãnescu (1998, 2000) and Drãgãnescu and Kafatos (2000a) have explored the thesis of a deep reality, paralleling the thesis of a conscious universe. Moreover, Drãgãnescu and Kafatos (2000a, 2000b) explored the possibility that foundational principles operate at all levels in the physical as well as beyond physical aspects of the cosmos. These go beyond the two principles revealed in studying the quantum and cosmological realms. In conclusion, New approach of starting with foundational principles is proposed (see Drãgãnescu and Kafatos, 2000a). The present science (which concerns itself with explanations of structural realities and as such can be considered to be a structural science ) cannot completely explain not only life, mind and consciousness, but the nature of matter and reality, in general. The approach here is to explore foundational principles as the underlying structures themselves (rather than relying on the physical structures to account for the underlying non-structural or phenomenological levels). Reductionism itself is a consequence of a generalized principle of Simplicity: A whole is composed of simpler parts yielding discreteness. Reductionism is then the methodology of exploring the discreteness and relationships arising from it. It may be supposed that existence itself consists of complementary principles in its utmost depths. It may be supposed that from the depths of existence a single universe (or world) manifests (or many universes). Different levels of existence or universes are possible. Drãgãnescu and Kafatos (2000a), propose the following set of foundational principles: Principle of complementarity is a foundational principle of existence

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Nature of existence is both physical and informational (sub-principles can be constructed from this basic principle) Ontological principle of self-organization is a foundational principle Fundamental Consciousness of Existence is a foundational principle The ultimate reality is the deep underlying reality or existence The universe generated from the deep reality is non-local The universe is quantum-phenomenological The objects with life, mind and consciousness in an universe are structural-phenomenological. Other Principles Guided from quantum theory one can perhaps extend the list of the above principles to include additional candidates such as: Correspondence Light as the “glue” of the universe We conclude here that foundational principles may be needed to begin to understand the physical universe, as well as the all-pervasive phenomenon of consciousness. These principles operate beyond or below the physical universe and as such are meta-mathematical or pre-mathematical in the sense that mathematical constructs of the physical universe emerge from them.

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References Aspect, A., Grangier, P. and Roger, G (1982), Phys. Rev. Lett. 49, 91. Bohr, N. (1961), Atomic Theory and the Description of Nature, 4, 34, Cambridge, Cambridge University Press. Dirac, P.A.M. (1937), Nature 139, 323. Dirac, P.A.M. (1938), Proc. Royal Soc. A165, 199. Drãgãnescu M., (1998), Constiinta fundamentala a existentei (The Fundamental Consciousness of Existence), Academica, ianuarie 1998, p.20-21 (p. I-a), Febr. 1998, p.20 (p. II-a), March 1998, p.III-a, p.28-29. Drãgãnescu, M. (2000), “The Frontiers of Science and Self-Organization”, comm.. at the IV-th Conference Structural-Phenomenological Modeling, Academia Romana, June 20-21 (in press). Drãgãnescu, M. and Kafatos, M. (2000a), in Consciousness in Science and Philosophy,Charleston, Ill., Noetic Journal, 2, 341-350. Drãgãnescu, M. and Kafatos, M. (2000b), “Towards an Integrative Science,” Noesis (in press). Dyson, F.J. (1972), in Aspects of Quantum Theory, ed. A. Salam and E.P. Wigner, Eddington, A.S. (1931), M.N.R.A.S. 91, 412. Eddington, A.S. (1939), The Philosophy of Physical Science, Cambridge, CambridgeUniversiry Press. Einstein, A., Podolsky, B. and Rosen, N. (1935), Phys. Rev. 47, 777. Geller, M.J. and Huchra, J. (1989), Science 246, 897. Guth A., 1981, Phys. Rev. D., 23, 347 Harrington and S.P. Maran, 198, Cambridge, Cambridge University Press. Harrison, E.R. (1981), Cosmology: The Science of the Universe, 329, Cambridge,Cambridge University Press. Kafatos, M. (1986), in Astrophysics of Brown Dwarfs, ed. M. Kafatos R.S. Kafatos, M. (1989), in Bell’s Theorem, Quantum Theory and Conceptions of the Universe, ed. M. Kafatos, 195, Dordrecht, Kluwer Academic Publishers. Kafatos, M. (1996), in Examining the Big Bang and Diffuse Background Radiations, ed. M. Kafatos and Y. Kondo, 431, Dordrecht, Kluwer Academic Publishers. Kafatos, M. (1998), in Causality and Locality in Modern Physics, ed. G. Hunter et al., 29, Dordrecht, Kluwer Academic Publishers. Kafatos, M. (1999) Noetic Journal, 2 21-27. Kafatos, M. and Kafatou, Th. (1991) Looking In, Seeing Out: Consciousness and Cosmos, Wheaton, Ill., Quest Books/The Theosophical Publishing House. Kafatos, M. and Nadeau, R. (1990), The Conscious Universe: Part and Whole in Modern Physical Theory, New York, Springer-Verlag. Kafatos, M. and Nadeau, R. (2000), The Conscious Universe: Part and Whole in Modern Physical Theory, second edition, New York, Springer-Verlag (in press). Kafatos, M., Roy, S. and Amoroso, R. (2000), in Buccheri et al., (eds.) Studies on the Structure of Time: From Physics to Psycho(path)logy, Kluwer Academic/Plenum, New York. Nadeau, R., and Kafatos, M. (1999), The Non-local Universe: The New Physics and Maters of the Mind, Oxford, Oxford University Press. Peebles, P.J.E. (1993), Principles of Physical Cosmology, Princeton, Princeton University Press. Roy, S., and Kafatos, M., (1999), “Bell-type Correlations and Large Scale Structure of the Universe” (preprint). Silk, J. (1989), The Big Bang, New York, W. H. Freeman. Smoot, G.F. (1996), in Examining the Big Bang and Diffuse Background Radiations, ed. M. Kafatos and Y. Kondo, 31, Dordrecht, Kluwer Academic Publishers. Struppa, D. C., Kafatos, M., Roy, S., Kato, G., and Amoroso, R. L. (2000), “Category Theory as the Language of Consciousness,” (submitted). Tittel, W., Brendel, J., Zbinden, H. and Gisin, N. (1998), Phys. Rev. Lett. 81, 3563. Wheeler, J.A. (1981) in Some Strangeness in the Proportion, ed. H. Woolf, Reading, Addison-Wesley Co.

ABSORBER THEORY OF RADIATION IN EXPANDING UNIVERSES

JAYANT V. NARLIKAR Inter-University Centre for Astronomy and Astrophysics Ganeshkhind, Pune 411 007, India

Abstract

The Wheeler-Feynman absorber theory of radiation of the symmetric combination of retarded and advanced potentials, originally developed in a static universe model, is applied to asymptotic boundary conditions for an action-at-a-distance electro dynamic framework of a Quasi-Steady State Universe; which as discussed is in opposition to the broad class of Bigbang cosmologies. 1. Introduction

The subject of electricity and magnetism started with Coulomb laws which were similar to the Newtonian inverse square law of gravitation. Both laws were action-at -a-distance laws and they worked well till the mid-nineteenth century when the studies of rapidly moving electric charges brought out the inadequacies of instantaneous action at a distance. In 1845, Gauss (1867) in a letter to Weber hinted that the solution to the problem may come via the concept of delayed action at a distance wherein the interaction travels with speed of light. In restrospect one can say that the concept of action at a distance as developed by Newton and Coulomb was not relativistically invariant and Gauss's idea was to make it so. Gauss's suggestion remained unattended for several decades and in the meantime in the 1860s a satisfactory picture of electrodynamics was given by the field theory of Maxwell. A relativistically invariant action at a distance formulation became available only in the early part of this century and it was given independently by K. Schwarschild (1903), H. Tetrode (1922) and A.D. Fokker (1929). While formally this met the required criteria and produced equations that resembled those of Maxwell and Lorentz, the theory had a major practical defect: it treated the advanced interactions on an equal footing with the retarded interactions. Thus electric charges interacted via past directed signals as well as the future directed ones, the field of a typical charge being described by a symmetric combination 81 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 81-84. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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instead of by the observed . The question was : how can such an acausal theory describe reality which seems to respect the causality principle.

2. The Wheeler-Feynman Theory In a couple of papers, J.A. Wheeler and R.P. Feynman (1945,1949) found an ingenious way out of this difficulty by appealing to thermodynamic and cosmological considerations. They demonstrated a general result that in a universe well filled with electric charges, where all locally produced and outward propagating electrodynamics effects get eventually absorbed, the net effect is to produce only the full retarded signals. Thus, we find in this type of universe, which these authors called “a perfect absorber", the net effect on a typical charge a of all other charges in the universe is

We may term this the “response" of the universe to the local acceleration of charge a. This is the field which acting on the charge a produces the well known radiative damping, as first appreciated by Dirac (1938). Further, when (2) is added to (1) we get the full retarded field in the neighborhood of charge a. This theory was called by the authors the absorber theory of radiation. Thermodynamics entered the absorber theory through time asymmetry of absorption process which in a subtle way introduced asymmetry of initial conditions.

3. The Asymmetry of Expanding Models Later Hogarth (1962) demonstrated that in an expanding universe the time asymmetry is automatically incorporated, a point missed in the Wheeler-Feynman discussion which was centered on static universes. Since the expanding world models are described in Riemannian spacetimes, it was necessary to express the absorber theory and the basic framework of action at a distance in such spacetimes. Further, as pointed out by Feynman, Hogarth's use of the collisional

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damping formula to decide the absorption properties was inappropriate as it depended on thermodynamic asymmetry that Hogarth was trying to avoid. It was thus necessary to do calculations with an absorption process whose origin was purely electrodynamic. Hoyle and Narlikar (1963) carried out these tasks and demonstrated that Hogarth's claims were broadly correct. These results go against the broad class of the popular big bang models in the sense that in these models the response of the universe does not have the correct value given by (2) above but it is the exact opposite! The result is that it is the advanced rather than the retarded signals that manifest themselves in all electrodynamic processes. On the other hand, the steady state model of Bondi, Gold and Hoyle and the recently proposed quasi-steady state model of Hoyle, Burbidge and Narlikar (1993) give the correct response.

4. The Quantized Version Later work by Hoyle and Narlikar (1969,1971) showed how these concepts can be extended from the classical to quantum electrodynamics. The notion of action at a distance can indeed be described within the path-integral framework mechanics. The following results then follow: (I) The phenomenon of spontaneous transition of an atomic electron can be described as the interaction of the electron with the response of the universe. Provided the universe gives the correct classical response, it will then give the correct result for this phenomenon also.

(ii) Instead of being independent entities called “fields” with uncountably infinite degrees of freedom, here we have only the degrees of freedom of the charges and the collective response of the universe. Thus the formal divergences associated with field quantization are avoided. (iii) When path integral formulation is extended to the relativistic domain, the above method can be generalized to include the full quantum electrodynamics of interacting electrons including such phenomena as scattering, level shifts, anomalous magnetic moment, etc. Recently Hoyle and Narlikar (1993,1995) have found that the infinities that require renormalization of integrals in quantum field theory do not appear in the quantum absorber theory provided we are in the right kind of expanding universe. Thus the event horizon in the future of the steady state or quasi-steady state theory produces a cut-off at high frequencies of the relevant integrals which therefore are finite. It is thus possible to talk of a finite bare mass and bare charge of an electron.

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5. Concluding Remarks These investigations of the absorber theory in the expanding universe therefore tells us that provided the universe has the right kind of asymptotic boundary conditions, the action-at-a-distance framework of electrodynamics has the following advantages over the field theory: (I) It links the time asymmetry in cosmology to time asymmetry in electrodynamics and thus helps us to better understand the local principle of causality as a consequence of the large scale structure of the universe. (ii) It explains quantum electrodynamics with fewer degrees of freedom. (iii) It is free from divergences that beset quantum field theory. There is an additional possibility not yet fully investigated, namely, the response of the future absorber to any microscopic experiment in the laboratory. Could it be that we are unable to predict the outcome of an experiment with classical certainty, because not all variables are local? As in spontaneous transition, there is the response of the universe which may enter into the dynamics in an unpredictable way. Thus concepts like the collapse of the wavefunction, Bell's inequality, the EPR paradox, etc. may receive alternative interpretation in this action-at-a-distance framework.

References Dirac P.A.M., Proc. Roy. Soc., A167, 148 (1938). Fokker A.D., Z. Phys., 58, 386 (1929). Gauss C.F., Werke, 5, 629 (1867). Hogarth J.E., Proc. Roy. Soc., A314, 529 (1962). Hoyle F., Narlikar J.V., Proc. Roy. Soc., A277, 1 (1963). Hoyle F., Narlikar J.V., Ann. Phys. (N.Y.) 54, 207 (1969). Hoyle F., Narlikar J.V., Ann. Phys. (N.Y.) 62, 44 (1971). Hoyle F., Narlikar J.V., Proc. Roy. Soc., A442, 469 (1993). Hoyle F., Narlikar J.V., Rev. Mod. Phys., 67, 113 (1995) Hoyle F., Burbidge G., Narlikar J.V., ApJ, 410, 437 (1993). Schwarzschild K., Gottinger Nachrichten, 128, 132 (1903). Tetrode H., Z. Phys., 10, 317 (1922). Wheeler J.A., Feynman R. P., Rev. Mod. Phys., 17, 157 (1945). Wheeler J.A., Feynman R.P., Rev. Mod. Phys., 21, 425 (1949).

BOHM & VIGIER: IDEAS AS A BASIS FOR A FRACTAL UNIVERSE

CORNELIU CIUBOTARIU, VIOREL STANCU Technical University Gh. Asachi of Iasi, Department of Physics, Bv. D. Mangeron No 67, RO-6600 Iasi, Romania, Email: [email protected]

and CIPRIAN CIUBOTARIU Al. I. Cuza University of Iasi, Faculty of Computer Science, RO-6600 Iasi, Romania, Email: [email protected]

Abstract. Bohm and Vigier introduced the notion of random fluctuations occurring from interaction with a subquantum medium. Fényes-Nelson's stochastic mechanics generalises these ideas in terms of a Markov process and tries to reconcile the individual particle trajectory notion with the quantum (Schrödinger) theory. Bohm-Vigier deterministic trajectories are in fact the mean displacement paths of the underlying Nelson's diffusion process. However, random paths of stochastic mechanics are quite akin to Feynman paths which are non-differentiable and thus have fractal properties in the Mandelbrot sense. How a random field makes particles to propagate? This is the question. Can we speak about a stochastic acceleration property of (vacuum) spacetime which has stochastic (and chaotic) features? Can this offer an explanation of the inertial properties of matter? What is the source of randomness? The present paper tries to find an answer to these questions in the framework of the universality of a fractal structure of spacetime and of stochastic acceleration. Some arguments in favour of a fractal structure of spacetime at small and large scales areas follows. (i) Fractal trajectories in space with Hausdorff dimension 2 (e.g. a Peano-Moore curve) exhibit both an uncertainty principle and a de Broglie relation. Quantum mechanical particles move statistically on such fractal (Feynman) paths. Thus, Schrödinger equation may be interpreted as a fractal signature of spacetime. (ii) The formal analytic continuation iD) which relates the Schrödinger and diffusion equations has a physical alternative: there exists a (classical or quantum) stochastic fluid which can be either a fluid of probability for a unique element or a real fluid composed of elements undergoing quasi-Brownian motion. A particle (corpuscule) may be one or a small cluster of stochastic elements. There is a sort of democracy (statistical self-similarity) between the stochastic elements constituting the particle. As regards the cause of the randomness, the parton model involves a fragmentation of the partons. (iii) Nature does not “fractalize” (and quantize); it is intrinsically fractal (and quantum). Wave function of the universe is a solution of the Wheeler-DeWitt equation of quantum cosmology and corresponds to a Schrödinger equation. This can be related to the fact that observations of galaxy-galaxy and clustercluster correlations as well as other large-scale structure can be fit with a fractal with which may have grown from two-dimensional sheetlike objects such as domain walls or string wakes. The fractal dimension D can serve as a constraint on the properties of the stochastic motion responsible for limiting the fractal structure. (iV) The nonlinear (soliton) equation corresponds to a (linear)

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Schrödinger equation coupled to a medium with a specific nonlocal response. Physically, this model is similar to a simple case of linear propagation of thin beams in a wave guide. Thus, a free photon in (fractal) space represents in fact a “bouncing ball” in a wave guide. In other words, spacetime is structured as a (fractal) web of optical fibers (channels) which represents the skeleton of spacetime. (V) The proper wave functions describing a hydrogen-like atom are similar to the electromagnetic modes in optical resonating cavities.

1. Introduction As Vigier emphasised [1], it appears that a common stochastic basis unifies at least the three attempts to reconcile the individual particle trajectory notion with the Schrödinger theory of quantum particle behaviour: (i) the Bohm-Vigier causal aproach [2]-[6]; (ii) Fényes-Nelson-Guerra stochastic mechanics [7]-[11]; and (iii) the Feynman path description of quantum mechanics [12], [13]. For example, the Bohm-Vigier deterministic trajectories of the causal approach are not unknown objects in the stochastic mechanics. They are the mean displacement paths in the framework of the Nelson's (stochastic) diffusion process [14]. A given solution of the Schrödinger equation corresponds to a stochastic diffusion process satisfying the Newton second law in the mean. The stochastic differential equation describes the paths of motion of a point particle through a non-dissipative random medium. These paths may be identified with true configuration space paths of physical particles. Furthermore, random paths of stochastic mechanics are similar to Feynman paths [15] which are (fractal) curves described by continuous but nowhere differentiable functions of time. For example, the measured (average) path of a nonrelativistic quantum particle in a harmonic oscillator potential is a fractal curve with Hausdorff dimension equal to two [16], [17]. This is the formal arena in which the present paper unfolds. Bohm and Vigier introduced also the notion of random (stochastic) fluctuations arising from the interaction with a (random, stochastic) subquantum medium. This aspect generates the physical arena of our work. What is this stochastic (subquantum) medium (fluid)? Sometimes, in quantum mechanics - hydrodynamical analogies, one introduces a fictious fluid coupled to a suitable stochastic fluid (Schrödinger fluid, Dirac fluid, quasibrownian fluid, Klein-Gordon fluid, etc) whose nature (probabilistic or real, physical or formal) is not yet clear [18]-[22]. The main purpose of the present paper is to propose an intrinsic fractal structure of a dynamical space as a substitute or merely a property of a general subquantum (vacuum, fluid) stochastic medium1.

1

All real (natural) systems are very high dimensional (in the sense that they have a very large number of degrees of freedom), such as well developed chaotic motion, stochastic acceleration or turbulence. However, the observed behaviour displays a few degrees of freedom. In this case we may represent the most important of these few degrees of freedom with a low dimensional dynamics and hide our ignorance about the other ones by treating them as a stochastic component. In some (macroscopic) cases we even may ignore the stochastic component. In other words, if given some irregular dynamics, one is able to show that the system is dominated by a low-dimensional chaos, then only a few nonlinear (collective) modes are necessary to describe the system dynamics. This means that one could substitute the original set of p.d. equations with a small system of o.d. equations. The behaviour of systems dominated by a very large number of excited modes is better described by stochastic or statistical models. Generally, one considers that systems whose dynamics are governed by stochastic processes have an infinite value for the fractal dimension. This means that random processes fill very large-dimensional subsets of the system phase space. A finite, non-integer value of the dimension is considered to be an indication of the presence of deterministic chaos. That is, the existence of a low-dimensional chaotic attractor implies that only a low-dimensional subset of the phase space is visited by the system motion.

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2. Feynman Paths And Fractalons The first direct fractal signature in elementary quantum mechanics is considered to be the fact that the 'observed' path of a particle may be interpreted as a fractal curve with Hausdorff dimension equal to two. The fractal nature of a quantum-mechanical path may be also related to at least the following results: the Einstein-de Broglie rule; Heisenberg uncertainty principle; self-similarity of a fractal is a reflection of the underlying (quantum) dynamics; the energy spectrum of a particle. The key to all these results consists in the concept of a Feynman propagator the physics of which may be explained on the basis of a two-slit experiment. The Feynman path integral approach is a reformulation of quantum mechanics in terms of classical quantities. A path integral can be expressed as a sum over a path in phase space or in configuration space. Thus, sometimes the path integral has no direct relation to the real trajectory of a quantum particle. We emphasize that there are several (generally, an infinity of) alternative paths (C) which the particle could have adopted between A and B, each of these possesses a (partial) probability amplitude, The Feynman propagator can be written as a (Feynman path) or (functional) integral

where the sum (integral) is extended over all paths

is an infinitesimal range of

connecting A with B, and

Generally,

represents a history of a

(dynamic) geometry, i.e. a spacetime incorporating the motions of a particle. From a classical point of view, a real history should yield an extremum of the corresponding actions as compared to all adjoining histories. Adopting a quantum point of view, one applies the Feynman principle of democracy of histories which asserts that all histories (i.e. all the world lines connecting A and B) possess an equal probability amplitude but differ in the phase of the complex amplitude. In the sum over all probability amplitudes, the destructive interference cancels the contributions from all the histories which differ too significantly from the optimal or classical history (Fresnel wave zone and Feynman's principle of a sum over histories). The main contribution to the Feynman propagator is deduced from a “strip” around the classical path where the variation of the action is small. For a classical system this strip is very “thin” and defines a classical path. In contrast, the strip of a quantum system is very “broad” and thus the path of a quantum particle is fuzzy and smeared out, as for an electron orbiting around a nucleus. In other words, when considering the sum over all histories (1D Feynman-path approach), a quantum particle moves along an ensemble of paths simultaneously [23], [24]. On the contrary, in a stochastic (3D) approach the motion of a

quantum particle is studied by considering individual time steps [25]. In this (in fact, 3+1 D) case the point represents a plane (wavefront) perpendicular to the direction of motion, and the 'corners' of the Feynman paths corespond to the reversals along the specified direction. We call fractalon a particle moving on a fractal trajectory and fractons the vibrational (or fluctuating) aspects of fractals. The Feynman path approach, its relativistic generalization (the Feynman chessboard model), and Ord approach [26] refer to fractalons. In the following section we will define fractons as constitutive elements of a fractal space. Thus, a Feynman strip of a quantum

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system can be in fact a fracton. In favour of a fractal space pleads also the possiblity to provide a stochastic mechanics derivation of the Feynman propagator formula. Feynman's path integral may be represented as the summation of phase contributions associated with random trajectories of a stochastic process.

3. Fractons We face now the question: does a fractal space mean a space fractalized by a fractal matter or an absolute (intrinsic) fractal space which exists prior to any matter? In the later case a (pure) space represents a (separate) entity which exists as any field (e.g. an electromagnetic field) independently of any matter. However, this pure space, in the presence of matter and fields, may interact with them. From this point of view, we can assert that Einstein identified a space-time field with the gravitational field. Why does space pervade everything? Why cannot we separate the matter from space? Because the source of space (i.e. of gravitational field, following Einstein) is the matter, the mass. These questions are equivalent to the following: why cannot we separate an electron (generally, a charge) from its electromagnetic field? Because the charge is the source of the electromagnetic field. If we want, however, to have a 'separation' source-field, we can conceive an electrical charge as a very dense (conned, concentrated, condensed) electromagnetic field. Similarly, a very dense matter (like the initial Big Bang singularity) can be considered as a confined space which after Big Bang will expand and generate (convert into) matter and field. This conjecture that space is a field generated by any matter, implies that an exact fractal trajectory of a particle must be disturbed by the proper space field of the particle. If indeed the space field can be identified with the gravitational field, the quantum of space is, of course, the celebrated graviton. Since space is the only omnipresent field in the universe, possibly under the influence of a suitable topology it may generate any kind of matter and fields. This may be an argument that a vacum space (pure space) is in fact full of virtual particles for any matter and field. To devoid a particle of a gravitational field is equivalent to devoiding this particle of its proper space field. Why does a space field (or gravitational field) generate only forces of attraction in our region of the universe? Possibly, because the topology of this region with positive masses is specific for such forces. Can we imagine a universe without space? Yes, the last moment of the Big Crunch and the first moment of the Big Bang mean singularities without any dimension. The space arises after the Big Bang. We cannot say that the Big Bang is located somewhere in a vacuum space because there is no space when the Big Bang took place. Now we are in a position to understand how it is possible that a point can hide an entire universe and the meaning of a compactified dimension. A remarkable property of a fractal is its (exact or statistical) self-affinity which means that, in order to 'reproduce' the fractal structure, the spatial coordinates of a point are scaled by a different ratio Thus, in order to extend the fractal features to space and time we assume that self-affinity is characterized by an anisotropic scaling whereas a self-similarity arises by an isotropic scaling. Does this mean that a fractal may be considered related to anisotropic media? At this point we shift our attention to the Finsler geometry which may offer an approximate mathematical approach to the study of fractal geometry. This proposal is based on the fact that all the Finslerian geometrical objects depend, generally, on both the point and the directional variable (a tangent vector at direction of motion or velocity). Thus, a fractal is identified with a macroscopic anisotropic medium like a crystal and its metric description can be associated with a Finsler geometry which appertains to a locally anisotropic (i.e. Finslerian) spacetime (see Fig. 1). The fractalization of space means that the particle can “choose” a fractal trajectory.

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Between point I and F there exists an infinite number of different fractal trajectories. For example, (a), (b), (c) can be the von Koch snowflake curves and (d) can be a Peano-Moore trajectory. Since, we do not observe in nature, in general, a mixing of different fractals, we assert that a fractal curve, once chosen as trajectory by a (free) particle, will be maintained until the final point F is reached. A (free) particle cannot jump from one fractal curve to another one as also in general relativity a free particle cannot change its geodesic without an external force. How does a particle choose a fractal trajectory? Who or what determines such a choice. At this point we remark that the form of the fractal curve does not only depend on the scale (precision) of observation or measurement, i.e. of resolution (or level of description). The scale is decided by the particle itself. If the particle changes its direction of velocity by at a point A, its trajectory will be a von Koch trajectory. If the angle is the trajectory will be a Peano-Moore curve. The lengths of such curves decrease if the particle delays the change of direction of its velocity [see curves (b), (c), (d)]. This dependence of trajectories on point and direction suggests that fractal geometry may be, for example, Finslerian. In a fractal spacetime, the fractal trajectories pre-exist as empty fractals (i.e., before the moving particle has entered). The fractals in question form a web of possible routes which we denote as fractal geodesics. (This idea is not so 'exotic' as it may appear if we recollect that in Riemannian space the geodesics pre-exist also as solutions of the equation of motion of free particles independently of their masses). The number of these fractal geodesics is, of course, infinite, the von Koch and Peano curves being only some particular examples.

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A particle may choose a certain fractal curve without any visible external cause. The real cause is represented by (vacuum) fluctuations which may impose a change in the direction of the velocity of a particle at an earlier instant (e.g. at a point A) or subsequently (at a point C). Furthermore, a particle which chooses a fractal trajectory may perturb this curve (a path in space) generating a fractal wave which accompanies the particle and may be identified with a de Broglie wave. The new idea which emerges from this discussion is that the finer details of a fractal trajectory are decided by the interaction of a triad: particle-vacuum fluctuation-fractal spacetime (fractal vacuum) and not only by the precision of measurement. A (free) particle may also choose a fractal trajectory with no finer details [see curves (c) and (d)] between the two points I and F. Thus, in this case the resolution cannot change the result of a measurement, and in fact there is no sense in increasing the precision in this situation. In a fractal spacetime, if the particle is stopped, the fractal trajectory excited by this particle may still operate and transport information about the status of the particle via an empty wave which represents in fact a fractal trajectory excited by a particle (see Fig. 2) [27]. We remind that, in order to find a causal interpretation of quantum mechanics, some suggestions about the existence of an empty wave (but without any reference to a fractal structure of spacetime or other physical argument) appeared long time ago [28]-[32]. In essence, an imaginative picture of this hypothesis is given by a beam-splitter (i.e. “choice-junction”) experiment in which a particle (e.g., an electron) with its accompanying wave chooses a specific path and thus the reality exists independently of the process of measurement. Another possible choice may, of course, exist but does not contain the particle. It may be occupied, however, by an empty wave, i.e. by a physical wave without any particle and devoid of usual energy and momentum. The first wave (which accompanies the electron) we might call the full wave. After the beam splitter has been traversed, at which instant the full and empty waves are supperposed, there emerges an interference pattern which acts on the particle. In terms of a fractal spacetime we can assert that an empty wave represents an excited empty fractal (a fracton).

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4. An Example Of A Quantitative Result: Field Equations For Fractalons Vigier et al [22], [33] obtained the Proca equation,

considering a particle as submitted to stochastic fluctuations described by Nelson's equation,

where is the position of the particle in Minkowski spacetime and D and represent, respectively, the total derivative with respect to the proper time and the stochastic derivative:

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is the drift velocity and

is the stochastic velocity, and represents a (field) density which satisfies the equation of continuity. The next idea is to introduce as stochastic elements the scalar or (suitably chosen) spinning particles in order to obtain the Klein-Gordon or Proca equation. The probability distributions corresponding to these equations correctly describe the stochastic distribution. In order to motivate the fractal space conjecture our task is now to obtain some of the well known equations on the basis of the hypothesis that free particles are obliged to move on (preexisting) fractal trajectories. For the time being, we are not waiting to obtain a general result because there is no generally accepted formal mathematical (satisfactory) definition of fractals which includes all physical systems which are commonly considered to be of a fractal nature and excludes all those which are considered to be non-fractals. The concept fractal is rather used as a general overall descriptive concept with a dubious mathematical status. For the time being, what we can offer is to describe a fractal as a special real physical system (physical fractal or natural fractal or quasi-fractal, for example, coastlines, mountains, clouds, the surfaces of solids, etc.) or a special idealized (abstract) mathematical set (exact ideal fractals or strictly fractals or abstract fractals or mathematical fractal, for example, attractors as subsets in a phase space associated with the time evolution of a nonlinear dynamical system). The mathematics used to describe fractals do not constitute a theory of fractals, but rather refer to some properties and numerical simulation of fractals. Thus, there exist algorithms for generating random fractals, for measuring fractal dimensions etc, but no elaborated theory of fractals. The so called fractal geometry does not yet constitute a modus operandi in the world of fractals. However, the fractals exist and their general presence at all scales of the universe is ubiquitous. A theoretical physical model restrained to a particular type of fractal can be constructed starting from Ord's approach [26]. We choose, for example, a Peano-Moore (PM) fractal trajectory on which a free particle (a fractalon) is conned to move. In a random walk model with drift, the geometry becomes complicated and we define a gauge-covariant-fractal derivative,

where represents a 1D fractal-gauge field generated by time-like and space-like vertices (corners, collision centers) of a fractal trajectory which can be also modeled by a (scattering) lattice. With this derivative we can obtain the principal equations of quantum phenomena (Proca, Klein-Gordon, Dirac equations) maintaining the single particle-continuous-fractal trajectory concept. The fact that the vertices of a fractal trajectory decide the physical motion of a quantum

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particle is equivalent with the old idea of de Broglie, Bohm and Vigier referring to the interpretation of the deviation from the Newtonian equations of motion as being due to a quantummechanical potential associated with the wave function.

5. Open Problems It is well known [34] that fractal objects can be used for optical diffraction studies. Thus, the transparent fractal curve (e.g., a Koch curve) in the photographic negative behaves as an aperture in diffraction experiments. Our hypothesis on the existence of fractons may be proved by an experiment of diffraction using vacuum fluctuations.

Aknowledgments The authors are thankful to Professor Richard L. Amoroso (Noetic Advanced Studies Institute, Orinda, CA, USA) for encouragement, and to Dr. Carmen Iuliana Ciubotariu (Mount Royal College, Calgary, Alberta) for discussions.

References [1] Vigier, J. P.: Real physical path in quantum mechanics. Equivalence of Einstein-de Broglie and Feynman points of view on quantum particle behaviour, in Proc. 3rd Int. Symp. on Foundations of quantum mechanics, Tokyo, 1989. [2] Bohm, D., Phys. Rev 85 (1952), 166; 85 (1952), 180. [3] Bohm, D. and Vigier, J. P., Phys. Rev 96 (1954), 208; 103 (1958), 1822. [4] Vigier, J. P., Physica B 151 (1988), 386. [5] Vigier, J. P., Ann. der Physik 7 (1988), 61. [6] Bohm, D. and Hiley, B., Physics Rep. 172 (1989), 94. [7] Fénies, I.: Eine wahrscheinlichkeitsteoretische Begründung und Interpretation der Quantenmechanik, Zeitschrift fürPhysik 132 (1952), 81-106. [8] Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev 150 (1966), 1079-1085. [9] Nelson, E.: Quantum Fluctuations, Princeton University Press, Princeton, 1985. [10] Guerra, F. and Loredo, M. I.: Stochastic equations for the Maxwell field, Lettere al Nuovo Cimento 27 (1980), 41-45. [11] Guerra, F.: Physics Rep. 77 (1981), 263. [12] Feynman, R. P.: Space-Time Approach to Non-Relativistic Quantum Mechanics, Rev. Mod. Phys. 20 (1948), 367-387. [13] Feynman, R. P. and Hibbs, A. R. Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965. [14] Garbaczewski, P.: On the concepts of stochastic mechanics: Random versus deterministic paths, Physics Letters A 143 (1990), 85; Accelerated stochastic diffusion processes, Physics Letters A 147 (4) (1990), 168174; Randomness in the quantum description of neutral spin 1/2 particles, Fortschr. Phys. 38 (l990) (6), 447475. [15] Garbaczewski, P.: Nelson's stochastic mechanics as the problem of random ights and rotations, Lecture delivered at the XXVII Winter School of Theoretical Physics in Karpacz, 18 Feb.-l March 1991. [16] Abbott, L. F. and Wise, M. B., Dimension of a Quantum-Mechanical Path, Am. J. Phys. 49 1981, 37-39. [17] Campesino-Romeo, E., D'Olivo, J. C. and Socolovsky, M., Hausdorff Dimension for the Quantum Harmonic Oscillator, Phys. Lett. 89 A(7) (1982),321-324. [18] Aron, J. C.: A stochastic basis for microphysics, Foundations of Physics 9 (3/4) (1979), 163-191. [19] Aron, J. C.: Stochastic foundation for microphysics. A critical analysis, Foundations of Physics 11 (9/10) (1981), 699-720. [20] Aron, J. C.: A Model for the Schrödinger zitterbewegung and the plane monochromatic wave, Foundations of Physics 11 (11/12) (1981), 863-872.

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[21] Aron, J. C.: The foundations of relativity, Foundations of Physics 11 (1/2) (1981), 77-101. [22] Cufaro Petroni, N. and Vigier, J. P.: Stochastic derivation of Proca's equation in terms of a fluid of Weyssenho tops endowed with random fluctuations at the velocity of light, Phys. Lett. 73 A(4) (1979), 289-291. [23] River, R. J., Path Integral Methods in Quantum Field Theory, Cambridge University Press, Cambridge, 1987. [24] El Naschie, M. S.: Chaos and Fractals in Nano and Quantum Technology. Chaos, Solitons & Fractals, 1999, 9(10), 1793-1802. [25] Ord, G. N. and McKeon, D. G. C.: On the Dirac Equation in 3+1 Dimensions, Annals of Physics 222(2) (1993), 244-253. [26] Ord, G. N.: Fractal space-time: a geometric analogue of relativistic quantum mechanics, J. Phys. A: math. Gen. 16 (1983), 1869-1884. [27] Argyris, J., Ciubotariu Carmen I. and Weingaertner, W. E.: Fractal Space Signatures in Quantum Physics and Cosmology. I. Space, Time, Matter, Fields and Gravitation, Chaos, Solitons and Fractals, in course of publication. [28] de Broglie, L., The Current Interpretation of Wave Mechanics, Elsevier, Amsterdam, 1964. [29] Hardy, L., On the existence of empty waves in quantum theory, Phys. Lett., 1992, A 167, 11-13. [30] Bohm, D. and Hiley, B. J., The Undivided Universe. Routledge, London, 1993. [31] Folman, R. and Vager, Z.: Empty wave detecting experiments: a comment on auxiliary “hidden" assumption. Found. Phys. Lett., 1995, 8 (1), 55-61. [32] Mac Gregor, M. H.: Model basis states for photons and “empty waves", Found. Phys. Lett., 1995, 8 (2), 135-160. [33] Vigier, J. P., Lett. Nuovo Cimento 24 (1979), 258, 265; 25 (1979), 151. [34] Uozumi, J. and Asakura, T.: Demonstration of diffraction by fractals, Am. J. Phys. 62 (3) (1994), 283-285.

A RANDOM WALK IN A FLAT UNIVERSE FOTINI PALLIKARI Athens University, Physics Department, Solid State Physics Panepistimiopolis, Zografos, 157 84 Athens, Greece

Abstract. Our experience of the geometry of physical space at small scales is one that is flat obeying Euclidean laws, as simple measurements confirm. At cosmological scales, on the other hand, space appears also to be flat according to newly acquired evidence on the cosmic microwave background radiation, even if it gets considerably curved near the presence of massive bodies. This paper argues that the ‘geometry’ of the representational space of a thermal electronic noise process in fractional Brownian motion (fBm) is simply the reflection of the geometry of our universe; that is, flat with scattered local regions of curvature.

1. Introduction

The long-awaited answer regarding the geometry of physical space containing our universe was given early this year [1]: It is flat obeying Euclidean geometry. The information was provided by a high-resolution temperature map of the cosmic microwave background (CMB) radiation, the radiation left over from the big bang, which was photographed in the process of an international balloon experiment over Antarctica (‘Boomerang’ collaboration). Additional evidence soon followed from another international balloon experiment, designed by the University of California at Berkeley (‘MAXIMA’ collaboration), to confirm the previous finding [2] as well as by another study referring to the detection of weak gravitational lensing distortions of distant galaxies [3]. According to the different solutions to the fundamental mathematical equation that governs the theory of gravitation [4], three space-Geometries are possible. This equation describes the space radii of curvature in terms of the spatial distribution of mass and leads to a flat, spherical or hyperbolic space [5]. Flat space obeys the rules of Euclidean geometry. Parallel lines never meet, triangles span 180 degrees, the circumference of the circle is and so on, within our familiar world of relatively small distances. This, however, could have been just a false conclusion based on insufficient information. For instance, although the earth is not flat it appears to be so at small scales. Alternatively, if the universe were spherical, at very large scales parallel lines would eventually meet triangles would span above 180 degrees and the circumference of the circle would be smaller than This would be a closed space possessing a finite volume and a positive radius of curvature. There is the third alternative, the hyperbolic universe, in which parallel lines would diverge, triangles would span less than 180 degrees and the circumference of the circle would be larger than Same as in the flat space, this hyperbolic universe would also be open 95 R.L. Amoroso et al (eds.), Gravitation & Cosmology: From the Hubble Radius to the Planck Scale, 95-102. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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resembling in two dimensions the geometry of the shape of a saddle. The universe is a vast sea of scattered lumps of matter of various sizes and the space (& time) curves near massive bodies. This space distortion would resemble in two dimensions to ‘pimples’ growing on the surface of a plane. Of course one would have to extend this visualization into three dimensions to appreciate the real thing. The brilliant scientist Richard Feynman in his 1963 ‘Lectures on Physics’ described the following picture of a possible universe [5], which coincides with our present image of the universe: “Suppose we have a bug on a plane, and suppose that the “plane” has little pimples in the surface. Wherever there is a pimple the bug would conclude that his space has little local regions of curvature. We have the same thing in three dimensions. Wherever there is a lump of matter, our three-dimensional space has a local curvature - a kind of three-dimensional pimple.” In this paper we shall compare two random-walk processes within their representational space comprised by Hurst exponents, H, [6-8]. The first process is a three-dimensional random walk in physical space. The second process is an imaginary one-dimensional random walk, the one dimension being time while the size of the steps are determined by the outcomes of a thermal electronic noise process following Gaussian statistics. The electronic noise process exhibits a characteristic fractal character [9]. We shall show that the power law formulation that connects the range of diffusion, R, with the number of steps, N, in both random walk processes occasionally deviates from the orthodox behaviour like a distortion to the geometry of its representational space: the Euclidean geometry in the first process and the ‘geometry’ that characterizes a random independent process in the second. This deviation introduces a fractional Brownian motion (fBm) character in the respective processes according to their associated H exponents, also found in an abundance of natural processes [10]. It will be shown here that the characteristic fractal behaviour that was observed in the electronic noise process -featured as windows of fBm character within an overall ordinary Brownian motion - is also characteristic of the geometry of the universe. It is depicted as an overall flat space containing localized areas of curvature. The discussion will make reference to the rescaled range (R/S) analysis, as it has been applied in the previous study [9].

2. Rescaled Range Analysis And Fractional Brownian Motion The characteristic parameter that the rescaled range analysis (R/S) determines is the Hurst exponent H. The hydrologist H.E. Hurst has argued [10-11] that independent random processes yield records in time that after appropriate transformation to

through the linear relation:

have a range R that obeys the Hurst’s law [10]:

A RANDOM WALK IN A FLAT UNIVERSE where S is the standard deviation in the distribution of

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and N is the number

of records. Hurst argued that many natural processes such as the influx of water in a river, the height of rain, or the size of tree rings, etc. within a given period of time, did not constitute random independent processes but they were better described through the power law

The R/S analysis showed that the average Hurst exponent for a wide number of natural processes studied was not 0.5 as expected by equation 2, but around the value 0.7. Mandelbrot and co-workers, on the basis of this result, developed the fractional Brownian motion to model such processes that obeyed equation 3. A transformed time series of records of a natural process is a one-dimensional fractional Brownian motion, fBm, if the increment of

has

a Gaussian distribution with a mean zero and variance which is proportional The Hurst exponent will vary between the values For the proper random walk case the H exponent takes the value according to equation 2. The generalized fBm determines three regions of H values, table 1 and two original fBm processes: The persistent and the anti-persistent fBm. The persistent fBm describes a process having records in time whose increments are positively correlated regardless of distance between neighbours within the time series. The opposite occurs in the anti-persistent fBm: the increments are negatively correlated, while the strength of correlation in both cases depends on the deviation of H exponent from the value 0.5. Only in an ordinary Brownian motion the increments of the records in time are random and independent and therefore their correlation is equal to zero. It was deduced from the scaling form of equation 3 that the range, R, in a one-dimensional generalized type of random walk1 depends on the number of steps according to [11]: The range R determines the range of diffusion in the motion of an object away from the origin, while the origin represents the average of the normal distribution of its steps. The coordinate of this moving object’s actual position is while the size of each of its steps is given by

as discussed above.

1 That is, a Brownian motion in a straight line with normally distributed steps about a fixed average and with a fixed standard deviation

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The R/S analysis has been applied on time series of converted records of Johnson electronic noise [9]. The electronic noise random process was manufactured such as to generate a broad white noise power spectrum and its records in time would, therefore, be represented by an ordinary Wiener Brownian motion [12] (Hurst exponent, H = 0.5 ). The R/S analysis of electronic noise, however, confirmed this on the one hand but also revealed that the ordinary Wiener Brownian process occasionally shifted to a persistent fractional Brownian motion of relatively weak strength (only 4% deviation from 0.5), which suggested a chaotic behaviour underway. A chaotic process may often appear as a Wiener Brownian motion displaying windows of disorder For the electronic noise studied by R/S analysis, the ordinary Brownian motion described the fundamental background structure of the process. Its shifting to the fractal regime was portraying a superimposed structural anomaly. In the next paragraph we shall argue that a random walk in a curved space obeys equation 4, same as a generalized random walk in a straight line for which Both in the three-dimensional space and the one-dimensional case of the electronic noise process, the background structure of their corresponding ‘Geometries’ is flat (H = 0.5).

1.

Random Walk In Flat, Or Curved Space

In a strict mathematical sense a curved physical space is one in which the orthodox laws of Euclidean geometry do not hold any more. We shall extend here this notion to include any system in their configurational space whose behaviour deviates from the valid laws. If the confirmed laws and principles in a given configurational space (the realm where the parameters through which this given law is manifested) are breaking down, we could denote this configurational space as ‘curved’. The electronic noise process referred to in the previous paragraph is manufactured to be a random independent process characterized by H = 0.5 and it was observed to behave as such (across the 70% of the data). However, a deviation from the norm occasionally occurred which collectively yielded an H exponent H >0.5 (H = 0.521 ±0.004), apparently breaking down the law that holds in theory as well as in practice for its behaviour. This observation is analogous, in terms of mathematical expression, to the cosmological observation of a flat universe containing scattered ‘pimples’ of curved space. Let us see why this is so. Let us consider that the lumps of celestial matter are distributed uniformly in space. Then we only need to count their number within certain volume to assess whether space is flat or curved. Without loosing accuracy of description, we can transfer this treatment from the three- to the two- or even to one-dimensional case (random walk in a straight line). We shall refer in this section to the two-dimensional case and then extend it to three dimensions. Consider a random walk of an imaginary particle on a flat surface of area figure 1. With each step the particle moves away from the origin and the range of its diffusion, R, increases as the square root of the number of steps, N. This is because all of the particle’s stopover-points, N, on the plane at time t, with steps independent from each other, are contained in a circle of radius R. The number N of uniformly distributed stopovers on the plane is proportional to the area of the circle, i.e.

A RANDOM WALK IN A FLAT UNIVERSE

and therefore To curve the plane surface into, say, a spherical surface of radius r, one would have to shrink it. The original circular area, is equal to, and it shrinks to a spherical cap of area Consider that the curving of the flat area follows these steps: First, it bends like a circular tablecloth falling over a sphere. The cloth will get folds around the circular path of points of contact with the sphere and shrinking along these circumferences will be required for a tight fit. Second, the circumference, a, of the flat tablecloth becomes after shrinking the circumference, b on the sphere. After fitting on the sphere, the radius R of the original circle becomes the arc (OA), subtending an angle

The angle will depend on the extent of bending. For reasons of simplicity let us assume that the bending is such that the original area of the circle becomes the area

of a hemisphere of radius r,

then

Assuming that the surface shrinking has not affected the original number of uniformly distributed points, these points will appear on the spherically curved surface as being closer than they were on the plane due to shrinkage and the point pattern will appear more dense than before. For instance, let the original number of points be which is now the number of points on the spherical cap that the tablecloth has become. If one tries to measure the number of points on the

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spherical area equal to the original area counted since

more points

will be

Unaware of the surface bending one would wonder why

the mathematics of flat space fail and that the number of points appear to grow faster with distance than before (R is estimated by area measurements2). They would appear to grow as: where

or equivalently:

where3 H<0.5 and The exponents related to the space curving involved.

and H are arbitrary parameters

It is clearer if we apply here a numerical example. A random walk on circle of radius R = 2, curved into a spherical cap on a sphere of radius

appears

to have a range that grows as rather than the expected (see appendix for more details). Similarly, N random events growing in a random walk process on a flat surface within a range R, would appear more distant should the flat surface become curved with a negative radius of curvature, a hyperbolic two-dimensional space, a surface resembling a saddle. This is because to curve a flat surface into a saddle-like surface one needs to stretch it. The points on the negatively curved surface will appear to grow slower than with the square of the range, as it was the case of random walk in a plane, i.e.

and The density of random walk stopovers will be lower than in the case of flat space. The ideas presented here on surface curvature and random walk are inspired by George Gamow [14]. In this lucid, admirable exposition of curved space geometry he argued that the analysis applicable to two-dimensional space will equally hold true for a three dimensional space, that is in other words for our universe. A random walk and a Brownian motion are frequently used under equivalent contexts. The scaling properties observed in a one-dimensional random walk having steps determined by the distribution of records in time of the converted Johnson electronic noise (equation 4), were shown above (equations 13 & 15) to also account for three dimensions as in the case of the distribution of matter in the universe. In both processes extending at diverse scales from each other, the world of the very large and the world of the very small, the representation of their Geometries are fabricated by an underlying flat ‘space’ (H=0.5)

2 3

as if the surface was assumed to be flat.

The same letter H is used as in fBm for simplicity of comparison.

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displaying localized regions of curvature At the cosmological scale the deviations from a flat space are the consequence of the presence of massive bodies and curve physical space spherically, while in the case of the electronic noise process the deviations result from an inherent chaotic element and curve their representational ‘space’ (described by equation 4) hyperbolically.

Appendix

Consider a circular area

of radius R=2, see figure 1,

curved by shrinking into a spherical cap on a sphere of radius r. Let us suppose that r, be such that the area of the hemisphere, equals the original area of the circle:

The two radii are,

therefore, related through: will be4, (equation 11):

then the angle The area of the spherical cap

through equation 7, 8, A-3 & A-5 is:

and in this

example is equal to:

The extent of

shrinking is therefore:

that is, about 16% of

the original area, due to curving. This could have occurred by a mass of the sphere comparable to the mass of Saturn5. Suppose that stopover points were expected to grow on by an ordinary random walk. After shrinking, the circle becomes the spherical cap of area The number of stopovers on the original area 4 5

since from the geometry of this example Assuming that the excess radius due to the bending of the surface

is

(now the area of the

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hemisphere) has increased to the number

as discussed in section

Being unaware of the surface curving we would think that our initial law: constant equal to 100/4

(where C is a in this example) has changed into The number of stopovers grows now faster with

range than in the case of the plane, or the range R varies with N as: noting that the exponent takes values below what is expected for flat space, (H= 0.5). Acknowledgements The electronic noise data were generated by Emil Boller and Holger Bösch of IGPP, Germany. Petros Belsis of Athens University created the original Hurst analysis software. A comment in a public e-mail posting by Jack Sarfatti has triggered my interest in the Hurstanalysis. 'This work was funded by the special research account of Athens University.'

References 1. 2. 3.

4.

5.

6. 7. 8. 9. 10. 11. 12. 13. 14.

De Bernandis P et al, (2000) A flat Universe from high-resolution maps of the cosmic microwave background radiation. Nature 404, 955-959 Hannany S et al, (2000) MAXIMA-1: A Measurement of the Cosmic Microwave Background Anisotropy on Angular Scales of 10 arcminutes to 5 degrees. Submitted to Astrophys. J. Lett. Wittman DM, Tyson JA, Kirkman D, Dell’Antonio I, & Bernstein G (2000). Detection of weak gravitational lensing distortions of distant galaxies by cosmic dark matter at large scales. Nature 405, 143-148 See for instance, Eddington AS (1920). Report on the Relativity Theory of Gravitation: Fleetway Press Ltd, London, and also Ciufolini I, Wheeler JA (1995). Gravitation and Inertia: University Press, Princeton. Feynman RP: (1963) Lectures on Physics. Prepared for publication in 1997 by Leighton RB and Sands M, under the title: “Six Not So-Easy Pieces, Einstein’s Relativity, Symmetry and Spacetime”. Addison-Wesley, USA. Mandelbrot, B. B. and Van Ness J. W. (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev., 10, 4: 422-437. Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. (W. H. Freeman, San Francisco). Mandelbrot, B.B. (1999). Multifractals and 1/f Noise. Wild Self-Affinity in Physics. (Springer, New York). Pallikari, F. (2000) A Study of the Fractal Character in Electronic Noise Processes. Chaos, Solitons & Fractals, 10 (8), in press. Hurst H. A. (1951) Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng., 116: 770-808. Feder J. (1988) Fractals. Plenum Press New York. Mandelbrot, B.B. (1999). Multifractals and 1/f Noise. Wild Self-Affinity in Physics. (Springer, New York). M. Schroeder (1991). Fractals, Chaos, Power Laws. Minutes from an Infinite Paradise: W. H. Freeman and Company, New York. Gamow G (1993). Mr. Tompkins in Paperback: University Press, Cambridge

MULTIPLE SCATTERING THEORY IN WOLF’S MECHANISM AND IMPLICATIONS IN QSO REDSHIFT SISIR ROY* & S.DATTA** Physics & Applied Mathematics Unit Indian Statistical Institute Calcutta 700035, India *[email protected] **[email protected]

Abstract. The theory of Correlation-induced spectral changes is becoming popular over the past decade and it has established its possibility in the field of QSO redshift. A brief review of its development including multiple scattering, no blueshift condition, effect on spectral width and the correlation between shift and width of a spectral line is presented. 1. Introduction Today one of the most controversial topics of astrophysics is the nature of quasar redshift. Some questions are raised against the well established BBC(Big Bang Cosmology)-black hole school of thoughts as follows: 1. The presence of a black hole, or of dark matter in general, is inferred from the study of the dynamics of the environment and it is expected that a black hole would pull matter, which will fall towards it. But unfortunately, in reality, we have yet to get a spectroscopic evidence of such inward flow. 2. The maximum luminosity that can be sustained by an accreting source is the Eddington luminosity [ Kembhabi & Narlikar ], given on the assumption that the size of the emitting region is hundred times the Schwarzschild radius:

For time variations observed on the scale of hours we should get at the order of and thus arrive at a maximum luminosity of order erg The black hole in this case has to have a mass being the solar mass. The most luminous quasars are however, are at least hundred times as powerful. Thus if the above scenario is correct, then we should not expect the time variability of the order of hours that is actually observed in some quasars. 3. Basically, the violation of cosmological hypothesis can be demonstrated by showing the existence of two extra-galactic sources in close proximity but with different 103 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 103-110. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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S.ROY & S.DATTA redshifts. The cosmological hypothesis requires a unique relation between redshift z and distance D, say, and hence if close neighbors have the same value of D, it is expected that their redshifts will be the same. An example of Quasar-galaxy pair is described below to explain the above.

A galaxy (NGC 4319) and a quasar (Markarian 205) are connected by the straight luminous link. However the redshift of the quasar is approximately twelve times greater than that of the galaxy, and consequently, according to Hubbies law the quasar should be twelve times further from Earth than the galaxy [ Wolf, 1998]. To solve this discrepancy, several explanations have been proposed specially to interpret redshift other than due to Doppler effect. Some of them are stated below. 1. Recently a mechanism has been proposed by L. Accardi and his coworkers modeling the interstellar medium by a low density Fermi gas [Accardi et al, 1995]. They proposed a general redshift theorem. 2. H.Arp has shown, over the past two decades, that the quasars appear to cluster near normal galaxies [Arp, 1987], 3. Another unexplained ( perhaps circumstantial ) result is by William Tifft who has suggested that redshifts are quantized and they are whole multiples of 72 km / sec! [Tifft , 1987]. But all of the above theories are neither well explained nor have any scope to be tested experimentally. After several such attempts to explain the redshift alternatively, a mechanism was proposed by E. Wolf in the mid-eighties [ Wolf, 1987] that has no connection with the relative motion and gravitation. The main features of this new mechanism for redshift, called the Wolf Effect are as follows. The Wolf effect is the name given to several closely related phenomena in radiation physics dealing with the modification of the power spectrum of a radiated field due to spatial fluctuations of the source of radiation [ James, 1998]. It was shown [ James & Wolf, 1990] that if an incident spectral line is of Gaussian profile and the scattering medium is also Gaussian, then the scattered field is also approximately Gaussian, but with different mean frequency and standard deviation . The changes are as follows.

where

and

are the mean frequency and the width of the incident spectral line, and

are medium parameters which mainly determine the angle of scattering. Consequently the relative frequency shift z is given by

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Let us now assume that the light, in its journey, encounters many scatterers described as in the previous chapter. What we observe at the end is the light scattered many times, with an effect as that stated in the last section of the previous chapter in every individual process. Further we assume that the scatterers are weak and incoherent in nature, i.e., scattered rays do not interfere with the incident ones. This assumption leads to the validity of first order Born approximation in the case of multiple scatterers too. The physical justification of this assumption lies in the fact that the dimension of the scatterers is small enough compared to the cosmological distance scale. 2. Effect Of Multiple Scattering On Broadening Of Spectral Lines

Since the z-number due to such effect is Doppler-like (i.e., does not depend on the central frequency of the incident spectrum), each depends on only, not on . Let us calculate the broadening of the spectral line after N number of scatterings [ Datta et. al., 1998B ]. From the equation of (1), we can easily write,

and from the

equation of (1)

Let us assume that the redshift per scattering process is very small, i.e., Then, after some calculations, we get

As the number of scattering, N, increases, the width of the spectrum obviously increases and the most important topic to be considered is whether this width is below some tolerance limit or not, from the observational point of view. There may be several measures of that tolerance limit. One of them is the sharpness ratio, defined as

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where,

S.ROY& S.DATTA

and

are the mean frequency and the width of the observed spectral line.

Obviously, Q increases when line-width decreases, but, the problem is elsewhere: whenever we employ any scattering mechanism, the spectral line must suffer from broadening [ Schrödinger, 1955], and consequently, the sharpness of a line fall gradually with the increasing z-number. 3. No-Blueshift Condition

Though we are familiar with galaxy-blueshifts, no direct evidence of quasar-blueshift is observed so far. Recent researches indicate some kind of co-existence of redshiftblueshift in a same spectrum, but that is totally in relative sense, i.e., one line is shifted towards the bluer end with respect to another line. Here we formulate a no-blueshift condition [ Datta et al, 1998A] using multiple scattering theory and study this condition in both isotropic and anisotropic cases. From multiple scattering theory we get

as a sufficient condition for no blueshift. It is interesting to note that if this condition (7) is violated, then z may be positive or negative but the corresponding broadening will be very high in the case of multiple scattering [according to equation (5)]. In other words equation (7) serves also as no blurring condition. The most interesting clue lies here that under multiple scattering the no blueshift and no blurring conditions coincide with each other. Actually general form of no blurring condition is

Then it follows that if any spectral line is blueshifted, due to the above coincidence of no blurring and no blueshift, the width of the line becomes too high to be analyzed.

4. Wolf Contribution In Observed Redshift

Some lines in observed spectra of astrophysical sources have their widths more than the expected values due to natural width and Doppler broadening (i.e., due to internal effects). Scattering in interstellar medium (external effects) might cause the excess width and if we consider multiple scattering under Wolf mechanism, it will help us to find the corresponding contribution in z-number. Then cosmological contribution of the spectral shift will be reduced and consequently it might force us to re-estimate the distances of those sources. We get the Wolf contribution of z-number from the relation

MULTIPLE SCATTERING IN QSO REDSHIFT

where

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is the relative frequency shift in an individual collission.

It is clear from the above consideration that

does depend on the spectral width and so

we can correlate spectral shift with its linewidth, and more detail analysis of that dependence will be discussed in the next section. Here in the following figure[Figure-l], we have shown the Wolf-contribution in z-number.

Clearly, for low z, the Wolf-contribution is negligible and for z > 1, a linear relationship between z and shows a possibility of re-estimation of cosmological distance.

5. Critical Source Frequency In Multiple Scattering Theory Though under Wolf mechanism it has been observed that the relative frequency shift of a spectral line does not depend on its source frequency, the source-contribution in the observed spectra cannot be neglected. The relative shift in the frequency does depend on the width of the line in the source spectra. However, under the assumption that the source is monochromatic, this contribution is very small. The width of the observed line on the other hand, is very much dependent on the source-width of the line and in this case the contribution is not small. The explicit dependence on the source width provides a quantitative measure of the deviation from the mean frequency [ Roy et. al., 2000]. After some algebraic manipulation we get

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We define the right side of this inequality to be the critical source frequency

i.e.,

Thus for a particular medium between the source and the observer, the critical source frequency is the lower limit of the frequencies of analyzable spectral lines coming from the source. Here the analyzability is in the sense that the shift of a particular spectral line dominates its width. In other words, if we say that the spectral line is not analyzable, we mean that its shift is completely masked by its broadening. So it will be very difficult, rather impossible, to detect the shift of the spectral line, coming from a source, whose central frequency will be less than the critical source frequency.

6. Luminosity-Line Width Correlation And Tully-Fisher Relations

In spite of the frequent use of the Tully-Fisher relation as a distance indicator, the physical origin of this relationship is still poorly understood. Although it is often assumed that the relation is strictly linear in a logarithmic sense [ Still & Israel 1998], it remains unclear whether all rotationally supported disk galaxies obey a single luminosity-line width correlation. Tully-Fisher relation can be written in a more generalized form as

Here the slope a is neither theoretically nor empirically a constant, but a continuous of the magnitude system, varying from function of the isophotal wavelength in the blue, to in the infrared. 7. Tully-Fisher Type Relations In Multiple Scattering Theory

According to the second equation of (1) and the relation

we

conclude that both the shift and the width of a spectral line depend on the medium parameters. In the following figure [Figure-2] the spectral broadening (due to multiple scattering) is shown to vary with the relative frequency shift due to same reason.

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We can write the relation between the width and the shift as

where K is a positive constant,

is the minimum (natural) broadening, and

is the

spectral width inherent to the source. Taking the logarithm of both sides we can write,

Now for small z, i.e., z<<1,

where d is the distance modulus. After simplification,

where The above relation between the distance modulus and the width has a striking similarity to the Tully-Fisher relation but without any angular dependence [Roy et. al., 1999].

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8. Possible Implications Unlike the Doppler effect where the width of a line is unaltered to the shift, the Wolf mechanism predicts a tight relationship between the width and the shift of a line. As such, it is evident from the above analysis that the multiple scattering theory within Wolf’s framework might play a significant role for QSO’s. References Accardi L, Laio A, Lu Y G, Rizzi G, Phys. Lett. A 209(1995) 2 Arp H C, Quasars, Redshifts and Controversies, Interstellar Media, Berkeley, 1987 Datta S, Roy S, Roy M, Moles M, Int. J. Theo. Phys. 37(1998)1469. Datta S, Roy S, Roy M, Moles M, Int. J. Theo. Phys., vol-37, No-4, 1998A. Datta S, Roy S, Roy M, Moles M, Int. J. Theo. Phys., vol-37, No-5, 1998B. James D. F. V. & Wolf E., Phys. Lett. A, vol-146 (1990) 167. James DFV, Journal of the European Optical Society A : Pure and Applied Optics (1998 ). Kembhabi A. & Narlikar J. V. in Quasars & AGN Roy S, Kafatos M & Datta S, Astronomy & Astrophysics, vol-353(2000)1134. Roy S, Kafatos M, Datta S, Phys. Rev. A, vol-60, No-1,1999. Schrödinger E, IL Nuovo Cimento 1955, vol-1, No-1, p-62. Still J M and Israel F P, (1998), communicated in Astronomy & Astrophysics Tifft W.G. in New Ideas in Astronomy, Cambridge University Press, 1987. Wolf E, Conf. Proc. Vol-60, Waves, Information & Foundation of Physics, SIF, Bologna (1998). Wolf E, Nature, 326(1987)363.

CONNECTIONS BETWEEN THERMODYNAMICS, STATISTICAL MECHANICS, QUANTUM MECHANICS, AND SPECIAL ASTROPHYSICAL PROCESSES

DANIEL C. COLE Dept. of Manufacturing Engineering, Boston University, Boston, Massachusetts 02215 USA

Abstract

This article discusses several diverse “zero-point" notions, ranging from early classical blackbody radiation analysis, to astrophysical and cosmological considerations now being contemplated. The now commonly accepted quantum mechanical meaning of “zeropoint" is compared with the early historical thermodynamic meaning. Subtle points are then reviewed that were implicitly imposed in the early thermodynamic investigations of blackbody radiation. These assumptions prevented this analysis from applying to the situation when classical electromagnetic radiation does not vanish at temperature T=0. These subtle points are easy to skip over, yet significantly change the full thermodynamic analysis. Connections are then made to some proposed mechanisms involved in various astrophysical processes. The possible connection with the observed increasing expansion of the universe is noted, and the increasing inclination of scientists to attribute this expansion to “vacuum energy." However, since the universe is not in a state of thermodynamic equilibrium, then the commonly accepted notion of “vacuum energy" or “zero-point" energy may not really be accurate. Altering this perspective may be helpful in coming to terms with the full physical description. 1. Introduction

The present article touches on a combination of several overlapping topics, with an attempt made to connect all these topics together at the end. To begin, a brief historical perspective is presented in Sec. II on the early meaning of “zero-point" (ZP), as in regards to the state of atomic systems near absolute zero temperature, and contrasted to the almost universal reference now used as the lowest quantized energy state of a system. A quantum mechanical perspective certainly provides a relation between these two concepts, but, as discussed here, the modern point of view is significantly different than the one first envisioned. Such thoughts naturally lead one to examine the meaning of thermodynamic equilibrium, both from the more macroscopic perspective of conventional 111 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 111-124. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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thermodynamics, but then as regards smaller and even “single" atomic systems in interaction with the infinite number of radiation modes. Such a discussion is indeed a fascinating one and requires a deep examination. Planck tackled aspects of these problems in his treatise in Ref. [1]. He introduced several devices for aiding the discussion of the thermodynamic behavior of radiation, such as a black carbon particle" and rough scattering walls. His treatment of rays of heat radiation in a cavity included the concept that each ray could be described by a separate temperature, where the state of maximum entropy occurred when all rays were at the same temperature. Many of these concepts are still useful constructs today, and indeed can provide means for experimental testing in quantum cavity electrodynamics. However, as discussed in Sec. III, some of these early ideas by Wien, Planck, and others, led to unnecessary restrictions on the thermodynamic analysis of electromagnetic radiation. Indeed, much of the early ideas implicitly restricted the Wien displacement analysis from applying to radiation that at absolute zero temperature did not reduce to zero radiation. These restrictions prevented the early analysis from being sufficiently general to take into account Casimir-like force considerations. These subtle points are easy to skip over, yet significantly change the thermodynamic analysis of a system. This article ends by making some connections to mechanisms in proposed astrophysical processes involving the secular acceleration of particles, such as may be a contributor for cosmic ray formation. The possible connection with the observed increasing expansion of the universe is noted, and the increasing inclination of scientists to attribute this expansion to “vacuum energy." This brings the present discussion back full circle, by bringing in the notion that the universe is not in a state of thermodynamic equilibrium, so the commonly accepted notion of “vacuum energy," or “zero-point" energy, may not really be accurate. Altering this perspective should be helpful in coming to terms with the full physical description. 2. Zero Point

It is interesting to note the original historical meaning of “zero-point" as having to do with the observed nonvanishing kinetic energy of systems near the point of absolute zero temperature [2]. In contrast, the modern perspective and emphasis on the term “zeropoint" is really quite different. Indeed, very often in the textbooks on quantum physics, as well as in the physics literature, the terminology of the “ZP state" and the “ZP energy" of a system is used interchangeably with the “ground state" and the “ground-state energy," without ever mentioning temperature or thermal equilibrium conditions [5]. Indeed, in keeping with this modern perspective, it's likely that most students in physics today directly associate ZP with the lowest (“zeroth") energy quantum state, rather than with the state of a system at zero temperature. Now, of course, in the quantum mechanics (QM) perspective, the ground state is equivalent to the state of the system at T= 0, so technically, this viewpoint is certainly correct. In QM, the ground state of a physical system is the state with the lowest possible quantized energy level. We assume that the ground state of a system is nondegenerate, which is the usual assumption made.

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Nevertheless, it is interesting to note that the thermodynamic significance of the ground state in QM plays a far less prominent role in the development of QM than does the property of quantization. Indeed, the thermodynamic role of the quantum-mechanical ground state is usually deduced as almost an afterthought, once the quantization of states is deduced, and the existence is established of a lowest energy level for the bound states of a system. Using statistical-mechanics notions, a thermal equilibrium state of the system at some temperature T is formed by taking an incoherent superposition of bound states, where the weighting factor is exp (-Ei/kT) for each quantized energy level Ei. As only the ground state remains in this summation, so that the ZP state is obtained.

3. Implicit Restrictions Imposed In Blackbodyradiation Analysis 3.1 TRADITIONAL THERMODYNAMIC BLACKBODY RADIATION ANALYSIS

Here we will discuss how the implicit assumption entered the blackbody radiation analysis by early researchers around 1900 that as the electromagnetic thermal radiation spectrum in a cavity reduced to zero radiation. This assumption was certainly not explicitly made, but was rather buried in other steps that resulted in this assumption being imposed. To emphasize the critical points, perhaps it is best to first review the steps that are traditionally followed in the derivation of the Stefan-Boltzmann relationship [8]. Reviewing these points is very quick and enables one to rapidly get to the heart of the matter. Reviewing similar points for Wien's displacement law is also very helpful and revealing, although more complicated; hence, we will only touch on these points here. More detailed information can be found in Ref. [12] In addition, a nice brief summary of the early historical work on the thermodynamics of radiation between 1859 and 1893, leading up to Planck's work, including the important work by Kirchhoff, Boltzmann, and Wien, is described in the first few pages of Ref. [3]. Planck covers the physical reasoning and assumptions behind this early work in beautiful detail in Parts I and II in Ref. [1]. If one considers a cylinder of volume V, with the walls of the cylinder maintained at temperature T, then blackbody radiation should exist within the cylinder. If one of the walls of the cylinder is taken to be a piston that can be moved, so as to change the volume of the cylinder, then work can be done as the piston is displaced. An apparently conventional thermodynamic analysis of this operation can then be carried out. One key assumption made in this early work was that the radiation in the cavity was uniform and isotropic within the cavity. Part of the reasoning for this assumption was based on arguments advanced early on by Kirchhoff. Planck touched on these assumptions, recognizing that if the cavity was sufficiently small, or if the surface and any objects in the cavity had spatial variations on the order of the wavelength of the light within the cavity, then one could not use the reasoning and justifications first introduced by Kirchhoff for investigating the radiation properties at such wavelengths. Other than this acknowledgement [13], however, that their analysis cannot apply to small spatial dimensions, this early thermodynamic analysis does not go into any more real detail regarding these points.

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Hence, one usually begins this analysis by assuming that the time average of the electromagnetic radiation energy is given by U=Vu, where V is the volume of the cavity or cylinder, and u is an electromagnetic energy density, that is independent of position within the cavity, and only dependent on temperature. Thus, U is taken to be an extensive thermodynamic quantity. The first law of thermodynamics is written as [14],

where infinitesimal quasi-static processes are carried out, d’Q is the heat flow into the cavity during such a process, -PdV is the work done on the radiation in the cavity, and P is the radiation pressure on the piston. A second key step followed in this traditional analysis is equating the radiation pressure P to u/3. Many references either simply note this point, or provide some rough reasoning to enable the relationship to be seen as reasonable. Planck in Ref. [1], Part II, Chap. I, shows the full reasoning that was initially put into establishing this relationship. He carried out the analysis of a plane wave incident at some angle to the normal of the plane conducting surface, where the material was taken to be composed of a perfect conductor that was nonmagnetizable. If one makes the further assumption that the density of plane waves incident on the plane surface is independent of direction, then one can show, as in Ref. [1], that P = u/3. The second law of thermodynamics allows us to equate that d’Q = TdS where dS is called the caloric entropy and is an exact differential. From Eq. (1),

Equating

, which is a consequence of the second law, then yields a simple

first-order differential equation of

that can be solved to yield the usual

form of the Stefan-Boltzmann relationship of [8]

In addition to the above relationship, the other major thermodynamic work pertaining to blackbody radiation, prior to Planck's involvement, included Kirchhoff's analysis on absorptivity and emissivity, and the Wien displacement law. The latter, in particular, requires a fair bit of further analysis and includes the examination of how the spectrum of the radiation in the cylinder is altered as the piston is slowly pushed in or out. Wien's analysis took into account the Doppler shift that occurs to the radiation as the piston is quasistatically moved, resulting in the deduction that the spectral energy density must be of the following functional form: where

is

the

angular

frequency

of

the

radiation,

and

Here, E and B are the electric and magnetic fields

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of the radiation within the cavity, and the angle brackets represent a time or ensemble average. Nevertheless, although Kirchhoff's and Wien's analyses are interesting and important, the above short review of the Stefan-Boltzmann relationship is probably quite adequate for our present purpose, namely, to quickly get to the heart of the early thermodynamic analysis of blackbody radiation to examine what assumptions were implicitly made, in rather subtle ways, that resulted in the presence of electromagnetic ZP radiation being missed in the analysis. For those interested, however, Refs. [15] and [12] goes into the Wien displacement law in considerable more detail.

3.2 SUBTLE ASSUMPTIONS Although it has taken some time for the following points to be recognized, the investigation into Casimir forces, begun in 1948 [16, 10, 17, 12] are what has now enabled the limitations of the early thermodynamic analysis to be better understood. As discussed in Refs. [15] and [12], a number of assumptions and steps were made in the early analysis that are not in general valid. One key point can immediately be brought out, namely, that the internal electromagnetic thermal energy of a cavity at temperature Tw was treated as being an extensive quantity that is proportional to the volume of the cavity (U = Vu). From the study of Casimir forces, and, indeed, from even the study of microwave resonators [18, 19], , we know that this assumption is not in general valid. Part of the reason for this is that zero point radiation and Casimir forces necessarily involves the consideration of wavelengths of radiation that violate the restriction Planck states in Ref. [13], that “... the linear dimensions of all parts of space considered, as well as the radii of curvature of all surfaces under consideration, are large compared with the wavelengths of the rays considered." When considering the full thermal radiation spectrum, some frequency components of the radiation will always violate this restriction for any cavity. In a microwave resonant cavity, where the dimensions of the cavity are typically close to the wavelength of the radiation being manipulated, the electromagnetic energy density inside the cavity is certainly not a constant at all points in the cavity, but varies depending on the location in the cavity. However, this point is only a part of the concern. Indeed, for cavities with good conducting walls, only a relatively small part of the total standing wave modes will violate the previous dimensional concern. For all the very high frequency modes, for which there are an infinite number, the wavelengths involved are all small compared to typical dimensions in a cavity. (Actually, this point is not quite true. If a cavity has sharp corners, such as in a rectangular parallelpiped, and if the surface is treated as being a continuum (as opposed to being composed of atoms), then the corners always violate the dimensional restriction [13] for any wavelength.) Nevertheless, as we know from the study of Casimir forces, the electromagnetic ZP energy within a cavity violates the extensive property assumption. If a cutoff is not imposed [20], the electromagnetic energy within a cavity due to the presence of electromagnetic ZP radiation is infinite. However, the change in total electromagnetic energy due to a change in volume of the cavity is finite. \ Calculating this finite change in energy cannot be done by treating the total electromagnetic energy within the cavity as

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being equal to the volume V times an electromagnetic energy density u that is independent of the size and shape of the cavity. Rather, the full normal mode sum must be retained, and the difference in energy between the sum of normal mode energies due to the change in volume must be calculated, before continuum approximations are made [12]. Another simple way to recognize that the early thermodynamic analysis was not sufficiently general to include the situation that might not equal zero, or, that ZP electromagnetic radiation might exist, is the following: The piston surface was only explicitly calculated in the case of a perfectly reflecting surface [1], and, in that case, only when an isotropic and uniform radiation was assumed. Researchers then made the assumption that the pressure on other walls in the chamber were equal to this calculated value, independent of the material and relatively independent of the geometry of the wall. At the end of Sec. 66 in Ref. [1], Planck said, “...it may be stated as a quite general law that the radiation pressure depends only on the properties of the radiation passing to and fro (within the cavity), not on the properties of the enclosing substance (i.e., the walls of the cavity).” However, this statement is not really accurate, as can be understood when normal modes of radiation within a cavity and different boundary conditions demanded by different materials are taken into account [18, 19]. If we compare two cavities of the same shape and size, but made of different materials, the pressure in the two cavities are in general different, even when the walls of both cavities are held at the same temperature. For example, the Casimir force between two conducting plates is attractive, but between a perfectly conducting and an infinitely permeable plate, the force is repulsive [21]. Changing the shape of the cavity can change the magnitude and sign of the radiation force even more dramatically. Of course, this difference in pressure from the one Planck refers to is really only largely noticeable when temperatures are sufficiently low and sizes sufficiently small, that is, when Casimir-related forces dominate over conventional thermal radiation pressure. Thus, although the theoretical importance of including the consideration of ZP radiation has numerous critical implications, to date the experimental importance has only been truly evident for sufficiently small cavities, at which point Casimir-like forces can become important. However, this brings up an even more noticeable point, namely, it is known that when parallel perfectly conducting plates are treated as continuous materials, then the radiation pressure due to the ZP radiation between the plates is formally infinite, as is the radiation pressure due to radiation impinging on the outside of the plates. The difference between the two is finite, however. The simple identification of

with

treated as finite and independent of the shape, volume,

and material wall properties of the cavity, is clearly inadequate for addressing such subtleties. Indeed, the early analysis only considered the radiation pressure from the inside of cylinder. Now, Refs. [12] and [22] show how to adequately account for radiation pressure when Casimir related forces are present by taking into account the normal modes, and Ref. [12] shows the relationship of this expression to the energy density. Moreover, Ref. [23] deduces the relationship between this radiation pressure and the energy density for

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cavities of arbitrary shape and size, when the cavity walls are composed of perfectly conducting material. Now let us turn to some specifics regarding the underlying mathematics that prevented the early thermodynamic analysis from being sufficiently general to include the consideration of ZP radiation. Perhaps the best way to make this point is to examine the force expression of Eq. (31) in Ref. [12] versus the electromagnetic energy expression of Eq. (37). It is then easy to see that in general, unless one follows the approximate reasoning surrounding Eq. (32) in that reference to make the connection. However, it is also interesting to make a more direct connection with the original statements surrounding the early blackbody thermodynamic analysis. The lines after Eq. (96) in Ref. [1] read, in our present notation,

The first line in the above equation comes from Eq. (94) in Ref. [1], which was deduced as part of the Wien displacement analysis. The second line above was obtained via partial integration. This is the step that concerns us presently. In the paragraph that follows the above line in Ref. [1], Planck assumed that the first term above equals zero, which is an invalid assumption for ZP radiation [24]. Indeed, if we insert Planck's final result of Eq. (1.4), we explicitly see that the assumption is made that

If nonzero radiation is present at T = 0, the above limit should not equal zero.

4. Correcting Analysis

The point of the present section of this article is not to go through all the details involved with generalizing the early blackbody thermodynamic analysis; much of this more detailed work was carried out in Refs. [15] and [12] (also see [23] for arbitrary cavity shapes with perfectly conducting walls). Rather, the intent here is simply to make the need and means for the required generalization more apparent. By doing so, the importance of including the concept of ZP radiation in thermodynamic analysis will be better brought out. We will then build on these points later in this article when examining recent observations and ideas concerning astrophysical phenomena. Summarizing previous work then, just like the early blackbody thermodynamic analysis involved quasistatic displacement operations of a piston in a cavity, Refs. [25, 15, 12, 26-28 and 23] also involved quasistatic displacement operations. References [15, 2527] largely involved the quasistatic displacements of simple harmonic electric dipole oscillators that were in thermodynamic equilibrium with stochastic electromagnetic radiation. \ The consideration of van der Waals forces, at all distances (i.e., including

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retardation effects), was taken into account here. Similarly, Ref. [12] most closely paralleled the early analysis by examining the displacement of two conducting parallel plates and a wall of a conducting parallelpiped, while taking into Casimir-like forces. References [23] and [29] extended much of this analysis to cavities of arbitrary shape. The key steps in the analysis were somewhat similar to the early analysis outlined here in Sec. III A for the Stefan-Boltzmann law, although considerably more involved. More specifically, the first law of thermodynamics was imposed, as in Eq. (1), but without the assumption of a uniform electromagnetic energy density. The ensemble average of the total internal energy was calculated for the fluctuating quantities involved, and the average work calculated for making a quasistatic displacement. The mathematics in the case of blackbody radiation, in Ref. [12], was actually considerably easier than the corresponding mathematics for the N oscillators considered in Refs. [25] and [15]. Restricting attention here only to the blackbody radiation analysis in Ref. [12], the real key here was to calculate internal energy and average forces involved by summing over all the normal modes of the radiation in the cavity, without imposing along the way that the radiation should somehow be independent of the material, shape, and size of the cavity walls. The next step was then to impose a consequence of the second law of thermodynamics, namely, just as in the Stefan-Boltzmann analysis reviewed earlier, that where dS is an exact differential. In the case of two walls separated by a distance L [12]}, or in the more general case of an arbitrary deformation or displacement of section of a wall of an arbitrarily shaped cavity [23], one can require that The result is a first order partial differential equation that must be satisfied by the radiation spectrum. The solution of this equation is satisfied by a functional form given by We can call this relationship a generalized Wien displacement law, since the functional form is the same, but now the steps in the derivation also applies in the situation where the restriction of is not imposed, so that the possibility of ZP radiation is taken into account. Having gone through this analysis, which really does parallel the much simpler analysis for the original Stefan-Boltzmann derivation in Sec. III A, now one can readily deduce what perhaps could be called a generalized Stefan-Boltzmann relationship. Upon recognizing that the really important factor in physics are the changes and comparisons of systems, it then becomes clear that heat flow, work done on systems, and temperature changes are what really are observed and measured. For a volume V in free space, the change in thermal electromagnetic radiation energy in going from T= 0 to T is given by

where now

takes into account the existence of ZP radiation, if

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Further thermodynamic analysis can also be carried out, and has been for the case of dipole oscillators and cavity thermodynamics. In both cases, one can derive the functional form for ZP radiation, based on the thermodynamic definition of no heat flow at T = 0 during reversible thermodynamic operations [12, 23, 25-26, 28]. Moreover, these systems were explored regarding the restrictions of the third law of thermodynamics, the requirement of a finite specific heat, and the ultraviolet catastrophe. In addition, one can examine other thermodynamic questions, such as whether extracting energy from ZP radiation violates any known physical laws [30-32]. Without question, work can be done on systems, or have systems do work, at or near T = 0; very large releases of energy are even possible if systems are taken out of thermodynamic equilibrium, so that irreversible processes ensue, either at T = 0, or at nonzero temperatures. Thus, at this point it should be apparent that the early blackbody thermodynamic analysis contained some very innocuous and physically appealing assumptions that resulted in significant differences than what has been observed in nature. In particular, by not taking into account the possibility of electromagnetic ZP radiation in the analysis, effects such as due to Casimir-like forces and van der Waals forces could not be taken into account, and investigations on the Stefan-Boltzmann law, the third law of thermodynamics, the ultraviolet catastrophe, specific heats, and other effects and properties, were significantly hindered in the early analysis. 5. Secular Acceleration And Astrophysical Processes Having now gone over some of the subtleties involved with the early thermodynamic analysis, let us now turn to investigations largely led by A. Rueda [33-35]. These investigations involve what has been termed the possible “secular acceleration" [35] of charged particles due to the presence of electromagnetic ZP radiation. Charged particles bathed in thermal radiation are constantly being accelerated, as well as constantly radiating energy, due to the stochastic interaction nature of the particles and fields in equilibrium. Under thermal equilibrium conditions, and when T > 0, the average energy picked up due to the fluctuating impulses from the radiation should roughly average out to equal the energy loss due to a velocity dependent “drag" force. Einstein and Hopf first investigated these aspects [36]. Boyer noticed, however, that at T = 0, the usual “drag force" is necessarily absent, thereby leaving the effect of the fluctuating radiation impulses due to ZP radiation uncompensated [37-38]. An average continual increase of velocity was then predicted, unless collisions with other matter occur to dissipate the increased kinetic energy. Rueda subsequently proposed [33] that these effects might be observable in astrophysical processes. The extensive review in Ref. [34], and the more recent reference [35], contain relevant work and references related to Rueda's investigations. In particular, though, it seems possible that this “secular acceleration" mechanism may contribute, or even be a very important aspect of, phenomena such as cosmic rays, cosmic voids, and the observed X-ray and gamma ray backgrounds. Thus, here we have yet another possible dramatic consequence of the existence of radiation being present at T = 0, i.e., ZP radiation. Naturally such possible phenomena brings up many issues on whether basic thermodynamic concepts and laws are being violated, such as the first and second laws. These questions were investigated in Ref. [31].

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The conclusion was that, no, if ZP radiation does indeed contribute to cosmic ray formation, then the secular acceleration phenomena should not violate either the first or second law of thermodynamics. Rather, the situation of “free" charged particles without a “container," situated in an environment with a near ZP-like spectrum, is really a system out of thermodynamic equilibrium. Indeed, if ZP-like fields are contributing to cosmic ray formation, then the particles being accelerating across the long “empty" (low mass regions) of intergalactic space, are clearly not in any sort of thermodynamic equilibrium with the matter they eventually encounter and strike. If secular acceleration is indeed the main contributor of cosmic rays, then the apparent free energy that has been acquired by these very high velocity particles is due to a finite change in the enormous amount of electromagnetic energy available in space. Energy can certainly still be conserved, as the kinetic energy picked up by the particles is lost by the radiation [27, 31], on average, but can be returned to the radiation upon the particle undergoing a deacceleration and colliding with other matter. As for the second law, at first glance it does indeed appear to be violated if ZP radiation provides secular acceleration effects, as the particles appear to be, roughly speaking, extracting energy from a heat reservoir. Clausius' statement of the second law is [14], “No process is possible whose sole result is the transfer of heat from a cooler to a hotter body." \However, a closer analysis of this phenomena reveals that this statement is not violated by this phenomena, since it is important to consider changes in thermodynamic equilibrium states. If one had a large enough container, so that the particles hit the walls, then it would be necessary to examine many traversals of such a system to really contemplate a system in equilibrium. The secular acceleration effect really only has to do with the average behavior of particles at different points in their trajectory. Upon averaging over the trajectories, the particles both pick up and lose energy, via collisions, in a natural stochastic behavior that needs to be considered in its totality. This point, and related ones, are discussed in more detail in Ref. [31]. However, one point should be brought out that was not fully emphasized in Ref. [31]. At a certain level, we cannot just take one system at some temperature T, and another system at the same temperature T, and put them together, and expect there to be “no changes." This is somewhat contradictory to our intuition, since we have long learned the “zeroth law of thermodynamics," namely: “Two systems in thermal equilibrium with a third are in thermal equilibrium with each other." Macroscopically, of course, that is what we see. For example, suppose we have a thermometer, or temperature gauge, and two substances. Suppose we can put the thermometer in either substance and never see a change in the thermometer's reading. We would then say that via the zeroth law, all three are at the same temperature. However, two systems that can effect each other always have an interaction energy. Separate systems, nominally at some temperature, will change in subtle and sometimes not so subtle ways as they are brought into contact with each other. At the very least, a microscopic thermocouple or small thermometer will experience van der Waals or Casimir-like forces as it is brought close to a substance that is nominally at the same temperature. If we were to take an atomic force microscope probe, and have it “walk" around the surface of a cavity, where the probe and the walls of the cavity are all nominally at the same temperature, the probe will experience changes in forces as it moves closer and farther from the walls, and as it examines a wall of one material versus

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another. Contrary to what one might intuitively suspect, such a probe can distinguish the different types of material forming the walls of a blackbody cavity. This is a quite different situation than what Kirchhoff, Wien, and Planck [1] originally described in their analysis, where the radiation and radiation pressure were treated as being independent of the material of the walls of the cavity. My reason for making these points is that it may be critically important to consider the concept of thermodynamic equilibrium as being the equilibrium state of systems in net equilibrium with each other. Probes, particles, radiation, etc., all need to be treated as part of the net system, and need to be analyzed as being in interaction with each other, even when they are in thermodynamic equilibrium with each other. Having a cavity of electromagnetic thermal radiation, without ionized atoms present in the cavity, can in many important ways be quite different than the situation where the ions are present, as we know from plasma studies. For systems not in thermodynamic equilibrium, the system is yet far more complicated. Extrapolating these ideas to the whole universe requires a careful examination of the basic assumptions, as will be discussed more in the following two sections. 6. Increasing Expansion Of The Universe

A fascinating phenomena has been observed and reported in recent years, that has attracted considerable attention in the astrophysical community. A major turning point occurred in 1998 [39-40], when two teams of astrophysicists reported on new studies involving the luminosity of a particular type of supernova in nearby and distant galaxies. The results provided surprising evidence that not only is the universe not slowing down its rate of expansion since the Big Bang, but the expansion is actually increasing its rate. In addition, very recent reports this year on the mappings of tiny fluctuations in the cosmic microwave background from the experiments BOOMERANG (Balloon Observations of Millimeter Extragalactic Radiation and Geophysics) [41] and MAXIMA (Millimeter Anisotropy Experiment Imaging Array) [42-43] have provided additional evidence that the universe is flat. Thus, we now have several sets of confirming evidence that the universe is expanding, and apparently at an increasing rate. However, the density of all visible matter and dark matter in the universe appears to be only about one third of what is needed to account for a flat universe, where expansion will continue forever. Many physicists are proposing that vacuum energy is a key component of the puzzle for understanding this phenomena. At a recent meeting of the American Astronomical Society (AAS 196, June 2000) in Rochester, N.Y., a team of astronomers reported they have new evidence for what makes up most of the mass of the universe, based on a survey of the redshifts of 100,000 galaxies. According to their analysis of this mapping of galaxies' redshifts, called the Two Degree Field (2dF) Galaxy Redshift, a universe composed of 2/3 vacuum energy contribution and 1/3 visible and dark matter contribution, would fit the observed astronomical data well [44].

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7. Concluding Comments On The State Of The Universe And Zero-Point

I will end this article with a few cautionary comments. In addition to well recognized microscopic phenomena, such as van der Waals and Casimir forces and the Lamb shift, which are commonly attributed to the effects of electromagnetic ZP radiation [10, 17], we have what appear to be very much macroscopic consequences of ZP fields. In particular, it appears that “zero-point" fields may play a role in astrophysical phenomena and cosmology. Without doubt, this comment is certainly speculative at this point, as both the theoretical and experimental aspects still require deeper investigation. However, an increasing number of physicists are beginning to examine the possible role of “ZP fields," particularly in light of the experimental data obtained during the past two years. One reason for speaking with caution about the possible role of ZP fields in astrophysics is undoubtedly the one most frequently cited, namely, the famous cosmological constant problem [45]. Formally, the vacuum is supposedly infinite in energy, although most researchers feel there must be some sort of effective cutoff [20]. Nevertheless, even if huge, how one can reconcile the enormous energies to the ones needed to provide some gravitational effects, but not enormous ones, is by no means clear. However, another reason for speaking with caution here, is one that I have not seen mentioned elsewhere, and which brings us back full circle to the initial discussion of this article on the thermodynamics of blackbody radiation. As mentioned in Sec. II, in quantum mechanics, “zero-point" is now commonly accepted as meaning the “zeroth," or lowest, quantized energy level of a system. In contrast, “zero-point" historically first referred to the properties of a system at absolute zero temperature. Specifying a temperature, even at T = 0, implies the system under discussion is in thermodynamic equilibrium. Our universe is not in thermodynamic equilibrium; indeed, in many ways, it is far from being in equilibrium. Thus, just as it was important to more carefully examine the meaning of ZP in the early classical analysis of blackbody radiation, so also it may be important to be careful regarding introducing “zero-point" notions in regard to the entire universe.

Acknowledgments

This work was supported in part by the California Institute for Physics and Astrophysics via grant CIPA-CD2999 to Boston University.

Annotated References [1] M. Planck, The Theory of Heat Radiation (Dover, New York, 1959). This publication is an English translation of the second edition of Planck's work entitled Waermestrahlung, published in 1913. A more recent republication of this work is Vol. 11 of the series The History of Modern Physics 1800-1950 (AIP, New York, 1988). [2] See, for example, Ref. [3], pp. 246 and 247, or the footnote on p. 73 in Ref. [4]. [3] T. S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894-1912 (Oxford University Press, New York, 1978). [4] L. Pauling and E. B. Wilson, Jr., Introduction to Quantum Mechanics (Dover, New York, 1985). [5] See, for example, pp. 134 and 170 in the undergraduate textbook of Ref. [6] or, p. 69 in the graduate

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textbook of Ref. [7]. [6] A. P. French and E. F. Taylor, An Introduction to Quantum Physics (Norton, New York, 1978). [7] L. I. Schiff, Quantum Physics, 3rd ed. (McGraw-Hill, New York, 1968). [8] The traditional reasoning for deducing the Stefan-Boltzmann relationship is widely available and can be found, for example, in Part II, Chap. II in Ref. [1], Appendix XXXIII in Ref. [9], pp. 5-6 in Ref. [3], p. 2 in Ref. [10], or Prob. 9.10 on p. 399 in Ref. [11]. For a close description of the underlying assumptions, Ref. [1]is particularly helpful. [9] .M. Born, Atomic Physics, 8th ed. (Dover, New York, 1969). [10] P. W. Milonni, The Quantum Vacuum. An Introduction to Quantum Electrodynamics, (Academic Press, San Diego, 1994). [11] F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965). [12] D. C. Cole, Phys. Rev. A 45, pp. 8471-8489 (1992). [13] For example, Planck writes on p. 2 in Ref. [1], Only the phenomena of diffraction, so far at least as they take place in space of considerable dimensions, we shall exclude on account of their complicated nature. We are therefore obliged to introduce right at the start a certain restriction with respect to the size of the parts of space to be considered. Throughout the following discussion it will be assumed that the linear dimensions of all parts of space considered, as well as the radii of curvature of all surfaces under consideration, are large compared with the wave lengths of the rays considered." Other similar statements are made elsewhere in his treatise in his coverage of the early thermodynamic investigations of blackbody radiation, including Kirchhoff's law, the St'efan-Boltzmann law, and the Wien displacement law. [14] M. W. Zemansky and R. H. Dittman, Heat and Thermodynamics, 6th ed. (McGraw-Hill, New York, 1981). [15] D. C. Cole, Phys. Rev. A 42, pp. 7006-7024 (1990). [16] H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, pp. 793-795 (1948). [17]L. de la Pena and A. M. Cetto, The Quantum Dice - An Introduction to Stochastic Electrodynamics (Kluwer, Dordrecht, 1996). A review of this book is given in D. C. Cole and A. Rueda, Found. Phys. 26, pp. 1559-1562 (1996). [18] J. C. Slater, Microwave Electronics, (D. Van Nostrand Co., Inc., New York, 1950). [19] H. B. G. Casimir, Philips Res. Rep. 6, pp. 162-182 (1951). [20]R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), p. 245. [21]T. H. Boyer, Phys. Rev. A 9, pp. 2078-2084 (1974). [22] P. W. Milonni, R. J. Cook, and M. E. Goggin, Phys. Rev. A 38, pp. 1621-1623 (1988). [23] D. C. Cole, Found. Phys. 30 (11), pp. 1849-1867 (2000). [24] A similar subtle restriction is made in Planck's analysis in [1] following Eq. (124) on p. 91; this restriction is invalid if ZP radiation is to be considered. [25] D. C. Cole, Phys. Rev. A 42, pp. 1847-1862 (1990). [26] D. C. Cole, Phys. Rev. A 45, pp. 8953-8956 (1992). [27] D. C. Cole, "Reviewing and Extending Some Recent Work on Stochastic Electrodynamics," in Essays on Formal Aspects of Electromagnetic Theory (refereed compendium), ed. by A. Lakhtakia, (World Scientific, Singapore, 1993). [28] D. C. Cole, Found. Phys. 29, pp. 1819-1847 (1999). [29]D. C. Cole, “Relating Work, Change in Internal Energy, and Heat Radiated for Dispersion Force Situations," Proc. Of Space Technology and Applications International Forum -- 2000 (STAIF 2000), AIP 504, ed. by M. S. El-Genk, pp. 960-967 (2000). [30] C. Cole and H. E. Puthoff, Phys. Rev. E 48 , pp. 1562-1565 (1993). [31] D. C. Cole, Phys. Rev. E 51, pp. 1663-1674 (1995). [32] D. C. Cole, "Energy and Thermodynamic Considerations Involving Electromagnetic Zero-Point Radiation," Proc. of Space Technology and Applications International Forum -1999 (STAIF 99), AIP 458, ed. by M. S. El-Genk, pp. 960-967 (1999). [33] A. Rueda, Nuovo Cimento A 48, 155 (1978). [34] A. Rueda, Space Sci. Rev. 53, 223 (1990). This review contains an extensive set of relevant references. [35] A. Rueda, B. Haisch, and D. C. Cole, Astrophys. J. 445, pp. 7-16 (1995). [36] A. Einstein and L. Hopf, Ann. Phys. (Leipzig) 33, 1096 (1910); 33, 1105 (1910). [37] T. H. Boyer, Phys. Rev. 182, 1374 (1969). [38] T. H. Boyer, Phys. Rev. A 20, 1246 (1979). [39] S. Perlmutter, G. Aldering, et al., Nature 391, pp. 51-54(1998). [40] B. P. Schmidt, P. Challis, A.V. Filippenko, et al., Astrophysical Journal 507 (Nov. 1, 1998). [41] Hu, Nature 404, pp. 939-940 (2000).

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[42] Hanany, et al., “MAXIMA-1: A Measurement of the cosmic microwave background anisotropy on angular scales of 10’ to 5 degrees," http://xxx.lanl.gov/abs/ number 0005123. [43] A. Balbi, et al., “Constraints on cosmological parameters from MAXIMA-1," http://xxx.lanl.gov/abs/ number 0005124. [44] Additional information about the 2dF Galaxy Redshift Survey is available at Digital Sky Survey, at http://www.sdss.org. For related information, see other background information, see preprint by M. Turner, http://xxx.lanl.gov/abs/astro-ph/9811364 (to be published in Proc. Astron. Soc. Pacific, Feb. 1999), “Cosmology Solved? Quite Possibly!” [45] Weinberg, “The cosmological constant problem," Rev. Mod. Phys. 61 (1), pp. 1-23 (1988).

NEW DEVELOPMENTS IN ELECTROMAGNETIC FIELD THEORY BO LEHNERT Alfvén Laboratory Royal Institute of Technology SE-100 44 Stockholm, Sweden

Abstract

Conventional electromagnetic theory and quantum mechanics have been very successful in their applications to numerous problems in physics. Nevertheless there are questions leading to difficulties with Maxwell s equations which are not removed by and not associated with quantum mechanics. As a consequence, several new developments have been elaborated which both contribute to the understanding of so far unsolved problems, and predict new features of the electromagnetic field to exist. In this review some examples are first given on the problems which arise in conventional theory. This is followed by short descriptions of the main characteristics of modified field theories being based on the hypothesis of additional electric currents in vacuo, as well as on more generalized forms including magnetic monopoles and a unification of electromagnetism and gravitation. The new aspects which result from these extended theories are manifold, including both time-dependent states with wave phenomena and timeindependent states with electromagnetic equilibria. The former states are discussed with emphasis on plane waves, axisymmetric models of the individual photon with its integrated parameters and unified wave particle concepts, the possible existence of a nonzero photon rest mass, the thermodynamics of a photon gas, the transition of a beam of photon wave packets into a plane wave, and superluminous phenomena including the basis of tachyon theory. The latter states are discussed with emphasis on particle-shaped electrically neutral and charged equilibria providing possible models for neutrinos and charged leptons, features of the electron behaving nearly as a point charge, string-shaped equilibria, and instantaneous interaction at a distance including longitudinal field components. 1. Introduction

Conventional electromagnetic field theory based on Maxwell’s equations and quantum mechanics has been very successful in its application to numerous problems in physics, and has sometimes manifested itself in an extremely good agreement with experimental results. Nevertheless there are are as within which these joint theories do not seem to provide fully 125 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 125-146. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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adequate descriptions of physical reality. Thus there are unsolved problems leading to difficulties with Maxwell’s equations which are not removed by and not directly associated with quantum mechanics [1-2]. Due to these circumstances a number of modified and new approaches have been elaborated during the last decades. Among the reviews and conference proceedings describing this development, those by Lakhtakia [3], Barrett and Grimes [4], Evans and Vigier [5]. Evans et al. [6-7], Hunter et al. [8] and Dvoeglazov [9] can be mentioned here. The purpose of these approaches can be considered two-fold: To contribute to the understanding of so far unsolved problems. To predict new features of the electromagnetic field. The present review summarizes these new developments in electromagnetic field theory. After a description of some cases where standard theory fails to give fully adequate solutions, a number of modified field theories will be shortly reviewed, then followed by the aspects which result from these developments. Within this review it has not been possible to do justice to all the relevant contributions, and the references have mainly been concentrated to some more recent investigations. 2. Unsolved Problems in Conventional Electromagnetic Theory

There are a number of cases which illustrate the failure of standard electromagnetic theory based on Maxwell’s equations: (I) Light appears to be made of waves and simultaneously of particles. In conventional theory the individual photon is on one hand conceived to be a massless particle, still having an angular momentum, and is on the other hand regarded as a wave having the frequency v and the energy whereas the angular momentum is independent of the frequency. This dualism is not fully understandable in terms of conventional theory [5]. (ii) The photon can also sometimes be considered as a plane wave, but there are also experiments which indicate it can behave like a bullet. In investigations on interference patterns created by individual photons on a screen [10], the impinging photons produce dotlike marks on the latter, as being made by needle-shaped objects. (iii) In attempts to develop conventional electrodynamic models of the individual photon, there is a difficulty in finding axisymmetric solutions which both converge at the photon centre and vanish at infinity. This was realized by Thomson [11], and later by other investigators[12]. (iv) During the process of total reflection at a vacuum boundary, the reflected light beam has been observed to be subject to a parallel displacement with respect to the incident beam. For this so called “Goos-Hanchen effect” the displacement was further found to have a maximum for parallel polarization of the incident electric field, and a minimum for perpendicular polarization [13-14]. At an arbitrary polarization angle the displacement does not acquire an intermediate value but splits into two values for parallel and perpendicular polarization. This behavior is not explained by conventional theory.

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(v) The Fresnel laws of reflection and refraction of light in non-dissipative media have been known for some 180 years. However, these laws do not apply to the total reflection of an incident wave at the boundary between a dissipative medium and a vacuum region [15]. (vi) In a rotating interferometer fringe shifts have been observed between light beams which propagate parallel and antiparallel with the direction of rotation [4]. This so called "Sagnac effect" requires an unconventional explanation. (vii) Electromagnetic wave phenomena and the related photon concept are even in our days somewhat of an enigma in more than one respect. Thus, the latter concept should in principle apply to wave lengths ranging from about of gamma radiation to about of long radio waves. This leads to a so far not fully conceivable transition from a beam of individual photons to a nearly plane electromagnetic wave. (viii) As the only explicit time-dependent solution of Cauchy’s problem, the LienardWiechert potentials are not adequate to describe the entire electromagnetic field [2]. With these potentials only, the part of the field is missing which is responsible for the interparticle long-range Coulomb interaction. (ix) There are a number of observations which appear to indicate that superluminal phenomena should exist [16]. Examples are given by fast mini-quasar expansion, photons tunneling through a barrier at speeds larger than c, and the propagation of so-called Xshaped waves. These phenomena cannot be explained in terms of purely transverse waves resulting from Maxwell’s equations, and they require a longitudinal wave component to be present in the vacuum [17]. (x) It has been found that Planck’s distribution law cannot be invariant to adiabatic changes of a photon gas, because such changes would then become adiabatic and isothermal at the same time [18]. To remove such a discrepancy, longitudinal modes have possibly to be present which do not exist in conventional theory [18-19]. (xi) It is not possible for conventional electromagnetic models of the electron to explain the observed property of a "point charge" with an excessively small radial dimension. Nor does divergence in self-energy of a point charge vanish in quantum field theory where the process of renormalization has been applied to solve the problem. 3. Main Characteristics of Modified Field Theories

Before turning to details, we shall here describe the main features of some of the modified theories. 3.1. THEORIES BASED ON ADDITIONAL VACUUM CURRENTS

The displacement current density in vacuo was introduced by Maxwell. In combination with the Lorentz condition, the basic equations of conventional electromagnetic field theory in vacuo can be cast into the four-dimensional form

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of the d’Alembert equation where

is a four-vector with A as the

magnetic vector potential in three-space. In a number of extended theories equation (1) has been modified into a Proca type with a four-current density

Part of these theories will be briefly outlined here.

3.1.1 Quantum Mechanical Theory of the Electron In the theory of the electron by Dirac[20] the relativistic wave function components in spin-space. With the Hermitian adjoint wave function mechanical form of the charge and current densities become [21-22]

has four

, the quantum

where are the Dirac matrices of the three spatial directions (x ,y, z). There is more than one set of choices of these matrices [23]. This form could be interpreted as the result of an electronic charge being “smeared out” over the volume of the electron. 3.1.2. Theory of the Photon with a Rest Mass At an early stage Einstein [24] as well as Bass and Schrodinger [25] have considered the possibility for the photon to have a very small but nonzero rest mass . Later de Broglie and Vigier [26] and Evans and Vigier [5] derived a corresponding form of the four-current in the Proca type equation (2) as given by

As a consequence, the solutions of the field equations were also found to include longitudinal fields. Thereby Evans [27] has drawn attention to a longitudinal magnetic field part, 3.1.3.

, of the photon in the direction of propagation.

Nonzero Electric Field Divergence Theory

One approach is based on a nonzero electric field divergence and an associated "space-charge current", in combination with Lorentz invariance [28-32]. The four-current then becomes

The divergence can be understood as the effect of a polarization of the vacuum state [28]. It introduces an extra degree of freedom, leading to new possibilities such as "free"

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dynamic states of wave phenomena including non-transverse and longitudinal modes, and "bound" electromagnetic equilibrium states. The velocity vector C corresponds in an axisymmetric case to the two values , a vanishing divergence, and a nonzero curl. With a time part of the four-current (5) resulting from the nonzero electric field divergence, the Lorentz invariance thus requires the space part to adopt the form

.

There is a certain analogy between the currents (3) - (5), but in the theory of the latter current (5) the electric charge and the mass should first come out of the integrated solutions. 3.1.4. Nonzero Electric Conductivity Theory Maxwell's equations in the vacuum were already proposed by Bartlett and Corle [33] to modified by assigning a small nonzero electric conductivity to the formalism. As pointed out by Harmuth [34], there was never a satisfactory concept of propagation velocity of signals within the framework of Maxwell's theory. Thus, the equations of the latter fail for waves with non-negligible relative frequency bandwidth when propagating in a dissipative medium. To meet this problem, a nonzero electric conductivity σ and a corresponding three-space current density in vacuo. was thus introduced. The concept of this electric conductivity was later reconsidered by Vigier [35] who showed that the introduction of the current density (6) is equivalent to adding a related nonzero photon rest mass to the system. The dissipative “tired light” mechanism underlying this conductivity can be related to a nonzero energy of the vacuum ground state, as being predicted by quantum physics [5, 36]. That the current (6) is related to the form (4) of a four-current can also be understood from the conventional field equations for homogeneous conducting media [37]. The effects of the nonzero electric conductivity were further investigated by Roy et al.38-40' 32, They have shown that the introduction of a nonzero conductivity yields a dispersion relation which results in phase and group velocities depending on a corresponding nonzero photon rest mass, due to a "tired light" effect. 3.1.5.

Single Charge Theory

A set of first-order field equations was proposed by Hertz [41-43] in which the partial time derivatives in Maxwell’s equations were substituted by total time derivatives

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denotes a constant velocity parameter which was interpreted as the velocity of

the ether. Hertz' theory was discarded and forgotten at his time, because it spoiled the spacetime symmetry of Maxwell's equations. Recently Chubykalo and Smirnov-Rueda [2, 44] have presented a renovated version of Hertz' theory, being in accordance with Einstein's relativity principle. For a single point-shaped charged particle moving at the velocity v, the displacement current in Maxwell’s equations is then modified into a "convection displacement current"

The approach by Chubykalo and Smirnov-Rueda further includes longitudinal modes and Coulomb long-range electromagnetic fields which cannot be described by the LienardWiechert potentials [2, 45]. 3.1.6.

Problems with the Approaches of Additional Vacuum Currents

Two objections may at a first glance be raised against the parts of the approaches which lead to a nonzero rest mass. The first concerns the problem of gauge invariance. An answer on this point has recently been given by an analysis [46] using covariant derivatives in connection with the electric field divergence theory of the form (5). This also becomes consistent with an earlier conclusion that gauge Invariance does not require the photon rest mass to be zero [47]. The second objection concerns the supposition that a nonzero rest mass would provide a photon gas with three degrees of freedom, i.e, two transverse and one longitudinal. This would alter Planck's radiation law by a factor of 3/2, in contradiction with experience [32]. There are, however, arguments which also resolve this problem, as shown later in this context. 3.2. GENERALIZED THEORIES

There have also been made approaches of a more general character than those based on the introduction of additional vacuum currents, as demonstrated here by a few examples. 3.2.1.

Magnetic Monopoles

One may raise the question why only the divergence of the electric field should be permitted to be nonzero, and not also the divergence of the magnetic field. In any case, there are a number of investigators who have included the latter and corresponding magnetic monopoles in their theories [34, 48, 49], also from the quantum theoretical point of view [32]. According to Dirac [50], the magnetic monopole concept is an open question. This concept also leads to a quantization condition for the electric charge, and a similar result is derived from the t'Hooft-Polyakov monopole [51].

NEW DEVELOPMENTS IN EM THEORY 3.2.2.

131

Unification of Electromagnetism and Gravitation

In attempts to make further generalizations, there is a profound difference between geometrical and Maxwellian theories [7]. It is stated by Ryder [51] that in electrodynamics the field is only an "actor" on the space-time "stage", whereas in gravity the "actor" becomes the "space-time stage" itself. There are several reviews on theories aiming at a unification of electromagnetism and gravitation, of which those recently made by Evans [52] and described by Roy [32] can be mentioned here among others. From the quantum theoretical point of view the Dirac magnetic monopole has been discussed in connection with the massive photon [53-55]. An attempt has been made by Israelit [55] to generalize the Weyl-Dirac geometry by constructing a framework which includes both a massive photon and a Dirac monopole, thereby offering a basis for deriving electromagnetism and gravitation from geometry. The most general equations of a unified field theory of electromagnetism and gravitation have been elaborated in a series of investigations by Sachs [56]. A proposal for unification is also due to Evans [52] who considers electromagnetism and gravitation as different interlinked parts of the Riemann tensor. Then electromagnetism can be regarded as a “twisting" and gravitation as a "warping" of space-time. The electromagnetic field tensor

is then made

equivalent to a space-time curvature expressed by an antisymmetric tensor obtained from contraction of the Riemann tensor. This approach further connects the longitudinal magnetic field [27] of Section 3.1.2. with the Riemannian space-time [52, 57]. The corresponding geodesic equations of a plane wave in vacuo then have a space part which is a helix [52, 12]. Analogous field equations of the Proca type have also been proposed to form a model of the universe as a collection of dust particles and nonzero rest mass gravitons which possess a collective mode behavior [58]. Such an approach could also connect the graviton mass with the problem of dark matter in cosmological models [58, 59]. 4. New Aspects Due to the Extended Theories

The extended theories of this review lead to results which both clarify some unsolved problems and end up with new concepts, thus having impacts on the following areas: Time-Dependent Stares with Wave Phenomena These include plane waves, individual photon models, the transition to a light beam, and superluminosity. Time-independent States with Electromagnetic Equilibria These include particle-shaped states, string-shaped states, and nonlocality effects.

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4.1. PLANE WAVES

The plane waves in vacuo become modified both by a nonzero electric conductivity and a nonzero electric field divergence. 4.1.1.

Nonzero Electric Conductivity

Staiting from the field equations with an additional small current (6), a plane wave of the form yields a dispersion relation with phase and group velocities which differ slightly from C due to a resulting photon rest mass, thereby leading to a dispersion effect [35, 38-40]. These properties provide an explanation of the cosmic redshift [35, 38-40] being an alternative to the conventional geometrical Hubble expansion, and giving rise to a frictional "tired light" effect. If the latter would dominate, it should be related to the Hubble parameter by where is the relative dielectric constant. Since there are many strong commonly accepted arguments in favour of the Big Bang model, the relative importance of this tired light effect remains an open question. 4.1.2.

Nonzero Electric Field Divergence

The field equations with the four-current (5) result in three wave types: When div E = 0 and curl the result is a conventional transverse electromagnetic wave, henceforth denoted as an "EM wave". When div and curl E = 0 a purely longitudinal electric space-charge wave arises, here denoted as an “S wave”. When both div and curl a hybrid non transverse electromagnetic space-charge wave appears, here denoted as an "EMS wave". For plane elementary modes of the form notation where the angle

we can introduce the represents the new degree of freedom.

The set of plane wave modes then ranges from the EMS modes for for the

to the S mode for and

modes, and

mode for

, via the

The group velocities are for the EMS mode. Poynting’s

theorem for the energy flow of plane waves in vacuo applies to the EM and EMS modes, but not to the S mode. The problem (v) of Section 2 concerning total reflection of an incident conventional EM wave at the vacuum boundary of a dissipative medium can now be

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133

tackled. Thus the matching of the field components, which is impossible for a conventional EM wave in the vacuum region 15, 29-32, can now be performed by EMS waves in the same region. 4.2. INDIVIDUAL PHOTON MODELS

We now turn to individual photon models being based on axisymmetric solutions of a Proca type equation. Thereby there are connections between the photon rest mass theory and the nonzero electric field divergence theory outlined in Sections 3.1.2 and 3.1.3. 4.2.1. Basic Features of Axisymmetric Model

With a cylindric frame

and considering axisymmetric modes of the form

propagating in the axial z direction, the basic equations of the system (2) and (5) give the following general results: For the conventional EM mode with div E = 0, the general solution either diverges at the origin r = 0 or at infinite values of r. Such a behaviour was already realized by Thomson [11], and further by Heitler [60] as well as by Hunter and Wadlinger [12]. This mode is henceforth excluded from the axisymmetric analysis. For the EMS mode with div the general solution can be shown to become derivable from a generating function which yields convergent solutions within entire space [29-32]. For this axisymmetric normal EMS mode the velocity vector of equation (5) is now written as

where

and the dispersion relation becomes

Here the phase and group ve]ocities both become equal to v when is a constant angle. With cos exactly equal to zero, we are back to a divergent situation ]ike that of the EM mode. With a nonzero there is also a nonzero rest mass. This makes it possible to introduce a "rest frame" K’ which moves at the velocity V < c with respect to the laboratory frame K. Applying a Lorentz transformation to the field strengths in K, the resulting fields in K' have forms which agree with a velocity vector

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in K'. This is expected, and it confirms the relevance of the adopted approach. As a following step, the axisymmetric normal modes are integrated to form a wave packet of narrow line width, being centred at the wave number and having the characteristic axial length The field of this analysis is reconcilable with that by Evans and Vigier [5-7, 27], in the sense that all three components are nonzero. A vanishing axial component would contradict the basic equations. The results of the four-currents (4) and (5) are ori the other hand not identical. The nonzero magnetic component

in the rest frame

K' further supports the existence of a static field part such as the field by Evans and Vigier. A nonexistence of the latter cannot be proved in terms of the conventional d'’Alembert equation (1) in which the underlying mechanism of equation (4) for is missing. It should finally be observed that Poynting's conventional theorem, which holds for plane EMS Waves, is not generally satisfied for the corresponding axisymmetric modes, this being in agreement with conclusions by Evans et al. [6] and by Chubykalo and Smirnov-Rueda [2]. 4.2.2.

Integrated Photon Parameters

The EMS wave packet is now integrated in space. Thereby a normalized and convergent generating function of the form

is chosen in the laboratory frame where and

stands for a characteristic radial dimension

The results of the integration, which can as well be performed in the rest frame K', are as follows [30-32]. The total electric charge vanishes. For electric neutrality it is then not necessary to assert that the photon is its own antiphoton. The total magnetic moment vanishes, but the local magnetic field is still nonzero. The energy density can be expressed in the two alternative forms

NEW DEVELOPMENTS IN EM THEORY

of a field energy density

135

and a source energy density

same integrated total energy

but the moments of

They give the and

with an

arbitrary function become different. The present theory thus results in a total mass

and a photon rest mass

being related by

where v is the frequency associated with the mean wave number of a wave packet of narrow line width. A Lorentz transforiination to the rest frame yields the mass

which confirms the relations (14).

With the form (14) the axial symmetry and the symmetry with respect to make the integrals of the forces and vanish as well as those of

It is then possible to use the conventional [61] form

for the density of the angular momentum, with S as the Poynting vector and r standing for the radius vector from the origin. Here the integral of S is equal to the total angular momentum for the photon as a boson. From

expression (14) with associated field quantities we obtain where

is a sharply defined radius for

1 in equation (12). This result

agrees within an error of 0.5 percent with microwave transmission experiments in presence of an aperture, as being conducted by Hunter and Wadlinger [12]. With the angular momentum of a fermion, there would arise a discrepancy by a factor of two. In earlier interference experiments with individual photons [10], dot-shaped marks were father observed at a screen. For impinging individual photons these marks seem to be consistent with the radial extension of the packet given by equation (16). The axial magnetic field nonzero rest mass

of the wave packet is thus associated with the

and the total angular momentum (spin)

The latter comes

out the same in the frames K and K'. The three axisymmetric field components of E and B form a helical structure. These properties are analogous to the helical spin field configuration of 4.2.3.

and its cyclic field relations [5-7, 27, 57].

The Physical Picture of the Individual Photon Model

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An essential feature of de Brogue's picture of the wave-particle duality consists of regarding the particle and the associated wave as simultaneously existing physically real entities. The present wave packet model is consistent with this picture. The total energy is in the laboratory frame K. Thereby the fraction

can be regarded

as the energy of a "free" pilot wave of radiation, and the fraction

as the energy of a

"bound" particle state of "self-confined" radiation. The rest mass

thus represents an

integrating part of the total field energy. The "bound" radiation moves around the axis of symmetry at the velocity according to equation (9), and the pilot wave propagates at the velocity

. With the sharply defined radius

of equation

(16) the "bound" part of the radiation field becomes associated with a frequency

of revolution around the z axis in the laboratory frame K. In combination with relations (16) and (14) this yields

as being supported by the idea that all parts of the field energy should be included in the same way in the total energy The relation

by de Broglie then comes out of equation (18). Its connection with equations (9) - (18) has been confirmed in terms of gauge theory [46]. In the rest frame K' the velocity vector (11) has a component only, and the photon radius in the perpendicular r direction remains the same as in equation (16) with v as the frequency in K. With a Lorentz transfoimation of time from K to K' the frequency (inverted time) should further change from v to thus making of equation (18). In K' the frequency of revolution is expected to become for the "bound" radiation. Combining this frequency with relations (16) and (18), the angular momentum in K' becomes which is the same as in the laboratory frame. This supports the analysis and its physical interpretation. Provided that is independent of the frequency v, the results (17) - (19) and (10) thus permit the angle also to be independent of v. There are two important consequences of this analysis:

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137

The two-frequency paradox by de Broglie[6] is resolved, in the sense that the frequency becomes coupled to the frequency v by relation (18). The phase and group velocities become independent of frequency for a constant angle , thereby resulting in vanishing dispersion effects. If there would arise a frequency dependence of , the extremely small rest mass would only give rise to small dispersion effects, even at large cosmical distances. The present physical picture thus increases the understanding of the points (i) - (iii) of Section 2. 4.2.4.

The Thermodynamics of a Photon Gas

With a nonzero rest mass one would at first glance expect a photon gas to have three degrees of freedom, two transverse and one longitudinal. This alters Planck's radiation law by a factor of 3/2, in contradiction with experience [32]. A detailed analysis shows, however, that the spin field cannot be involved in a process of light absorption [5]. This agrees with the present analysis where the spin field is "carried away" by the pilot field, and Planck's law should be recovered in all practical cases [32], Moreover, transverse photons cannot penetrate the walls of a cavity, whereas this is the case for longitudinal photons which would not contribute to the thermal equilibrium [25]. The equations of state of a photon gas have been considered by Mezaros [18] and Molnar et al. [19]. They find that Planck's distribution cannot be invariant to an adiabatic change occurring in an ensemble of photons. This dilemma is due to the fact that the changes cannot be adiabatic and isothermal at the same time. Probably this contradiction is due to the lack of longitudinal magnetic flux density. Thus, in an adiabatically deformed photon gas the intensity will change in time, and so will a field like . This result can contribute to the clarification of point (x) in Section 2, but further analysis may become necessary. 4.2.5.

The Photon Rest Mass

The possible existence of a nonzero photon rest mass was already proposed by Einstein [24], Bass and Schrodinger [25], de Broglie and Vigier [26] and further by Evans and Vigier [5] among others. It includes crucial points such as the relation to the Michelson-Morley experiment, and the so far undetermined magnitude of the mass and its experimental determination. The velocity w of the earth around the sun is . If this would be the velocity with respect to a stationary ether, and if massive photons would move at the velocity in the same ether, then the velocity u of photons recorded at the earth's surface would become

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B. LEHNERT

For the magnitude of the rest mass and the velocity according to equation (14), the following conditions will apply: For

, corresponding to a rest mass

a

change in the eighth decimal of the recorded velocity of light can hardly be detected. With the same assumption, turning from a direction where opposite direction where become less, i.e.

the change in

c to the would even

. Such a value hardly becomes detectable.

Consequently, there should not arise any noticeable departure from the Michelson-Morley experiments when the rest mass is being changed from zero to about or less.

The quantum conditions and are satisfied for a whole range of small values of the corresponding rest mass. As pointed out by de Broglie and Vigier [14], this indeterminableness appears to be a serious objection to the underlying theory. The problem is that the derivations depend simply on the existence of the mass, but not on its magnitude. To motivate the analysis, de Broglie and Vigier take examples on other "macroscopic quantum effects" in theoretical physics, such as ferromagnetism and the indefinite precision in measuring Planck's constant. Additional examples can be given by the magnitudes of the electron and neutrino masses and the electronic charge which so far have not come out of pure theoretical deductions. Thus, the uncertainty in the magnitude of the photon rest mass does nor necessarily imply that the theory is questionable, but could just be due to some hidden extra condition or theoretical refinement which may has to be added to obtain fully specified solutions. For further analysis the following considerations have been made: From the hypothesis of a nonzero electric conductivity and the associated dispersion relation [32, 35, 38-40], the concepts of “tired light" and cosmic redshift could be related to a rest mass of about The latter would lead to anisotropy in velocity and frequency, thus resulting in anomalies of the redshift and the cosmic microwave background radiation [62, 63-64]. The Vigier mass of the photon being associated with the de Broglie wave length i s given through equation (19) by where when being put equal to the radius of the universe [35, 63-64].

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139

A deeper understanding of certain phenomena such as the Goos-Hanchen and Sagnac effects mentioned in Sections 2(iv) and (vi) could provide estimates of the rest mass, The latter can give explanations of these effects as shown by de Brogue and Vigier [26] and by Vigier [65]. 4.3.

TRANSITION TO A LIGHT BEAM

In many cases the photon can be represented by the two alternative models of a plane wave and a particle-like wave packet. This should apply to interference phenomena with individual photons [10]. For a given point at the screen of an experiment with two apertures, the resulting interference pattern could thus be interpreted in two ways: The photon as a plane wave is divided into two parts which pass either of the apertures, and then interfere at the screen. The photon as an axisymmetric wave packet is also divided into two parts which interfere at the screen to form a common dot-shaped mark. The photon has the energy hv both at its source and at the screen. The division into two parts appears to be in conflict with the quantization of energy. A solution is, however, found in terms of the Heisenberg uncertainty principle [31]. A tentative approach will now be made on the transition from a photon beam to a plane wave [31], as mentioned in Section 2(vii). A beam is considered which consists of a stream of individual wave packets, and where the macroscopic breadth of the beam is much larger than the photon radius of equation (16). The volume density of the wave packets is uniform in space. The mean distance between the wave packet centra then becomes

The energy flux per unit area is

Combination of relations (21) and (16) then yields a ratio

of the mean transverse separation distance d and the photon diameter . Analogous relations can be obtained in the longitudinal direction, but these become less critical to the present discussion [31]. Since is a sharply defined radius [12], there is a corresponding critical transverse ratio As long as the axisymmetric photon model applies, there is a range However, when

of negligible transverse overlapping between the individual photon fields. and in particular for

there arises overlapping

which would make the field vectors multivalued at every point in space, and therefore

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B. LEHNERT

conflicts with the basic field equations in the axisymmetric case. As seen from expression (22), such transverse overlapping does not occur for visible light at moderately large energy fluxes, whereas it arises for strong laser light in the visible regime and for radio frequency waves. To preserve the quantized beam energy flux, we now assume the deficit of energy due to overlapping to be compensated by the energy of a simultaneously appearing plane wave of the EM or EMS type, i.e. by the ansatz where

and

are contributions from the individual axisymmetric EMS fields

and from the plane wave, respectively. For then have

of strong overlapping we should

In this way the transition can possibly take place, from a stream

of individual photon wave packets to a plane wave. The geometry of the considered configuration, and its initial and boundary conditions, are also important to the analysis.

4.4. SUPERLUMINOSITY The superluminal phenomena mentioned in Section 2(ix) are characterized by propagation at a finite speed being larger than but the investigations and their interpretation are still at a preliminary stage. There seems to be observational evidence for such phenomena, as well as indications thereof in the theoretical analysis, as described in a review by Recami [16]. Thus the squared-mass of muon-neutrinos is found to be negative. Further there are observations which may be interpreted as superluminal expansions inside quasars and in galactic objects. Also so called"X-shaped waves" have been observed [66] to propagate at a velocity larger than c. Finally Nimtz et al. [67] have performed an experiment where Mozart’s Symphony No.40 was transmitted at a speed of 4.7 through a barrier bf 114 mm length. There is a difficulty in the interpretation of tunneling experiments, because no group velocity can be defined which is associated with evanescent waves. Reviews of the theoretical analysis have further been presented by Barut et al.[68] on tachyons and by Olkhovsky aud Recami [69] on tunneling processes. A special study is due to Walker [70] on the propagation speed of a longitudinally oscillating electric field, generated along the axis of vibration of an electric charge. The central assumption underlying tachyon theory is that the Lorentz transformation also applies to the superluminal case. One therefore simply takes the Lorentz factor and substitutes into it [71,16]. This leads to an imaginary rest mass, with difficulties of interpretation [71]. These central concepts also come out of the approach of Section 3.1.3. To satisfy the Lorentz condition one can thus replace the form (9) of the velocity vector by

NEW DEVELOPMENTS IN EM THEORY

For propagating axisymmetric normal modes of the form tachyon mode yields the dispersion relation

141

an EMS-like

which replaces equations (10). If we further assume relations (14) to be adopted and modified to apply to the tachyon case, the result would become

or

Consequently, this leads to what could be conceived as an imaginary rest mass, as is also obtained in current tachyon theory. 4.5. STEADY ELECTROMAGNETIC EQUILIBRIA

With equations (2) and (5) steady equilibrium states are determined by

We limit ourselves to axisymmetric states of two kinds: "Particle-shaped" equilibria where the configuration becomes bounded both in the radial and axial directions. "String-shaped" equilibria where the configuration is uniform in the axial direction. For both these kinds the general solution can be expressed in terms of a generating function where the vector potential A only has a component in the direction of a spherical frame Separable functions are considered where and 4.5.1.

with

as a characteristic amplitude,

as a characteristic radius. Particle-Shaped States

By integration of the obtained field quantities, expressions for a net electric charge magnetic moment mass and angular momentum (spin) are obtained. There are two classes of solutions being of possible interest to leptons [28-32];

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B. LEHNERT

For a convergent radial part moment

there is a vanishing charge

A nonzero and very small mass

can be made equal to

and magnetic

can be obtained, and the spin

of a fermion. These solutions could become

models for the neutrino. The latter moves nearly at the speed of light, has weak interaction with its surroundings, and has no color charge. For a divergent radial part nonzero

at the origin

solutions can be obtained for

and

by permitting the

and finite

characteristic radius

to approach the very small range of a "point charge".

Thereby the parameter ranges of

can be chosen to include

of the

elementary charge, as well as to satisfy the value of the magnetic moment by Dirac and Feynman [72]. On pure physical grounds it appears to be unacceptable to have a radius being exactly equal to zero. By applying a small general relativistic correction, a nonzero radiu5 of the order of can be obtained for the electron. Here we turn to the question of Section 2(xi) on the infinite self-energy of a point charge in classical and quantum theory. The latter uses a renormalization procedure, by subtracting two "infinities" to end up with a finite result. Despite of the success of such a procedure, there ought to be searched for a more satisfactory way from the physical point of view [51]. The present theory may possibly provide such an alternative, where instead a product of an "infinity" with a "zero" yields a finite result. 4.5.2.

String-Shaped States

The string-shaped equilibria which result from equations (27) can serve as an analogous model which reproduces several of the desirable features of the earlier proposed string configuration of the hadron color field structure. These equilibria have a constant longitudinal stress which tends to pull the ends of the configuration towards each other. The magnetic field is thereby located to a narrow channel, and the system has no net electric charge. Since the divergence of the magnetic field is zero, no model based on magnetic poles is needed. 4.6. NONLOCALITY EFFECTS

An increasing interest has recently been taken in instantaneous long-range interaction. The investigations within this field are yet at a preliminary stage, also including difficult concepts and interpretations.

NEW DEVELOPMENTS IN EM THEORY 4.6.1.

143

General Questions

It was already pointed out by Dirac 50] that, as long as dealing only with transverse waves, we cannot bring in the Coulomb interactions. There must then also arise longitudinal interactions between pairs. In fact, as already argued by Faraday and Newton and further stressed by Chubykalo and Smirnov-Rueda [2] among others, instantaneous long-range Interaction takes place not instead of but along with the short-range interaction in classical field theory. This point of view has also been expressed by Pope [73] in stating that instantaneous action-at-a-distance and the finite speed of light are generally considered as antithetical, but it is well known that in relativistic physics light has both finite and infinite speed. Thereby c is not a velocity but is a space-time constant having the dimension of velocity. In this connection Argyris and Ciubotariu [58] point out that the unquantized longitudinal-scalar part of the field yields the Coulomb potential, and that transverse photons transport energy whereas longitudinal (virtual) photons do not carry energy away. Thus, there is direct interaction between a transverse photon and the gravitation field of a black hole, but not with a longitudinal photon. The Coulomb field is therefore able to cross the event horizon of a black hole. 4.6.2.

Instantaneous Long-Range Interaction

In considerations on a single moving charged particle, Chubykalo and Smirnov Rueda [2, 44-45] have shown the Lienard-Wiechert potentials mentioned in Section 2(viii) to be incomplete, by not being able to describe long-range instantaneous Coulomb interaction. In a modified version of the earlier theory by Hertz [41-43], such an interaction is included by Chubykalo and Smirnov-Rueda. The same authors have also made an analysis [2, 44-45] in which a Proca type equation is divided into two pairs. The first manifests the instantaneous and longitudinal aspects of electromagnetic nature, as represented by functions of an implicit

time dependence. For a single charge system this would then lead to the form where

is a fixed vector from the point of observation to the origin, and

is the position of the moving charge. The implicit time dependence then implies that all explicit time derivatives disappear from the basic equations of the first part of the divided pairs. The second part is responsible for transverse wave phenomena, as represented by functions g(r , t) of an explicit time dependence. There are a number of arguments which could support the existence of long-range instantaneous interaction. Thus, superluminal phenomena cannot, be explained from Maxwell's equations without longitudinal wave concepts [17]. The energy of "longitudinal modes" cannot be stored locally in space but can be spread by an arbitrary velocity [57]. Moreover, non-locality behavior is supported by observations as a fundamental property of the universe [74]. There are also several quantum mechanical arguments in favor of long-range interaction, such as that of the Aharonov-Bohm effect [75] and those related to the discussion on the Einstein-Podolsky-Rosen paradox [76-77]

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Instantaneous action at a distance as represented by longitudinal components can thus be interpreted as a classical equivalent ofnonlocal quantum interactions [2]. With these arguments, and with the similarity between electromagnetic and gravitational long-range interaction, it should be justified to represent the former with the implicit time dependence part in the basic equations. There are similarities between electromagnetism and gravitation which could be Important to the investigations on long-range interaction. Thus, there is a resemblance between the Coulomb and Newton potentials [58]. A holistic view of electromagnetism and gravitation would imply that action at a distance occurs in a similar way in gravitation [17] and vice versa. This is supported by the general principle that there exists no screen against gravitational forces acting between distant massive sources [35, 62]. In other words, the position, velocity and acceleration of a source of gravity would then be felt by the target body in much less than the light-time between them. 5. Conclusion The new developments in electromagnetic field theory have here been seen to provide a number of contubutions to the improved understanding of problems which have so far not been clarified in terms of conventional analysis. Examples have thereby been given in connection with the points (i) - (xi) of Section 2. This development has also resulted in a number of predicted features of the electromagnetic field, in the form of new models of the photon and of leptons. The areas of new development do not seem to be exhausted by far, and there are several important questions which require further investigations. Examples are here given by the magnitude of the photon rest mass, the tired light hypothesis versus the Big Bang concept, superluminosity, the Sagnac effect, the light beam transition, the absolute values of the elementary electronic charge and of the masses of leptons, long-range interaction and the general concepts of causality and locality. References 1. R.P. Feynman, Lectures on Physics: Mainly Electromagnetism and Matter, Addison-Wesley, 1964. 2. A.E. Chubalo and R. Smirnov-Rueda, in The Enigmatic Photon, 4 (1998) 261, Edited by M.W. Evans, J.-P. Vigier, S. Roy and G. Hunter, Kluwer Academic Publishers, Dordrecht/Boston/London. 3. A. Lakhtakia, Editor, Essays on the Formal Aspects of Electromagnetic Theory, World Scientific Publishers, Singapore, 1993. 4. T. Barrett and D.M. Grimes, Editors, Advanced Electromagnetics, World Scientific Publishers, Singapore, 1995. 5. M. Evans and J.-P. Vigier, The Enigmatic Photon, 1 (1994) and 2 (1995), Kluwer Academic Publishers, Dordrecht/Boston/London. 6. M.W. Evans, J.-P. Vigier, S. Roy and S. Jeffers, The Enigmatic Photon, 3 (1996), Kluwer Academic Publishers, Dordrecht/Boston/London. 7. M.W. Evans, J.-P. Vigier, S. Roy and G. Hunter, The Enigmatic Photon, 4 (1998), Kluwer Academic Publishers, Dordrecht/Boston/London 8. G. Hunter, S. Jeffers and J.-P. Vigier, Causality and Locality in Modern Physics, Kluwer Academic Publishers, Dordrecht/Boston/London, 1998. 9. V.V. Dvoeglazov, Editor, Contemporary Fundamental Physics, Nova Science Publishers, Commac, 1999.

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COMPARISON OF NEAR AND FAR FIELD DOUBLE-SLIT INTERFEROMETRY FOR DISPERSION OF THE PHOTON WAVEPACKET

R. L. AMOROSO Noetic Advanced Studies Institute 120 Village Square, MS 49 Orinda, CA 94563-2502 USA J-P VIGIER Pierre et Marie Curie Université Gravitation et Cosmologie Relativistes Tour 22 – Boite 142, 4 place Jussieu, 75005 Paris, France M. KAFATOS Center for Earth Observing and Space Research George Mason University Fairfax, VA 22030-4444 USA G. HUNTER Department of Chemistry York University, 4700 Keele St. Toronto, Canada M3J1P3

Abstract. Extending EM and Quantum Theory suggests the possibility of photon mass, additional terms for Maxwell‘s equations, reality of de Broglie-Bohm causality and the Vigier model of extended charged particles. Experimental tests indicative of these hypotheses can be performed with double-slit interferometry of single visible wavelengths comparing near and far field sources over laboratory and various cosmological distances to observe the possibility of spreading of the photon wavepacket during propagation. These observations could determine whether nonlinearities causing non-dispersivity are associated with Maxwell’s equations. If so, this may be an indirect determination of nonzero restmass photon anisotropy, de Broglie photon piloting and vacuum permittivity reincarnating the Michelson-Morley experiment in terms of a Dirac covariant ether.

1. Introduction

Traditional thinking suggests that EM radiation coming from a point source is subject to spreading of the photon wave-packet over x = -ict. Recently however, families of 147 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 147-156. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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nondispersive waves have been found for Maxwell, Klein-Gordon, Dirac, Weyl and Schrödinger field equations [1-7, 31-33]. Courant and Hilbert [8] were probably among the first to make this distinction. In this paper we briefly review classical and extended theoretical predictions of photon propagation, present an experimental design for possible empirical tests and relevant discussion of the physical consequences for the de Broglie, Bohm and Vigier formalisms [9] of extended electromagnetic theory that suggest the photon wavepacket is piloted and does not spread over cosmological distances as suggested by current classically oriented interpretations of Maxwell’s field equations. The experiment is accomplished by comparison of the double-slit fringes of monochromatic light from near and far field monochromatic emission sources. Recent EM-Theory discussions on the possible existence of photon mass and U(1) group invariance of EM-Theory in view of the putative

implies the

introduction of.new terms for Maxwell-Lorentz equations. This cannot be considered as purely theoretical as in the past because of recent new experimental evidence [10]. The aim of the present text is to discuss the two possible interpretations of Maxwell’s equations: 1. Photon propagation without trajectory - random probability distributions [classical]. 2. Photon propagation with trajectory - piloted with no or minute spreading of the wavepacket [extended]. The former is in accord with the standard model of classically oriented Copenhagen approaches to the EM and quantum formalism; and the latter the Vigier-Bohm-de Broglie extended charge particle causal approaches.

1.1 STAGES IN DEVELOPMENT OF THE THEORY OF LIGHT As generally known the present status of the theory of light is the result of three stages of development. a) The Maxwell – Heaviside – Hertz stage which developed (in the middle of the last century) the Maxwellian linear equations of light (and Hertzian waves) with transverse continuous waves (with zero-mass and U(1) invariance) of velocity with the separate existence of instantaneous Coulomb interaction. Technically this led to the huge development of electrical/photographic devices in the modern world. b) The Einstein stage (1905) with 1) The discovery of the photon Lichtquanten carrying an energy 2) The idea (1916) that photons carry the observed electromagnetic energy in the form of unidirectional Nadelstrahlung , of oriented beams and not in the form of spherical emission Kugelstrahlung which did not exist [ 12]. 3) The observed quantized electromagnetic potential for inertial massive charges only emit energy when accelerated; except in the case of Bohr orbits, where it

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was postulated if one introduced Poincaré forces to keep the sub-charges together and that the inertia of the gravitational and EM charged part in any particle was separately covariant [10]. c) The quantum stage (1920-2000) where the Copenhagen proponents dropped the stepby-step description of the propagation of light only discussing the statistical properties. Later photon emission and absorption was shown experimentally by Aspect [13] to correspond to (individual photon emission followed by absorption) linear recoils of the corresponding sources. Two theories developed in parallel : The Copenhagen interpretation. The QED version suggests probability waves or quantized photons ; never the two simultaneously. The Einstein – de Broglie interpretation (dropped between 1925 and 1950) allowing simultaneous real waves and piloted photon particles where both carry energy momentum distributions mostly concentrated on the latter. Both interpretations recover FAPP the same experimental facts. d) The recent period in which new properties of EM waves have been discovered that are generating renewed discussion of differences between the interpretations. We only mention here the simultaneous existence of two different electron radii (the Compton and charge radii) in scattering experiments [14]. observed anomalies in scattering [10]. Recent observations in EM theory [15] and excess energy in scattering [16].

2. General Nature Of The Photon Wave-Packet A wave-packet is a balanced superposition of waves of one predominant wave number with phase amplitudes that interfere constructively over the small region or outside of which the amplitude reduces to zero quickly by destructive interference. This is not true for ideal monochromatic wavepackets; Figure 1A shows a wave function or amplitude that depends on space coordinates and time t simplified to with average wavelength over the limit defined by a light pulse passing through a shutter, since an ordinary plane wave would be spread over all space [17,18].

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Figure 1A is plotted in Figure 1B according to equation 1 [17] as a function of reaching a maximum at to zero where ; thus obtaining a wave function concentrated in a packet where

The Fourier transform of

which, as shown in Figure 1C, is the wavepacket for a single photon. Equation 2 is a general type of wavepacket for any function thus defined [17, 18].

2.1 REVIEW OF THE CLASSICAL DOUBLE-SLIT EXPERIMENTAL FORMALISM A description of Young’s 1801 classic fringe experiment for two slits [19] of width a, at a distance A apart as shown in Figure 2 is described by [20]:

where s(x) is the transmission function for one slit of width a. For very narrow slits a goes to zero and the incident amplitude is proportional to 1/a so that s(x) becomes .

The intensity of the diffraction pattern (Figure 2B) [20] is modulated by a function going to zero for known fringe experiment [19,20].

of period 1/2A This is Young=s well

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For a coherent wavepacket passing through a double-slit interference creates phase difference such that a maximum occurs at where and a minima at where m = 1,2,3 represents d is the distance between slits (A in fig. 2) and is the angle made by an arbitrary point P on the screen, the central point between the slits q(x) and the corresponding center of the screen. The distance between the two screens D must be much larger than d so that the distances from the slits to P can be considered parallel [21].

2.2 CURRENT CLASSICAL INTERPRETATION OF DOUBLE-SLIT PHENOMENA Historically Einstein first proposed in 1905 that radiant electromagnetic energy should appear as About 10 years later he assumed that these quanta should also be spatially quantized with a unique orientation. The essential conclusion of his research was that Maxwell’s - spherical radiation around a source – does not exist but that elementary light energy h v always appears in a unique direction creating a recoil upon emission which he called or ‘needle radiation’, giving deference to Einstein’s original term [12]. This is also responsible for radiation pressure. According to classical interpretations of Maxwell’s theory dispersion of the wave-packet over distance is expected [17,18]. This is partly due to the fact that in the Copenhagen view emission is point-like so any packet should expand. The uncertainty principle can be derived from propagation of the wavepacket which in classical terms propagates according to the Schrödinger equation. This is described by Bohm [17] as :

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where a wavepacket with an initial diameter of

will spread to

as t

becomes limitless. The narrower the wavepacket originally the more rapid the spreading. (see Figure 3) According to Bohm [17] the reason for the spread is in terms of the uncertainty principle. The region confining the packet has a number of wavelengths

near

so that even though the average velocity of the wavepacket is equal to the

group velocity, the actual velocity will fluctuate, and the distance propagated by the packet isn’t fully determined. It can fluctuate by

According to Bohm [17] photons have momentum as evidenced in the radiation pressure during absorption , such that the energy and momentum of light quanta is the same as a zero mass particle in 3D space. It is the wave properties of a wavepacket that produces the that allows spreading because a particle will never spread ; but a collection of particles because of uncertainty in velocity gradually spread with (x) Thus although equations (5) & (6) used for illustration of classical wavepacket spreading are Schrödinger type equations, Maxwellian equations give a similar result for photons in the classical limit – i.e. – spreading. In our extended theoretical approach that includes photon mass [10,22]; piloting effects prevent spreading as There may be an infinitesimal spreading over cosmological distance. This is the current empirical knowledge limit; where further understanding can only be achieved by experimentation like the suggestion in section 3.

2.3 EXTENDED THEORETICAL APPROACH TO PHOTON PROPAGATION De Broglie wave mechanics creates a relationship between wave numbers and momentum not considered in classical mechanics. In the de Broglie mode a classical wave of a wave number can be of arbitrary amplitude and momentum; and whenever position or

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momentum is measured a definite number results. The de Broglie relation implies a definite wave number for a definite momentum. This is contrary to a classical description of a wavepacket, which suggests a range of wave numbers and positions [17]. Concurrently defined position and momentum values is considered equivalent to the assumption of that constantly determine these values. This is inconsistent with the standard Copenhagen interpretation of quantum theory which is statistical and not causal. 2.4 RECENT WORK ON NON-DISPERSIVE PROPAGATION MODES

Rather than the classically oriented Schrödinger equation which suggests spreading of the wavepacket (in a variation of Figure 3), non-dispersive wave modes would be expected to propagate according to the de Broglie relativistic Klein-Gordon type equation [1,24]. Recent work on non-dispersive modes of the wavepacket by numerous authors [1-8] has demonstrated mathematically the possibility for the existence of ‘real’ non-dispersive modes of the photon wavepacket satisfying linear Maxwell equations but considered contrary to the prevailing opinion. This has urged empirical testing of the issues at the heart of the matter and is our main inspiration for writing this paper. We give here only the very briefest review of this recent theoretical work and refer parties interested in deeper analysis to the main references [1-8]. According to established wave mechanics a de Broglie wave with infinite wavelength is said to be associated with all particles and have a wave function uniform throughout all space. A particle’s internal vibration and infinite de Broglie wave stays in phase at the particle’s location. This suggests how the de Broglie wave pilots a particle’s motion with no spreading; whereas a Schrödinger wavepacket spreads because of uncertainty in momentum [1]. Because the de Broglie relation is relativistic a Lorentz transformation might be involved between the particle’s point of origin and present position during propagation, canceling insertion of any would be classical uncertainty effects and maintaining phase coherence between the particle’s internal motion and the wave function in the de Broglie relativistic-piloted regime. This might be considered reminiscent of error correction modes discussed in terms of quantum computing: and If then for all therefore

With these conditions de Broglie theory yields [1] or also in the form

Using a different technique, Hillion [2] uses electromagnetic theory to derive nonhomogenious nondispersive waves from Maxwell’s equations. With the variables the wave equation becomes

This was first shown by Courant and Hilbert [8] and has

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nondispersive solutions of the type a solution of the characteristic equation

where the phase is where F is arbitrary with

continuous partial derivatives and g is an attenuation factor [2]. Brittingham [7] derives on the other hand homogenous nondispersive solutions to Maxwell’s equations in the soliton regime with both linear and nonlinear parameters. Beil [31-33] has applied the Brittingham solutions to modeling of the photon as a specific realization of Nadelstralung, which Einstein conjectured was the only kind of radiation that is consistent with relativistic dynamics. Finally Shaarawi [6] derives Brittingham like nondispersive solutions for the wavepacket applicable to Klein-Gordon equations which can be used as local de Broglie scalar wave particles.

3. Proposed Experimental Design A. Stellar objects with emission spectra compatible with the telescopes instrumentation are chosen for observation. As a baseline fixed sources in our galaxy of about 100 light years distance are selected for each of 3 wavelengths (red, yellow/green, blue). Narrow pass filters of single wavelengths are used with 3 far-field stars of about 2 million light years from the local group in the Coma cluster and beyond for comparison with the control stars from our galaxy for each of the 3 wavelengths. B. The comparisons with near and far-field stars are made for possible spreading during propagation of the photon wavepacket over cosmological distances in the fringe patterns of standard double-slit interferometry. It is suggested that 3 control stars be compared with 3 far-field stars for a 12 star database. To ensure uniformity of stellar types, we suggest bright Cepheid’s or M-giants and O-supergiants in the local group of galaxies, and the Virgo and Coma clusters. Or sources of greater observational ease such as emission nebulae with Balmer and HII and lines

[25]. C. Because of the foregoing discussions on the nature of the photon wavepacket during propagation the experiment might optimally be performed with additional filters allowing passage of only single photons. However in case any physical parameters might be missing from current theoretical predictions it would be useful if practical to also perform the experiment with continuous wave trains of multi photon wavepackets for experimental diversity and exploration of group dynamics. As in figure 2 and accompanying discussion it is deemed important to perform the experiment with double-slit designs with 3 dimensions or spacing; 1. Optimally maximum, 2. Median and 3; Optimal minimum. Account of work on measurements of photon radius [26] should probably also be taken into account for optimal a distance as also shown in figure 2A. D. To simplify the experiment for preliminary results the model can be done initially with the near-field control group performed in an Teran laboratory setting. The far field sources could then be emission lines in the solar chromosphere. E. Anticipated results. The current model formalism suggests significant spreading of the wavepacket over cosmological distance because of uncertainties in momentum.

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However our view according to extended theoretical models of Vigier, de Broglie, and Bohm that there might be infinitesimal spreading of the wavepacket because of de Broglie-Bohm piloting. Until we have preliminary tests we are unsure of sufficient limits in discerning the degree of spreading within the current instrumentation limits of CCD cameras and computer analysis of the data. The mathematical predictions for spreading will be included in the proposals for telescope time. By the time of publication we anticipate having at least preliminary data. F. Comparison of dispersion for dust-free and clear spacetime regions as test of gravity effects and redshift from TIFFT.

4. Conclusion And Summary Superficially the nature of a wavepacket and its spreading during propagation seems straight forward; but the subtleties involved are at the heart of wave mechanics and quantum theory which is by no means complete and entails persistent discussion on the merits of Copenhagen vs. extended forms of quantum theory. A definitive delineation is not possible in terms of any type of current theoretical discussion alone. Therefore if technically feasible experiments on the nature of the wavepacket and its propagation like those proposed here might advance our understanding of Quantum Theory. Classical approaches predict wavepacket spreading because of uncertainty relationships. The de Broglie-Bohm approaches predict coherence over all space and time in view of putative causal action of the pilot wave or quantum potential. This is the well known assumption of hidden variables deemed inconsistent with the classically oriented Copenhagen model. A final understanding of the photon and its propagation is far from being understood [27]. To understand the anticipated experimental results photon propagation may not only have to be perceived in terms of internal de Broglie Lorentz transformations [1] but also deeper aspects of nonlocality [28] which might only be clarified in terms of a post big bang cosmology and the attendant understanding of spacetime hyperstructure [29] which non-zero photon restmass seems to demand. A Newtonian ether was disallowed by Einstein’s relativistic dynamics and the Michelson-Morley experiment. Einstein himself said that relativity did not preclude an ether. We revisit this issue in terms of a Dirac covariant subquantum stochastic ether with correspondence to relativity and inclusive of de Broglie Bohm Vigier charged particle models [30].

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References 1. Mackinnon, L. 1978, A nondispersive de Broglie wave packet, Found. Phys. 8:3/4, 157-176. 2. Hillion, P. 1991, Nonhomogenious nondispersive electromagnetic waves, Phys. Rev. A, 45:4, pp.26222627. 3. Peshkin, M. 1999, Force-free interactions and nondispersive phase shifts in interferometry, Found. Phys. 29:3, pp. 481-489. 4. Rodrigues, W.A., & Lu, J-Y., 1997, On the existence of undistorted progressive waves, Found. Phys. 27:3, pp. 435-508. 5. Ignatovich, V.K., 1978, Nonspreading wave packets in quantum mechanics, Found. Phys. 8:7/8, pp. 565571. 6. Shaarawi, A.M. & Ziolkowski, R.W., 1990, A Novel approach to the synthesis of nondispersive wave packet solutions to the Klein-Gordon and Dirac equations, J. Math. Physics, 3:10, pp. 2511-2519. 7. Brittingham, J.N. 1983, Focus waves modes in inhomogenious Maxwell=s equations: Transverse electric mode, J Appl. Phys. 54:3, 1179-1189. 8. Courant, R. & Hilbert, D., 1962, Methods of Mathematical Physics, Vol. 2, Interscience :New York. 9. Gueret, Ph. and Vigier, J-P, 1982, de Broglie’s wave particle duality and the stochastic interpretation of quantum mechanics, Found. Phys. 12:12, pp. 1057-1083. 10. Vigier, J-P, 2000, Photon mass and Heaviside force, Phys. Let. A, 270, 221-231. 11. Einstein, A. 1905, Ann. Phys. 7,132. 12. Cormier-Delanoue, C., 1988, Sur l’emission d’une radiation electromagneticique par une charge electrique en movement rectiligne accelere, Annales de la Fondation Louis de Broglie, 13 :1, pp. 43-63. (Translation by JP Vigier, edited by R.L. Amoroso) 13. Aspect, A., 1978, Phys Rev D, 14,1944; Phys Rev Let, 1980, 47,480. 14. Anderson, J.D., Lain, P.H., Lau, E.L., Liu, A.S., Nieto, M.M., & Turyshev, S.G. 1988, Phys. Rev. Let. 81,2858. 15. Vigier, J-P, 1997, Phys Let A, 234,75. 16. Vigier, J-P, 1993, New Hydrogen (Deuterium) Bohr orbits in quantum chemistry and cold fusion processes. Proc. ICC 54 Hawaii. 17. Bohm, D. 1963, Quantum Theory, Prentice-Hall: Englewood Cliffs. 18. Schiff, L.I., 1987, Quantum Mechanics, London : McGraw-Hill. 19. Young, T., 1803, Philosophical Transactions. 20. Cowley, J.M., 1975, Diffraction Physics, Amsterdam : North Holland. 21. Halliday, D., & Resnick, R., 1963, Physics for Students of Science & Engineering, New York: Wiley & Sons. 22. Amoroso, R.L., Kafatos, M. & Ecimovic, P., The origin of cosmological redshift in spin exchange vacuum compactification and nonzero rest mass photon anisotropy, in G.Hunter, S. Jeffers & J-P Vigier (eds.) Causality and Locality in Modern Physics, Dordrecht: Kluwer. 23. Jackson, J.D., 1999, Classical Electrodynamics, New York : Wiley & Sons. 24. De Broglie, L. 1925, Ann. Phys. 3,22. 25. Kaufmann, W.J., 1988, Universe, New York : W.H. Freeman. 26. Hunter, G, & Wadlinger, R.L.P., 1989, Physics Essays, Vol. 2. 27. Whitney, C.K., 1998, The mass-connected photon, in G.Hunter, S. Jeffers & J-P Vigier (eds.) Causality and Locality in Modern Physics, Dordrecht: Kluwer. 28. Nadeau, R. & Kafatos, M., 1999, The Non-Local Universe, London: Oxford. 29. Amoroso, R.L. 2001, The continuous state universe, in R.L. Amoroso, G. Hunter, M. Kafatos, & J-P Vigier (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale Kluwer Academic Publishers. 30. Vigier, J-P, 1983, Dirac’s ether in relativistic quantum mechanics, Foundations of Phys. 13:2, 253-285. 31. Beil, R.G. 1993, Found. Phys. 23, 1587. 32. Beil, R.G. 1995, Found. Phys. 25, 717. 33. Beil, R.G. 1997, pp. 9-16, in The Present Status of the Quantum Theory of Light, S. Jeffers et al, (eds.) Dordrecht: Kluwer Academic.

PHOTON DIAMETER MEASUREMENTS

GEOFFREY HUNTER*, MARIAN KOWALSKI REZA MANI, ROBERT L.P.WADLINGER** 0 Centre for Research in Earth and Space Science York University, Toronto, Canada M3J 1 P3. FRITZ ENGLER Itibar Industries (Division of Radareseareb Inc.) 3650 Weston Road, Weston, Ontario, Canada MDL 1 W2. TIM RICHARDSON Bio-Microtech Inc., 670 Hardwick Road, Unit 4 P.O.Box 23, Bolton, Ontario, Canada L7E 5T1.

Abstract. The 1985-89 Hunter-Wadlinger electromagnetic theory of the photon predicted that the photon is a soliton-wave with the shape and size of a circular ellipsoid of length with a diameter of

the wavelength),

This prediction is being tested by three diameter measurements: 1) those

carried out in 1985-86 with microwaves, 2) in progress measurements on 10 micron photons from a laser, and 3) an imminent experiment with monochromatic visible light (400-800 nm).

1. Introduction In 1985-86 we measured the transmission of X & K-band microwaves through circular and rectangular apertures. The transmitted intensity plotted as a function of window area (slit width) was linear with the slit-width intercept at zero transmission as the photon diameter. The measurements confirmed the theoretically predicted photon diameter of with an 1 experimental error of 0.5%. In a 1999-2001 experiment with photons the

*Corresponding author, email: gbunter~yorku.ca **Deceased 1991 1 Note that

differs from

by 5% - 10 times the experimental error.

157 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 157-166. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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radiation impinges upon gratings of slits (from

to

with slit-spacing

proportional to slit width so that all gratings have the same window area. Thus transmitted intensity is expected to be constant with a sharp cut-off for slit-widths less than In July 2000 T. Richardson realized the possibility of photon diameter measurements with visible light by illuminating a Bio-Microtech microscope “test slide" with variable wavelength light from a monochromator to observe the wavelength at which slits on the slide no longer transmit light. The slide has several gratings with slit-widths from 100 nrn to 500 nm. This yield s a direct measurement of the diameter of the photons that will just pass through the slits of each grating.

2. Motivation for the Experiments Our theory of a freely propagating photon [1, 2] yielded a soliton (limited by the principle of causality) with the photon's electromagnetic field contained within a circular ellipsoid of length and cross-sectional circumference both equal to the wavelength the ellipsoid's long axis being the axis of propagation. That the photon is about one wavelength long is supported by laser pulses less than two wavelengths long [3]. The soliton model predicts that a beam of monochromatic, circularly polarized photons will readily pass through apertures whose smallest linear dimension is greater than the soliton's diameter of For smaller apertures one would expect strong attenuation/diffraction based upon the simple mechanical notion that the soliton cannot pass undeflected through an aperture that is smaller than its own diameter of For larger, uniformly illuminated apertures one would expect the transmitted power to be proportional to the difference between the area of the aperture and the soliton's cross-sectional area. In 1985 our literature search covering electromagnetic radiation from to radio waves failed to reveal definitive measurements to confirm or deny the above hypothesis. However, microwaves having wavelengths of 1-10 cm seemed to be most amenable to laboratory scale experiments, and several reported measurements with microwaves broadly supported the hypothesis. Most of these measurements were of electric field intensity measurements in the vicinity of apertures, designed to investigate diffraction patterns. The post-war work of Andrews is extensive [4]. Diffraction measurements were made on slits by Hadlock [5], on circular holes by Robinson N] and on a variety of other shapes by Buchsbaum et al. [7]. Theoretical work by Meixner and Andrejewski [8] based upon plane waves (reviewed in English by King and Wu [9], also predicted strong attenuation of the transmission coefficient for slits and circular apertures smaller than Despite this supporting evidence, no direct measurements of the transmission coefficient could be found, and so we decided to do them ourselves.

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3. Summary of the Experiments

In all of the experiments a beam of monochromatic photons impinges upon one or more holes or slits in an otherwise opaque screen; the measurements consist of observing those photons which are transmitted through the screen undeflected. Thus the observed photons are those that have not interacted with the screen, which means that we are truly measuring a property of freely propagating photons. The principle of the original experiments was to allow a beam of circularly polarized microwaves to impinge upon a circular or rectangular aperture in a metal screen, and then measure the transmitted power on the far side of the screen. The microwave generator, the centre of the aperture, and the receiver were coaxial, and the screen was perpendicular to this axis. For a uniform flux of infinitesimal particles, one would expect the transmitted power to be proportional to the area of the aperture. For finite-sized particles the effective width of the aperture is the difference between its actual width and the particle diameter; extrapolation of measurements of the transmitted power to zero as a function of the aperture width will yield the particle diameter. There were, however, a number of experimental exigencies that had to be taken into account; the requisite technique for an accurate measurement evolved through a series of experiments carried out over a 6-month period. In the light of our experience with these experiments, the relevant experimental exigencies are as follows: The transmitted power decreases towards zero as the aperture size approaches the critical size discriminating low power transmitted radiation from instrumental noise is error prone. The fraction of transmitted radiation that is diffracted (i.e. bent through an angle as it passes through the aperture) increases as the aperture size is reduced towards the critical size of In terms of the theory diffraction occurs when a soliton impinges upon the wall of the aperture, the angle of bending being a function of the impact parameter of the collision. The proportion of incident photons that collide with the aperture walls will increase towards 100% as the aperture size approaches the critical size, For the purpose of the experiment we want to detect only the non- diffracted light; i.e. the solitons that pass through the aperture without colliding with its walls. Separating diffracted from non- diffracted light requires an appropriate experimental arrangement. Currents induced in the nominally opaque screen may cause some radiation to appear on the far side of the screen. One must be careful to separate this re-radiation through the screen from the radiation that passes through the aperture. While the relative intensity of re-radiation tends to be small, it can be significant if the detector is close to the aperture.

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G. HUNTER

The extrapolation to zero transmitted power assumes monochromatic radiation. Any higher frequency radiation in the incident beam (harmonics in the microwave experiments) will dominate the transmitted power for aperture sizes close to the critical size, This must be considered in analyzing the measurements.

4. 1985-86 Experimental Results The experiments were carried out in the research laboratory of Tribar Industries over a 6-month period from October 1985 to April 19862. As in most experimental studies, experience gained in the early experiments led to an improved arrangement designed to improve the accuracy of the measurements in the later experiments. The microwave generators were standard production models based upon a turnstile junction and a horn antenna; they produced a beam of circularly polarized microwaves. The transmitted power was measured initially with a receiving antenna coupled to an amplifier and millivolt meter, and later with a Hewlett-Packard power meter.

The first set of measurements were made with circular apertures cut in a thin aluminum screen with compasses; we aimed to control the diameter to within 0.1 mm. The measurements with X-band radiation (10.525 GHz., are shown in Table 1. The non-linearity for the large holes may be due to non-uniformity of the beam intensity over the area of the hole; this was also observed in some of the later experiments. The useful data (3 points) is minimal for calculating statistics. This first experiment is recorded here to show that even a very simple set-up yields the result that there is practically no transmission through holes smaller than

2 Previous attempts (since 1986 to publish these experimental results were thwarted by referee resistance to the intepretation of the experimental measurements.

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In experiments recorded in Tables 2 and 3 many more holes were used to improve the statistics, and the diameters of the drilled holes were accurately measured to the

the nearest thousandth of an inch. A disadvantage of this technique is that the drilling necessitated using thicker aluminum than in the first experiment; diffraction (scattering) from the internal walls of the hole shouW increase with screen thickness. IdeaHy one would use an infinitesimally thin screen. The calibrated Hewlett- Packard power meter was the dectector in these experiments. Table 2 records the results of the X-band radiation experiment, arid Table 3 the similar experiment with K-band radiation Although the results were satisfactory the errors (computed from the least squares straight line) were quite large (5% and 8% respectively), and hence it was felt that the accuracy could be improved by working with a slit, because the aperture area is proportional to the slit width, whereas for a circular aperture it is proportional to the square of the diameter. The last experiment (Table 4) employed a slit instead of a circular hole; the slit width was varied by moving the aluminum plates that formed its edges. After experience with previous slit experiments the exit side of the slit was shielded with absorbing foam to prevent diffracted radiation from entering the receiving antenna. After correcting for harmonic content of the beam a satisfactory result was

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G. HUNTER

obtained; the theoretical photon diameter of was confirmed within t experimental error of 0.5%. It is note-worthy that this result discriminates between effective diameters of and for differs by 5% from - ten times the experimental error of 0.5%.

The experiments confirm the theoretical prediction [1, 2] that in transmission of light (experimentally microwaves) through apertures in a coil- duct mg screen, the light behaves as would a stream of particles (photonsolitons) of diameter This is experimental confirmation of our photon-soliton model [1, 2], in which the photon's electromagnetic field is contained within a circular ellipsoid whose length and cross-sectional circumference are both equal to the wavelength The precision ofthese experiments (0.5%) is not sufficient to discriminate between our photon model and the classical, continuous field theory of Meixner and Andrejewski [8]. However, the classical prediction of strong attenuation below about results from an elaborate algebraic and numerical analysis; the classical theory fails to provide a simple explanation for the phenomenon. In contrast, the finite-photon-soliton model provides a very

PHOTON DIAMETER

163

simple explanation for a phenomenon that is now commonly witnessed whenever a person looks through a metal screen into a microwave oven to see food cooking; the visible photons readily pass through the 2 mm holes in the screen to allow the observer to see inside, while the microwave photons do not.

5. The Photon's Intrinsic Intensity

The theory [1, 2] predicts that the intrinsic intensity of the photon,

is given by:

and at beam intensities substantially (say by an order of magnitude) lower than

the beam

is believed to consist of separately propagating spheroidal solitons; this accords with and quantifies Einstein's original concept of a beam of light as consisting of spatially separated photons [11]. At beam intensities higher than the spheroidal solitons necessarily overlap; this occurs in focused laser beams and is the essential physical condition required for multiphoton absorption to take place.

164

G. HUNTER The overlapping also explains (in view of the

dependence of

in (1)) why

long wavelength (radio) radiation behaves as a classical electromagnetic field rather than a beam of separately propagating photons; e.g. for which is considerably smaller than typical operating intensities of radio waves.

For X-band

and K-band

photons

has the

values and respectively. These intrinsic photon intensities are just about the same as the working beam intensity of about employed in the experiments recorded in Tables 2 and 3, the implication being that the beams consisted of closely bunched photon-solitons with some overlapping of them.

6. The

Experiment

In view of the above analysis it is desirable to repeat the experiments with shorter wavelength radiation. However, the wavelength must be long enough for fabrication of holes or slits as small as (somewhat smaller in practice). These considerations led to the prospect of repeating the photon diameter measurements with the radiation from a carbon dioxide laser. In continuous wave (CW) mode this laser is highly monochromatic, and beams of lower intensity than are readily produced and measured. Furthermore the fabrication techniques employed in the microelectronics industry can fabricate holes/slits in a layer of metal at about the 1 micrometer level, thus satisfying the theoretical expectation of needing a range of slits from somewhat smaller than to about 10 times this value. Thus the availability of the with

CW laser and VLSI fabrication technology makes the experiment

jim radiation feasible at a working beam intensity less than the intrinsic

photon intensity of An improvement to having a single hole or a single slit was suggested by Dr. Tyler Ivanco [12]. His idea was to construct a regular array of holes or slits in a screen that is larger than the beam diameter. The window (non-metal) area of the array would be say 10-50% of its overall area. The array would be drawn using VLSI design software, and a set of screens with differently sized holes/slits would be produced by (photographic) reduction of the original array by different scale factors. Thus all of the different screens would have the same ratio of window area to beam area, and hence if photons were infinitesimal one would expect all the screens to transmit the same radiation intensity. The different screens constructed in this way would all have the same effective

PHOTON DIAMETER

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window area (10-50% of the cross-sectional area of the beam); they would only differ in the hole diameter (or slit width). Thus all the screens having holes substantially larger in diameter than should transmit the same beam intensity. At smaller hole diameters one would expect a sharp attenuation in transmission as the hole diameter drops below even though the window area is unchanged. For the above reasons we anticipate that this experiment should yield a measurement of the effective diameter of photons that is more accurate than the 0.5% of our March, 1986 experiment. It may even discriminate between the finite photon-soliton model [1, 2] and the classical, algebraic theory [8]. Four gratings (with slit widths of 2.5, 3.2, 3.8 & and slitspacing =10 x slit-width) were manufactured early in the year 2000 by the Canadian Microelectronics Corporation3 as 3mm x 3mm silicon chips with the required pattern of slits in one of the metal layers of a VLSI circuit. Attempts to observe the transmission of laser radiation through these gratings were carried out in the laboratory of Reza Mani during the summer of 2000, but after much experimentation we realized that a layer of titanium oxide covering the whole area of the chip was preventing any transmission. Negotiations with CMC for the manufacture of chips having only the designed slotted metal layer on the silicon wafer base broke down because it would entail non-standard processing in Mitel's silicon foundry, and because CMC's mandate is to facilitate the manufacture of VLSI circuits for electrical engineering researchers at Canadian universities.

7. Visible Photon Measurements

These measurements were conceived by T. Richardson in July 2000; the Bio-Microtech microscope, test slide, and the monochromator and adapters required for the experiment were assembled in August 2000 with the intent ion of doing the measurements on August 28, 2000. However, due to an email miscommunication this plan was not realized and the experiment is pending - dependent upon an available time-slot in the commercially active laboratories of Bio Microtech; our hope is that this will occur in December 2000 or January 2001. The experiment will be conducted by T. Richardson and G. Hunter. White light will be input to a monochromator having a wavelengthca]ibrated output slit. The monochromatic light will be directed into the light-source condenser of the high-performance microscope, where it will illuminate the test-slide containing a pattern of 5 gratings (metal lines on silica). The gratings ranged from 100 nrn wide lines on 200 nrn centres, to 500 nm wide lines on 1000 nm centres. The wavelength of the monochromatic light will be increased to determine the wavelength at which a specific grating no longer transmits any light; i.e. appears to be black -- the lines disappear; a photo-electric camera will be used to allow measurements in the ultra-violet arid infra-red as well as at visible

3 CMC - an agency of the Canadian Natural Sciences and Engineering Research Council located at Queens' University, Kingston, Ontario, Canada

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G. HUNTER

wavelengths. The effective diameter of the photons of known wavelength will be equated with the slit-width of the dark grating. The expectation is these measurements will also yield an effective photon-diameter of the theoretical prediction for circularly polarized radiation. The experimental measurements will also investigate whether the effective diameter depends upon the polarization (linear or circular) of the radiation.

References [1] G. Hunter and R.L.P. Wadlinger, Physical Photons: Theory, Experiment, Interpretation, in: Quantum Uncertainties: Recent and Future Experiments and Interpretations: Proceedings of tile NATO Workshop, University of Bridgeport, Connecticut, USA 1986, NATO ASI Series B, Vol.162 (Plenum Press, 1987). [2] G. Hunter and R.L.R Wadlinger, Physics EssQvs, Vol. 2(1989) 158-172. [3]C. Spielman, C. Ran, N.H, Burnett, T. Brabec, NI. Geissler, A. Scrinzi, M Scinjirer, and F. Krausz, IEEE J.Selected Topics in Quantum Electronics, Vol.4 (1998) 249-264. [4]C.L. Andrews, Optics of the Electromagnetic Spectrum (Prentice-Hall, Englewood Cliffs, N.J., 1960) p.328. [5] R.K. Hadlock, J.APPI.PIkvs. Vol.629 (1958) 918. [6] H.L. Robinson, J.Appl.Phys. Vol.624 (1953) 35. [7]S.J. Buchsbauni, A.R. Mime, D.C. Hogg, 0. Bekefi, and C.A. \Voonton, J.Appl.Phys. Vol.626 (1955) 706. [8] J. Meixner and W. Andrejewski, Ann. Physik, 7,157 (1950). [9]R.W.P. King and T.T. Wu, The Scattering and Diffraction of Waves (Harvard University Press, Cambridge, Massachusetts, U.S.A. 1959). [10] J.D. Kraus, Antennas (McGraw-jim, New York, 1950) p.178. [11] S. Diner, D. Irague, 0. Lochalt and F. Selleri (Editors), The Wave-Particle Dualism (D.Reidel Company, Dordrecht, Holland, 1984). [12] T.A. rvanco, Ph.D. Dissertation, York University, Toronto, Canada, 1987.

WHAT IS THE EVANS-VIGIER FIELD?

VALERI V. DVOEGLAZOV Escuela de Física, Universidad Autónoma de Zacatecas Apartado Postal C-580, Zacatecas 98068, Zac., México E-mail: [email protected] URL: http://ahobon.reduaz.mx/ ~ valeri/valeri.htm

Abstract. We explain connections of the Evans-Vigier model with theories proposed previously. The Comay’s criticism is proved to be irrelevant.

The content of the present talk is the following: Evans-Vigier definitions of the field [1]; Lorentz transformation properties of the field and the B-Cyclic Theorem [2, 3]; Clarifications of the Hayashi and Kalb-Ramond papers [4, 5, 6]; Connections between various formulations of massive/massless J = 1 field; Conclusions of relativistic covariance and relevance of the Evans-Vigier postulates. In 1994-2000 I presented a set of papers [7] devoted to clarifications of the Weinberg (and Weinberg-like [8, 9]) theories and the concept of notoph. In 1995-96 I received numerous e-mail communications from Dr. M. Evans, who promoted a new concept of the longitudinal phaseless magnetic field associated with plane waves, the field (which is later obtained the name of M. Evans and J.-P. Vigier). Reasons for continuing the discussion during 2-3 years were: 1) the problem of massless limits of all relativistic equations does indeed exist; 2) the dynamical Maxwell equations have indeed additional solutions with energy (apart of those 167 R.L Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 167-182. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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VALERI V. DVOEGLAZOV

with see [10, 11, 12, 13, 14];1 3) the concept met strong non-positive criticism (e. g., ref [15, 16, 17]) and the situation became even more controversial in the last years (partially, due to the Evans’ illness). What are misunderstandings of both the authors of the model and their critics? In Enigmatic Photon (1994), ref. [1], the following definitions of the longitudinal Evans-Vigier field have been given:2 Definition 1. [p.3,formula (4a)]

Definition 2. [p.6,formula (12)]

and Definition 3. [p. 16,formula (41)]

The following notation was used: is the wave number; is the phase; and are usual transverse modes of the magnetic field; and are usual transverse modes of the vector potential. The main experimental prediction of Evans [1a,b] that the magnetization induced during light-matter interaction (for instance, in the IFE)

has not been confirmed by the North Caroline group [18]. As one can see from Figure 4 of [18] “the behaviour of the experimental curve does not match with Evans calculations”. Nevertheless, let us try to deepen understanding of the theoretical content of the Evans-Vigier model. In their papers and books [1] Evans and 1

If we put energy to be equal to zero in the dynamical Maxwell equations

we come to and i. e. to the conditions of longitudinality. The method of deriving this conclusion has been given in [19]. 2 I apologize for not citing all numerous papers of Evans et al and papers of their critics due to page restrictions on the papers of this volume.

THE EVANS VIGIER FIELD

169

Vigier used the following definition for the transverse antisymmetric tensor field:

If this formula describes the right-polarized radiation. Of course, a similar formula can be written for the left-polarized radiation. These transverse solutions can been re-written to the real fields. For instance, Comay presented them in the following way [16c] in the reference frame

and analized the addition of to (9). Making boost to other frame of reference he claimed that a) is not parallel to the Poynting vector; b) with the Evans postulates has a real part; c) transverse fields change, whereas is left unchanged when the boost is done to the frame moving in the direction. Comay concludes that these observations disprove the Evans claims on these particular questions. Furthermore, he claimed that the model is inconsistent with the Relativity Theory. According to [20, Eq.(11.149)] the Lorentz transformation rules for electric and magnetic fields are the following:

where

with

being

the parameter of the Lorentz boost. We shall further use the natural unit system After introducing the spin matrices and deriving relevant relations:

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VALERI V. DVOEGLAZOV

one can rewrite Eqs. (10,11) to the form

Pure Lorentz transformations (without inversions) do not change signs of the phase of the field functions, so we should consider separately properties of the set of and which can be regarded as the negative-energy solutions in QFT and of another set of and the positive-energy solutions. Thus, in this framework one can deduce from Eqs. (12,13)

and

(when the definitions (7) are used). To find the transformed 3-vector is just an algebraic exercise. Here it is

THE EVANS VIGIER FIELD

171

We know that the longitudinal mode in the Evans-Vigier theory can be considered as obtained from Definition 3. Thus, considering that transforms as zero-component of a four-vector and as space components of a four-vector: [20, Eq.(11.19)]

we find from (22) that the relation between transverse and longitudinal modes preserves its form:

that may be considered as a proof of the relativistic covariance of the B(3) model. Moreover, we used that the phase factors in the formula (7) are fixed between the vector and axial-vector parts of the antisymemtric tensor field for both positive- and negative- frequency solutions if one wants to have pure real fields. Namely, and As we have just seen the field in this case may be regarded as a part of a 4-vector with respect to the pure Lorentz transformations. We are now going to take off the abovementioned requirement and to consider the general case:

Our formula (26) can be re-written to the formulas generalizing (6a) and (6b) of ref. [2] (see also above (18,19)):

One can then repeat the procedure of ref. [2] (see the short presentation above) and find out that the field may have various transformation laws when the transverse fields transform with the matrix which can be extracted from (12,13). Since the Evans-Vigier field is defined by the

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VALERI V. DVOEGLAZOV

formula (3) we again search the transformation law for the cross product of the transverse modes with taking into account (27,28).

We used again the Definition 3 that One can see that we recover the formula (8) of ref. [2] (see (22) above) when the phase factors are equal to In the case and the sign of is changed to the opposite one.3 But, we are able to obtain the transformation law as for antisymmetric tensor field, for instance when Namely, since under this choice of the phases

the formula (30) and the formula for opposite choice of phases lead precisely to the transformation laws of the antisymmetric tensor fields:

B(0) is a true scalar in such a case. What are reasons that we introduced additional phase factors in Helmoltz bivectors? In [21] a similar problem has been considered in the (1/2, 0) (0, 1/2) (cf. also [7, 22]). Ahluwalia identified additional phase factor (s) with Higgs-like fields and proposed some relations with a gravitational potential. However, the E field under definitions becomes to be pure imaginary. One can also propose a model with the corresponding introduction of phase factors in such a way that 3

By the way, in all his papers Evans used the choice of phase factors incompatible with the B-Cyclic Theorem in the sense that not all the components are entries of antisymmetric tensor fields therein. This is the main one but not the sole error of the Evans papers and books. 4 In the case and the sign in the third term in parentheses (formula (30) is changed to the opposite one.

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173

to be pure imaginary. Can these transverse fields be observable? Can the phase factors be observable? A question of experimental possibility of detection of this class of antisymmetric tensor fields (in fact, of the antihermitian modes on using the terminology of quantum optics) is still open. One should still note that several authors discussed recently unusual configurations of electromagnetic fields [23, 24]. Let us now look for relations with old formalisms. The equations (10) of [4] is read

for antisymmetric tensor expressed through cross product of polarization vectors in the momentum space. This is a generalized case comparing with the Evans-Vigier Definition 2 which is obtained if one restricts oneself by space indices. The dynamical equations in the approach are

and the new Kalb-Ramond gauge invariance is defined with respect to transformations

It was proven that the equations are related to the Weinberg formalism [25, 26] and [7b-f,i]. Furthermore, they [4] also claimed “In the limit the helicity becomes a relativistic invariant, and the concept of spin loses its meaning. The system of states is no longer irreducible: it decomposes and describes a set of different particles with zero mass and helicities (for integer spin and if parity is conserved; the situation is analogous for half-integer spins)5.” In fact, this hints that actually the Proca-Duffin-Kemmer theory has two massless limit, a) the wellknown Maxwell theory and b) the notoph theory The notoph theory has been further developed by Hayashi [5] in the context of dilaton gravity, by Kalb and Ramond [6] in the string context. Hundreds (if not thousands) papers exist on the so-called Kalb-Ramond field (which is actually the notoph), including some speculations on its connection with Yang-Mills fields. 5

Cf. with [27]. I am grateful to an anonymous referee of Physics Essays who suggested to look for possible connections. However, the work [27] does not cite the previous statement.

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In [28] I tried to use the (32)) to construct the “potentials”

definitions of (see We can obtained for a massive field

This tensor coincides with the longitudinal components of the antisymmetric tensor obtained in refs. [9a,Eqs.(2.14,2.17)] (see also below and [7i, Eqs.(16b,17b)]) within normalizations and different forms of the spin basis. The longitudinal states reduce to zero in the massless case under appropriate choice of the normalization and only if a particle moves along with the third axis OZ.6 Finally, it is also useful to compare Eq. (35) with the formula (B2) in ref. [29] in order to realize the correct procedure for taking the massless limit. Thus, the results (at least in a mathematical sense) surprisingly depend on a) the normalization; b) the choice of the frame of reference. In the Lagrangian approach we have

and

where (if one applies the duality relations). Thus, we observe that a) it is important to consider the parity matters (the dual tensor has different parity properties); b) we may look for connections with the dual electrodynamics [30]. The above surprising conclusions induced me to start form the basic group-theoretical postulates in order to understand the origins of the results. The set of Bargmann-Wigner equations, ref [31] for is written, e.g., ref. [32]

6 There is also another way of thinking: namely, to consider “unappropriate” normalization N = 1 and to remove divergent part by a new gauge transformation.

THE EVANS VIGIER FIELD

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where one usually uses

In order to facilitate an analysis of parity properties of the corresponding fields one should introduce also the term In order to understand normalization matters one should put arbitrary (dimensional, in general) coefficients in this expansion or in definitions of the fields and 4-potentials [28]. The matrix is

Matrices are chosen in the Weyl representation, i.e., be diagonal. The reflection operator has the properties

is assumed to

They are necessary for the expansion (41) to be possible in such a form, i.e., in order the and (if considered) to be symmetrical matrices. I used the expansion which is similar to (41)

and obtained

If one renormalizes or one obtains “textbooks” Proca equations. But, physical contents of the massless limits of these equations may be different. Let us track origins of this conclusion in detail. If one advocates the following definitions [33, p.209]

176

and

VALERI V. DVOEGLAZOV

ref. [33, p.68] or ref. [34, p. 108],

for the field operator of the 4-vector potential, ref. [34, p.109] or ref. [35, p.129]7,8

the normalization of the wave functions in the momentum representation is thus chosen to the unit, We observe that in the massless limit all defined polarization vectors of the momentum space do not have good behaviour; the functions describing spin-1 particles tend to infinity. This is not satisfactory, in my opinion, even though one can still claim that singularities may be removed by rotation and/or choice of a gauge parameter. After renormalizing the potentials, e. g., we come to the field functions in the momentum representation:

7

Remember that the invariant integral measure over the Minkowski space for physical particles is

Therefore, we use the field operator as in (54). The coefficient can be considered at this stage as chosen for convenience. In ref. [33] the factor was absorbed in creation/annihilation operators and instead of the field operator (54) the operator was used in which the functions for a massive spin-1 particle were substituted by which may lead to confusions in searching massless limits for classical polarization vectors. 8 In general, it may be useful to consider front-form helicities (or “time-like” polarizations) too. But, we leave a presentation of a rigorous theory of this type for subsequent publications. 9 The metric used in this paper is different from that of ref. [33].

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(N = m and which do not diverge in the massless limit. Two of the massless functions (with are equal to zero when the particle, described by this field, is moving along the third axis The third one is

and at the rest also vanishes. Thus, such a field operator describes the “longitudinal photons” which is in complete accordance with the Weinberg theorem for massless particles (let us remind that we use the D(l/2, 1/2) representation). Thus, the change of the normalization can lead to the change of physical content described by the classical field (at least, comparing with the well-accepted one). Of course, in the quantum case one should somehow fix the form of commutation relations by some physical principles.1 If one uses the dynamical relations on the basis of the consideration of polarization vectors one can find fields:

and

1

I am very grateful to the anonymous referee of my previous papers (“Foundation of Physics”) who suggested to fix them by requirements of the dimensionless nature of the action (apart from the requirements of the translational and rotational invariancies).

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VALERI V. DVOEGLAZOV

where we denoted, as previously, a normalization factor appearing in the definitions of the potentials (and/or in the definitions of the physical fields through potentials) as N . E(p, 0) and B(p, 0) coincide with the strengths obtained before by different method [9a,28], see also (35). identically. So, we again see a third component of antisymmetric tensor fields in the massless limit which is dependent on the normalization and rotation of the frame of reference. However, the claim of the pure “longitudinal nature” of the antisymmetric tensor field and/or “Kalb-Ramond” fields after quantization still requires further explanations. As one can see in [5] for a theory with the application of the condition (in our notation see the formula (18a) therein, leads to the above conclusion. Transverse modes are eliminated by a new “gauge” transformations. Indeed, the expanded lagrangian is

Thus, the Lagrangian is equivalent to the Weinberg’s Lagrangian of the theory [36] and [7a-e],2 which is constructed as a generalization of the Dirac Lagrangian for spin 1 (instead of bispinors it contains bivectors). In order to consider a massive theory (we insist on making the massless limit in the end of calculations, for physical quantities) one should add as in (37). The spin operator of the massive theory, which can be found on the basis of the Nöther formalism, is

2

The formal difference in Lagrangians does not lead to physical difference. Hayashi said that this is due to the possiblity of applying the Fermi method mutatis mutandis.

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In the above equations we applied dynamical equations as usual. Thus, it becomes obvious, why previous authors claimed the pure longitudinal nature of massless antisymmetric tensor field after quantization, and why the application of the generalized Lorentz condition leads to equating the spin operator to zero.3 But, one should take into account the normalization issues. An additional mass factor in the denominator may appear a) after “re-normalization” (if we want to describe long-range forces an antisymmetric tensor field must have dimension [energy]2 in the unit system, and potentials, [energy] 1 in order the corresponding action would be dimensionless; b) due to appropriate change of the commutation relations for creation/annihilation operators of the higher-spin fields (including c) due to divergent terms in E, B, A in under certain choice of N . Thus, one can recover usual quantum electrodynamics even if we use fields (not potentials) as dynamical variables. The conclusions are: While first experimental verifications gave negative results, the construct is theoretically possible, if one develops it in a mathematically correct way; The model is a relativistic covariant model. It is compatible with the Relativity Theory. The field may be a part of the 4-potential vector, or (if we change connections between parts of Helmoltz bivector) may be even a part of antisymmetric tensor field; The model is based on definitions which are particular cases of the previous considerations of and Polubarinov, Hayashi and Kalb and Ramond; The Duffin-Kemmer-Proca theory has two massless limits that seems to be in contradictions with the Weinberg theorem Antisymmetric tensor fields after quantization may describe particles of both helicity and in the massless limit. Surprisingly, the physical content depends on the normalization issues and on the choice of the frame of reference (in fact, on rotations). Acknowledgments. I am thankful to Profs. A. Chubykalo, E. Comay, L. Crowell, G. Hunter, Y. S. Kim, organizers and participants of the Vigier2K, referees and editors of various journals for valuable discussions. I acknowledge many internet communications of Dr. M. Evans (1995-96) on the concept of the field, while frequently do not agree with him in many particular questions. I acknowledge discussions (1993-98) with Dr. D. Ahluwalia (even though I do not accept his methods in science). 3

It is still interesting to note that division of total angular momentum into orbital part and spin part is not gauge invariant.

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VALERI V. DVOEGLAZOV

I am grateful to Zacatecas University for a professorship. This work has been supported in part by the Mexican Sistema Nacional de Investigadores and the Programa de Apoyo a la Carrera Docente. References 1.

Evans M. W. (1992) Physica B182, 227; ibid. 237; (1993) Modern Non-linear Optics. [Series Adv. Chem. Phys. Vol. 85(2)], Wiley Interscience, NY; (1994-1999) in Evans M. W., Hunter G., Jeffers S., Roy S. and Vigier J.-P. (eds.), The Enigmatic Photon. Vols. 1-5, Kluwer Academic Publishers, Dordrecht. 2. Dvoeglazov V. V. (1997) Found. Phys. Lett. 10, 383. This paper presents itself a comment on the debates between E. Comay and M. Evans and it criticizes both authors. 3. Dvoeglazov V. V. (2000) Found.Phys. Lett. 13, 387. 4. V. I. and Polubarinov I. V. (1966) Yad. Fiz. 4, 216 [Translation: (1967) Sov. J. Nucl. Phys. 4, 156]. 5. Hayashi K. (1973) Phys. Lett. B44, 497. 6. Kalb M. and Ramond P. (1974) Phys. Rev. D9, 2273. 7. Dvoeglazov V. V. (1994) Rev. Mex. Fis. Suppl. 40, 352; (1997) Helv. Phys. Acta 70, 677; ibid. 686; ibid. 697; (1998) Ann. Fond. L. de Broglie 23, 116; (1998) Int. J. Theor. Phys. 37, 1915; (1999) ibid. 38, 2259; (1998) Electromagnetic Phenomena 1, 465; (2000) Czech. J. Phys. 50, 225; (2000) in V. V. Dvoeglazov (ed.), Photon:Old Problems in Light of New Ideas, Nova Science Publishers, Huntington. 8. Sankaranarayanan A. and Good R. H., jr. (1965) Nuovo Cim. 36, 1303; (1965) Phys. Rev. 140, B509; Sankaranarayanan A. (1965) Nuovo Cim. 38, 889. 9. Ahluwalia D. V. and Ernst D. J. (1993) Int. J. Mod. Phys. E2, 397; Ahluwalia D. V., Johnson M. B. and Goldman T. (1993) Phys. Lett. B316, 102. 10. Oppenheimer J. R. (1931) Phys. Rev. 38, 725. 11. Good R. H., Jr. (1957) Phys. Rev. 105, 1914 (see p. 1915); (1959) in Lectures in theoretical physics. University of Colorado. Boulder, Interscience, p. 30 (see p. 47); Nelson T. J. and Good R. H., Jr. (1969) Phys. Rev. 179, 1445 (see p. 1446). 12. Gianetto E. (1985) Lett. Nuovo Cim. 44, 140; (1985) ibid. 145. See pp. 142 and 147, respectively. E. Recami et al. were very close to re-discover this solution in the papers (1974) Lett. Nuovo Cim. 11, 568 and (1990) in M. Mijatovic (ed.), Hadronic mechanics and nonpotential interactions, Nova Science Pubs., New York, p. 231 but in the analysis of the determinant of the Maxwell equations they put additional constraint of transversality on the solutions of Eq. (1,2). 13. Ahluwalia D. V. and Ernst D. J. (1992) Mod. Phys. Lett. A7, 1967; Ahluwalia D. V. (1996) in G. Hunter et al. (eds.), Proceedings of The Present Status of Quantum Theory of Light: A Symposium to Honour Jean-Pierre Vigier. York University, Toronto, Aug. 27-30, 1995, Kluwer Academic Publishers, p. 443. 14. Dvoeglazov V. V., Tyukhtyaev Yu. N. and Khudyakov S. V. (1994) Izv. VUZ: fiz. 37, 110 [Translation: (1994) Russ. Phys. J. 37, 898]. This is a version of the Saratov University preprint of 1992, which was revised in order to include references to the works appeared in 1991-1994. 15. Barron L. D. (1993) Physica B190, 307; Lakhtakia A. (1993) Physica B191, 362; (1995) Found. Phys. Lett. 8, 183; Grimes D. M. (1993) Physica B191, 367. 16. Comay E. (1996) Chem. Phys. Lett. 261, 601; (1996) Physica B222, 150; (1997) Found. Phys. Lett. 10, 245; (1997) Physica A242, 522; (1999) Apeiron 6, 233. 17. Hunter G. (1999) Chem. Phys. Lett. 242, 331; (2000) Apeiron 7, 17. 18. Akhtar Raja M. Y., Sisk W. N., Yousaf M. and Allen D. (1995) Appl. Phys. Lett. 67, 2123; (1997) Appl. Phys. B64, 79. 19. Dvoeglazov V. V. et al (1997) Apeiron 4, 45.

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20. Jackson J. D. (1980) Electrodinámica Clásica. Spanish edition, Alhambra S. A. 21. Ahluwalia D. V. (1998) Mod. Phys. Lett. A13, 3123; see also V. V. Dvoeglazov’s comment on this paper, to be published. 22. Dvoeglazov V. V. (1997) in J. Keller and Z. Oziewicz (eds.), Advances in Clifford Algebras – Proceedings of the Int. Conference on the Theory of the Electron. Cuautitlan, Mexico, Sept. 27-29, 1995. Vol. 7C, UNAM, México, pp. 303-319. 23. In (1993) A. Lakhtakia (ed.), Essays on the Formal Aspects of Electromagnetic Theory, World Scientific, Singapore. 24. In (1995) T. W. Barrett and D. M. Grimes (eds.), Advanced Electromagnetism: Foundations, Theory and Applications, World Scientific, Singapore. 25. S. Weinberg, Phys. Rev. B133 (1964) 1318; ibid B134 (1964) 882; ibid 181 (1969) 1893. 26. Dvoeglazov V. V. (1997) Weinberg Formalism and New Looks at the Electromagnetic Theory, in J.-P. Vigier et al. (eds.), The Enigmatic Photon. Vol. IV, Kluwer Academic Publishers, Dordrecht, Chapter 12, and references therein. 27. Kirchbach M. (2000) Rarita-Schwinger Fields without Auxiliary Conditions in Baryon Spectra, in A. Chubykalo, V. Dvoeglazov et al. (eds.), Lorentz Group, CPT and Neutrinos. Proceedings of the International Workshop, World Scientific, Singapore, pp. 212-223. 28. Dvoeglazov V. V. (1998) Photon-Notoph Equations, physics/9804010, to be published. 29. Ahluwalia D. V. and Sawicki M. (1993) Phys. Rev. D47, 5161. 30. Strazhev V. I. and Kruglov S. I. (1977) Acta Phys. Polon. B8, 807; Strazhev V. I. (1978) ibid. 9, 449; (1977) Int. J. Theor. Phys. 16, 111. 31. Bargmann V. and Wigner E. P. (1948) Proc. Nat. Acad. Sci. (USA) 34, 211. 32. Lurié D. (1968) Particles and Fields, Interscience Publisher. New York, Chapter 1. 33. Weinberg S. (1995) The Quantum Theory of Fields. Vol. I. Foundations, Cambridge University Press, Cambridge. 34. Novozhilov Yu. V. (1975) Introduction to Elementary Particle Theory, Pergamon Press, Oxford. 35. Itzykson C. and Zuber J.-B. (1980) Quantum Field Theory, McGraw-Hill Book Co., New York. 36. Dvoeglazov V. V. (1993) Hadronic Journal 16, 459.

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NON-ABELIAN GAUGE GROUPS FOR REAL AND COMPLEX AMENDED MAXWELL’S EQUATIONS

E.A. RAUSCHER Tecnic Research Laboratory, 3500 S. Tomahawk Road, Bldg. 188, Apache Junction, AZ 85219, USA, Email: [email protected]

Abstract. We have analyzed, calculated and extended the modification of Maxwell’s equations in a complex Minkowski metric, in a space using the gauge, SL(2,c) and other gauge groups, such as for n>2 expanding the gauge theories of Weyl. This work yields additional predictions beyond the electroweak unification scheme. Some of these are: 1) modified gauge invariant conditions, 2) short range non-Abelian force terms and Abelian long range force terms in Maxwell’s equations, 3) finite but small rest of the photon, and 4) a magnetic monopole like term and 5) longitudinal as well as transverse magnetic and electromagnetic field components in a complex Minkowski metric in a space.

1. Introduction

We have developed an eight dimensional complex Minkowski space composed of four real dimensions and four imaginary dimensions which is consistent with Lorentz invariance and analytic continuation in the complex plane [1]. The unique feature of this geometry is that it admits of nonlocality consistent with Bell’s theorem, (EPR paradox), possibly Young’s double slit experiment, the Aharonov – Bohm effect and multi mirrored interferometric experiment. Additionally, expressing Maxwell’s electromagnetic equations in complex eight space, leads to some new and interesting predictions in physics, including possible detailed explanation of some of the previously mentioned nonlocality experiments [2]. Complexification of Maxwell’s equations require a non-Abelian gauge group which amend the usual theory, which utilizes the usual unimodular Weyl group. We have examined the modification of gauge conditions using higher symmetry groups such as and other groups such as the SL(2,c) double cover group of the rotational group SO(3,1) related to Shipov’s Ricci curvature tensor [3] and a possible neo-aether picture. Thus we are led to new and interesting physics involving extended metrical space constraints, the usual transverse and also longitudinal, non Hertzian electric and magnetic field solutions to Maxwell’s equations, possibly leading to new communication systems 183 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 183-188. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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E.A. RAUSCHER

and antennae theory, non zero solutions to and a possible finite but small rest mass of the photon. Comparison of our theoretical approach is made to the work of J. P. Vigier, [4] T.W. Barrett [5] and H.F. Harmuth’s [6] work on amended Maxwell’s theory. We compare our predictions such as our longitudinal field to the term of Vigier, and our NonAbelian gauge groups to that of Barrett and Harmuth. This author interprets this work as leading to new and interesting physics, including a possible reinterpretation of a neoaether with nonlocal information transmission properties. 2. Complexified EM Fields In The

Minkowski Space And Nonlocality

We expand the usual line element metric

in the following manner.

We consider a complex eight dimensional space, and likewise for

constructed so that

where the indices V and

run 1 to 4 yielding

Hence, we now have a new complex eight space metric as

(1, 1, 1, -1). We have

developed this space and other extended complex spaces (1) and examined their relationship with the twister algebras and asymptotic twister space and the spinor calculus and other implications of the theory [7]. The Penrose twister SU(2,2) or is constructed from four space – time,

where

is the real part of the space and

is the imaginary part of the space, this metric appears to be a fruitful area to explore. The twister Z can be a pair of spinors and which are said to represent the twister. The condition for these representations are 1) the null infinity condition for a zero spin field is conformal invariance and 3) independence of the origin. The twister is derived from the imaginary part of the spinor field. The underlying concept of twister theory is that of conformally invariance fields occupy a fundamental role in physics and may yield some new physics. Since the twister algebra falls naturally out complex space. Other researcher have examined complex dimensional Minkowski spaces. In reference [10], Newman demonstrates that M4 space do not generate any major “weird physics” or anomalous physics predictions and is consistent with an expanded or amended special and general relativity. In fact the Kerr metric falls naturally out of this formalism as demonstrated by Newman [11]. As we know twisters and spinors are related by the general Lorentz conditions in such a manner that all signals are luminal in the usual four N Minkowski space but this does not preclude super or trans luminal signals in spaces where N>4. H. Stapp, for example, has interpreted the Bell’s theorem experimental results in terms of trans luminal signals to address the nonlocality issue of the Clauser, et. al and Aspect experiments. C.N. Kozameh and E.T. Newman demonstrate the role of non local fields in complex eight space [16]. We believe that there are some very interesting properties of the space which include the nonlocality properties of the metric applicable in the non-Abiliass algebras

GAUGE GROUPS FOR MAXWELL’S EQUATIONS

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related to the quantum theory and the conformal invariance in relativity as well as new properties of Maxwell’s equations. In addition, complexification of Maxwell’s equations in space yields some interesting predictions, yet we find the usual conditions on the manifold hold [2,8]. Some of these new predictions come out of the complexification of four space 2 and appear to relate to the work of Vigier, Barrett, Harmuth and others [4, 5, 6]. Also we find that the twister algebra of the complex eight dimensional, space is mapable 1 to 1 with the twister algebra, space of the Kaluza-Klein five dimensional electromagnetic - gravitational metric [17, 18]. Some of the predictions of the complexified form of Maxwell’s equations are 1) a finite but small rest mass of the photon, 2) a possible magnetic monopole, 3) transverse as well as longitudinal B(3) like components of E and B, 4) new extended gauge invariance conditions to include non-Abelian algebras and 5) an inherent fundamental nonlocality property on the manifold. Vigier also explores longitudinal E and B components in detail and finite rest mass of the photon [19]. We consider both the electric and magnetic fields to be complexified as are real quantities. Then substitution of these two equations into the complex form of Maxwell’s equations above yields, upon separation of real and imaginary parts, two sets of Maxwell-like equations. The first set is

the second set is

The real part of the electric and magnetic fields yield the usual Maxwell’s equations and complex parts generate “mirror” equations; for example, the divergence of the real component of the magnetic field is zero, but the divergence of the imaginary part of the electric field is zero, and so forth. The structure of the real and imaginary parts of the fields is parallel with the electric real components being substituted by the imaginary part of the magnetic fields and the real part of the magnetic field being substituted by the imaginary part of the electric field. In the second set of equations, (2), the I’s, “go out” so that the quantities in the equations are real, hence and not zero, yielding a term that may be associated with some classes of monopole theories. See references in ref. [2].

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We express the charge density and current density as complex quantities based on the separation of Maxwell’s equations above. Then, in generalized form and where it may be possible to associate the imaginary complex charge with the magnetic monopole and conversely the electric current has an associated imaginary magnetic current. The alternate of defining and using, which Evans does and would not yield a description of the magnetic monopole in terms of complex quantities but would yield, for example

in the second set of equations.

for Using the invariance of the line element and for for the distance from an electron charge, we can write the relation,

3. New Gauge Conditions, Complex Minkowski For Physics

Space And New Implications

In a series of papers, Barrett, Harmuth and Rauscher have examined the modification of gauge conditions in modified or amended Maxwell theory. The Rauscher approach, as briefly explained in the preceding section is to write complexified Maxwell’s equation in consistent form to complex Minkowski space [2]. The T.W. Barrett amended Maxwell theory utilizes non-Abelian algebras and leads to some very interesting predictions which have interested me for some years. He utilizes the non commutive gauge symmetry rather than the symmetry. Although the Glashow electroweak theory utilizes and but in a different manner, but his theory does not lead to the interesting and unique predictions of the Barrett theory. T.W. Barrett, in his amended Maxwell theory, predicts that the velocity of the propagation of signals is not the velocity of light. He presents the magnetic monopole concept resulting from the amended Maxwell picture. His motive goes beyond standard Maxwell formalism and generate new physics utilizing a non-Abelian gauge theory.[5] The group gives us symmetry breaking to the group which can act to create a mass splitting symmetry that yield a photon of finite (but necessarily small) rest mass which may be created as self energy produced by the existence of the vacuum1. This finite rest mass photon can constitute a propagation signal carrier less than the velocity of light. We can construct the generators of the SU2 algebra in terms of the fields E, B, and A. The usual potentials,

is the important four vector quality

where the index runs 1 to 4. One of the major purposes of introducing the vector and scalar potentials and also to subscribe to their physicality is the desire by physicists to

GAUGE GROUPS FOR MAXWELL’S EQUATIONS avoid action at a distance. In fact in gauge theories

187

is all there is! Yet, it appears

that, in fact, these potentials yield a basis for a fundamental nonlocality! Let us address the specific case of the group and consider the elements of a non-Abelian algebra such as the fields with (or even symmetry then we have the commutation relations where or Which is reminiscent of the Heisenberg uncertainty principle non-Abelian gauge.2 Barrett does explain that fields can be transformed into fields by symmetry breaking. For the gauge amended Maxwell theory additional terms appear in term of operations such and A × B and their non Abelian converses. For example no longer equals zero but is given as

where

for the dot product of A and B and hence we have a magnetic monopole term and j is the current and g is a constant. Also Barrett gives references to the Dirac, Schwinger and G. t Hooft monopole work. Further commentary on the gauge conjecture of H.F. Mamuth [6] that under symmetry breaking, electric charge is considered but magnetic charges are not. Barrett further states that the symmetry breaking conditions chosen are to be determined by the physics of the problem. These non Abelian algebras have consistence to quantum theory. In this author’s analysis, using the group there is the automatic introduction of short range forces in addition to the long range force of the group. is one dimensional and Abelian and is three dimensional and is non-Abelian. is also a subgroup of The group is associated with the long range force and such as for its application to the weak force yields short range associated fields. Also is a subgroup of the useful SL(2,c) group of non compact operations on the manifold. SL(2,c) is a semi simple four dimensional Lie group and is a spinor group relevant to the relativistic formalism and is isomorphic to the connected Lorentz group associated with the Lorentz transformations. It is a conjugate group to the group and contains an inverse. The double cover group of is SL(2,c) where SL(2,c) is a complexification of Also LS(2,c) is the double cover group of related to the set of rotations in three dimensional space [3]. Topologically, is associated with isomorphic to the three dimensional spherical, (or three dimensional rotations) and is associated with the group of rotations in two dimensions. The ratio of Abelian to non Abelian components, moving from to gauge is 1 to 2 so that the short range components are twice as many as the long range components. Instead of using the gauge condition we use SL (2,c) we have a non-Abelian gauge and hence quantum theory and since this group is a spinor and is the double cover group of the Lorentz group (for spin ½) we have the conditions for a relativistic formalism. The Barrett formalism is non-relativistic. SL (2,c) is the double cover group of but utilizing a similar approach using twister algebras yields relativistic physics. It appears that complex geometry can yield a new complementary unification of quantum theory, relativity and allow a domain of action for nonlocality phenomena, such as displayed in the results of the Bell’s theorem tests of the EPR paradox [22], and in which the principles of the quantum theory hold to be universally. The properties of the nonlocal connections in complex four space may be mediated by non -or low dispersive loss solutions. We solved Schrödinger equation in complex Minkowski space [25].

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In progress is research involving other extended gauge theory models, with particular interest in the nonlocality properties on the S pact-time manifold, quantum properties such as expressed in the EPR paradox and coherent states in matter. Utilizing Coxeter graphs or Dynkin diagrams, Sirag lays out a comprehensive program in terms of the and and Lie algebras constructing a hyper dimensional geometry for as a classification scheme for elementary particles. Inherently, this theory utilizes complexified spaces involving twisters and Kaluza-Klein geometries. This space incorporates the string theory and GUT models [27]. 4. Conclusion

It appears that utilizing the complexification of Maxwell’s equations with the extension of the gauge condition to non-Abelian algebras, yields a possible metrical unification of relativity, electromagnetism and quantum theory. This unique new approach yields a universal nonlocality. No radical spurious predictions result from the theory, but some new predictions are made which can be experimentally examined. Also, this unique approach in terms of the twister algebras may lead to a broader understanding of macro and micro nonlocality and possible transverse electromagnetic fields observed as nonlocality in collective plasma state and other media. References 1. P. Penrose and E.J. Newman, Proc. Roy. Soc. A363, 445 (1978). 2. E.T. Newman, J. Math. Phys. 14, 774 (1973). 3. R.O. Hansen and E.T. Newman, Gen. Rel. and Grav. 6, 216 (1975). 4. E.T. Newman, Gen. Rel. and Grav. 7, 107 (1976). 5. H.P. Stapp Phys. Rev. A47, 847 (1993) and Private Communication. 6. J.S. Bell, Physics 1, 195 (1964). 7. J.F. Clauser and W.A. Horne Phys. Rev. 10D, 526 (1974) and private communication with J. Clauser 1977. 8. A. Aspect, et. al. Phys. Rev. 49, 1804 (1982) and private communication. 9. E.T. Newman and E.T. Newman, third MG meeting on Gen. Rel., Ed. Ha Nang, Amsterdam Netherlands, North-Holland, pgs 51-55 (1983). 10. Th. Kaluza, sitz. Berlin Press, A. Kad. Wiss, 968 (1921). 11. O. Klein Z. Phys. 37, 805 (1926) and O. Klein, Z. Phys. 41, 407 (1927). 12. J.P. Vigier, Found. Of Phys. 21, 125 (1991). 13. M.W. Evans and J.P. Vigier “the enigmatic photon” 1 and 2 “Non-Abelian Electrodynamics”, Kluwer Acad. Dordrecht (1994, 1995, 1996). 14. E.A. Rauscher, Bull. Am. Phys. Soc. 21,1305 (1976). 15. E.A. Rauscher, J. Plasma Phys. 2. 16. T.T. Wu and C.N. Yang, “Concepts of Nonintergreble phase factors and global formulation of gauge fields”, Phys. Rev. D12, 3845 (1975). 17. E.A. Rauscher, “D and R Spaces, Lattice Groups and Lie Algebras and their Structure”, April 17, 1999. 18. E.A. Rauscher “Soliton Solutions to the Schrödinger Equation in Complex Minkowski Space”, pps 89-105, Proceeding of the First International Conference , 19. A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935). 20. E.A.Rauscher, Complex Minkowski Space & Nonlocality on the Metric & Quantum Processes, in progress. 21. S.P. Sirag “A Mathematical Strategy for a Theory of Particles”, pps 579-588, The First Tucson Conference, Eds. S.R. Hameroff, A.W. Kasniak and A.C. Scott, MIT Press, Cambridge, MA (1996). 22. T.T. Wu & C.N. Yang, (1975), Phys. Rev. D12, 3845. 23. N. Gisin, Phys. Lett. 143, 1 (1990). 24. W. Tittel, J. Bredel, H. Zbinden & N. Gisin, Phys. Rev. Lett. 81, 3563. 25. E.A. Rauscher, Proc. 1st Int. Conf. , Univ. of Toronto, Ontario, Canada, pp. 89-105, (1981). 26. T.R.Love, Int. J. of Theor. Phys. 32, 63 (1993).

EXPERIMENTAL EVIDENCE OF NEAR-FIELD SUPERLUMINALLY PROPAGATING ELECTROMAGNETIC FIELDS

WILLIAM D. WALKER Royal Institute of Technology, KTH-Visby Department of Electrical Engineering Cramérgatan 3, S-621 57 Visby, Sweden Expanded paper at: http://xxx.lanl.gov/abs/physics/0009023 [email protected]

1.

Theoretical analysis of an oscillating electric dipole

Numerous textbooks present solutions of the electromagnetic (EM) fields generated by an oscillating electric dipole. The resultant electrical and magnetic field components for an oscillating electric dipole are known to be [1, 2]:

It should be noted that all of the above solutions are only valid for distances (r) much greater than the dipole length In the region next to the source the source cannot be modeled as a sinusoid: Instead it must be modeled as a sinusoid inside a Dirac delta function: The solution of this problem can be calculated using the Liénard-Wiechert potentials [3, 4, 5]. It is noted from the above analysis that the field solutions of the electric dipole can be written as a sum of sinusoidal waves, which travel away from the dipole source at the speed of light. Even if the waves are generated by unique physical mechanisms, only the superposition of the waves is observable at any point in space. These wave components in effect form a new wave which may have different properties from the original components. Only the longitudinal and transverse wave components are real since they can be decoupled by proper configuration of a measurement antenna. The following analysis derives general relations that are used to determine the instantaneous phase and group speed vs distance graphs for the field components. 189 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 189-196. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

190 1.1.

WILLIAM D. WALKER DERIVATION OF PHASE SPEED AND GROUP SPEED RELATIONS

In several papers written by the author [3, 4, 6] and in reference [7], the instantaneous phase speed has been determined to be: Using the following relations: and the instantaneous phase speed becomes:

This relation (ref. Eq. 4) shows that the phase speed is inversely proportional to the slope of the phase vs distance curve. Note, zero slope would imply infinite phase speed. In addition, in the same papers [3, 4, 6, 7] it is determined that the instantaneous group speed is: This relation can be made a function of (kr) by multiplying the numerator and the denominator by yielding: Using the relation: and the group speed becomes: Using the following relations: and that the instantaneous group speed becomes:

The above relation (ref. Eq. 5) indicates that the instantaneous group speed is inversely proportional to both the curvature and the slope of the phase vs distance curve. Note that if the denominator of the above equation is zero, the group speed will be infinite. Also note that if the curvature is zero, the group speed equation (ref. Eq. 5) will be the same as the phase speed equation (ref. Eq. 4). The group speed is thought to be the speed at which wave energy and wave information propagate [8, 14 (p.268)]. 1.2. TRANSVERSE ELECTRIC FIELD

Applying the phase to the transverse electrical field [6]:

and group speed relations component (ref. Eq. 2) yields the following results

SUPERLUMINAL EM FIELDS

Plots of the longitudinal electrical field found at the following reference [6].

and transverse magnetic field

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are

1.2.1. Interpretation of theoretical results The above theoretical results suggest that longitudinal electric field waves and transverse magnetic field waves are generated at the dipole source and propagate away. Upon creation, the waves (phase and group) travel with infinite speed and then rapidly reduce to the speed of light after they propagate about one wavelength away from the source. In addition, transverse electric field waves (phase and group) are generated approximately one-quarter wavelength outside the source and propagate toward and away from the source. Upon creation, the transverse waves travel with infinite speed. The outgoing transverse waves reduce to the speed of light after they propagate about one wavelength away from the source. The inward propagating transverse fields rapidly reduce to the speed of light and then rapidly increase to infinite speed as they travel into the source. In addition, the above results show that the transverse electrical field waves are generated about 90 degrees out of phase with respect to the longitudinal waves. In the near-field the outward propagating longitudinal waves and the inward propagating transverse waves combine together to form a type of oscillating standing wave. Note that unlike a typical standing wave, the outward and inward waves are completely different types of waves (longitudinal vs transverse) and can be separated by proper orientation of a detecting antenna. In addition, it should also be noted that both the phase and group waves are not confined to one side of the speed of light boundary and propagate at speeds above and below the speed of light in specific source regions. The mechanism by which the electromagnetic near-field waves become superluminal can be understood by noting that the field components can be considered rectangular vector components of the total field (ref. Fig. 5). For example, the vector diagram for the longitudinal electric field is (ref. Eq. 1):

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From this vector diagram it can be seen that the phase of the longitudinal electric field is: Also it can be seen that angle: Combining these relations yields phase relation: Note that for small the angle bisector: has about the same length as vector (kr). Therefore when (kr) is small the two vector components add together to form a longitudinal electric field vector which has nearly zero phase. Note that the angle bisector approximation is valid for several values of (kr) when (kr) is small. This result can also be seen by Taylor expanding the phase relation for small (kr) yielding: where These results show that very near the dipole source the phase of the longitudinal electric field is zero, causing both the phase speed and the group speed to be nearly infinite. In the near-field the phase increases to causing the phase speed to be: and the group speed to be: [ref. 3, 4, 6]. In the far-field the phase becomes: causing both the phase speed and the group speed to be equal to the speed of light. The other components of the electromagnetic field can also be analyzed in the same way yielding similar results. It should be noted that arguments against the superluminal interpretation presented in this paper have appeared in the literature [9, 10]. R. Feynman’s anlaysis should also reviewed [11]. 2.

Experiment to measure phase and group speed of transverse EM fields

A simple experimental setup (ref. Fig. 6) has been developed to qualitatively verify the transverse electric field phase vs distance plot predicted from standard electromagnetic theory (ref. Fig. 2).

The experiment setup consists of a high frequency UHF FM transmitter (Hamtronics model no. TA451) [12] which generates a 437MHz (68.65cm wavelength), 2 watt sinusoidal electrical signal. The output of the transmitter is connected with a RG58 coaxial cable to a vertical mobile telephone antenna (Tx) designed for the carrier frequency (model no. RA3126) [13] The output of the transmitter is also connected to channel 1 of the input of a high frequency 500MHz digital oscilloscope (model No. HP54615B). The transmitter output, cable, antenna, and oscilloscope input all have 50 Ohm impedance in order to minimize reflections. A second identical receiver antenna (Rx) is connected to channel 2 of the high frequency oscilloscope and the antenna is positioned parallel to the vertical transmitting antenna. The sinusoidal

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signals from the two antennas are monitored with the oscilloscope, triggered to channel 1. The phase difference between the signals is measured using the oscilloscope measurement cursors as the antennas are slowly moved apart from 5 cm to 70 cm in increments of 5 cm (measurements made with a ruler). The oscilloscope calculates the phase from the measured time delay and the measured wave period (T): The phase vs distance data is analyzed using HPVEE (Ver. 4.01) PC software. The data is then curvefit with a order polynomial and the data is superimposed to visually verify the accuracy of the curvefit. The phase speed vs distance curve and the group speed vs distance curve are then generated by differentiating the resultant curvefit equation with respect to space and using the transformation relations (ref. Eq. 4, 5). 2.1.

EXPERIMENTAL RESULTS

The following graph (ref. Fig. 8) is a plot of the phase vs distance data (ref. Fig. 7) taken during one experiment. The instantaneous phase and group speed graphs were generated by curvefitting the experimental data and inserting the curvefit equation into the phase and group speed transformations: (ref. Eq. 4, 5). The first data point is not real and was added to improve the polynomial curvefit. The curvefit yielded the following polynomial:

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It should be noted that these experimental results are only qualitative due to electromagnetic reflections from nearby walls and objects. Quantitative measurements can only be attained in an anechoic chamber. The experiment has been repeated several times in different parts of a 4 x 4m (area) x 2m (height) room at different angular orientations to the walls and the phase vs distance curve always appears the same within 10%. It is also observed that changing the scope input impedance from 50 Ohms to 1M Ohm input impedance does not noticeably affect the phase vs distance curve. Since no effect is observed it is concluded that the Tx antenna to Rx antenna variable capacitance combined with the scope input impedance (thereby forming a high pass filter) is not the cause of the phase change. Experimentally it is observed that the electrical field near the source (less than 0.6 ) is at least one order of magnitude greater than the electric field several wavelengths from the source, which may be reflected. It is concluded that the observed field near the source is predominantly due to near-field effects thereby making the observed results qualitatively reliable. The experimental results (ref. Fig. 8, 9, 10) are qualitatively similar to the electric dipole solution presented (ref. Fig. 2, 3, 4). 2.2.

INTERPRETATION OF EXPERIMENTAL RESULTS

Analysis of the experimentally derived phase vs distance curve (ref. Fig. 8) indicates that the phase vs distance curve generated from the experimental data is very similar to the curve predicted from electric dipole theory (ref. Fig. 2). Performing the experiment in an anechoic chamber and improving theoretical model for the dipole antenna should yield a better match between theory and experiment. The phase speed (ref. Fig. 9, 3) and group speed (ref. Fig. 10, 4) vs distance curves do not match theory as well as the phase vs distance curves (ref. Fig. 8, 2). This difference is due to the fact that small errors in the experimental data and curvefit become magnified after differentiating the data, which is required by the phase and group speed transformation relations (ref. Eq. 4, 5). Although the experimental results are not as accurate as they can be, it can be qualitatively seen from the results that transverse electric field waves (phase and group) are generated approximately one-quarter wavelength outside the source and propagate toward and away from the source (ref. Fig. 9, 10). Upon creation, the transverse waves travel with infinite speed. The outgoing transverse waves reduce to the speed of light after they propagate about one wavelength away from the source. Note that infinite phase speed and group speed is expected since the phase vs distance curve (ref. Fig. 8) has a minimum at about one quarter wavelength distance from the source (ref. Eq. 4, 5). 3. 3.1.

Discussion RELATIVISTIC IMPLICATIONS

Although superluminal phase speeds are known to exist in other physical systems (e.g., electromagnetic wave propagation in the ionosphere [14]), group speeds exceeding the speed of light are not known to exist and are thought to conflict with relativity theory [15]. Simple relativistic analysis shows it unlikely that superluminal

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near-field electromagnetic waves generated by an electric dipole can be used to violate Einstein causality [16]. To demonstrate this property, an example using the near-field superluminal longitudinal electric field will be analyzed. Assume that a stationary electric dipole transmits a longitudinal electric field with an amplitude-modulated signal (modulation wavelength: The signal is then reflected back toward the dipole by an electron moving away at constant velocity (v). If the dipole and the moving electron are separated by distance when the end of the signal arrives, then the group speed of the signal will be nearly instantaneous[6]. Using Lorentz transformations it can be seen that from the moving electron’s perspective, the end of the signal propagates backward in time: where When the moving electron reflects the signal, from its perspective, the electric dipole moves with velocity (v) and sees a contracted distance The end of the signal therefore arrives at the electric dipole at time: where Therefore one obtains Since the velocity of the moving electron (v) can be at most This result indicates that although the signal can be transported backward in time, it cannot be transported before the same signal was initiated, thereby prohibiting one from changing the signal that was sent. 3.2.

SPEED OF INFORMATION PROPAGATION AND DETECTION

Although the group speed may be superluminal, the speed of information propagation and detection may be less. If an amplitude modulated carrier signal propagates with a group speed and the sinusoidal modulation propagates a distance in time detection of the signal may require several cycles The speed of information propagation and detection can then be modeled: In the farfield of an electric dipole the propagation time can be much larger than the number of cycles needed to decode the signal, therefore: In the superluminal nearfield region of an electric dipole the propagation time can be much smaller than the number of cycles needed to decode the signal, therefore: Inserting the following known relationships: and using the fact that the superluminal region is known to be less than one wavelength yields: It is known from Fourier theory that several cycles (n >1) of a sinusoid are required for the information (frequency) to be determined. Therefore, if information detection is based on Fourier decomposition of the signal, the speed of near-field information transmission and detection will be less than the speed of light. It is also known from information theory that only two points of the modulated sinusoid signal are required to determine its frequency, amplitude and phase. If the signal noise is small, these points can be very close together (n << 1) and a curvefit can be performed to detect the signal. If information detection is based on this method, the speed of near-field information propagation and detection may be greater than the speed of light but less than 3.3.

MAGNETIC DIPOLE AND OSCILLATING GRAVITATIONAL MASS

Two other physical systems are noted to generate similar superluminal waves. Mathematical analysis of a magnetic dipole and a gravitationally radiating oscillating mass [4, 5, 6] reveals that they are governed by the same partial differential equation as

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the electric dipole. For the magnetic dipole, the only difference is that electric and magnetic fields are reversed. Consequently all of the analysis presented in this paper also applies to this system, and therefore similar superluminal wave propagation near the source is also predicted from theory. For a vibrating gravitational mass, the difference is that electric (E) and magnetic (B) fields are replaced by analogs: electric (G) and magnetic (P) component of the gravitational field [17]. In addition, a second mass vibrating with opposite phase is required to conserve momentum thereby making the source a quadrapole. But very close to the source, the effect of the second mass is negligible and can be neglected in the analysis. Consequently superluminal wave propagation is also predicted next to the source. Further away from the source the fields tend to cancel. Evidence of nearly infinite gravitational phase speed at zero frequency has been observed by a few researchers by noting the high stability of the earth’s orbit about the sun [18, 19]. Light from the sun is not observed to be collinear with the sun’s gravitational force. Astronomical studies indicate that the earth’s acceleration is toward the gravitational center of the sun even though it is moving around the sun, whereas light from the sun is observed to be aberrated. If the gravitational force between the sun and the earth were aberrated then gravitational forces tangential to the earth’s orbit would result, causing the earth to spiral away from the sun, due to conservation of angular momentum. Current astronomical observations estimate the phase speed of gravity to be greater than 2xl010c. Arguments against the superluminal interpretation have appeared in the literature [9, 10]. 1

Panofsky, W., Philips, M. (1962) Classical electricity and magnetism, Addison-Wesley, Ch. 14 Lorrain, P., Corson, D. (1970) Electromagnetic fields and waves, W. H. Freeman and Company, Ch. 14 3 Walker, W. D. (1998) Superluminal propagation speed of longitudinally oscillating electrical fields, Conference on causality and locality in modern physics, Kluwer Academic Pub. 4 Walker, W. D., Dual, J. (1997) Phase speed of longitudinally oscillating gravitational fields, Edoardo Amaldi conference on gravitational waves, World Scientific. Expanded paper found at electronic archive: http://xxx.lanl.gov/abs/gr-qc/9706082 5 Walker, W. D. (1997) Gravitational interaction studies, ETH Dissertation No. 12289, Zürich, Switzerland 6 Walker, W. D. (1999) Superluminal Near-field Dipole Electromagnetic Fields, International Workshop: Lorentz Group, CPT and Neutrinos, Zacatecas, Mexico, June 23-26, to be published in Conference proceedings, World Scientific. Expanded paper found at electronic archive: http://xxx.lanl.gov/abs/physics/0001063 7 Born, M. and Wolf E. (1980) Principles of Optics, 6th Ed., Pergamon Press, 15-23 8 Gough, W., Richards, J., Williams R. (1996). Vibrations and waves, Prentice Hall, 123 9 Carlip, S. (2000) Aberration and the speed of gravity, Phys. Lett. A267 81-87, Also ref. electronic archive: http://xxx.lanl.gov/abs/gr-qc/9909087 10 Ibison, M., Puthoff, H., Little, S. (1999) The speed of gravity revisited, Also ref. electronic archive: http://xxx.lanl.gov/abs/physics/9910050 11 Feynman, R. P. (1964) Feynman lectures in physics, Addison-Wesley Pub., Vol. 2, Ch. 21 12 Reference website: www.hamtronics.com 13 Reference website: www.elfa.se - part no. 78-069-95 14 Crawford, F. (1968) Waves: berkeley physics course, McGraw-Hill, Vol. 3, 170, 340-342 15 Einstein, A. (1907) Die vom relativitätsprinzipgeforderte trägheit der energie, Ann. Phys., 23, 371-384. Also ref: Miller A. (1998) Albert Einstein’s special theory of relativity, Springer-Verlag, New York 16 Walker, W. D. (2000) Analysis of Causality Issue in Near-field Superluminally Propagating Electromagnetic and Gravitational Fields. Ref. elect. archive: http://xxx.lanl.gov/abs/physics/0009076 17 Forward, R. (1961) General relativity for the experimentalist, Proceedings of the IRE, 49 18 Laplace, P. S. (1966) Mechanique, English translation, Chelsea Pub., New York, 1799-1825 19 Van Flandern, T. (1998) The speed of gravity – what the experiments say, Phys. Lett. A 250 2

THE PHOTON SPIN AND OTHER TOPOLOGICAL FEATURES OF CLASSICAL ELECTROMAGNETISM

R. M. KIEHN http://www.cartan.pair.com Email: [email protected]

1. Introduction

Cartan’s methods of exterior differential forms can be used to demonstrate that the laws of classical electromagnetism are topological constraints on a domain of independent variables and are independent from the geometric constraints of a metric, or a connection. In Cartan’s language of differential forms and exterior differential systems [1], the two topological constraints, F-dA=0, and J-dG=0 represent the PDE,s of Maxwell’s equations in a manner free from any choice of coordinates, metric or connection. The differential forms that make up the two postulates can be used to construct other topological statements that depend upon two independent 3-forms: the 3-form of topological torsion, A^F, and the 3-form of topological spin, A^G. The exterior derivative (divergence) of these two 3-forms creates the two familiar Poincare deformation invariants of an electromagnetic system, valid in the vacuum or plasma state. In domains where the Poincare invariants vanish, the closed integrals of A^F and A^G exhibit topological invariant or coherent properties similar to the "quantized" helicity and spin properties of a photon. The possible evolution of these and other topological properties can be studied with respect to equivalence classes of processes that can be defined in terms of singly parameterized vector fields. Non-zero values of the Poincare invariants are the source of topological change and non-equilibrium thermodynamics. Example solutions to the Maxwell exterior differential system exhibiting the properties of topological torsion and topological spin are given in terms of 1-forms of Pfaff dimension 4, deduced from coupled Hopf maps that have spinor and minimal surface properties. A non-zero topological spin for the photon implies the existence of a longitudinal component to the electromagnetic field. 2. Topological Torsion and Topological Spin 3-forms

The two fundamental postulates of an electromagnetic system, F-dA = 0, and J-dG=0, require the existence of four fundamental exterior differential forms, {A,F,G,J}, which form a differential ideal on the variety, {x,y,z,t}. The postulates constrain the topology of the variety. For example, the first postulate implies that domain of support for the E and 197 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 197-206. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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B fields cannot be compact without boundary (like the surface of a sphere). The elements of the ideal can be used to construct the complete Pfaff sequence of forms {A, F=dA, G, J=dG, A^F, A^G, A^J, F^F, G^G} by processes of exterior differentiation and exterior multiplication. A (Cartan) topology constructed on this system of forms has the useful feature that the exterior derivative may be interpreted as a limit point, or closure, operator in the sense of Kuratowski [2]. The complete Maxwell system of differential forms (which assumes the existence and physical importance of the potentials, A, also generates two other exterior differential systems: d(A^G) – F^G + A^J = 0 and d(A^F)-F^F= 0. These equations introduce the (apparently novel to many researchers) 3-forms of Topological Spin Current density, A^G [3] and Topological Torsion-Helicity, A^F [4]. For an electromagnetic system, the Action 1-form, A, which has the physical dimensions of the flux quantum, h/e. The 2-form, G has the physical dimensions of charge, e. The 3-form, A^G has the physical dimensions of angular momentum, h, and the 3-form A^F has the physical dimensions of spin (angular momentum) multiplied by the Hall impedance, [5]. By direct evaluation of the exterior product on a domain of 4 independent variables, each 3-form (A^G, A^F, and the charge-current J) will have 4 components that can be symbolized by the 4-vector arrays given in engineering format below. Each 3 form can be composed by the equivalent contraction process with the 4D volume element, Topological Spin current, Topological Torsion vector, Charge – current density, Note that the ubiquitous helicity density, is merely the fourth component of A^F. The vanishing of A^G is an additional topological constraint on the domain that defines topologically transverse electric (TTE) waves: the vector potential, A, is orthogonal to D in the sense that The vanishing of is an additional topological constraint on the domain that defines topologically transverse magnetic (TTM) waves: the vector potential, A, is orthogonal to B in the sense that When both 3-forms vanish, the topological constraint on the domain defines topologically transverse (TTEM) waves. For classic real fields this double constraint would require that the vector potential, A, is collinear with the field momentum, B. and in the direction of the wave vector, k. Such constraints permit the definition of singular solutions of propagating discontinuities, or electromagnetic "signals" [6], and in the general case lead to the result that signal propagation has 4 different phase speeds dependent upon both polarization and direction. Note that if the 2-form F was not exact, such topological concepts of transversality would be without distinct meaning, for the 3-forms of Topological Spin and Topological Torsion depend explicitly upon the existence of the 1-form of Action. For future developments, also observe that the topological torsion vector and the topological Spin vector are associated vectors to the 1-form of Action, in the sense that The two distinct concepts of Spin Current and the Torsion vector have had almost no utilization in applications of classical electromagnetic theory, for they are explicitly dependent upon the potentials, A. Examples, both novel and wellknown, of vacuum and plasma solutions to the electromagnetic system which satisfy (and

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which do not satisfy) these topological constraints are given elsewhere [7] 3. The Poincare Invariants

The exterior derivatives of the 3-forms of Spin and Torsion produce two 4-forms, F^GA^J and F^F whose closed integrals are deformation invariants for any continuous evolutionary process that can be defined in terms of a singly parameterized vector field. These topological objects are related to the conformal invariants of a Lorentz system as discovered by Poincare and Bateman [8]. Note that their topological properties are valid even in the plasma domain of dissipative charge currents and radiation, as well as in the vacuum. In the format of independent variables {x,y,z,t} the exterior derivative corresponds to the 4-divergence of the 4-component Spin and Torsion vectors, and Poincare 1 = Poincare2 For the vacuum state defined by J = 0, zero values of the Poincare invariants require that the magnetic energy density is equal to the electric energy density and, respectively, that the electric field is orthogonal to the magnetic field Note that these constraints often are used as elementary textbook definitions of what is meant by electromagnetic waves. Consider the definitions: Definition 1: Spin-chirality is defined as the closed integral of the 3-form A^G Topological Spin-chirality = Definition 2: Torsion-helicity is defined as the closed integral ofthe 3-form A^F Topological Torsion-helicity = By using Cartan's magic formula [9] it is possible to prove Theorem 1: If the First Poincare Invariant vanishes, topological Spin is an evolutionary deformation invariant with values whose ratios are rational. Theorem 2: If the second Poincare Invariant vanishes, topological Torsion is an evolutionary deformation invariant with values whose ratios are rational. The quantized (integer) ratios comes from the deRham cohomology theorems on closed integrals of closed p-forms [10]. All of the above development has been without the constraint of a metric and without the choice of a connection, in the spirit of Van Dantzig [11]. It is important to realize that these topological conservation laws are valid in a plasma as well as in the vacuum, subject to the conditions of zero values for the Poincare invariants. On the other hand, topological transitions between "quantized" states of Spin or Torsion require that the respective Poincare invariants are not zero. On space time domains where both 3-forms are closed, the ratio of the two integrals of topological torsion and topological spin have the physical dimensions of the fractional Hall impedance. [12] 4. Why Use Differential Forms?

The reasons for studying electromagnetism in the Cartan's language of exterior differential forms can be listed as follows: 1. Differential Forms are independent of the choice of coordinate system (they are diffeomorphic invariants).

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2. Constrained Differential Forms contain topological information as well as geometrical (tensor) information. 3. The exterior derivative, d, generates Limit Points, the exterior product, ^, generates Intersections; both are topological properties. 4. ddA=dF=0 generates the Maxwell-Faraday PDE's, dG=J generates the MaxwellAmpere PDE's, for any coordinate system. 5. The topological properties of dimension, orientability, intersections, connectedness (components, holes and handles) induced by the electromagnetic field have explicit realization in terms of exterior differential forms. 6. Closed integrals of closed exterior differential forms are evolutionary invariants. 7. When the 3-form of topological torsion A^F = 0 the electromagnetic system can exhibit enantiomorphic pairs. 5. The Pfaff Dimension One of the basic objects in a topological study of physical systems is the 1-form of Action, which for electromagnetism is constructed from the vector and scalar potentials, The limit sets (exterior derivatives) of the 1-form A the Field Intensities E and B. The question can be asked: What is the minimum number of functions and t required to represent the exterior features of the 1-form of Action? The answer can be determined by computing the number of non-zero terms in the Pfaff sequence, {A, dA, A^dA, dA^dA....}. Most historical investigations are constrained to the sequence, {A, dA, 0, 0,....} so that the minimum Pfaff topological dimension required is two. That is, there exists a submersive map from the original space of 2k + 2 variables to a space of dimension 2. Why is this the usual case studied? The answer is that when A^dA = 0 the system satisfies the Frobenius unique integrability theorem. This result enables predictive determinism. For a fluid A^dA = 0 defines a laminar flow, and must be a property of a turbulent flow, which is not laminar [13]. In another example, the work 1-form, W = i(V)dA for all Hamiltonian reversible evolutionary processes is of Pfaff dimension 1 [14]. Such is not the case for irreversible processes where the work 1-form is of Pfaff dimension 4 [15]. In this article the two non-zero 3-forms of topological torsion, A^ F , and topological spin, A^G, are of special interest. When the Pfaff dimension of the Action, A, is therefore 3 or more, and global unique integrability of the Action is not possible. The implication is that there does not exist a single function on the domain whose gradient defines the covariant direction field of the form, or a unique subspace foliation that is global. Instead when there can be enantiomorphic pairs and Faraday Rotation. In addition, the 3-form of topological spin, A^G, which has the units of angular momentum, is important to electromagnetic systems that involve enantiomorphic pairs and Optical Activity. As mentioned above, the ratio of the closed integrals of the two 3-forms yields the Hall Impedance. 6. Examples In Terms Of The Hopf Map: Why Use The Hopf Map? 1. The Hopf map is a map from 4 to 3 (real or complex) dimensions that has interesting and useful topological properties related to links and braids and

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other forms of entanglement. The Hopf map is a projection which can be used to determine a global basis frame for the variety in terms of 3 exact 1-forms and 1 adjoint 1-form which is of Pfaff dimension 4. The Frame field so defined has non-zero affine torsion. 3. The Hopf adjoint field can be used to represent, within a factor, the 1-form of Action (potentials) for a certain class of electromagnetic fields that exhibit propagating non-zero topological torsion and non-zero topological spin. 4. The Hopf map yields two pairs of orthogonal 3 vectors, one which is lefthanded and the other which is right handed. The 4 form of topological parity, dA^dA, constructed from the respective adjoint fields is either negative or positive. 5. The complex sum of two Hopf vectors generates a Cartan spinor. 2.

6.1 THE HOPF MAP AND ITS ADJOINT FIELD OF PFAFF DIMENSION 4

Consider the map from R4(X,Y,Z,S) to R3(u,v,w) given by the formulas

These formulas define the format of a Hopf map. The 3 component Hopf vector, is real and has the property that . Hence a real (and imaginary) 4 dimensional sphere maps to a real 3 dimensional sphere. If the functions [u1,v1,w1] are defined as [x/ct,y/ct/,z/ct] then the 4D sphere implies that the Hopf map formulas are equivalent to the 4D light cone. The Hopf map can also be represented in terms of complex functions by a map from C2 to R3, as given by the formulas

The variables and also can be viewed as two distinct complex variables defining ordered pairs of the four variables [X,Y,Z,S]. For example, the classic format given above for can be obtained from the expansion, Other selections for the ordered pairs of [X,Y,Z,S] (along with permutations of the 3 vector components) give distinctly different Hopf vectors. For example, the ordered pairs,

which is another Hopf vector, a map from R4 to R3, but with the property that Similarly, a third linearly independent orthogonal Hopf vector H3 can be found,

such that

and

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The three linearly independent Hopf vectors can be used as a basis of R3 excluding those points where the quartic form vanishes. The mapping functions [u,v,w] of any Hopf vector can be differentiated with respect to [X,Y,Z,S] to produce a set of three exact 1- form whose coefficients form 3 independent 4 component vectors on R4. A 4th linearly independent vector can be created algebraically and forms the "adjoint" field for the given Hopf vector. This direction field can be used to construct a non-integrable 1-form, A, of Pfaff dimension 4. These three exact 1-forms and the nonintegrable 1-form can be used as a basis frame for the domain of support for induced topology. The exterior derivatives of the basis frame produce the usual Cartan connection which is not affine-torsion free in its subspaces. By this mechanism the differential structure of R4 as induced by the Hopf map is determined. For the 4 independent 1 forms are given by the expressions (where A(X,Y,Z,S) is an arbitrary scaling function):

A Frame Matrix F can be generated by the coefficients of the 4 independent 1-forms, such that Det It is some interest to examine the properties of the adjoint 1-form, A, defined hereafter as the Hopf 1-form. For it follows that the Hopf 1-form is of Pfaff dimension 4. It is also of interest to consider factors that are of the format of the Holder norm, where n and p are integers, and (a,b,k,m) are arbitrary constants.

The exponents n and p determine the homogeneity of the resulting 1-form, which is given below in an ambiguous format (the plus or minus sign) For example, for n = p = 2, the scaling factor becomes related to the classic quadratic form of a vector. The scaled Hopf 1-form, A, is then homogeneous of degree zero. For arbitrary n and p, the 3-form of topological (Hopf) torsion becomes: where the Torsion 4 vector is equal to The Torsion vector, for the Hopf map is proportional to the position vector from the four dimensional origin and represents an expansion or a contraction process. The factor upon the integers n and p as well as the constants ( a,b,k,m). The Topological Parity 4-form, whose coefficient is the 4 divergence of the Torsion vector, becomes: It is most remarkable that for n = 2, any p and any (a,b,k,m), the topological parity 4 form vanishes; the scaled Hopf 1-form is of Pfaff dimension 3, not 4. In such cases the ratios of integrals of the topological torsion 3-forms over various closed manifolds are rational and the closed integrals are topological deformation invariants. (coherent structures). Also note that if the scaling factor is restricted to values such that n = 4, p = 2, a = b = k = m = 1, then the Frame matrix is unimodular, and the scaled Hopf 1-form is homogeneous of degree -2, relative to the substitution (A somewhat different definition of homogeneity relative to the volume element will be given below.)

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Emphasis here will be placed on examples for which n = 4, p =2, a = b = k = 1, m = ±1. 7. Spinors As Linear Combinations Of Hopf Maps

The 3D isotropic (null) complex position vector, [z1, z2, z3] can be decomposed into a real and a imaginary part, such that both parts have the same magnitude and are orthogonal. In short, the Cartan Spinor, [16] can be represented as the complex sum of two Hopf vectors. The spinors come in two triples of the form and The inner product of each element of a triple (and the complex conjugates) is an isotropic vector as for These complex combinations of Hopf vectors can be used to generate solutions for which the topological torsion vanishes, and yet the topological spin is finite and quantized. 7.1 ELECTROMAGNETISM OF INDEX ZERO HOPF 1-FORMS

Guided by prior investigations, it is of interest to use the scaled Hopf 1-form as the generator of electromagnetic field intensities. When the number of minus signs in the quadratic form is zero (index 0), and the exponents are n = 4, p = 2, define and The coefficients of the scaled Hopf 1-form can be put into correspondence with the classic vector and scalar potentials, using S = CT where C is a constant. The Action for the first example is then of the format, For this choice it is remarkable that the derived 2-form F = dA has coefficients E and B that are proportional to the same Hopf Map. The adjoint 1-form generated from one Hopf map has a limit set which is another Hopf map. Using the minus ambiguity (parity) sign, leads to the classic result that but with the not-usual result that the E field is anti-parallel to the B field (If the positive ambiguity (parity) sign is used, the E and B fields are parallel.). The explicit formulas for the field intensities are: It is natural to ask if these E and B fields admit a Lorentz symmetry constitutive constraint such that vacuum state is charge current free. Recall that a constitutive constraint is a relationship between a 2-form, F and a 2-form density G such that the coefficients of G(D,H) are related to the coefficients of F{E,B). A Lorenz vacuum condition implies that the fields are solutions of the vector wave equation. The question becomes, "If is presumed that and do the Maxwell Ampere equations generate a zero 3 form of charge current? ". Direct computation for the field intensities generated by the index zero Hopf 1- form indicates that unless Hence the scaled Hopf Action, where the scaling is of signature zero, does NOT describe a charge current free vacuum, for real positive values of and C. On the other hand, if it is presumed that the domain is such that say or is negative, then the Hopf Map, scaled as above, does generate charge-current free wave solutions. Negative epsilon appears to hold in metals and the Earth's ionosphere. Recent announcements indicate constructions that yield negative However, for situations where or are negative, the Hopf wave solutions imply that the Topological Spin angular

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momentum, A^G is not a deformation invariant (hence Spin angular momentum of the field is not conserved.) 7.2 ELECTROMAGNETISM OF INDEX ONE HOPF 1-FORMS When the number of minus signs in the quadratic form is one (index 1), and the exponents are n = 4, p = 2, define (lower case letters will be used for Index one Hopf 1-forms). For his choice, it is remarkable that the derived 2-form has coefficients E and B that are proportional to different Hopf Maps. The Action 1-form is the same as above, but with a different denominator. This fact leads to the classic result that but now the E field is not co-linear with the B field. Using negative ambiguity (parity) sign leads to fields (from F – dA = 0):

The Spin current density for this first non-transverse wave example is evaluated as: and has zero divergence. The Torsion current may be evaluated as and has a non-zero divergence equal to the second Poincare invariant The solution has magnetic helicity as and is radiative in the sense that the Poynting vector, It is possible to construct the 3-form of Topological Torsion, and its exterior derivative defined as Topological Parity. The Topological parity can be either positive, zero, or negative. For the example Hopf 1-form given above (using the negative ambiguity sign), the Topological Torsion is represented to within a factor by a position vector [-x-y,-z,-t] in 4 dimensions, and has a negative divergence or parity. If the positive sign of the ambiguity factor is changed, then the parity of the form changes sign. For example, for the 1-form, the 4 -form of topological parity is positive, and the topological torsion is represented by an outbound position vector (to within a factor). Similar to the investigation described above for the index zero Hopf vectors, it is natural to ask if these E and B fields admit a Lorentz symmetry constitutive constraint such that vacuum state is charge current free. Again, such a condition implies that the fields are solutions of the vector wave equation. Direct computation of the Maxwell Ampere equations indicates that dG = J = 0 if the phase velocity constraint vanishes, Hence the scaled Hopf Action, where the scaling is of index one, does describe a charge current free vacuum, for real positive values of and C. It is some interest to give the charge current solutions to show how the "phase factor establishes the vacuum charge free conditions. The example results for the components of the current density are

It is conjectured that fluctuations of the “perfect” vacuum phase relations, where are associated with ZPF. Note that there are possible charge-current free (

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singular wave solutions) that are governed by curves in space time. There curve are generated by the intersection of the three surfaces created by setting each of the coefficients of the current density equal to zero. These solutions are valid for any phase velocity and could be a source of “needle” radiation. The solution given above is not free of Topological Torsion, A^F, and there is a non-zero value of the second Poincare invariant, However, the Spin 3-form, A^G is also non-zero, but it has, subject to the phase constraint, a zero 4-divergence.(the first Poincare invariant is zero.) The divergence of the Spin 3-form, has 2 parts. The first part is twice the conventional Lagrange density of the fields, The second part is the interaction between the potentials and the charge currents, When the divergence of the 3-form is zero, then the closed integrals of Topological Spin are deformation invariants, and have closed integrals with rational (quantized) ratios. That is, with regard to any singly parameterized vector field, V, describing an evolutionary process, The function is an arbitrary deformation parameter. 7.3 LACK OF TIME REVERSAL SYMMETRY It should be noted that if the Action 1-form in the above example is subjected to the time reversal operation in its coefficients, then the new Action 1-form does NOT generate a charge current free vacuum (for real positive values of and C and the Lorentz constitutive constraint). 7.4 TWISTORS COMPOSED BY SUPERPOSING TWO INDEX 1 HOPF 1-FORMS By superposing (adding or subtracting) two different, index 1, Hopf 1-forms (which can be shown to be similar to a Penrose twistor solution) it is possible to construct a vacuum (charge current free wave) solution to the Maxwell system, subject to the constraint that the phase speed satisfies the phase velocity equation, As an example consider another Hopf 1-form of index 1 formulated as

Similar formulas for the field intensities can be determined as in the example above. Note that the parity of the Hopf forms to be superposed can be the same or different. If the parity of the two superposed Hopf 1-forms are opposite, then without consideration of the phase constraint, the Topological Torsion of the "twistor" 1-form vanishes, A^F=0. Yet the quantized topological spin 3-form, A^G, does not vanish, and moreover, subject to the phase constraint, the closed integrals of the Spin 3 form are conserved. This result implies that such a construction yields "quantized" values for the Spin integrals. These formulations can be compared with the Penrose twistor definitions in terms of differential forms [18] 8. The Classical Photon When the spinor solution of two Hopf vectors of opposite parity are combined, the

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resulting wave solution is transverse magnetic (in the topological sense that Not only does the second Poincare invariant vanish under the superposition, but so also does the Torsion 4 vector. Conversely, there can exist transverse magnetic waves which can be decomposed into two non-transverse waves. A notable feature of the superposed solutions is that the Spin 4 vector current does not vanish, hence the example superposition is a wave that is not transverse electric (in the topological sense that For the superposed example, the first Poincare invariant vanishes, which implies that the Spin integral remains a conserved topological quantity, with values proportional to the integers. The non-zero Spin current density for the combined examples is given by the formula:

while the Torsion current is a zero vector, A^F = 0. In addition, for the superposed example, the spatial components of the Poynting vector are equal to the Spin current density vector multiplied by such that

These results seem to give classical credence to the Planck assumption that the vacuum state of Maxwell's electrodynamics supports quantized angular momentum, and that the energy flux must come in multiples of the spin quanta. In other words, these combined solutions to classical electrodynamics have some of the experimental qualities of the quantized photon. References [1] R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffths, Exterior Differential Systems (Springer Verlag, 1991). [2] S. Lipschutz, General Topology (Schaum, New York, 1965) p. 72. [3] R. M. Kiehn and J. F. Pierce, "An Intrinsic Transport Theorem" Phys. Fluids 12, 1971 (1969) [4] R. M, Kiehn, "Topological Torsion, Pfaff Dimension and Coherent Structures", in: H. K. Moffatt and T. S. Tsinober eds, Topological Fluid Mechanics (Cambridge University Press, Cambridge, 1990) p. 449. [5] E. J. Post, Quantum Reprogramming, (Kluwer, Dordrecht 1995) [6] R. M. Kiehn, G. P. Kiehn, and R. B. Roberds, "Parity and Time-reversal Symmetry Breaking, Singular Solutions", Phys Rev A 43, 5665 (1991). [7] http://www.cartan.pair.com, especially http://www22.pair.com/csdc/pdf/helical6.pdf [8] H. Bateman, Electrical and Optical Wave Motion, (Dover, New York, 1914, 1955) p.12. [9] J. E. Marsden and T. S. Riatu, Introduction to Mechanics and Symmetry (Springer-Verlag, 1994) p. 122 [10] G. deRham, Varieties Differentiables (Hermann, Paris, 1960). [11] D. Van Dantzig, Proc. Cambridge Philos. Soc. 30, 421 ( 1934). Also see: D. Van Dantzig, "Electromagnetism Independent of metrical geometry", Proc.Kon. Ned. Akad. v. Wet. 37 (1934) [12] R. M. Kiehn, "Are there three kinds of superconductivity" Int. J. Mod. Phys B 5 1779 (1991) [13] R. M. Kiehn, NASA Ames NCA 2 –OR-295-502 (1/20/76) [14] E. Cartan, Lecons sur les invariants integraux (Hermann, Paris, 1958) [15] http://www.cartan.pair.com, especially http://www22.pair.com/csdc/pdf/irrev1.pdf [16] E. Cartan, The theory of Spinors, (Dover, New York, 1966) [17] Physics Today, p17 May 2000. [18] R. Penrose, "The Central Program of Twistor Theory", Chaos, Solitons, Fractals, 10 2-3, p581-611, 1999)

THE PROCESS OF PHOTON EMISSION FROM ATOMIC HYDROGEN

MARIAN KOWALSKI York University 4700 Keele Street, Petrie Bldg, Rm 206 Toronto, On M3J 1P3, Canada

Abstract. Ultra-fast lasers generating pulses as short as one or two photon periods raise the question: “How long does the process of photon emission take”? This question has been answered by modeling photon emission by an atom in terms of classical radiation theory. This theory involves the Coulomb force and a radiation resistance force. The change of energy and angular momentum and the transition of the electron between atomic states is considered. The transition time is treated as a distinct concept from the lifetime of the excited state. In this semi-classical model the calculated transfer of the electron’s energy and angular momentum is in accord with spectroscopic data. The emitted radiation is monochromatic. 1. Review Of Atomic And Radiation Theories And Data With Conclusions 1.1. THE BOHR THEORY2). The energy (E), radius (r), circumference (l), velocity (v), orbital period (T) and orbital angular momentum (L), of the orbit for quantum number n are:

where

are the values for the first Bohr orbit. 1.2. THE SOMMERFELD THEORY2). The energy and orbital period of the elliptical orbit, the semi-major axis semiminor axis the radii and the electron’s acceleration of the orbit are: 207 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 207-222. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

208

where

MARIAN KOWALSKI

and

are the radial and angular quantum numbers and

eccentricity of the ellipse; the circular Bohr orbits correspond to

is the The

orbital angular momentum of the electron in the elliptical orbit is where l is orbital angular momentum number: In the orbits elliptical.

states the remaining orbits

and

and

the electron is moving in circular are

1.3. REVIEW OF EXPERIMENTAL FEATURES OF PHOTONES EMITTED FROM ATOMIC HYDROGEN

The experimental energy and the angular momentum of photons are:

where R is Rydberg constant, atomic hydrogen.

is the ionization energy of

1.4. CONCLUSION FROM COMPARISON OF EXPERIMENTAL DATA WITH BOHR-SOMMERFELD THEORY

Comparison of photon’s wavelengths

and periods

(see tables 1 and 2.).

THE PROCESS OF PHOTON EMISSION FROM ATOMIC HYDROGEN

The electron’s orbital circumferences

The photon’s experimental periods

209

and periods

are in Tab. 2:

We can see that photon’s and electron’s periods are in relation:

for 2p-1s, 3d-2p, 4f-3d,.., see diagonal values of Tab. 2 and 2a. For non-diagonal values of n and n’ (see Tab. 2 and 2a) the average of and increases rapidly, whereas decreases slightly. However, the transition time of the electron is close to the photon’s period if the transition is assumed to start when the electron on its elliptical orbit crosses the lowest Bohr orbit with the same l, and with lower value of n equal to

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In order to compare the lengths, we estimate the distance traveled by the electron during a time equal to as the product of the average of its velocities in the initial and final states and the photon’s period: The relation between the distance traveled by the electron and the wavelength of the photon (see tables 3 and 3a) is:

It is seen from the tables 3 and 3a that the distance traveled by the emitting electron during one photon period are close to the average values of Bohr circumferences (especially for Rydberg states (in a case of elliptical orbits the situation is similar):

Thus, a photon can be emitted from a hydrogen atom during one complete rotation of the electron around the nucleus. 1.5. REVIEW OF CLASSICAL ELECTRODYNAMICS OF RADIATION THEORY. Around a charge moving with velocity v there exists electromagnetic fields connected

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with the acceleration of this charge1): where R is the electron radius to the observer. This electromagnetic field causes the electron’s deceleration The emitted radiation power1) is obtained from the Poynting vector:

where is the angle between and the radius of the observer R. The total emitted power is equal to: The radiation resistance force is associated with the average work done by this force upon the electron in the time interval T:

In the initial and the final states of the transition process, the electron is in BohrSommerfeld periodic orbits, for which hence1): where m is the mass of the electron and is the characteristic time. However, there are known difficulties connected with self-dispersed solutions of the Abraham-Lorentz equation: for There is defined in the literature,1,7) called the damping constant, where is the angular frequency of the electron oscillating along the x axis, under a restoring force (not Coulomb force), the radiation resistance force is The resistance force with damping constant1) for non-radiative dissipative processes is The electron is changing energy in accord with the equation: where total damping constant parameter are very small and time of the radiation is very big.

The values of

and

1.6. REVIEW OF QUANTUM THEORY.

The total electromagnetic field for the radiation and the particles is:

and the total momentum is introduced with the gauge invariance fulfilled with the minimal coupling: The interaction operator between the particles

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and the electromagnetic field is:

The first part induced transitions for which one photon is either created or annihilated:

The total system (atom + radiation field) can decay from the initial state: to the final state: by emitting a photon. If no photon is present at the beginning

state

the lifetime of the initial atomic

is:

where

is the transition frequency.

This relation is the quantum-mechanical analogue to the classical Larmor equation (15) considered in this paper. The lifetime of the state in the hydrogen atom with respect to decay into state in spherical coordinates and is equal to:

Knowing

that

and

we

obtain that

This lifetime is much larger than the photon period and much larger than very short laser pulses obtained recently as well as much larger than very short X-ray pulses (shorter than 10 optical cycles) emitted from the atoms after excitation by these short laser pulses14). This experimental evidence indicates that the transition can take place in a time much shorter than the lifetime of the exited state. 2. Classical Calculations Of Photon Emission From Hydrogen

During the action of the non-conservative force and angular momentum L according to formulae:

on the electron it is losing energy E

where is the total energy of the electron, , r is its radius centered at the nucleus and v is its velocity. We used equation (13) for where in the cylindrical coordinates we have:

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The loss of the energy and orbital angular momentum by the radiated electron is:

and we obtain the Newtonian form in cylindrical coordinates:

which has the vector form: Finally, using (22) we obtain equations:

However, these equations are self-dispersed, as for the Abraham-Lorentz equation. The approximation which provide non-dispersive solutions of the resistance force is:

It was found numerically that the factor ‘3’ is not so important. This suggests that the radiation resistance force can be anti-parallel to the electron’s velocity, as in dissipative processes: where

is angular velocity of the radiating electron. Parameter C ought to

be close to 1. We can define damping parameter From the equation:

like in (15). we have in cylindrical coordinates:

Assuming that according to known radiation theory1), we obtain correct values of the electron’s radius, energy and orbital angular momentum in the final state. However, because of small value of the parameter the electron travels many times around the nucleus during the transition process. This result is not satisfactory because the time needed by the radiating electron for one orbit is changing continuously from in the initial state to

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in the final state. This implies that the emitted radiation would not be monochromatic (not have gaussian line’s shape) contrary to what is observed. Our estimates show that for Rydberg states the photon’s period is close to the Bohr orbital period and for lower states the photon’s period is close to the average of the Bohr periods of the initial and final states. We can obtain a monochromatic photon if the electron makes only one orbit around the nucleus during the transition; i.e. within a time equal to the photon’s period This proposed mechanism is supported by a recent experiment with an ultra-fast laser that generates light pulses shorter than two photon wavelengths8, 14). When we assume that we obtain only one electron orbit, a transition time equal to the photon’s period and the correct energy and angular momentum in the final state. It is also possible to use the resistance force to be anti-parallel to the velocity of the electron v but with average resistance parameter for transitions between the two neighboring atomic states. Equation (34) suggests that can be similar to average of between two atomic states, see table 4. The orbiting electron (in transiting between the and levels) looses energy and angular momentum (see (21)) equal to

where is the time of the electron transition. The QED zero-point electromagnetic radiation can balance the radiation resistance force on orbits and as a result stable orbits are created, see H. M. Franca et al5). There are two ways of treating the transition processes: by using the radiation resistance force or the circular electromagnetic field (in the (x, y) plane):

where is the resistance parameter and is the angle between the electron’s and velocities. A similar electric (electromagnetic) field is carried by the photon itself. Using the resistance force (34) and equations (21) we have,

Remembering that there exists the Coulomb conservative force also acting on the electron (the electron’s energy is: we obtain two equations:

where

is the initial value of the electron angular momentum. In angular coordinates for the energy and angular momentum

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we have:

Then, (40-42) leads to the equation:

which is difficult to solve analytically. The formula is only an approximate solution. Accurate solution was obtained by integrating (43) numerically. In previous treatments1, 7) (see (15)) the electron’s motion is oscillatory in the radiating atom. This is possible, because both the centrifugal and Coulomb forces acting on the orbiting electron may be written approximately as

where is the equilibrium distance of the electron from the nucleus, is the angular velocity of the radial oscillating electron and is its displacement from equilibrium. This radial approximation neglects circular aspects of the process - the angular momentum of the electron and the emitted photon. The electron must lose energy given by equation (4) and must lose angular momentum equal to during the emission process during one circulation around the nucleus. Knowing the energy of the radiating electron in the initial state we stop the numerical integration when the electron has the final state energy equal to Stopping the numerical integration in this way yields a time for the emission process, the distance traveled by the electron and the loss of angular momentum, by the electron: In accord with equation (39) the duration of the process depends upon the value of the damping parameter With damping constant used in the literature1): the electron completes many orbits around the nucleus and the transition time is many times the photon period, However, this model of radiation is physically incorrect, because the radiation is not monochromatic as is observed. In order to obtain monochromatic radiation the electron has to make only one orbit during the transition process, since the period of the radiation is equal to the period of rotation of the electron. In order to fulfill this requirement equation (39) was used to calculate the value of the stifling parameter, with the time t, equal to the photon’s period: for the emission between the n and states, with and L equal to the

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electron’sangular momentum in the initial and final states:

where the latter expressions are obtained from equation (4). The values of this stifling parameter derived from (46) between every pair of neighboring states are shown in table 4.

is in principle a function of and like in In the numerical calculations we introduced a parameter k equal to the number of electron orbits around the nucleus, and then solved the equations with a damping parameter However, k had to be greater or equal to 1 in order to obtain a good value for the change in angular momentum consistent with the known transition energy. For the change of the electron’s energy was correct, but the change in its angular momentum was too large. These numerical experiments persuaded us that the process is best described by k=1, giving almost monochromatic photons. 3. Differential Equations Used For Numerical Calculations

We have used the two-body differential Hamilton’s equations12) with the Coulomb interaction potential V and additionally with the resistance force described in terms of the conjugate variables – as position and momentum) defined in the center of mass of hydrogen. The relative distance between the nucleus and the electron, its relative velocity and the relative general momentum are: and where the reduced mass of the electron is: derivatives of and are given by Hamilton’s equations:

The

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for the hydrogen where the hamiltonian atom and radiation is:

This yields a set of differential equations for the derivatives of the conjugate momentum:

If the resistance force is equal to then So, we have a set of six differential equations for

and

in space.

In the numerical program we defined these equations in the matrix form as:

We used dimensionless (Bohr atomic) units of length: of time: of energy: where of angular momentum: We have to define initial conditions for

(the Bohr radius), or (the first Bohr orbit circumference), or (the period of the first Bohr orbit),

and

corresponding to the

hydrogen atom in the initial state

where the is the arbitrary phase angle. We stop the calculations when the electron‘s energy is equal to the energy of the final state after emission of one photon. The output from the calculation are: the time of the radiation process, the radius of the electron, and the loss of energy and angular momentum of the electron, etc, as a function of time. 4. Results Of Calculations Obtained Using Resistance Force

In accord with the choice of (i.e. that the transition time is equal to one photon period), we used the values of obtained from (46), given in Table 4. The values of the damping parameter are much larger than the values of the damping constant assumed in the literature1). For example is

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much larger than:

The relationship between them is:

For many orbits of the electron around the nucleus (i.e. would be equal to but as already noted, this fails to result in the emission of monochromatic (even having gaussian line’s shape) radiation. In accord with our earlier estimations and calculations, supported by a recent experiment with an ultra-fast laser that generates light pulses shorter than two photon wavelengths8, 14), the electron can make only one orbit during the transition process. The lifetimes (20) of excited states are a few orders of magnitude larger (just about ) ) than the periods of photons. This suggests that the electron executes many orbits in the quasi-stable excited state before spontaneously emitting a one-wavelength long photon in the period of time This means that the lifetime of the excited state can be much longer than the time in which the photon is emitted. Figure 1 shows the trajectory of the electron during the H(2p-1s) transition. The resistance force with (46) causes the spiral movement of the electron between the circular states.

An observer will see two peaks (the beginning and ending) of energy (electromagnetic field13) ) emitted by the electron during one of its orbits around the nucleus; see figure 2. The distance between the peaks emitted during a time equal to the photon’s period is equal to one wavelength of emitted photon. The resistance parameter (46) give good results for the change in energy and angular momentum and also for the change in the radius of the electron during one photon period and during one orbit of the electron around the nucleus. The loss of the angular momentum by the electron is close to like for the emitted photon. The

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calculated time of the transition H(2p-1s) is equal to which is close to the period of photon The distance traveled by the electron (45) is equal to , which is close to the value calculated approximately from equation (9); see table 3. The distance traveled in the same time by the emitted photon equal to is close to the experimental photon wavelength Figures 3 and 4 show behaviours of other variables during the transition: the radius of the electron the angular momentum of the electron the derivative of the angular momentum of the electron the total electron energy etc. in atomic units.

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For the transition H(3d-2p) between the Bohr orbits, the parameter is about 10 times smaller than see Tab.4. The obtained from (45) transition time agrees very well with the experimental photon-period see Tab.2. The distance traveled by the electron is close to the value obtained approximately from equation (9), see Tab.3. The transfer of energy is exact and the transfer of angular momentum from the electron to the photon is close to The final electron radius is equal to For transitions from elliptical to circular orbits (i.e., n=3,4,5,6... to n'=1), the stifling parameter is different between every pair of adjacent states. When the radiation process starts from the point where the electron passes by the state, the transition time is comparable with the experimental period of the photon in conformity with the values in Tab. 2. 5. Results Of Calculations Using Rotational Electric Field

After placing the electron in such rotational electric field

and assuming simultaneous existence of the resistance force and circular Bohr orbits are obtained; in the case the parameter; see Fig. 5.

the stable

The Bohr electron orbits (n) have also been obtained by neglecting the resistance force but with the electron in a rotational circular electric field, whose direction reverses periodically twice during the time

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where for and When values of the parameters and were changed slightly, annular-circular orbits were obtained, whose average radius was equal to the Bohr orbit radius; see. Fig.6. For the transition of the electron from the n=2 to the n=1 state, using the resistance force with and the circular electric field for n=1, with and the calculation yields a final stabilized n=1 Bohr orbit; see Fig.7. Fig.8 shows the electron excitation calculated between the n=1 and n=2 levels, after placing the electron in the circular electric field only (i.e. neglecting the resistance force The behavior of a circulating charge in the constant electric field is totally different, see Fig. 9.

The major axis of the final ellipse is perpendicular to the electric field lines. 5.1. FEATURES OF THE ROTATIONAL ELECTRIC FIELD. The circular, real electric field E of the equation (54), has interesting features: - it satisfies Maxwell’s equation; i.e. d’Alembert’s wave equation in the real space, - it satisfies the Schroedinger equation with E written in the imaginary space. Schroedinger introduced the wave function which may represent the circular electric field:

where

It is in accord with Majorana’s idea9): “if the Maxwell theory of electromagnetism has to be viewed as the wave mechanics of the photon, then it must be possible to write Maxwell equations as a Dirac-like equation for a probability quantum wave

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where is proportional to the probability density function for a photon”. 6. Conclusions

Emission of a photon during a transition between Bohr atomic states can take place over a time interval equal to one period of the photon’s wave. This time corresponds to the time interval needed for one full orbit of the radiating atomic electron around the nucleus. Assumed values of the resistance parameter guarantee correct simulation of the simultaneous loss of both energy and angular momentum by the electron together with the emission of almost monochromatic photon with the proper wave period of time The recent experiment with an ultra-fast laser that generates light pulses shorter than two photon wavelengths8, 14), indicate that the electron can make only one orbit during the transition process. Especially in Rydberg states the photon’s period is almost equal to the period of the Bohr orbit. The lifetime of states is much larger than the photon period and much larger than very short laser pulses obtained recently as well as much larger than very short X-ray pulses (shorter than 10 optical cycles) emitted from the atoms after excitation by these short laser pulses14). This experimental evidence indicates that the transition can take place in a time much shorter than the lifetime of the exited state. It is possible to calculate the same transition processes of the electron by using circularly polarized rotational electric (electromagnetic) fields. Acknowledgments This work was partially supported by the Institute for Nuclear Studies, Swierk, Poland. Professor Geoffrey Hunter, York University (Toronto) provided advice and editorial help.

References 1. Jackson J. D. (1962) Classical Electrodynamics, John Wiley and Sons, New York, pp. 662-770. 2. Sscholz O. (1968) Atomphysik Kurz und Bündig, Würzburg, Germany. 3. Schiff L. I. (1968) Quantum Mechanics, Stanford University, pp. 30-50, 350-370. 4. Hunter G. and Wadlinger R. L. P. (1989) Physics Essays, V2, pp. 158. 5. Franca H. M., Marshall T. W., and Santos E. (1992) Phys. Rev. A45, No.9, pp. 6436. 6. Santos E. (1974) Il Nuovo Cimento 19B, No.l, pp. 57. 7. Dalibard J., Dupont-Roc J., and Cohen-Tannoudji C. (1982) J. Physique 43, pp. 1617. 8. Krausz F., Spielmann C., Brabec T., Geissler M., Scrinzi A.and Schnürer M., (Technische Universität Wien), Kan C. and Burnett N. H. (University of Alberta Canada) (1998) IEEE Journal of Selected Topics in Quantum Electronics, Vol. 4, No. 2, March/April. 9. Esposito S. (1998) Foundation of Physics, Vol. 28, No. 2. 10. Rueda A., Haisch B. (1998) Phys. Lett. A240, pp. 115. 11. Greiner W. (1994) Quantum Mechanics – Introduction, 3rd ed. Springer, Berlin, Heidelberg. 12. Peach G., Willis S. L., et al. (1985) J. Phys. B18, pp. 39 13. Tsien R. Y. (1972) AJP vol. 40, pp. 46. 14. Kapteyn H.and Murnane M. (1999) Physics World, January, 31.

HOLOGRAPHIC MIND – OVERVIEW: THE INTEGRATION OF SEER, SEEING, AND SEEN

EDMOND CHOUINARD Measurements Research Inc, 485 River Ave, Providence, RI 02908 USA [email protected]

To Expand Measurement Techniques On Mind-Matter Inter-Relationships To Build A Physical Model To Unveil Psychic & Religious Experiences

Abstract. This theoretical holographic model integrates mind and matter where consciousness and physics are merged. It is a continuing development of Vedic models of creation that derive from ancient esoteric texts, but it is based on modern day physics. Any useful model must support all knowledge and all experiences and it must be consistent with the dictates and substance of modern science. As this project is so overwhelming in scope, many facets of critical expertise are yet far from complete. Ongoing refinements of this model continue concurrently with a corroborating research program that involves both consciousness and physics. Extraordinary experiences as reported by modern day folks from diversified orientations constitute a beginning 1st level of corroboration. An ongoing scientific mind-matter interrelations measurements program, using sensors and tools of the physicist, constitutes a 2nd level of corroboration. The knowledge and experiences recorded by the ancients constitutes a third level of corroboration. This model correlates and integrates experiential phenomena with subtler and ever subtler states of physical particulant transformations. Particularly, this model unveils most of the baffling and scientifically isolated ideas surrounding faith, belief, suggestion, invocation, hypnosis, god, miracles, psychic phenomena, extraordinary visual perceptions, and mystical experiences. Similarly, the model reveals much of the operational mechanics behind the elusive Sanskrit expressions: mantra (connecting thread of lively flowing energy), tantra (phenomenal scheme of creation), tattvas (fundamental structures of creation), samyama (experiencing the merging of mind and matter), siddhis (extraordinary or 'miraculous' type experiences), samskâras (residual scars or deep impressions), and sutras (short fragments of holistic wisdom passed down verbally from the ancients). The mind is considered to be independent of the brain though it is normally closely tied to it. Consciousness is considered to be independently functioning in a subtle body that is totally separate from the autonomic neuronal activities of the brain. Communications and energy transfer between the subtle mind and the physical brain is conjectured to occur via a holographic connection by some kind of boson resonance. Consciousness is considered to be light, large formations of photon packets. The associated photon frequencies and wavelengths are considered to move up and down in correspondence with the many levels of consciousness that can be experienced. It is postulated that con223 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 223-232. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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sciousness, the SEER, is construed of the same reality underlying matter, the SEEN. The connecting process of SEEING is associated with spatial Fourier transforms that relate the SEER with the SEEN and the SEEN with the SEER. Beginning at vacuum state wavelengths such SEER-SEEING-SEEN relationships are found continually repeating, over and over again at progressively longer wavelengths until the SEEN phenomenal physical world appears. The low pass Fourier filtering that is inherent within this unfoldment of creation provides us with seemingly infinite sets of individual minds and particulant matter manifestations. Mind unfolds through all of its variously named transcendental states of floating consciousness as matter unfolds through all of its miniscule, nuclear, atomic, and molecular states of increasingly complex particulants. This holographic model serves as an initial theoretical structure, no matter how yet incomplete, upon which to explain a host of enigmas that traditional science ignores or forcefully represses. This theoretical platform not only is able to consider such enigmas but also quickly comes up with a host of reasonable, rational, and rather scientifically related answers to virtually all known anomalies, enigmas, and questions among the consciousness mystery paradigms. 1. Spatial Fourier Transforms And SEER-SEEING-SEEN

The connecting thread between mind and matter is here related in terms of spatial Fourier transforms. Whenever we take a picture with a camera, a double set of spatial Fourier transforms can explain the behavior of the light that traverses to the film. On top of next page, the figure entitled Optical Transfer Functions in Emerging Consciousness lists 5equations that relate the dynamics of these Fourier transforms. The bottom figure entitled SEER-SEEING-SEEN Transformation Mappings indicates a specialized camera to take pictures of the unfoldment of creation. We will refer to these 5-equations in the following text by their phenomenal names: 1 SEEN bindu = source = initial emerging particulants in configuration space-time 2 SEER sakti = consciousness = lively awareness in momentum space-time 3 FILTER holo = diversified color processing filters in momentum space-time 4 SEEN bîja = infinitely varied unfoldings of creation in configuration space-time 5 NORMALIZE = mathematics assuring proper calibration of instrumentation We expand the structural identification of spatial Fourier transformations to identify with the many floating states of consciousness that are available to all minds and we call this Holographic Mind. We will find that the relational concepts among the SEER, process of SEEING, and the SEEN explain the unfoldment of awareness of the various floating states of consciousness. Bindu is the first matter particulant condensation to emerge from vacuum state. Bindu dots fill the star map at the far left of the SEER-SEEING-SEEN Transformation Map. The SEEN bindu star function represents a spatial contrast distribution of bindu energy densities. We visualize the bindu dots tightly packed in a honeycomb like structure within the star outline, each with its own brightness level. They are matter particulants that have energy distributions in ordinary relativistic configuration space-time. A Fourier transformation of this star map is represented by a different set of variables in photon momentum space-time, that is, consciousness itself along with its experiential constituent of vital energy, sakti. The Sanskrit word sakti references subtle

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phenomenal energies that can be felt, perhaps as joy, mirth, exhilaration or grace. This transformed map is the result of an optical sorting process according to the size and contrast ratios of the elements on the bindu dot map. It is a spatial frequency spectrum Optical Transfer Functions in Emerging Consciousness 1 SEEN bindu = = ultimate initial groups of vacuum state particulants 2 SEER sakti = 3 FILTER holo =

= individual filters - holographic samskâra processing

4 SEEN bîja= 5 NORMALIZE:

SEEN bindu map

for calibration of instruments

SEER sakti FILTER holo map

SEEN bîja map

Vacuum State of Floating Consciousness of Seed Condensations Pure Intelligence Photon Momentum Space of Individualization analysis of the bindu dot map. It is not a photo for we are not in ordinary configuration space. It is a portion of a hologram for we are in photon momentum space, the SEER holographic space. Indeed, it is consciousness itself, and as of yet, uncolored. A simple, ordinary camera would create a film image that is quite similar to the SEEN bindu object being photographed. However, this special lens assembly creates a highly modified and transformed photo image of the original bindu dot map because of its extensive selective filtering. Light must first pass through the FILTER holo located in holographic space-time inside the lens assembly before forming the 'happy face' image arriving at the SEEN bîja film site of the SEER-SEEING-SEEN Transformation Map.

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The word bîja means seed. A bîja seed is larger and structurally more dense and complex than a bindu seedling. The 'happy face' is truly a 'new born child' of one sort or another, whether it is a spontaneous new particle creation for physics to discern, or a new subtle body creation for some happy parents to eventually materialize. The function SEER sakti is the arriving sakti flux at holographic space-time and it is the spatial Fourier transform of SEEN bindu map as previously discussed. The spatial filter can adjust, limit, enhance, or otherwise change the many characteristics that define the SEEN bîja map. The spatial FILTER holo is seen in the middle of equation 4 as part of the transformation properties leading to the bîja map. This spatial filter is located in photon momentum space-time, in floating consciousness itself, and it can filter or particularize any desired aspect of the ensuing bîja map in ordinary space-time. The FILTER holo function can be renamed at will among infinite varieties, as samskâras (personality traits), soul, god, or even God to name but a few. We can consider a lord (îsvara) filter. The lord is a spatial filter in holographic space-time leading to certain qualities of god consciousness that have been experienced by numerous folks. This lord filter (which might also include a stand alone anthropomorphic individualistic consciousness) can now adjust the bindu dot light rays of sakti coming into the domain of îsvara consciousness. Light rays anywhere across the optical aperture can thus be made brighter or dimmer, affecting a resulting apperception of god consciousness. Some prefer calling this lord filter 'creative intelligence' or perhaps 'god consciousness'. Whatever the name, having awareness at this level of consciousness might intercede in further coloring the ensuing bîja maps, as the sakti modulated light stria again coagulate in configuration space. Of particular interest is the bindu map's high degree of spatial resolution because of the incredibly small bindu dots. The SEER sakti consciousness photons passing through the first lens also have this same high resolution. However, FILTER holo dramatically cuts this resolution by juxtaposing the higher resolution bindu dot stria in various simple or complex spatial frequency arrangements. For instance, a lower resolution bîja map is obtained by simple inserting a donut like washer in the holographic plane. Unfolding of creation implies that SEEN bîja will have a lower spatial resolution its sourcing creator, SEEN bindu. FILTER holo reorganizes, redistributes, and compiles among many bindu dots to create unique and individualistic bîja seeds. This is simply the nature of double spatial Fourier transforms. The bîja seedlings are larger than the bindu dots and have a more complex structure. We iteratively reapply the bindu-bîja transformation mappings. Vacuum state energy wavelength dimensions thereby multiply downwards until we reach dimensions of physical world energy wavelengths. The names of the unfolding new maps reflect the new unfoldings being perceived. The names 'bindu' and 'bîja' are of a grouping called tattvas. The tattvas are fundamental subtle building blocks of creation that have been experienced by sages throughout the ages. In recent times, physicists have unveiled much about the distinct states of matter. Now there is a theoretical structure under which mind phenomena and matter phenomena can be linked together. 2. Tattva Dimensions See chart, Unfoldment of Creation via SEER-SEEN Transformations, on the following page. We now apply real dimensional values to the tattvas (fundamental structures of creation). Theoretical physics suggests that the smallest meaningful dimension associ-

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ated with quantum energy packets in vacuum state is of the order of centimeters, a Planck length. This is an infinitesimally small dimension where the so called 'random quantum fluctuations' of empty space-time start to get so large that even the idea of a smooth continuum in space-time begins to fall apart. We presume that siva-sakti space is coincident with vacuum space in unity consciousness. Siva and sakti have meanings quite analogous to physics' potential and kinetic energies, respectively. A first unfolding of creation develops when SEEN bindu first appears as kinetic energy, the sakti aspect of the ultimate witnessing siva consciousness. Vedic science and channeling literatures are full of cases of 'mystic' seers who have 'seen' or otherwise witnessed such unfoldments of early creation. We set each new unfolding bîja dot diameter 1 -million times larger than the sourcing bindu dot diameter. This magnification factor is a manageable pragmatic unfoldment rate of the universe that easily correlates with well-known realities of physics and with many well-documented classifications of subtle and extraordinary experiences. Starting at the minimum Planck length, we first look at the unfolding of material nature in configuration space. We move downwards on the SEEN-Condensations column from centimeters, to , to , to (nuclear dimensions), etc, again and again, until we reach the common physical world dimensions of centimeters and kilometers. Similarly, Unity consciousness is found to unfold in the SEER-Fluctuations column. Starting at centimeter, we move downwards through increasingly more dense stages of consciousness until we again reach everyday world dimensions of centimeters and kilometers. However, and with utmost significance, we now have two seemingly independent realities that are also connected together via Fourier transformations. The SEEN-Condensations considered realities of matter apparently unfold all by themselves. The SEER-Fluctuations considered realities of consciousness also apparently unfold all by themselves. They appear independent of one another until we look with greater and greater resolution. Moving up or down on either column, the SEER-Fluctuations or the SEEN-Condensations, implies the possibility of becoming aware of passing through the other column via a forward or inverse Fourier transform. The probability of becoming aware of such possibilities is a function of the instantaneously available level of subtlety that consciousness is willing and able to witness and this depends on the physiological culturing of the nervous system. Whether being witnessed or not, the Fourier filters can dramatically color "the eye of the beholder" and awareness is modified. Such coloring in momentum space-time might then color transformations to configuration space-time (movement and/or energy transfers), which again might color another unfolded level of consciousness, ad infinitum. We postulate that the magnitudes of colorings are further perturbed by the instantaneous spectrum of subtlety levels available from the combined influence of both personal and environmental group thought forms. We begin to catch better glimpses of the inherent SEER-SEEING-SEEN paradoxes. The holographic connection is indicated near the bottom of the Unfolding Creation via SEER-SEEN Transformations chart, between Subtle Consciousness and World Consciousness on the SEER-Fluctuations column. The subtle body is said to make bona fide

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EDMOND CHOUINARD Unfolding Creation via SEER-SEEN Transformations

SEER-Fluctuations

SEEN-Condensations

Consciousness of Floating Awareness

Transformations of Unfolding Matter

Filters in

space-time

Unity Consciousness (siva-sakti) Pure Intelligence

Particulants in (x, y) space-time

Vacuum State Causal Seedlings (bindu dots) Information Bits for Creation

God Consciousness (îsvara-nâdamantra) Lively Movement Creative Seed (bîja) Self Referential Set of Particulants Soul Consciousness (purusa) Individualized Causal Body Subtle Particles (prakrti and citta) Finest matter and mind Self Consciousness (ahamkâra) Expanding Ego and Will Nuclear Particles (tanmâtras) Emerging Subtle Senses Subtle Consciousness (âtman) Awareness of Subtle Body *holographic connection*

Atomic Structures (mâhabhutas) Notice of Physical Elements

World Consciousness (sarîra) Identity with Physical Body World Structures, Brain, Physical Senses, and SEEN Things Global Consciousness (devas) Correlation with Archetypes Global Weather, Calmness, Storms, and Violent Tornadoes

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contact with the brain of the physical body via a holographic connection. The associated energy wavelengths appear to be around We visualize the subtle mind as plasma like fog haze made up of large assorted clusters of photons. Structural resonance points, forming holographic connections, wrap themselves into precision superposition over corresponding locations of the physical brain, presumably located at appropriate specially evolved bio-potential sites and they may well be the tiny tubule structures at the ends of dendrites that have more recently become uncovered by brain physiologists. Whatever the detailed construction of holographic connections, their functional and operational features seem obvious. The SEER resides in the subtle mind and awareness results from stimulation of either the subtle senses or the physical senses. The holographic connection is quite transparent and we don't usually recognize it and thus we don't easily accept its existence. For the author, several quite extraordinary phenomenal experiences, some within the confines of a very high power plasma physics research laboratory, have given unusual relevance and persuasive meaning to this theoretical model. Activations from the subtle senses are many and they are variously called: imaginations, hallucinations, dreams, suggestions, hypnosis, paranormal sightings, psychic experiences, spiritual revelations, and religious personages. Awareness of the subtle senses is greatly diminished when the physical senses are active. When, and if, subtle sensory visual images reach the same magnitude as physical sensory visual images, then SEER awareness will experience two superimposed images, simultaneously. 3. Relativity And Consciousness QUESTION: So where are these worldwide pair of conjugate space-time lenses that allow the spirit essence (undoubtedly to eventually be stated in terms of non-local quantum phenomena) of SEER-Fluctuations to coagulate into SEEN-Condensations, and that allow SEEN-Condensations to reappear as new floating forms of SEER-Fluctuations? ANSWER: The SEER-SEEN space-time lenses, as depicted by forward and inverse spatial Fourier transforms, are a direct consequence of general relativity. It predicts that light is bent in near proximity of inertial mass. This is a well-proven experimental fact. The ability of a media to bend light is the definition of a lens. To get around the paradox that gravity cannot act on a photon whose 'rest' mass is zero, it is said that light bends because space-time is curved. Space-time is curved because of broken symmetries resulting from the occasional appearances of mass objects. The amount of curvature in any region of space-time is proportional to the amount of mass in that region. Nonlinear relationships among many competing forces and galactic movements give rise to inherently rich lens structures. As the universe is in continual expansion, the focusing power and nonlinear properties of these space lens structures are continuously varying. This helps explain the simultaneously arising, continuously changing states of world consciousness. Because of the vastness of this curved space-time that is spread out among various galaxies, clusters, super clusters, black holes, and dark matter, there is always a rich lens structure (heavy mass objects) to be found at some location that is deemed necessary for some specific transformation of awareness. Our consciousness comes to lively existence because of these lenses - because of the workings of general and special relativity. Minkowski 'time' of special relativity is the time 'experienced' by a swiftly moving object relative to laboratory clock. This 'time' becomes smaller and smaller, approaching

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zero, as the object's speed approaches the speed of light. A photon itself then apparently does not 'experience' any passage of time in traveling anywhere on the surface of the space-time cone. If we take a ride on a photon (and consciousness is here considered to be photon clusters in momentum space-time), our conscious awareness should then take NO time to move around the universe, and of course, this is our common experience. Instantaneously consciousness can move from New York to the Moon or anywhere else. Thus, it will also take No time to arrive at any desired Fourier filter space lens anywhere in the universe. Consciousness is omnipresent. 4. Summary Though there are yet innumerable aspects that need considerable attending, this model has been most effective in answering a number of daunting questions. It is much easier to understand paranormal phenomena now, for the subtle mind is completely free to go anywhere at all times. The subtle mind can spread itself thin, by expanding itself throughout the universe (sleep) or it can be very localized (focusing in the waking state). When two subtle minds so agree, a momentary superposition of their plasma fog haze structures superimpose and resonances among their respective holographic connections can communicate with one another (as in dreams, precognition, intuition, telepathy and other such expressions). Hypnosis now has meaning in that the subtle body does indeed have a methodology, via the holographic connections, of momentarily touching some other physical brain and/or subtle mind in a true parallel processing arrangement. Coma is now viewed as a subtle body that is mostly disengaged from the physical body but with autonomic brain functions continuing to maintain body survival functions. A drunken stupor can now be viewed as a wishful disengagement of the subtle body from the physical body, for whatever reason. Religious phenomena can become explained as delightful superimpositions upon varied forms of highly developed intelligences. Eternal life must be the reality if the subtle body continues on when holographic connections are completely severed from the physical body. The meaning of invocation now becomes extremely literal, not simply metaphorical. DNA coding probably derives from a causal body (soul) sourcing filter, which eventually coagulate to a subtle body and which eventually cohabitates a newly born physical body. Even Einstein's supreme observation that might now be recognized through direct personal experience, for indeed, energy-matter transformations are part and parcel of the continuing witnessing of the SEER-SEEING-SEEN dynamics of floating consciousness. I know of no scientific anomaly where this simple model does not supply reasonable and rational answers that are potentially verifiable with the tools of science. Most such answers can be experientially confirmed (if we allow it) and they do not disturb the integrity of existing scientific structures. Indeed, this model enhances the scientific culture. It further provides a rich basis under which to perform research in such mind-matter related topics, and an exciting laser metrology based experimental program, using sophisticated measurement tools of physics, is now quite productively under way. Bibliography Alper, Harvey; Mantra (State Univ of N Y Press, Albany, 1989) Anderson, James A; Practical Neural Modeling (MIT Press, 1989)

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PHOTONS FROM THE FUTURE

RALPH G. BEIL 313 S. Washington Marshal, TX 75670 USA

Abstract. What would be the consequences of assuming the existence of antiphotons which propagate backward in time? This is obviously permitted mathematically. There is also physical justification in the Wheeler-Feynman advanced radiation, the Einstein Nadelstrahlung, the Brittingham solutions of the wave equation, and a photon model of the author (presented at Vigier I). Interesting consequences of this assumption include explanations of the attractive Coulomb force as an exchange of antiphotons, quantum entanglement and ”teleportation,” the problem of action at a distance, and some characteristics of emission and absorption of radiation.

1. A Photon Model In order to achieve some idea of the possible nature of an antiphoton one should have in mind a fairly clear idea of a model of the photon itself. This is not as easy, however, as might be thought. As Einstein wrote in 1951 [1]: “All the fifty years of conscious brooding have brought me no closer to the answer to the question, ‘What are light quanta?’ Of course today every rascal thinks he knows the answer, but he is deluding himself.” If one is willing in spite of this to risk self-delusion, a good starting point for a comparison of various photon models is a review article [2] which appeared about ten years ago. This article compares many of the photon models which have been proposed both before and after Einstein’s remark and concludes that none is entirely satisfactory. Of course, many people take the position that photons do not exist as individual particles and thus no model at all is appropriate. I can only respond that, while many experiments can be explained without explicit reference to the photon, there are many other experiments, ranging from the Compton effect to measurements of atomic recoil at emission of radiation [3] to antibunching effects [4, 5], where the photon concept gives the most plausible explanation. 233 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 233-240. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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There is also an extensive set of experimental results on the inelastic scattering of electrons on photons [6]. These experiments show a definite particle-like structure for the photon. So for the sake of argument I will assume a certain model and see what the consequences might be. The model chosen here falls into the general category of electromagnetic wave bundle. There are two main features which distinguish this photon from any previous models. The first feature is that the photon is assumed to be of limited extent and to exist only along a single null path instead of over an entire spherical or plane wave front. In other words, I adopt Einstein’s preference for Nadelstrahlung, that is, needle radiation, over the traditional Kugelstrahlung which is spherical or plane wave radiation. Einstein proved in 1917 [7] that only needle radiation could satisfy the requirements of relativistic dynamics. This has received experimental verification [3] in results which show that a single atom has a certain direction of recoil as it radiates. This would not be the case if the radiation were spherical. The second feature of this photon model is the use of the Brittingham solutions to the Maxwell equations [8]. These are localized bundles of energy which can propagate in a null direction. They give a specific mathematical realization of wave packets which are needle radiation. The electromagnetic potential vector of the photon is taken to be a linear combination of these solutions. A version of this model was described in my presentation at the Vigier I meeting in 1995 and also appeared in the Proceedings volume [9], so I will only give a brief outline here. I use a set of rectilinear coordinates in the frame of the paricle. These coordinates are related to the laboratory coordinates by a Lorentz transformation. The transformation matrix forms an orthonormal tetrad I also use null coordinates U = X + T, V = X – T. The particle frame is taken to be transporting in the U direction along V = 0. The space part of the propagation is in the +X direction with Y and Z in the transverse plane. The photon resides on the moving frame and the frame transports with speed c in a null direction. This motion is determined by a little group which is a subgroup of the Poincaré group. The little group is composed of a rotation in the Y-Z plane and a null translation along X and T. This little group is described in detail in the book by Kim and Noz [10] who label it as “E(2)-like.” The group leaves invariant four-vectors in the U direction. The development of the model starts with a general assumption for the electromagnetic potential of the particle,

where the are components of the polarization, which is taken to be in the transverse or Y and Z directions.

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The functions F are solutions of the wave equation,

Note that the wave bundle solutions which are used are soliton-like solutions of the linear wave equation. A catalog of these solutions has been given by Donnelly and Ziolkowski [8]. The particular solutions used here are obtained by summing over advanced and retarded component waves and also by superposing waves in the transverse cylindrical direction. The form obtained for the electromagnetic potential of the photon is

This has a one-dimensional extension in the U direction and is easily seen to be a solution of (2). The constants k and m are wave numbers for propagation in the U direction so that the arguments kU and mU are invariant under Lorentz transformations in the X direction. It is easily seen that m determines the characteristic size of the photon while k is the wave number of the oscillation. This potential has a circular polarization. The form (3) allows for wave effects such as interference through the trigonometric part, but also can evidence effects of small extension through the exponential factor. The constant is shown in the previous work to be

The four-momentum has been computed to be

The helicity is calculated as The physical quantities computed in this model correspond to measured values of the photon. So the photon model combines both wave and particle features and represents a sort of compromise between point particles and plane waves. In a way, its one dimensional structure is reminiscent of string particle models. But the main point is that it is a specific mathematical realization of needle radiation. It is also somewhat of a synthesis of classical and quantum concepts. By the way, I deliberately do not specify how a quantum wave function is constructed from this potential, since there are differing prescriptions for doing so. Indeed, it is sometimes claimed that there is no consistent quantum wave function for the photon.

2. What is an Antiphoton? With a model in mind I now turn to the question of the antiphoton.

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In order to define an antiphoton one should first be clear what is meant by the antiparticle of a particle with mass To accomplish this, consider the four-momentum vector.

Now, very simply, antimatter is defiined as the result of the action of the charge conjugation operator C on the physical quantities of matter. For the four-momentum, by the CPT theorem, C is just reversal of both time and space components:

So antimatter is matter going backwards in time (and also space). This interpretation goes back to Feynman [11] and is the point of view taken in most standard works on quantum field theory, for example, the book by Kaku [12]. Antimatter is also described as negative energy matter, which is compatible with (7). A definition of the antiphoton consistent with the above picture of antiparticles is that the antiphoton also results from the action of C on the fourmomentum. Recalling (5):

This is a photon travelling backwards in time. It is a wave bundle moving in an advanced direction along a single null line. Note again that the bundle itself is a superposition of advanced and retarded component waves [8]. The antiphoton, since it carries negative energy backwards in time, lowers the energy of the source system. The energy change is the same as the change which would have taken place if the system had emitted a photon. This is consistent with quantum field theory [12]. One should get accustomed to the idea that the antiphoton is absorbed before it is emitted. How does this antiphoton compare with previous theories? First, like the old Wheeler-Feynman picture the new theory uses advanced waves. However, the Wheeler-Feynman advanced waves are emitted over an entire spherical surface and going backward in time. Here, I have individual photons as needlelike radiation emitted at points of limited extent (say, atoms) in the future. The time symmetry of the new theory is point to point rather than sphere to point. Instead of the implausible assumption of a large number of emitters coordinating to send a coherent wave which converges at a point in the past, there are simply single antiphotons propagating from future emitter to past absorber. A wave of any shape could be formed by superposition of individual antiphotons. An even earlier precedent for the antiphoton is the work of G. N. Lewis [13] who actually invented the name of the photon in 1926.

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Lewis emphasized the time symmetry of radiation and also its point to point nature. Actually, the new theory is very similar to the direct particle interaction theory of Hoyle and Narlikar [14]. The means of direct interaction is here specifically identified as the photon. In another sense one could claim that both photons and antiphotons only exist as entities with emission at one end and absorption at the other. Second, a similar comparison could be made with the photons of QED which are plane waves, a limiting case of spherical waves. It might be interesting to try to use the needle radiation wave bundles as the photon states in QED instead of plane waves, each of which extends over the whole universe and has infinite energy. It could be expected that states of limited extent and finite energy might solve the divergence problems of QED without having to use renormalization. Third, it can be noted that the Evans-Vigier B(3) theory also makes use of antiphotons [15]. However, in their theory the photon has mass and so the antiphoton is an antiparticle of the conventional type. Finally, I mention some very recent results where materials have been fabricated which have a negative index of refraction [16]. This can be interpreted as light going backwards in time. One explanation is that photon-antiphoton pairs are created at the surface of the material.

3. Some Consequences of the Antiphoton

The assumption that the antiphoton actually exists leads to several interesting consequences. For example, it is usually said that the photon is the mediating particle for the Coulomb interaction. It is easy to see that an exchange of photons could produce a repulsive force. But what about the attractive force? It is difficult to visualize how a photon with positive momentum could transfer negative momentum to a charged particle. But the mechanism is much more plausible if the mediating particle is an antiphoton. Each bounce of an antiphoton between two oppositely charged particles transfers negative momentum. This explains the attractive force. The negative energy of the mediating photons explains why the Coulomb potential well has negative energy. The radiation from an atom as an electron changes from an excited state to a state of lower energy is also easily explained. The mechanism is the creation of a photon-antiphoton pair. The photon appears as an ordinary quantum of radiation; the antiphoton remains in the atom as a virtual particle and lowers the energy level by an amount equal to the energy of its photon counterpart. If the atomic state is stable, the antiphoton exchange is an equilibrium process and there is no radiation. When a photon is absorbed by an atom it annihilates a virtual antiphoton and raises the energy of the system.

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The same processes of emission and absorption also apply to atoms of antimatter since the force is still attractive and is still mediated by antiphotons. For repulsive systems the exchange particles are photons. Radiation could be photons which are just exchange photons escaping from the system. It could, however, also be antiphotons resulting from annihilation of pairs with the exchange photons. The energy level in both cases is lowered due to a decrease in the number of exchange photons. Absorption could either be of photons (which add to the number of exchange photons) or of antiphotons in a pair creation. In both cases the energy level is raised due to an increase in exchange photons. Another area to which the antiphoton concept can be applied is recent results in the foundations of quantum theory. It is not difficult to see that the existence of photons going backward in time can explain a whole range of experiments involving quantum entanglement, spooky action at a distance, or even quantum “teleportation.” To illustrate the basic process imagine a point A in space-time and two other points B and C both on null paths going forward from A but separated from each other by a spacelike distance. Now imagine a photon and an antiphoton created simultaneously at A with the photon propagating forward in time to B in the “normal” fashion, but the antiphoton propagating backward in time from C to A and arriving at A at just the right instant to participate in the pair creation. Another way to think of this process is that there is only one photon involved. It starts at C and is an antiphoton during its journey from C backwards in time to A. At A it becomes a photon and travels forward in time to B. This interpretation was given long ago for massive particles by Feynman [11]. Now it is clear that the states of this photon and antiphoton can be correlated or entangled so the quantum information at B can be correlated with the quantum state at C even though B and C are separated by a spacelike distance. Thus spooky action at a distance not only can occur, but actually does occur, as measured in many experiments. This nonlocality must be taken into account in fundamental quantum theory. So if antiphotons exist, then the whole EPR/Bell Theorem discussion will have to be reopened. This has been recognized for a long time [17]. The basic assumptions of causality and locality can no longer be taken for granted. This might even give new life to hidden variable theories.

4. Conclusion The concept of the antiphoton has been proposed as an alternative which could apply to both classical and quantum theory. This antiphoton is derived from a specific photon model which uses the ideas of needle radiation, Brittingham wave bundles, and moving frames. An idea as simple as the application to quantum theory of a photon prop-

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agating backward in time should have been proposed before now. However, no precedents have been found. There is a large body of literature on time reversal and time symmetry, but nothing specifically on antiphotons. A final question: What would be a convincing experimental proof of the existence of the antiphoton? There is a general type of experiment which is technically feasible. Take a system which allegedly emits antiphotons and measure the time of arrival of these quanta at a second system. If this event lies on a forward light cone from the point of emision, the quantum is a photon. If it is on a backward light cone from the emission event it is an antiphoton. This is basically a nonlocal measurement of the type mentioned which involves quantum action at a distance. In fact, one could almost infer that the results of those experiments have already proven the existence of the antiphoton. References [1] Einstein, A. (1951) Letter to M. Besso, quoted in R. McCormmach, ed. Historical Studies in the Physical Sciences Vol. 2 (1970), University of Pennsylvania Press, Philadelphia, pp. 38-39. [2] Kidd, R., Ardini, J., and Anton, A. (1989) Am. J. Phys. 57, 27. [3] Piqué, J. L., and Vialle, J. L. (1972) Opt. Commun., 402. [4] Walls, D. F. (1979) Nature 280, 451 [5] Diedrich, F. and Walther, H. (1987) Phys. Rev. Lett. 58, 203. [6] Nisius, R. (2000) Phys. Rep. 332, 165. [7] Einstein, A. (1917) Phys. Z. 18, 121. [8] Donelly, R. and Ziolkowski, R. (1993) Proc. Roy. Soc. London A 440, 541. [9] Beil, R. G. (1997) in S. Jeffers et al (eds.) The Present Status of the Quantum Theory of Light, Kluwer, Dordrecht.

[10] Kim, Y. S. and Noz, M. E. (1986) Theory and Applications of the Poincaré Group, Reidel, Dordrecht. [11] Feynman, R. P. (1949) Phys. Rev. 76, 749. [12] Kaku, M. (1997) Quantum Field Theory, Oxford, pp. 75 and 91. [13] Lewis, G. N. (1926) Nature 118, 874. [14] Hoyle, F. and Narlikar, J. V. (1974) Action at a Distance in Physics and Cosmology, Freeman, San Francisco.

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[15] Evans, M. and Vigier, J.-P. (1994) The Enigmatic Photon Vol. 1, Kluwer, Dordrecht. [16] Smith, D. et al (2000) Phys. Rev. Lett. 84, 4124. [17] Sutherland, R. I. (1983) Int. J. Theor. Phys. 22, 377.

CAN ONE UNIFY GRAVITY AND ELECTROMAGNETIC FIELDS ? J-P. VIGIER Université Paris VI - CNRS Gravitation et Cosmologie Relativistes Tour 22-12 4 ème étage - Boîte 142 4, place Jussieu, 75252 Paris Cedex 05 R.L. AMOROSO Noetic Advanced Studies Institute – Physics Lab 120 Village Square MS 49, Orinda, CA 94563-2502 USA [email protected]

Abstract. This paper presents an attempt to unify gravity and electromagnetism associated with and in the covariant density distribution of a real average covariant Dirac aether built with extended random elements filling flat spacetime. Some possible experimental tests are also discussed. 1. Introduction The problem of the unification of gravity and electromagnetism into a single theory is as old as Modern Science itself and it has not been solved until now. Despite the initial discovery of similar forms of the Newton and Coulomb potential the two theories are still developping independently. Until the present, unification has been attempted mainly (as a consequence of Einstein’s discoveries) by Einstein himself [1], following Schrödinger [2], Maxwell [3] (and their present successors) within a frame associating electromagnetism with new geometrical properties of spacetime. The aim of the present paper is different. Following MacGrégor [4], Puthoff [5], and others, both fields are represented by fourvector field densities and one considers both types of phenomena as different types of motions within the same real physical zero-point field in flat spacetime, i.e. as two different aether types of collective perturbations carried by a single aether field moving in such a space. Since this approach suggests new types of experiments and yields an interpretation of unexplained new effects it will (perhaps), if confirmed, help to disantangle the present theoretical discussion. This model has the following experimental basis : I) The first basis (observational) is that the observable universe apparently does not change with distance [15] (as it should with big-bang type theories) and the ratio of 241 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 241-258. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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the local 2.7° microwave radiation is only isotropic in a specific absolute inertial frame so that the velocity of light not only changes with its direction (which suggests a non-zero photon mass

0) but is also isotropic in

, in time.

II) The second basis is that our essential instrument of (distant) observation (i.e. electromagnetic waves) is more complex than its initial discoverers (Maxwell and Ampere) thought. Newtons initial guess that light was both waves and particles (photons) was later confirmed by Einstein in 1905. The discovery by Fresnel that these waves were essentially transverse (i.e. with possible zero mass and invariant velocity of propagation) was later completed by de Broglie’s and Einstein’s discovery that one could write E = h v = (with so that individual massive photon’s can be considered as piloted by real non zero-mass Maxwellian waves i.e. by new properties of the Sagnac effects in a recent experiment of Levit et al. [7] which shows that the electromagnetic field should be represented by a vector density As shown by Aharonov-Bohm effect, this implies that the electromagnetic field is not completely represented by the fields [6,7]. III) The third basis has its theoretical origin in the introduction by Dirac et al. of a real covariant chaotic physical aether which fills space-time, carries real physical observable wave-like and particle like (soliton-like) perturbations or local extended elements, whose four momenta and angular momenta are statistically and evently distributed on specific hyperbolic surfaces, at each given point, in all given inertial frames. This vacuum distribution thus appears, FAPP, as invariant isotropic chaotic and undetectable (except in specific physical cases) for all inertial observers. The form taken by an aether within Relativity Theory carries both particles and waves is now discribed in terms of collective motions on the top of a real essentially stochastic covariant background. Such an aether theoretically justifies the statistical productions of Quantum Mechanics (in its causal stochastic interpretation) and SED theory, and has a direct experimental justification in the Casimir effect. This implies a background friction (associated with absolute local conservation of total momentum and angular momentum) and collective motions which provide a new interpretation of the observed cosmological red-shift [22, 23] and yields new possibilities to interpret (also in terms of local frictions) the anomalous red-shifts observed by Arp, Tifft and other astronomers [8]. On these bases, we shall, in section 3, recall results showing that one can describe the gravitational results of General Relativity in Maxwellian terms. In section 4 we develop a possible unification model of both theories. Section 5 then contains a brief discussion of possible consequences of the preceding attempt. This aether is locally defined by a particular real Poincaré frame in which (measured with real physical instruments) the velocity of light is identical in all directions at all observable frequencies. All observers tied to other frames passing through local inertial motions will see (measure) different space-time properties (associated with their velocity and

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orientations) defined by the corresponding Poincaré transformations.1 The local variations of physical properties of the aether correspond to local transitions relating differential inertial frames at neighbouring points. 2. A Real Physical Aether In Flat Spacetime Since the starting point of this model is the existence of a real physical vacuum (or zero point field) built with extended wave-like individual elements[9, 10]centered on points in an external flat space-time, such elements can overlap and interact (i.e.carry) collective motions corresponding to excess (electromagnetic ‘bumps’) or defects (gravitational ‘holes’) in the average density of the local aether elements. The model can be described F.A.P.P. as a gas of extended elements within flat space-time. These elements can interact locally (i.e. carry collective motions) and the gas’ local scalar density thus carries waves (and solitons) associated with excess (electromagnetic) or defects (gravitational) in density, with respect to the average local vacuum density. One thus defines field variables associated with these two possible (excess or defect) local density variations. The vector fields, for example, in this paper, represent localized excess or density defects w.r.t. the local vacuum density. This model thus implies: a) a description of real physical vacuum properties in terms of real extended vacuum elements average behaviour. b) a description of the behaviour of its collective defects (below average) associated with observed gravitational effects c) a description of the behaviour of its collective excess (above average) associated with recently observed electromagnetic effects. The introduction of such new concepts into Maxwell’s equations and the description of gravitational fields along the same lines (in terms of vector fields suggests (as we shall now see) a new type of unification of both theories. We shall discuss some of its prospects keeping in mind the restriction that, since new experiments are under way, it cannot yet be given a complete form. Instead of looking for a common geometrization of gravity and light (i.e. their unification within a unique form of extended space-time geometry) one could assume following Newton and Lorentz : A) That the evolution of extended (fields) and of localized (sources) in terms of 1) vacuum (aether) 2) gravitational fields, 3) the electromagnetic field, reflects the time evolution (motions) and interactions of perturbations of a real material substance moving in a 3-dimensional flat space. This means that all three field and particle subelements are localized at given points, at each instant, in this 3-space and move continuously (i.e. locally transform) according to causal laws2

1

To quote Kholmetsky In order to pass from one arbitrary inertial frame to another one it is necessary to carry out the transformation from to the absolute frames and then from to 2 As a consequence of the failure of the geometrical unification program Einstein himself was still obliged in 1954 to consider the electromagnetic field as filling curved space-time. He never reached a final satisfying model.

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This assumption (distinction of space and fields) is now supported by the existence of a special particular experimental inertial cosmological frame in which - the 2.7°K microwave radiation frame is isotropic and non rotating - The average distribution of different types of galaxies (spiral, elliptical, Q.S.O’s) is isotropic and does not change with distance [15]. - The observable anisotropy of the velocity of light propagation in different directions and around massive objects reflects the real motions of real fields described w.r.t. the frame in any real inertial Poincaré frame by covariant (local) four-vector scalar chaotic average density i.e. by average four-vectors

around each absolute space-time point where

denotes average measures taken in

in 3

B) That all real physical observations rest on : 1. The utilisation of real physical apparatus based on electromagnetic fields and gravitational material with charged (or uncharged) particles. 2. On observers also built with the same material i.e. influenced by the said fields and particles. In other terms all observers (and their observations, inertial or not) are an integral part of fields and particles since they are part of the same overall real field and particle distribution. This fact determines their relation with all real phenomena. A physical theory should explicitly provide (within its context) a definition of the means whereby the quantities with which the theory is built and can be measured. The properties of light rays and massive particles are thus sufficient to provide the means of making basic measurements. Since real clocks and rods are the real instruments utilized in physics, we shall thus first define, for an individual inertial observer, the behaviour of such instruments with respect to each other: since this determines, for every inertial observer possessing them, the behaviour, with respect to ,of the material fields around him. As a consequence of the covariant distribution character observed in the very small resistance to motion and assumed non-zero photon rest mass, real spin of possible extended vacuum sub-elements and their internal possible motions (and associated local interactions) one can describe the four-momenta and angular momenta of all extended subelements passing through a small four-volume with a constant average density on a hyperboloid The four-momenta and angular momenta of extended elements are distributed at each point

with constant density

on

space-like hyperboloids. C) Following an idea of Noether the local analysis of moving fields and extended particles at each point by real observers tied to this point, is defined by local clocks and 3

This implies 1) the existence of a basic high density of sub-elements in vacuum, 2) the existence of small density variations above (for light) and below (for gravity) the average density with the possibility to propagate densityvariation on the top of such a vacuum model as initially suggested by Dirac.

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rods which move with the corresponding element. It is thus locally performed at each point of coordinates which follows a world-line L.. To this point are attached local (in

internal

variables

properties and thus depend on

which describe its neighbourhoods physical

The evolution is given by

denotes the proper time dertivative w.r.t.

when

where

describes a world-line L.. A

scalar Lag-rangian thus represents the evolution of the real physical medium in which depends on a local Lagrangian L and is thus given by Poisson brackets. This description on is assumed to correspond to local space-time translations and four dimensional rotations which are determined by a Lagrangian L invariant under the local group of Poincaré transformations (i.e. the inhomogeneous Lorentz group). They contain [15] : 1) the operators

of infinitesimal translations of

only

and

can

be

described by 2) The operators

of infinitesimal four rotations in

which act simultaneously on

and on the internal variables. We have at

Their action on internal local variables depends on their choice. 3) A choice of L leads to the momenta

yielding a constant impulsion vector

and the total angular momentum:

so that with These quantities satisfy the Inhomogeneous Lorentz group commutation relations

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i.e. Poisson Group Relations :

With these quantities one can also define local conservation laws for free elements i.e.

and introduce a constant local mass term 4) An associated center of gravity

associated with

with

is defined by the introduction of the four-vector

i.e.

which implies that locally extended real media in first suggested by Yukawa. 5) An inertial mass (usually not constant)

can also be attributed to

are described by pairs of points as

defined by

being located at

since one has:

GRAVITY AND ELECTROMAGNETIC FIELDS so that the motion of

is locally rectlilinear and

(with

247

has a proper time

,

and we have :

and

w.r.t. the center of gravity. Local instantaneous four rotations are described by : A specific

beigrössen four-frame

with and

A specific four-frame along

centered on

with

for

and

This set of relations must be completed by relations which will define the interactions between the extended elements i.e. the propagation in the aether of collective motions corresponding to observed gravitational and electromagnetic phenomena. Before the introduction of such interactions one must recall that such proposals have already been made in the past. We only mention here: - Weyssenhof’s proposal [9] extensively discussed in the literature. - Nakano’s proposal [12] - Roscoe’s proposal with photon mass [13]. 3. Polarizable Vacuum Representation Of General Relativity Since all observed effects of gravity in distant space rest on light observation (including and radio em waves coming through space from distant sources) a simple model endows the polarizable vacuum with properties that might account for all the phenomena in terms of distorsions. This initial proposal of Wilson and Dicke has been recently revived with astonishing success by Puthoff [5] and Krogh [14]. We first summarize their model and will complete it with a supplementary mass term in electromagnetism. One starts from the idea that in flat space the electric field moves in a real vacuum medium with a point varying dielectric constant K: so that this D field satisfies the vacuum equation: This corresponds to a variable fine structure constant

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so that the vacuum has permittivity and permeability constants given by

and an impedance

to satisfy Eötvos-type experiments. The

local velocity of light for a given frequency v varies like i.e like The corresponding principle of equivalence implies that the self energy of a system changes when K changes; so that a flat-space energy in flat space changes into

and one has

As a consequence the condition

along with the time and length variations

becomes

and

given by the relations:

These relations are evidently equivalent to a local curvature of space. Indeed a length rod shrinks to and would measure where the rod remains rigid, is now expressed in terms of dx-length rod as Using the same argument for dt and we find that one can write:

which transforms into

i.e.

with In the case of a spherically symmetric mass distribution one writes

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where G is the gravitational constant, M the mass and r the distance from its origin located at the center of mass. Puthoff [5] has recently shown that this model accounts (sometimes with better precision) for all known experimental tests of General Relativity in a simple way i.e. one can describe The gravitational redshift given by (so that has a 1/100 precision). The bending of light rays by the sun and stars. The advance of the Perihelion of Mercury. He has also shown that one can derive the form of (22) from a general Lagrangian with a variable K i.e. leaving aside vacuum interaction,

in

This association of gravitational theory with electromagnetic theory based on the introduction of a variable dielectric vacuum constant K has recently been made more explicit by Krogh [14]. Noting that: a) Electromagnetic theory implies the effects of electromagnetic vector fourpotential vectors on the phases S of quantum mechanical waves so that one has

for charged particles moving under the influence of the four vector, b)

If

is the mass term introduced into Maxwell’s equation) the force

on charged particles takes the form

where the first term is the usual transverse Poynting force on currents and the second a longitudinal force along currents (resulting from non zero photon mass) recently observed by Graneau [11] and Saumont [16].

250 c)

J-P VIGIER & R. L. AMOROSO One can describe gravity with a four-vector density

so that the

gravitational (Newton) and electromagnetic (Coulomb) potentials have the same form, but different coupling constants. This suggests that both wave fields and singularities are just different aspects of the same fundamental field. 4. Extension of Maxwell’s Equations This discussion opens the possibility to test new types of extensions of Maxwell’s equations in the laboratory. Since this has already been attempted some results (derived within the frame of the model) are given here: a) From a non-zero vacuum conductivity coefficient we have in vacuum div E =0 with curl and div H = 0 with curl E b)

From an associated non-zero photon mass term

F.A.P.P.) where

(with

denotes the total four-potential density in Dirac’s aether model.

This introduces a non-zero fourth component of the current

(where

into the vacuum corresponding to a real detectable space. Within the present technology this implies that the present really carries space-charge currents [17] (so that the divergence of the electric field is different from zero and the corresponding existence of a displacement current (i.e. a curl of the magnetic field) and its associated current density4. 4.1 Massive Photons A unification of massive spin 1 photons piloted by electromagnetic waves built with massive extended sub-elements has been developed in a series of books by Evans, Vigier et al. [6] The model implies the introduction of spin and mass with an associated energyless magnetic field component in the direction of propagation and a small electrical conductivity in the Dirac vacuum also implying a new mechanism [6, 22]. Corresponding equations will be given below. In the inertial frame all massive particles are governed by a gravitational potential four-vector associated with a small mass which can be decomposed into transverse, longitudinal and gradiant potentials. We can thus associate the relations

4

Such attempts have been recently published in a book by Lehnert & Roy [18] so we shall only present a summary of some results and assumptions.

GRAVITY AND ELECTROMAGNETIC FIELDS

which represent the electromagnetic field in vacuum in any inertial frame relations:

which represent the gravitational field in the same vacuum; where density,

to the mass current and

mass (both very small in

and

grams) and

251

the

refers to the mass

to electromagnetic and gravitational in the ~ terms

represents the corresponding wave velocities (which except depend on the directions in flat space-time) so that one has:

where is the value in the absence of a gravitational potential

In this model, one

assumes, with Sakharov, that the gravitational field corresponds to local depressions in the immensely positive energy of the zero-point field; and gravitational fields represent regions of diminished energy (i.e. that their momentum gravity corresponds to holes in vacuum energy or local defects of vacuum elements). Their effective momentum is thus opposite and corresponding gravitational forces are attractive. Such an association also suggests that although measuring devices (i.e. observations) in local inertial Poincaré frames are altered by gravitational potentials (they are part of the same real physical background in this model). There is no effect on the geometry of flat space and time. For any given real inertial local Poincaré frame real space is Euclidean and one uses Poincaré transformations between and to describe real motions which include consequences of gravitational potentials. For example a reduction of the velocity of quantum mechanical waves, including light, is taken as a fundamental effect of gravitational potentials. Clocks are slowed and measuring rods shrink in such potentials by a factor 4.2 Divergence of the Electromagmetic Field

A non-vanishing divergence of the electric field given below, can be added to Maxwell’s equations which results in space-charge distribution. A current density arises in vacuo and longitudinal electric non-transverse electromagnetic terms (i.e. magnetic field components) appears (like ) in the direction of propagation. Both sets of assumptions were anticipated by de Broglie and Dirac. They imply that the real zero-point (vacuum) electromagnetic distribution

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- is not completely defined by four-vector density

but by a four-vector field distribution given by a

associated with a de Broglie-Proca equation i.e.

and its complex conjugated equation. - that the field potential equation also contains a gradient term so that one has in vacuum (20): with

(F.A.P.P) and a small electrical conductivity in vacuo.

5. New Possible Consequences

Since such models evidently imply new testable properties of electromagnetic and gravitational phenomena we shall conclude this work with a brief discussion of the points where it differs from the usual interpretations and implies new possible experimental tests. If one considers gravitational and electromagnetic phenomena as reflecting different behaviours of the same real physical field i.e. as different collective behaviour, propagating within a real medium (the aether one must start with a description of some of its properties. We thus assume A) that this aether is built (i.e. describable) by a chaotic distribution of small extended structures represented by four-vectors round each absolute point in This implies - the existence of a basic local high density of extended sub-elements in vacuum - the existence of small density variations above for light and below for gravity density at - the possibility to propagate such field variations within the vacuum as first suggested by Dirac [17]. One can have internal variations: i.e. motions within these sub-elements characterized by internal motions associated with the internal behaviour of average points (i.e. internal center of mass, centers of charge, internal rotations : and external motions associated with the stochastic behaviour, within the aether of individual sub-elements. As well known the latter can be analyzed at each point in terms of average drift and osmotic motions and distribution. It implies the indtroduction of non-linear terms. Tysis has been developed by MacGregor [4], Guerra and Pusterla and Smolin. To describe individual non-dispersive sub-elements within where the scalar density is locally constant and the average equal to zero, one introduces at its central point

a space-like radial four-vector

(with

GRAVITY AND ELECTROMAGNETIC FIELDS = constant) which rotates around

253

with a frequency

At both

extremities of a diameter we shall locate two opposite electric charges and (so that the subelement behaves like a dipole). The opposite charges attract and rotate around with a velocity The and electromagnetic pointlike charges correspond to opposite rotations (i.e ± perpendicular to

located at

and

rotates around an axis

and parallel to the individual sub-element’s four

momentum If one assumes electric charge distributions correspond to and gravitation to one can describe F.A.P.P. such sub-elements as holes around a point 0 around which rotate two point-like charges rotating in opposite directions as shown in Figure 1 below.

These charges themselves rotate with a velocity

at a distance

(with

= Const.). From 0 one can describe this by the equation

with force equation

along with the orbit equations for

and

we get the

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J-P VIGIER & R. L. AMOROSO

and the angular momentum equation:

Eliminating the mass term between (31) and (33) this yields

where is the electrostatic energy of the rotating pair. We then introduce a solitontype solution

where

satisfies the relation (31) with

so that one can add to

a linear wave

i.e.

(satisfying

which

describes the new average paths of the extended wave elements and piloted solitons. Within this model the question of the interactions of a moving body (considered as excess or defect of field density, above or below the aether’s neighbouring average density) with a real aether appears immediately5. As well known, as time went by, observations established the existence of unexplained behaviour of light and some new astronomical phenomena which led to discovery of the Theory of Relativity. In this work we shall follow a different line of interpretation and assume that if one considers particles, and fields, as perturbations within a real medium filling flat space time, then the observed deviations of Newton’s law reflect the interactions of the associated perturbations (i.e. observed particles and fields) with the perturbed average background medium in flat space-time. In other terms we shall present the argument (already presented by Ghosh et al. [19]) that the small deviations of Newton’s laws reflect all known consequences of General Relativity The result from real causal interactions between the perturbed local background aether and its apparently independent moving collective perturbations imply absolute total local momentum and angular momentum conservation resulting from the preceding description of vacuum elements as extended rigid structures. 5 As remarked by Newton himself massive bodies move in the vacuum, with constant directional velocities, i.e. no directional acdeleration, without any apparent relation friction or drag term. This is not the case for accelerated forces (the equality of inertial and gravitational masses being a mystery) and apparent absolute motions were proposed by Newton and later contested by Mach.

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6. Inertia And Vacuum Drag As Possible Extension of Newton’s Model

If one starts from an aether built with moving small extended structures with an average real distribution isotropic in an inertial frame i.e. examine the effects in a given inertial frame I centered on a point of the real vacuum distribution on a test particle moving with absolute velocity

and angular momentum

one can

evaluate more precisely, the collective interactions carried by this aether between two extended neighbouring regions centered on points A and B with two centers of mass situated at and If we start with i.e. gravitational effects, it appears immediately a) that if one assumes the gravitational potential is spherical in the rest frame of its source B, b) that the motion of A undergoes a velocity dependent inertial induction w.r.t. A i.e. a friction depending on the velocity v of A w.r.t. B c) that this motion is also submitted to an acceleration dependent inertial w.r.t. i.e. also an acceleration depending on its acceleration a measured in d) possible terms depending on higher order time derivations which we will neglect in the present analysis we can write (19) the force on A due to B in in the form where

The terms G, G’, G” are scalars possibly dependent on v. The terms and the gravitational masses in Û, is the unit vector along and

are must

have the same form i.e. 1/2 or cos If we also accept the preceding velocity dependent analysis for contracting rods and retarded clocks then we should write G = G’ in (38) and take as done by Ghosh [19]. Moreover, if we compare the form given by Weber to the repulsion of two electric charges of the same sign:

corresponding to electromagnetism, with the recent form given by Assis [18] to attracting interacting masses and i.e.

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J-P VIGIER & R. L. AMOROSO

We see they have exactly the same form; the difference of their coefficients being compatible (within our interpretation) since they correspond to opposite variations of the average vacuum density. Their interpretation in terms of (for electromagnetism) and (for gravitation) also explains (at last qualitatively) why extended depressions repel or attract when they rotate through parallel or antiparallel directions and only attract when This also explains why a reduction of attraction between two masses has been observed when one puts another mass between them (the LAGEOS satelite). In this model this similarity is indeed comparable to similar behaviours of vortices for gravitation and Tsunamis for electromagnetism on an ocean surface. If one assumes the absolute local conservation of four-momentum and angular momentum in regions containing the preceding aether carrying its associated collective electromagnetic and gravitational motions one can evaluate the effects of their interactions. With a real physical aether there is no such thing as free electromagnetic or gravitational phenomena. Drag theories (described as inertial induction are always present and responsible for Casimir type effects in the microscopic domain. Real consequence of the aether appear, at various levels, in the macroscopic and cosmological domains... as has already been suggested in the literature and tested in laboratory or astronomical phenomena. We only mention here: 1) The possible consequences of modifications of the Newton and Coulomb forces testable in the laboratory. 2) The redshift and variable velocity of electromagnetic waves which results from the rotational inertial drag of extended photons moving in vacuum: an effect already observed in light traversing around the earth [20]. 3) The possible measurable existence of the redshift of transverse gravitational waves... possible in the near future. 4) Observational redshift variations of light emited by Pioneer close to the solar limb i.e. also of photons grazing a massive object [20]. 5) The observed anisotropy of the Hubble constant in various directions in the sky [20] associated with various galactic densities. 6) Observed torques on rotating spheres in the vicinity of large massive bodies. This also appears in some experiments, i.e.: a) Secular retardation of the earth’s rotation. b) Earth-moon rotation in the solar system etc. 7) Apparent evolution with time of angular momentum in the solar-planetary system. 8) Different variation of redshift of light travelling up and down in the Earth’s gravitational field... Which also supports existence of photon mass.

7. Conclusions

As stated in the introduction of this paper, this model exploits : a) the analogy (underlined by Puthoff) between the four vector density representation of gravity and electromagnetism in flat space-time [5]

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257

b) the possibility of describing the causality of quantum mechanical phenomena in terms of extended solitons piloted i.e. by quantum mechanical potentials, by real guiding collective waves on a chaotic, polarizable Dirac-type aether - both moving in a flat space-time [20]. c) the representation of this real vacuum (Dirac aether) in terms of the chaotic distribution of real extended elements moving in the flat space-time. d) the introduction of internal motions within extended sub-elements and their relation with local collective motions i.e. the relation e) the representation of the electron (and its associated pilot-wave) in terms of extended elements with a point-like charge rotating around a center of mass [20]. These assumptions yield realistic physical characteristics to known empirical properties and predict new testable relations besides known properties of elementary particles. The present model must thus be extended, by associating new internal motions to these known properties and interpret them in terms of new strong spin-spin and spin-orbit interactions. The new predictions should be confronted experimentally to see if they represent a valid starting point for further theoretical and experimental developments. One of the justifications of the present attemp is the existence of electromagnetic phenomena not explained by Maxwell’s equations. As discussed by Barrett [21] Maxwell's theory does not explain the Aharonov-Bohm (AB) effect and Altahuler-Aharonov-Spivak (AAS) effects. It does not cover the topological phase question i.e. the Berry-Aharonov-Anandan, Pancharatnam and Chio-Wu phase-rotation effects. An inclusion of Stoke’s theorem is necessary and results of Ehrenberg and Siday must be analysed. The quantum results of Josephson, Hall, de Haas and van Alphen Sagnac-type experiments also need clarification. In other words Maxwell and his direct followers have discovered a continent. Its exploration of still lies in front of us ! Appendix The unification of gravity with electromagnetism attempted by Sakharov and Zeldovich rests on the following ideas : a) Gravitation results from perturbations of the zero point field (Z.P.F.) b) It results from a radiation pressure of the zero-point field: i.e. in this paper, - Sakharov has proposed to describe (Newton-Covendish constant) by

where G is the gravity vector,

is the Planck length and

258

J-P VIGIER & R. L. AMOROSO - Gravity thus corresponds to variations ofthe

drift field

Where S = Poynting vector c) A perturbed Z.P.F. modifies gravity. References 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23.

A. Einstein, Geometry and Experience Sidelights on Relativity, Denver, 1922 L. Bass and E. Schrödinger, Proc.Roy.Soc.A 232,1955 J.C. Maxwell, A Treatise on Electricity and Magnetism, Dover (1954). H. Mac Gregor, Stationary Vacuum Polarization, (2000). H.E. Puthoff, Polarizable-Vacuum, Representation of General Relativity, 1999, Austin and Phys.Rev.A, 1989 T.B. Andrews, Observed Tests and Theory of the Static Universe, 2000, preprint. M.W. Evans and J.P. Vigier, The Enigmatic Photon, Kluwer, 1996 L. de Broglie, Mécanique Ondulatoire du Photon, Paris: Gauthier-Viilas (1955). L. de Broglie and J.P. Vigier, Phys. Rev. Let. D. Bohm & J-P Vigier, Phys. Rev. 109 (1958), p. 1882 C. Levitt et al. Unpublished. H. Arp, Quasar Redshift and Controversies Interstellar Media, USA, 1987 J.V. Weyssenhof and A. Raabi, Act.Phys.Pol. 9, Kluwer, Ed. J.P. Vigier, Phys.Lett.A., 235, 1997 P. Graneau, 2002, Why does lightning explode and generate MHD, preprint I. Nakano, Progress in Theoretical Physics, (1956), 15. D.F. Roscoe, Maxwell’sequations, 1997, preprint. K. Krogh, Gravitation without Curved Space -Time, 2002, Submitted to General Relativity & Gravitation. F. Halbwachs, J.M. Souriau and J.P. Vigier, Journal de Physique et de Radium, 1922, 22, p. 26. See also F. Halbwachs, Théorie Relativiste. Fluides . Paris: Gauthier-Villars, 1960. F. Saumont, Proc. Cold Fusion and Now Energy Symposium, Manchester, 1998 P.A.M. Dirac, Nature, 1951,906 B. Lehnert, & S. Roy, Extended Electromagnetic Theory, World Scientific, 2000. A. Ghosh, Origin of Inertia, Apeiron, Montreal, 2000. R. Pound and G.A. Rebka, Physical Review Letters 4, 337, 1960. J-P Vigier & S, Maric, to be published. T.W. Barrett, Electromagnetic Phenomena not Explained by Maxwell’s Equations: Essays on the Formal Aspects of Electromagnetic Theory. Singapore: World Scientific, (1993). Amoroso, R.L., Kafatos, M. & Ecimovic, P. 1998, The origin of cosmological redshift in spin exchange vacuum compactification and nonzero restmass photon anisotropy, in G. Hunter, S. Jeffers & J-P Vigier (eds.), Causality & Locality in Modern Physics, pp. 23-28, Dordrecht: Kluwer. R.L. Amoroso, The Origin of CMBR as Intrinsic Blackbody Cavity-QED Resonance Inherent in the Dynamics of the Continuous State Topology of the Dirac Vacuum, in R.L. Amoroso , G. Hunter, M. Kafatos & J-P Vigier (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 2002 Dordrecht: Kluwer.

THE DIPOLAR ZERO-MODES OF EINSTEIN ACTION An Informal Summary With Some New Issues

GIOVANI MODANESE California Institute for Physics and Astrophysics 366 Cambridge Ave. Palo Alto, CA 94306 USA

And University of Balzano - Industrial Engineering Via Sernesi 1, 39100 Bolzano, Italy Email: [email protected]

Abstract.

In this note we describe a set of gravitational field configurations, called ”dipolar zero modes”, which have not been considered earlier in the literature. They give an exactly null contribution to the pure Einstein action and can thus represent large vacuum fluctuations in the quantized theory of gravity. The basic idea behind dipolar fluctuations was discussed for the first time in our earlier work on stability of Euclidean quantum gravity [1]; the Lorentzian case was treated in [2]. This year we made the first explicit computations and we were able to set some lower bounds on the strength of the fluctuations [3, 4], Also we gave for the first time in Ref.s [3, 4]: (1) an estimate of possible suppression effects by cosmological or (2) a computation of the total ADM energy of the zero modes; (3) a clarification (in the Lorentzian case) of the influence of matter fields on the fluctuations, with possible anomalous coupling. Here, after a few general remarks about vacuum fluctuations and ”spacetime foam” in quantum gravity, we shall set out the general features of the dipolar fluctuations (Section 1) and give some explicit order of magnitude calculations (Section 2). Then we shall show that a term cuts, to some extent, the dipolar fluctuations (Section 3); this can lead in certain cases to an anomalous coupling to matter (Section 4). In conclusion, we shall discuss a number of new topics not addressed in Ref.s [3, 4]. 259 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 259-266 © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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1. General Features Of The Dipolar Fluctuations

The functional integral of pure Einstein quantum gravity can be written as with The ”spacetime foam” [5] consists of fluctuations whose action does not exceed a quantity of order This implies, for curvature fluctuations on a scale d, that (according to naive power counting) or (according to numerical lattice estimates [6]). Therefore, large fluctuations are expected to take place only at very small distances. Since, however, the Einstein action is not positive definite, one can also expect some fluctuations due to peculiar cancellations of distinct contributions to the action, which are by themselves larger than In order to work this out explicitely, let us consider the Einstein equations with an auxiliary source

and their trace

Then consider a solution condition

of (1.1) with any source satisfying the

In view of (1.2), this metric has zero action. We have constructed in this way a zero mode of the pure Einstein action. The source is unphysical, but it is ”forgotten” after obtaining the metric. Condition (1.3) means in fact that it is a ”dipolar” source, with a compensation between regions having positive and negative mass-energy density. Since this auxiliary source is used to construct a virtual field configuration, we shall sometimes call it a ”virtual source”. 2.

Explicit Computation To Order

G2

We have then found the following ”recipe” for constructing dipolar zero-modes of the pure Einstein action: given any source with zero integral (condition (1.3)), one solves the Einstein equations and finds the corresponding metric. Note, however, that the condition on the source already contains furthermore, exact solutions are in general not known, and the approximation error could be such that the corresponding error on S is

ZERO-MODES OF EINSTEIN ACTION

261

larger than In this case we could still imagine that the zero-mode can be computed in principle, but we would not have in practice any definite idea of its properties. An explicit evaluation is therefore needed. To this end we consider static sources, with some free parameters (typically and their sizes), in the weak field approximation. We have in this static case

Using the Feynman propagator one finds to first order in G (compare [3])

It is straightforward to check from this expression that This means that the action of the metric generated by a static source is simply

and provided the integral of the mass-energy density vanishes, the field action is of order i.e. practically negligible, as shown in the following numerical example. Consider a static dipolar source which is adiabatically switched on/off with a lifetime of the order of 1 s. Suppose that the spatial size of the source is of the order of 1 cm and the two masses are of the order o f i.e. in natural units. Note that with this mass we have, for the ratio between the Schwarzschild radius and This implies that we can compute in the weak field approximation, with negligible error. More precisely, the residual of second order in G is found to be

Therefore the field of a static virtual source of this magnitude order, satisfying the condition has negligible action even if corresponding to an apparent matter fluctuation with density This should be compared to the action of the corresponding ”monopolar fluctuation”, namely

262 3.

GIOVANI MODANESE A

-Term In The Action Cuts-Off the Fluctuations

The most recent estimates of the Hubble constant support a non-zero value of the cosmological constant of the order of The cosmological term in the gravitational action, to be added to the pure Einstein term, is

It is possible to evaluate the contribution of the dipolar fluctuations to this term. This is easier for fluctuations with spherical symmetry, like those generated by virtual sources having the shape of ” +/- shells” (compare [3]). In this case one can use the exact Schwarzschild metric outside the source and the spherically symmetric Newtonian field inside it. To leading order one finds that

where Q is an adimensional factor which can be negative or positive, depending on the distribution of the positive and negative mass inside the virtual source. . Inserting for and the same values as before, we find Remembering that the lower bound for pure gravity was we see that the cosmological term cuts, to some extent, the dipolar fluctuations. This works even better for larger values of 4. Matter Coupling VS. Local Changes In Let us consider now a scalar field

To lowest order in

coupled to gravity

the interaction action can be rewritten as

ZERO-MODES OF EINSTEIN ACTION

263

On the other hand, the cosmological action is, still to lowest order in and expanding

Therefore the sum of the two terms can be rewritten as

We see that to leading order the coupling of gravity to gives a typical source term and subtracts from the local density This separation is arbitrary, but useful and reasonable if the lagrangian density is such to affect locally the “natural” cosmological term and change the spectrum of gravitational vacuum fluctuations corresponding to virtual mass densities much larger than the real density of Let us give an example. Suppose that represents a coherent fluid with the density of ordinary matter At the scale of 1 with the observed value of the lower bound on virtual source density is ~ which is much larger than the real density. If is comparable to in some region, an inhomogeneity in the cut-off mechanism of the dipolar fluctuations will follow, and this effect could dominate the effects of the coupling to real matter. In our opinion, this dynamical mechanism could be the basis for an explanation of the weak gravitational modification by superconducting spinning disks which E. Podkletnov claimed to have observed under very special conditions [7, 8]. A NASA/Argonne experimental team is at work to replicate the original results, which at this time are neither confirmed nor confuted. From the very beginning of our theoretical analysis [9] we maintained that the disks described in Ref. [7] cannot be a source of gravitational field or perturbate it in a classical sense. This is because the gravitational coupling to matter is far too small, and neither the presence of Cooper pairs inside the disk nor its fast rotation help much under this respect. We have then been looking for an interaction process not constrained by the coupling, and a candidate for this are large quantum fluctuations (compare our phenomenological model in [10]). More recently, anomalous gravity changes have been observed during a solar eclipse [11]. In this case the coherent matter which couples to the gravitational fluctuations could be the native iron which is abundant on the Moon.

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5. Some Remarks And New Topics

(1) The fluctuations maintain Even in the presence of strong quantum fluctuations, we expect that the vacuum average of computed through the functional integral, is zero. This is defined as the average of More physically, there must be no apparent mass-energy- momentum density, on the average, in the vacuum. An analogous property holds in QED. Even in the presence of vacuum fluctuations, the vacuum average value of the four-current, defined through Maxwell’s equations as vanishes. In fact, some general properties of the functional integral ensure that the condition above on is respected, without any need of restricting the integration space [12]. In other words, the fluctuations always average out in such a way to give a zero total virtual mass density at any point. (2) The dipolar zero-modes have S = 0 but are not extrema of S. The dipolar zero-modes are not a minimum of the pure Einstein action, because this would be equivalent of being a solution of the vacuum Einstein equations; but the dipolar zero-modes do not satisfy the Einstein equations in vacuum, even through they have the same action as the vacuum solutions (S = 0). We can understand better the situation at hand through a bidimensional analogue (while, of course, the space of the possible metric configurations is infinite-dimensional). Let us consider the function f ( x , y ) = and draw its cartesian plot. We find that it resembles a paraboloid, except for the fact that the axes x = 0 and y = 0 are a sort of ”cuts”; along these axes the function takes the value f = 0, which is also the value at the origin. However, the origin is a minimum, while the points on the axes distinct from the origin are not minima. (3) How much is the phase space volume of dipolar fluctuations ? The example above suggests another possible property of the dipolar zero-modes: in the same way as the ”zero lines” of the function above have null measure with respect to its full bidimensional domain, one may think that the condition for the dipolar zero-modes defines a subspace of all field configurations having lower dimension and thus null measure. In this case the dipolar fluctuations would be suppressed because they have zero volume in phase space. This would be true if the dipole condition had to be satisfied exactly. In fact, however, we considered weaker conditions. We saw, for instance, that up to values of the virtual mass of the order of

ZERO-MODES OF EINSTEIN ACTION

265

the difference between and i.e., the width of the ”line in phase space”, is of order where is the duration of the fluctuation. The appearance of in this relation could also provide an upper bound on the duration of the fluctuations, making closer to the minimum duration allowed by the Heisenberg principle, like it happens for electromagnetic or scalar fields (compare [3, 4]). In other words, a long lifetime requires a very precise compensation between the positive and negative masses of the virtual dipole, thus implying a very small phase space volume for that configuration. (4) Strong dipolar fluctuations are not allowed in QED. This is an immediate consequence of the quadratic form of the electromagnetic lagrangian density: The contribution to the action of the +/- charges in a virtual dipole is the same, therefore there are no cancellations. The dipolar fluctuations are not favoured with respect to the monopolar fluctuations. The reasoning above also leads to the following statement. (5) The null-action property of the dipolar fields is a non-perturbative feature. In fact, suppose we limit ourselves to consider, as usual in perturbation theory, the part of the action quadratic in h in an expansion around the minimum. We then would be in the same situation as for the electromagnetic field. It is easy to see that the quadratic part of the gravitational action is not positive definite and has positive and negative eigenvalues in the Euclidean formulation, unlike the electromagnetic action [1]. Nevertheless, any single term would be the same for positive or negative virtual sources (see eq. (1-5)), without any cancellation. In other words, the null action property of the dipolar zero modes cannot be obtained just considering the expansion in powers of h. This not because we are in strong field conditions, but because the zero-modes are not local minima of the action (compare Point 2). (6) The of any real pure electromagnetic field generates gravitational zero-modes, up to terms of order Since the trace of the energy-momentum tensor of any electromagnetic field is zero, it is possible to use, instead of virtual unphysical dipolar mass sources, a real electromagnetic field as source of gravitational zeromodes. It needs to be a pure electromagnetic field in vacuum, thus any plane waves or wave packets are admissible. We stress again, however, that the real or virtual character of the source is irrelevant in order to obtain vacuum fluctuations. Acknowledgments - This work was supported in part by the California Institute for Physics and Astrophysics via grant CIPA-MG7099. The

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author is grateful to C. Van Den Broeck, M. Gross and J. Brandenburg for useful discussions and remarks. References

[1] Modanese, G. (1998) Stability issues in Euclidean quantum gravity, Phys. Rev. D 59, 024004. [2] Modanese, G. (1999) Virtual dipoles and large fluctuations in quantum gravity, Phys. Lett. B 460, 276-280. [3] Modanese, G. (2000) Large “dipolar” vacuum fluctuations in quantum gravity, report gr-qc/0005009, to appear in Nucl. Phys. B. [4] Modanese, G. (2000) Paradox of virtual dipoles in the Einstein action, Phys. Rev. D 62, 087502. [5] Wheeler, J.A. (1957), Ann. Phys. 2, 604. [6] Hamber, H.W. (2000) On the gravitational scaling dimensions, Phys. Rev. D 61, 124008. [7] Podkletnov, E. and Nieminen, R. (1992) A possibility of gravitational force shielding by bulk Y Ba 2 Cu 3 O 7–x superconductor, Physica C 203, 441; Podkletnov, E. (1997) Weak gravitational shielding properties of composite bulk Y Ba2Cu3O7–x superconductor below 70 K under e.m. field, report cond-mat/9701074. [8] Modanese, G. and Schnurer, J. (1996) Possible quantum gravity effects in a charged Bose condensate under variable e.m. field, report gr-qc/9612022. [9] Modanese, G. (1996) Theoretical analysis of a reported weak gravitational shielding effect, Europhys. Lett. 35, 413-418. [10] Modanese, G. (1999) Gravitational anomalies by HTC superconductors: a 1999 theoretical status report, report physics/9901011. [11] Qian-shen Wang et al. (2000) Precise measurement of gravity variations during a total solar eclipse, Phys. Rev. D 62, 041101. [12] Collins, J.C. (1984) ”Renormalization” (Cambridge University Press, Cambridge, 1984). Modanese, G. (1994) Vacuum correlations at geodesic distance in quantum gravity, Riv. Nuovo Cim. 17, Vol. 8, 1-62.

THEORETICAL AND EXPERIMENTAL PROGRESS ON THE GEM (GRAVITY-ELECTRO-MAGNETISM) THEORY OF FIELD UNIFICATION

J.E. BRANDENBURG, * J.F. KLINE, AND *VINCENT DIPIETRO Aerospace Corporation Chantilly Virginia USA *Research Support Instruments Lanham-Seabrook Maryland USA

Abstract. Theoretical and experimental progress on the GEM theory is summarized. A portion of the Kaluza-Klein action is shown to form a “Vacuum Bernoulli Equation” showing Gravitational energy density to be equated to an EM dynamic pressure that is quadratic in the local Poynting Flux: = Constant where G and S are the local Gravity and Poynting vector magnitudes, respectively, G is the NewtonCavendish constant and is a local magnetic energy density. This relation satisfies the Equivalence Principle. It is shown that this equation predicts that gravity modification can occur through a Vacuum Bernoulli Effect or VBE by creating a perturbing Poynting flux by a rotating EM field, a “Poynting Vortex”, and that this effect can lead to a lifting force for human flight applications. The theory is then applied to experiments involving EM driven gyroscopes with some success. Explorations of the possibility of a GEMS theory, including the strong force, are briefly discussed. I Introduction: The GEM Theory The GEM theory1, 2 is a geometric theory for the unification of Gravity and Electromagnetism. The theory is an attempt to attack the problem of field unification by separating out the two long-range forces of nature, gravity and EM, and considering that the short range forces of nature, the Weak and Strong forces, are higher order corrections important in strong fields. The theory limits itself to protons and electrons. The theory is in an early state of development, being described as a "Bohr Model" of unification by analogy with the earliest quantum mechanical model of the hydrogen atom, and is based on an extension of the work of Einstein, Kaluza and Klein3, and Sahkarov4 and Zeldovich5. The theory, in its present state of development, will only be summarized here and will be discussed at length in a future article. The GEM theory begins with the Hilbert action principle, which allows the derivation of the fundamental equations for vacuum fields from the extremization of the action integral: 267 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 267-278. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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J.E. BRANDENBURG, J.F. KLINE, AND V. DIPIETRO

where is the Ricci curvature scalar of General relativity and is the Newton Gravitation constant. This integral was used by Sahkarov to relate the force of gravity to a radiation pressure, or Poynting flux, due to the ZPF (Zero Point Fluctuation). This concept of gravity being a manifestation of the ZPF has also been pursued by Puthoff (1989) and Haish Rueda and Puthoff (1989). In the GEM theory the ZPF is proposed to create a set of "ExB drift" cells that produces a gravity acceleration. So it can be said that Sahkarov showed that the existence of gravity could imply an EM field. In the GEM theory the EM fields are postulated to be arrays of ExB drift cells. The ExB drift is well a known phenomena in plasma physics and affects all particles , regardless of charge or mass. We can write the definition of gravity fields within GEM as ExB drifts in the special case of constant background B field where everywhere:

where the dot signifies a time derivative. Thus gravity implies EM. Kaluza and Klein extended the Hilbert action principle by adding an additional dimension that is "compactified " to small fixed length so that we have for an action principle

where the energy density

can be written

where E and B are the electric and magnetic field strengths and is the parameter that is the 5,5 element of the metric tensor in the five dimensional space and gives the product of the compactified fifth dimension with itself. Kaluza-Klein theory allowed, by a small increase in dimensionality, the explicit appearance of EM fields along with gravity. In the GEM theory the Kaluza-Klein parameter varies from 0 to 1 and controls not only the separate appearance of EM fields but also the separate appearance of matter fields of protons and electrons. The appearance of the fifth dimension, in addition to the four dimensions of space-time, allows then the appearance of both proton-electron matter fields and the EM fields.

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269

The signature of this coupled appearance in the GEM theory is the relation between curvature parameters of the Planck Length and what is termed the "mesoscale”, typical of classical particle radii

So that after a full development of the compact fifth dimension:

This leads to a formula for G (in esu units)

where

is the fine structure constant. This formula gives the value of the Newton gravitation constant to a within its presently accepted accuracy of 1.5 parts per thousand. This relationship allows a process called "vacuum decay" which leads to the appearance of hydrogen from the vacuum. This found to lead to a steady state universe with the Hubbell time or c times the Hubbell radius 2 of approximately

where is the electron classical radius. Where this is consistent with the Dirac expression for the Hubbell time. Thus the GEM theory literally relates the Planck length to the Hubbell radius. However, for laboratory experiments the most important GEM relation is the Kaluza-Klein action, from which be extracted a portion here written in terms of the local gravity acceleration, which must vanish in an inertial reference frame where is the

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J.E. BRANDENBURG, J.F. KLINE, AND V. DIPIETRO

Poynting vector,

and we define the basis stress

where g is the gravity acceleration, and E is an electric field associated with the portion of the ZPF giving rise to the ExB model of GEM gravity and is a constant magnetic field assumed to exist in the GEM ZPF. This extracted portion of the Kaluza-Klein action vanishes in a inertial reference frame, because E vanishes for an ExB drift, and thus satisfies the equivalence principle. This equation is termed the Vacuum Bernoulli Equation. A similar equation has been derived previously by Vallee8. The GEM theory thus combines Sahkarov's physical interpretation of the Hilbert action principle with the Kaluza-Klein extension of the action principle to produce a theory of field unification. This theory, though it involves the unification of two classical fields , requires field quantization to give rise to a ZPF. It is interesting to note that while the GEM theory ignores the Weak and Strong forces explicitly, they are present implicitly because of the requirement of the equivalence principle that all matter, including neutrons, must respond to gravity. In particular, the requirement in any Sahkarov type gravity theory, that the constituents of the heavy particles, quarks, be both charged and move freely in their deep confinement within the nucleons, has been noted by Puthoff (1989), and is amplified here. In order for the GEM theory to give a physical description of gravity, assumptions must be made about the dynamics of Strongly interacting particles. Thus, though the GEM effort is presently focused on gravity field control through EM fields, an area of interface with the Strong force investigations has developed.

2. Hints Of A GEMS (GEM + Strong Force) Theory Because of the focus of this paper lies elsewhere the recent explorations of the possibility of a GEMS theory will only be summarized here. Fundamentally, GEM implies GEMS because of the central appearance of the electron-proton mass ratio, which is the signature of the Strong force, however, this ratio must be connected to geometry at the mesoscale level. The GEM theory is fundamentally a geometric theory of fields and particles. Thus it is not surprising that the first progress on a GEMS theory comes from an examination of a peculiar shared geometric scale of both EM and strongly interacting particles, which is the classical radius of the electron. A simple electrostatic model of an electron gives its size as very close to the approximate radius of the proton or neutron.

This appears to imply that the entities making up the nucleons must have charge to mass ratio of If we assume that the EM and Strong force have some deep connection, then this approximate equality of geometric curvature parameter is its first confirmation.

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We can further pursue this trail a short distance into the realm of the Strong force when we further note that when our formula for the Newton constant is inverted to give the classic Kaluza-Klein expression for the value of the fundamental charge, the rationalized expression is written most naturally as

where

is the size of the Kaluza-Klein dimension, approximately Using the geometric approach of the GEM, where the classical radius of the electron is a controlling parameter, one can obtain the value for the electron-proton mass ratio by modeling the proton9 as a "water bag" full of saturated quark -color field as the maximum value of mass for the given value of the proton charge. When this is done the mass of the proton is then found to be of the form

where is a saturated color-quark field energy density. When this is done the protonelectron mass ratio is found be a dimensionless mass-charge ratio whose value is fixed as an extreme point of a Lagrange multiplier function10, written as

where is a dimensionless charge value parameter and corresponds to q = e. In this formalism is recovered as the value of the Lagrange multiplier required when the proton mass is maximized as function of charge at q = e, yielding the highly accurate Weiler formula

Thus some reason for optimism exists regarding a GEMS theory.

3. Gravity Modification And The Question Of Parity Violation By Gravitation

We can perturb the existing gravity field by perturbing the ZPF fields that create it, using Eq. 9.

The gravity field energy density must be that of an EM field that has “scattered” off matter once already. Thus we should have

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where the first term represents the energy density in the unscattered ZPF EM field and the second term is the energy density in the scattered or gravitational field, where is the ZPF electric field and is the corresponding magnetic field. When this is done with rotating EM fields ( a "Poynting vortex")it is found that gravity modification may occur as an interference pattern that depends on the rotation sense of the laboratory field and the ZPF field supporting gravity. If the gravity-ZPF rotates in only one direction, then the gravity modification has the form, first due to Kozyrev11,

where dv is the laboratory rotating field velocity, v, is the unit vector of the ZPF rotation velocity, is the angle between the spin vector of the rotor and the gravity vector. Thus gravity change occurs for spin vector parallel to the gravity vector, but no gravity change occurs for spin vector opposite the gravity vector, This effect is the correlation integral of two rotating fields. This sense of rotation effect appears mathematically only if a preferred sense of rotation is assumed for the ZPF, otherwise all angular dependence disappears. This effect, suggesting parity violation by gravity, appears in some experiments and not in others and remains mysterious. The formula also predicts increases in weight, not decreases.

II Application To Gyro Experiments 4. Motivation Previous researchers, Kozyrev11 and Hayasaka and Takeuchi,12 have observed decreases in the weights of small gyroscopes as a linear function of frequency of rotation. They operated their gyroscopes using 400 Hz three-phase power, and measured the weight decreases on chemical balances. The weight decrease had a magnitude as high as 10 mg. Polarity – spin-vector down (clockwise rotation viewed from above) operation resulted in weight decrease, while spin-vector up operation was a null result, spin-vector horizontal resulted in one-half the maximum decrease11. The results were strongly criticized in the literature, not on an experimental basis, but on the inability of some researchers to replicate the results. An unsuccessful replication of the results13 used compressed-air jets to accelerate the gyroscopes, and ignored the electromagnetic aspect of the experiments. The opinion that those researchers may have left out some fundamental features motivated the present projects. Two experiments were carried out – one at NASA Goddard Space Flight Center (GSFC), and one in-house at Research Support Instruments, Inc. (RSI).

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5. Materials and Methods

The investigations used gyroscopes from Sperry Flight Systems - Gyro-Horizon Indicator Model The rotors had a weight of 289.6 g, and a radius of 2.25 cm. The investigations did not seal the gyroscope in vacuum; based on the experimental work of Kosyrev, hermetic sealing rather than vacuum sealing was deemed important. The gyro was mounted on a small aluminum bracket that could itself be mounted in two positions to give the gyro spin vector either an up-down orientation (parallel or anti-parallel to gravity vector) or a horizontal orientation (perpendicular to gravity vector). The whole assembly of the gyroscope, photocell rotation speed monitor, and mount had a mass of approximately 580 g. A Jack and Hientz model F20-5 inverter supplied three-phase power with frequency proportional to voltage (at full 115V, frequency was 400 Hz). The wires to power the gyroscope were run along the fulcrum of the balances to minimize their effect on the experiments. A CdS photocell and a red LED were added to the gyroscope to observe the modulation of reflected light from a black stripe painted on the rotor. A Tektronics model TDS 210 digital oscilloscope provided a measurement of rotor frequency. The NASA GSFC experiment used a Mettler mechanical balance with a 1 kg capacity and 0.1 mg resolution. It was difficult to use – the readout on a Mettler balance is mechanical, and as such, has a slow response time. The weights had to be adjusted by hand (using front panel knob controls), so it took some time after turning on the gyroscopes to get a reading. The in-house RSI experiment, on the other hand, used an electronic balance, carefully separated from the gyroscope by building a mechanical beam balance that rested on the electronic scale and a knife-edge fulcrum. By counterbalancing the gyroscope with almost equal weight on the other end of the beam, and by locating the scale at ¼ the distance from the fulcrum as the gyroscope, a 4x resolution magnification was accomplished. A photograph of the balance is shown in Figure 1. Interestingly enough, this follows a similar setup by Podkletnov and Nieminen12 where weight changes above a superconducting disk were observed. The similarities are intriguing; the superconductor was levitated by a toroidal magnet, rather than bearings, the disk was spun using 400 Hz 3-phase power, much like the gyro experiments, and the balance was separated from the electronic scale. The significant difference is that gravitational shielding was observed; that is to say, it was a sample located above the superconductor that lost weight. The Podkletnov apparatus is shown in Figure 2 to illustrate the similarities.

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Two approaches were used for recording weight and frequency. For the GSFC experiment, due to problems with measuring weight changes on a Mettler balance in any quick fashion, the gyroscope frequency was measured during each experimental run, and an average frequency and scale weight change were recorded. In RSI experiment, the frequency and weight were recorded in real-time as the gyroscope spun up and spun down, with second time resolution. This allowed the measurement of frequency dependence of weight change in real-time, rather than accumulated average.

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6. Results The results for the GSFC experiment are given in Figure 3. The results were a weight loss approximately 1/5 of the values reported by Hayasaka and Takeuchi. The best explanation at this time is that the effect of apparatus design on the gravity modifications is poorly understood. If the local gravity field is indeed modified, various parts of the equipment will lose or gain weight, and this means that a slightly different experimental configuration may produce significantly different results. When the experiment was replicated at RSI, using the mechanical balance addition to an electronic scale, the results shown in Figure 4 were obtained.

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In this case, there was excellent magnitude agreement with Hayasaka and Takeuchi, though the spin vector orientation was opposite. Their results are shown in Figure 5.

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7. Discussion From the results above, it is clear that the Hayasaka and Takeuchi experiment is repeatable if their experimental methods are followed carefully, and in other circumstances also. These experiments appear to support the prediction of the GEM theory that gravity fields can be modified directly by EM fields. To avoid uncertainty in the interpretation of balance results, a balloon-based neutral buoyancy experiment has been attempted with intriguing results. Improved balloon are planned for the future. Further development of the GEM theory will also be pursued. Thus it appears possible that the GEM theory, unifying the Planck and Cosmic scales, may allow gravity modification and ultimately human flight technology.

Acknowledgements The authors wish to thank Joe Firmage for his kind support of our research along with Creon Levitte, Fausten, Cheryl my muse, and the rest of the ISSO merry band. Special thanks goes also to Michael Corson of RSI and Robert Feddes of Aerospace for their interest and encouragement in this research. Special thanks also to Richard F. Post and Edward P. Lee for their early encouragement of this research.

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References

1. Brandenburg, J. E. (1992) Unification of Gravity and Electromagnetism in the Plasma Universe, IEEE Transaction in Plasma Science Vol. 20, 6, pp 944 2. Brandenburg, J. E. (1995) A Model cosmology based on Gravity-Electromagnetism Unification, Astrophysics and Space Science, 227, pp133 3.Klein, Oskar, (1926) "Quantum Theory and Five-Dimensional Relativity Zeitschrift fur Physik, 37, 895. 4. Sakharov A.D. (1968) "Vacuum quantum fluctuations in curved space and the theory of gravitation" Soviet 5. Zel’dovich, Ya. B. (1967), Cosmological constant and elementary particles, Sov. Phys.--JETP Lett. 6, 316317. 6. Puthoff, H.E., (1989) "Gravity as a zero-point fluctuation force", Phys. Rev. A 39, 2333 7. Haisch, B. , Rueda, A., and Puthoff, H.E., "Advances in the Proposed Electromagentic Zero-Point Field Theory of Inertia", AAIA 98-3143. 8. Vallee, Rene-Louise,(1970) Gravitational and Material Electromagnetic Energy Synergetic Model, (unpublished) 9. F.E.Close (1979)"An Introduction to Quarks and Partons" Academic Press, New York, p411 10. Brandenburg, J.E. (2000) "The GEMS theory" Manuscript in preparation. 1 l.Kozyrev, N., A., (1968) Joint Publications Research Service # 45238 12.Hayasaka, Hideo, and Takeuchi, Sakae, (1989) Anomalous Weight Reduction on a Gyroscope’s Right Rotations around the vertical Axis on the Earth, Phys. Rev. Letters, Vol. 63, 25,pp 2701 13.Faller J.E., et al. (1990) Gyroscope Weighing Experiment With a Null Result, Phys. Rev. Lett. ,64, 8, pp 825 14. E. Podkletnov and R. Nieminen, Physica C 203 (1992) 441-444

CAN GRAVITY BE INCLUDED IN GRAND UNIFICATION?

PETER ROWLANDS IQ Group and Science Communication Unit, Department of Physics, University of Liverpool, Oxford Street, Liverpool, L69 7ZE, UK. JOHN P. CULLERNE IQ Group, Department of Computer Science, University of Liverpool, Peach Street, Liverpool, L69 72F, UK.

Key words: Grand Unification, Planck mass, U(5) Abstract:

A model of quarks, based on integral charges, suggests a value of with a consequent prediction of Grand Unification at the Planck mass, and a value of at 14 TeV.

1. A quark model with integral charges Grand unification of the electromagnetic, weak and strong interactions is at present believed to occur at an energy of order 1015 GeV, about four orders below the Planck mass at which quantum gravity becomes significant. However, there are serious problems with the current minimal SU(5) model, which fails to predict a full convergence of the three interactions. Also, the assumed electroweak mixing parameter, is at total variance with the experimental value of 0.231. A ‘renormalization’ procedure has been devised to reduce the GU value to about 0.21 at the Z boson mass However, applying the renormalized value to the equations for the coupling constants leads to at GU, in complete contradiction to the 0.375 assumed. In addition, though the weak and strong coupling constants are assumed to be exactly unified at GU, the incorporation of the SU(2) × U(1) electroweak model appears to require a modified value of the electromagnetic coupling, and so GU is not exact between the interactions but occurs only through an assumed group structure. The problem, we believe, lies in the current, rather ‘mechanistic’, understanding of the fractional values observed for quark charges. Experiments, such as the ratio of hadron / muon production in electron-positron annihilation events, and the rate of decay of neutral pions to two photons, have, of course, repeatedly shown that quarks carry charge units that are fractions of those of the charge on the electron e, either 2/3 in the 279 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 279-286 © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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case of u, c, t, or –1/3 in the case of d, s, b. Though experiments do not access individual quarks, they show, quite clearly, that only a fractionally-charged quark model can be constructed to explain the phenomenology of the interactions of baryons and mesons. There are, however, logical problems with the position that the experimental results require us to take. One is that it either makes charge imperfectly quantized or it suggests that quarks (or leptons) are not truly fundamental. Of course, we can redefine the fundamental unit of charge to be e/3, rather than e, but this would mean that a particle with two units of charge (u) would be as fundamental as one with a single unit (d); it would also mean that electrons, which show no sign of being composite, would nevertheless somehow acquire three units of the new fundamental charge. A second problem is that grand unified theories imply that quarks and leptons can be combined in a single overall unification scheme, with possible quark-lepton transitions; different units of charge, however, would make it difficult to see how this could be accomplished. In developing a model for an SU(5) group structure, and its breakdown to SU(3) × SU(2) × U(1), based on a symmetry with the Dirac algebra, we have found that fractional quark charges are best treated as an ‘emergent’ phenomenon, as suggested by Laughlin (1983, 1999). In the case of the fractional quantum Hall effect, in condensed matter, ensembles of particles with only exact units of e somehow acquire the characteristics of perfect fractions of this unit. Laughlin (1999) has explained this, with reference to Anderson (1972), as ‘a low-energy collective effect of huge numbers of particles that cannot be deduced from the microscopic equations of motion in a rigorous way and that disappears completely when the system is taken apart’. In his 1998 Nobel Lecture, he states the following opinion: ‘The fractional quantum Hall effect is fascinating for a long list of reasons, but it is important in my view primarily for one: It establishes experimentally that both particles carrying an exact fraction of the electron charge e and powerful gauge forces between these particles, two central postulates of the standard model of elementary particles, can arise spontaneously as emergent phenomena. Other important aspects of the standard model, such as free fermions, relativity, renormalizability, spontaneous symmetry breaking, and the Higgs mechanism, already have apt solid-state analogues and in some cases were even modeled after them (Peskin, 1995), but fractional quantum numbers and gauge fields were thought to be fundamental, meaning that one had to postulate them. This is evidently not true.’ The difficulty of establishing a universal charge quantization suggests that there is something not quite fundamental about the present status of the quark theory – just as the widely different masses for the negatively-charged electron and the positivelycharged proton suggested that there was something not quite fundamental about the theory of matter at an earlier epoch. Laughlin’s proposal indicates that the redefinition of quark fractional charges as an emergent phenomenon might be a way forward, but still leaves us without a mechanism for achieving this. There is, however, an older theory which might be of some use in this context. This theory has often been seen as an ‘alternative’ to the standard theory with fractional charges, and, although not completely discarded as a possible model, has gradually fallen into disuse, because of its apparent

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disagreement with accepted experimental results. If, however, we regard it, not as an alternative theory, but as a possible description of a mechanism for the standard theory, then it might again be seen to have value. In introducing the concept of colour to the quark theory in their well-known paper of 1965, Han and Nambu proposed that the differently coloured quarks carried either unit or zero values of electric charge, in preference to the fractional charges assumed in the version of the quark theory proposed in the previous year by Gell-Mann and Zweig. In the Han-Nambu theory, the charge assignments were of the form:

rather than the more familiar

required by the Gell-Mann-Zweig version. One of the advantages of this theory was that it made ‘colour’ differences a natural result of the existence of different charge structures and not an arbitrarily added extra property. It is widely assumed, and this may have been Han and Nambu’s original intention, that such a theory implies that, at some sufficiently high energy, the integral nature of the charges will become manifest and colours directly reveal themselves. However, if we assume perfect gauge invariance for the strong interaction, and hence perfect infrared slavery, then no such transition will occur; there will be no finite energy range at which integral charges or colour properties will emerge. The charges will be exactly distributed between the quark components of baryons and mesons, and will be exactly fractional, in every way identical to the fractional charges in the standard theory. The Han-Nambu model will become just another way of representing the theory of fractional charges. Close (1979) expresses it in the following way: ‘Imagine what would happen if the colour nonsinglets were pushed up to infinite masses. Clearly only colour 1 [singlets] would exist as physically observable states and quarks would in consequence be permanently confined. At any finite energy we would only see the “average” quark changes and phenomenonologically we could not distinguish this from the Gell-Mann model where the quarks form three identical triplets.’ The fractional charges would not, in fact, even be ‘averages’; they would be exact because of the effectively infinite rate of 'rotation' between the coloured states or phases. On such a model, the ‘intrinsic’ charges would be integral, but the observed charges ‘fractional’. Although this would not make fractional charges ‘emergent’ in quite the same way as Laughlin has demonstrated for the fractional quantum Hall effect, involving the collective action of a large ensemble of particles, they could be described as such on the grounds that they would be derived from the collective action of a multiplicity of possible states.

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In earlier work (Rowlands, 1998, Rowlands and Cullerne, 2000), we have found that, assuming that the ultimate charges are integral, but become fractional through perfect gauge invariance that cannot be broken down at finite energies, we are able to generate both the overall SU(5) and its breakdown to SU(3) × SU(2) × U(1). In principle, we derive five representations of the electric, strong and weak charge states (A-E), which map onto the charge units (e, s, w), and the five quantities (m, p, E) involved in the Dirac equation. The 24 SU(5) generators can be represented in terms of any of these units, for example, in the form:

The Standard Model disregards what would be the generator in a U(5) representation on the grounds that it is not observed; it may be significant that, if such a particle existed, it would couple to all matter in proportion to the amount, and, as a colour singlet, would be ubiquitous.

2 Grand Unification calculations

Of course, a model with integral charges would appear to be identical in all effects to the standard representation, and, though perhaps more satisfying from a logical point of view, would contribute no new physics and no new testable predictions. There is, however, another possibility, and its effects would be perfectly testable, and relatively easy to calculate. This is that, although all charge-related phenomenology would remain exactly as assumed at present, the gauge relations between interactions, being of a more fundamental nature, would reflect the newly-assumed underlying structure producing the observed fractional charges. Such processes would most likely be evident in the GSW electroweak theory or in Grand Unification, where we would be comparing the contributions of different interactions at a fundamental level on the basis of group parameters independent of the phenomenology. One possibility would be the weak mixing angle in the GSW SU(2) × U(1) theory, calculated from

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For the weak component, with only left-handed contributions to weak isospin, from 3 colours of u, 3 colours of d, and the leptons e and v,

This should be unchanged for intrinsic integral charges. For the electromagnetic component, however, this is not the case. Intrinsic fractional charges, with both left- and right-handed contributions, would lead to

from which

With intrinsic integral charges, we have

leading to

Now, it has been observed (Weinberg, 1996) that the value 0.375 for is in ‘gross disagreement’ with the experimental value of 0.231. However, 0.25 is much closer, and second order corrections, as we shall see, may suggest a reason for the relatively small discrepancy. In standard theory (Weinberg, 1996, Georgi and Glashow, 1974, Georgi et al., 1974, Masiero, 1984), to obtain anything like the ‘correct’ value for we take the renormalization equations for weak and strong couplings:

and

and combine these with a gauge-related modification of the equation for the electromagnetic coupling based on an assumed grand unified gauge group structure: where

From (2), (3) and (4), we obtain GeV, and then apply (2) and

(the Grand Unified Mass scale) of order

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to give ‘renormalized’values of of order 0.19 to 0.21. This procedure, in fact, not only assumes a particular value for the Clebsch-Gordan coefficient, but also proposes that the final unification between the three interactions is not an absolute equalization, but is achieved only through the mechanism of a group structure. The coupling constants for the strong and weak interactions are assumed to achieve exact equalization with each other, but not with that for the electromagnetic interaction. The calculation also proposes that the exact value for at Grand Unification becomes renormalized at lower energies, but does not suggest a mechanism for this renormalization or an independent way of calculating it. It should be stressed here that, while (2), (3) and (6) are well-established results, (4) and (5) are speculative, and are not supported by the experimental evidence. Though the calculation clearly has several rather ad hoc aspects, and its assumptions, especially those referring to the Grand Unified gauge group, are still only plausible conjectures, it is regarded as giving the best available fit to the data. It is not, however, a very good fit, for when the Grand Unification constants are substituted into the equations for the individual couplings (2), (3), (4), these manifestly fail to converge to a single value for the Grand Unified coupling In addition, recalculation of the value of at GeV gives 0.6 rather than the 0.375 initially assumed in setting up the equations! The equations not only fail to provide answers consistent with experiment, but are also massively inconsistent with each other. The discrepancy is so serious that a supersymmetric solution has been suggested (Amaldi et al., 1991). If, however, we have an independent value for of the right order, we can perform a much simpler calculation for which makes no assumptions about the group structure. We combine (2), (3) and (6) to give:

GeV, (or 1/129), (Weinberg, 1996, Novikov et al., 1999), and we obtain GeV for the Grand Unified mass scale Intriguingly, this is of the order of the Planck mass If we assume that is the Planck mass (on the basis that purely first-order calculations of tend to overestimate its value (Kounnas, 1984)), we obtain (the Grand Unified value for all interactions) = 1 / 52.4, and which is, of course, the kind of value we would expect for the weak coupling with We also obtain unit strength for the strong interaction at the energy level of baryonic and mesonic structure There is, as we have implied, some justification in using such precise values for as and GeV and 0.25, in the results of the higher order calculations which have been performed for the intrinsic fractional charge model. Kounnas’ two-loop approximation (1984) reduces the value of by a factor of about 0.64, while the presence of massive gauge bosons depresses the effective values of and in the energy range where they are normally measured. Kounnas’ plots for against show a distinct dip at against an overall upward trend. The actual Using typical values for

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decrease is from about 0.22 to 0.21, representing a possible decrease in what would be otherwise expected of up to about 0.02. Of course, these calculations are based on a model assuming intrinsic fractional charges, but a similar depression of the measured values of and might be expected on any related model. The renormalization equations for weak and strong couplings require no modification as a result of changes in the quark model, but the U(1) electromagnetic coupling requires alteration in the hypercharge numbers. In particular, changes from 1 / 6 to 1 / 2, while goes from –2 / 3 to –1, –1 or 0, depending on the colour, and from 1 / 3 to 0, 0 or 1. The fermionic contribution to vacuum polarization is, conventionally (Masiero, 1984),

where is the number of fermion generations. However, when modified for integral charges, this becomes

corresponding to the change from to when changes from 0.375 to 0.25. By our understanding, 0.25 is specifically the value for a broken symmetry, produced by asymmetric values of charge, whether or not it is contained within a larger group structure such as SU(5), and would be the value expected at the mass scale appropriate to the electroweak coupling, that is at We do not expect it to be valid at Grand Unification. An analysis of the behaviour of the electromagnetic coupling may indicate the value we should expect. Here, we find that the addition of the term to leads directly to the coupling for the electromagnetic interaction, and not to the modified coupling normalized to fit an overall gauge group, assumed in most Grand Unification schemes, for when GeV, GeV, and

This is of particular interest, since it suggests, a little unexpectedly, but quite plausibly, that the ‘unification’ at might involve a direct numerical equalization of the strengths of the three, or even four, physical force manifestations, without reference to the exact unification structure, unless, at Grand Unification, and in a group structure such as U(5), involving gravitation and a generator. We may, therefore, propose that the actual symmetry group, incorporating gravity, is U(5), with the additional generator, coupling to all others, representing the gravitational interaction (at least in numerical terms), and SU(5) being the first stage of the symmetry breakdown. Such a unification would be far more exact than one dependent on the constants of a particular group structure, and would confirm the interpretation of as the electroweak constant for a specifically broken symmetry, taking the value of 0.25 at the energy range where the symmetry breaking occurs.

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The particular value of this theory is that it is eminently testable and the testing procedure is quite straightforward, with the virtue of not requiring any additional experimentation beyond the programme already laid out for experimental high energy physics. We simply take the equations (2), (3) and (8) (of which the last is new), and the values for and as calculated in this paper, and determine the three interaction strengths at increasing energies. As the energy of our experiments increases, the divergences between the values found from this model and those from the version based on intrinsic fractional charges will become increasingly apparent. A particularly striking case occurs with which, on this model would be 1/118 at 14 TeV (the maximum energy of the proposed LHC), in comparison with the 1/125 predicted by minimal SU(5).

3. References Amaldi, U., Boer, W. de and Fürstenau, H. (1991) Phys. Lett. B, 260, 447. Anderson, P. W. (1972) Science, 177, 393. Close, F. E. (1979) An Introduction to Quarks and Partons (Academic Press), p 167. Georgi, H. and Glashow,.S. L. (1974) Phys. Rev. Lett., 32, 438. Georgi, H., Quinn, H. R. and Weinberg, S. (1974) Phys. Rev. Lett., 33, 451. Han, M. Y. and Nambu, Y. (1965) Phys. Rev. B, 139, 1006. Kounnas, C. (1984) 'Calculational schemes in GUTs', in C. Kounnas, A. Masiero, D. V. Nanopoulos and K. A. Olive, Grand Unification with and without Supersymmetry and Cosmological Implication (World Scientific), pp. 145-281, 188-227. Laughlin, R. B. (1983) Phys. Rev. Lett., 50, 1395. Laughlin, R. B. (1995) Rev. Mod. Phys., 71 (1999), 1395. Masiero, A. (1984) ‘Introduction to Grand Unified theories’, in Kounnas et al. (op. cit.), pp. 1-143, 20-25. Peskin, M. E. and Schroeder, D. V. (1995) Introduction to Quantum Field Theory (Addison-Wesley). Rowlands, P. (1998) ‘The physical consequences of a new version of the Dirac equation’, in G. Hunter, S. Jeffers, and J-P. Vigier (eds.), Causality and Locality in Modern Physics and Astronomy: Open Questions and Possible Solutions (Kluwer Academic Publishers), 397-402 Rowlands, P. and Cullerne, J. P. (2000) ‘The connection between the Han-Nambu quark theory, the Dirac equation and fundamental symmetries’, Nuclear Physics A (in press). Weinberg, S. (1996) The Quantum Theory of Fields, 2 vols. (Cambridge University Press), Vol. II, pp. 32732. Novikov, V. A., Okun, L. B., Rozanov, A. Z. and Vysotsky, I. (1999) Rep. Prog. Phys., 62, 1275.

GRAVITATIONAL ENERGY-MOMENTUM IN THE TETRAD AND QUADRATIC SPINOR REPRESENTATIONS OF GENERAL RELATIVITY

ROH S. TUNG California Institute for Physics and Astrophysics (CIPA) 366 Cambridge Avenue Palo Alto, California 94306, USA and Enrico Fermi Institute, University of Chicago 5640 South Ellis Avenue, Chicago, Illinois 60637, USA Email: [email protected] JAMES M. NESTER Department of Physics and Center for Complex Systems National Central University Chungli 320, Taiwan, ROC Email [email protected]

Abstract

In the Tetrad Representation of General Relativity, the energy-momentum expression, found by Moller in 1961, is a tensor wrt coordinate transformations but is not a tensor wrt local Lorentz frame rotations. This local Lorentz freedom is shown to be the same as the six parameter normalized spinor degrees of freedom in the Quadratic Spinor Representation of General Relativity. From the viewpoint of a gravitational field theory in flat space-time, these extra spinor degrees of freedom allow us to obtain a local energy-momentum density which is a true tensor over both coordinate and local Lorentz frame rotations.

1. Introduction

Conservation of energy-momentum, which is associated with spacetime symmetry, plays an important role in physics. When we trace back the history, we find that a new physics has usually been born with a violation of conservation of energy-momentum. Perhaps the only exception was Einstein's radical idea: general relativity --- a theory of spacetime itself. 287 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 287-294. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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The problem of determining the gravitational energy-momentum arose immediately after Einstein’s formulation in 1915; attempts looking for a local energymomentum only resulted in a set of pseudotensors. After much effort, people concluded that there is no proper physical local energy-momentum density for the gravitational field. The situation was gradually clarified to the following conclusions: (i) The energy-momentum concept in a gravitational field can be introduced if we replace the spacetime symmetry in ordinary relativistic field theory by the concept of asymptotic flatness, i.e. total energy is well-defined, (ii) Because of the equivalence principle, gravitational energy-momentum is not localized. As the famous textbook of Misner, Thorne and Wheeler teaches [1]: “anybody who looks for a magic formula for local gravitational energymomentum is looking for the right answer to the wrong question ”.1 Newtonian gravity theory is based on action-at-a-distance, so in that case we expect only a total energy for a gravitating system. But for relativistic gravity theories we expect meaningful local quantities since the interactions are local and they exchange energy-momentum locally. Roger Penrose was not satisfied with only the total energy for gravitation being defined; he stated “It is perhaps ironic that energy conservation, a paradigmatic physical concept arising initially from Galileo’s (1638) studies of the motion of bodies under gravity, and which now has found expression in the (covariant) equation cornerstone of Einstein’s (1915) general relativity — should nevertheless have found no universally applicable formulation, within Einstein's theory, incorporating the energy of gravity itself.” and then proposed the idea of quasilocal (i.e. associated with a closed 2-surface) energy-momentum [2]. There have been several proposals (an extensive literature was given in Ref.1 of [3] ) for quasi-local energy-momentum. They need either a reference background [3, 4] or a globally defined spinor field [5]. The meaning for a reference background was in fact pointed out by Poincaré [6, 7]—that the physical description is often based on a priori conventions. For spacetime geometry, two points of view are possible, (i) According to general relativity, the line element between neighboring events is measured by using rods or clocks with the same length or rate which are independent of the field present. The resulting spacetime is curved in general, (ii) On the other hand, one can define the line element to be of Minkowskian form. Accordingly, the rods and clocks are affected by the gravitational field. Apart from global topological questions the two complementary points of view are equivalent. The origin of Weyl’s gauge idea is in fact to abandon the idea of adding lengths in general relativity, in Weyl’s opinion, keeping a rod to have the same length is a concept which involves actionat-a-distance [8]. The same applies to clocks. 1

In order to be a good student, we should probably not ask such a “wrong question”. But in this paper, we are being naughty students.

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Microscopically, it is very difficult to have classical concepts such as rods and clocks giving a simple microscopic understanding of gravitation. Hence the main effort of current quantum gravity is to find new concepts for the space-time at the Planck scale, there have been many stimulating ideas, e.g., strings (p-branes), twisters, or loops. These concepts provide a rich structure for space-time at the Planck scale. From the field theory point of view, microscopically space-time geometry enters only as a background concept necessary to defining a field theory. The search for a good expression for local energy-momentum is thus especially important since the energy concept is associated with the fundamental structure of space-time. Since for Einstein’s general relativity, there is no space-time symmetry in general, we do not expect a local conserved energy-momentum. In this case, we expect at most a quasi-local definition. However, considering Einstein gravity as an ordinary field theory in flat Minkowski background space-time, we do have a Minkowski space-time symmetry. Can we then obtain an energy-momentum density tensor in this case? In this paper we will obtain such a quantity by a change of variables and by adding extra spinor gauge variables. 2. Metric Representation

The Hilbert Lagrangian density for General Relativity is The traditional approach uses the metric coefficients in a coordinate basis as the dynamic variables, so Because of the second derivatives, this is not suitable for getting an energy-momentum density. However a certain (noncovariant) divergence can be removed (without affecting the equations of motion) leading to Einstein’s Lagrangian One can now apply the standard procedure and get the canonical energy-momentum density. It is known as the Einstein pseudotensor; its value depends to a large extent on the coordinate (“gauge”) choice. No satisfying technique has been found to separate the “physics” from the coordinate gauge. 3. Tetrad Representation

An alternative is to use an orthonormal frame (tetrad), a pioneer of this apand proach was Møller [9]. Let with regard the Einstein-Hilbert Lagrangian as a function of the tetrad A suitable total divergence can again be removed yielding a Lagrangian density which is first order in the derivatives of the frame. Now there is an associated energy-momentum density which is a tensor (density) under coordinate transformations, but it depends on the choice of orthonormal frame (Lorentz gauge) [9, 10, 11]. 4. Quadratic Spinor Representation

Let

be any Dirac spinor field and let

where

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is a Clifford algebra 1-form.2 The covariant differential, includes the Clifford algebra valued connection one-form Now consider the second covariant differential on using (i.e., the torsion 2-form vanishes for the Levi-Civita connection), we obtain

Here we have introduced the convenient (Hodge) dual basis and used the identity The first term vanishes by the first Bianchi identity and the second term,

is proportional to the Einstein tensor. This provides a succinct representation of the Einstein equation. The Quadratic Spinor Lagrangian (QSL) [15, 16, 17, 18, 19] is given by

This QSL satisfies the spinor-curvature identity [20]

where is the Clifford algebra valued curvature 2-form. The field equations and 0 0 are equivalent to the Einstein equation and torsion free equation, respectively. The metric is given by The rhs of (4) expands to

Since

for a spinor field

normalized according to

we find that this QSL differs from the standard Hilbert scalar curvature Lagrangian only by an exact differential,

In the action this corresponds to a boundary term which does not affect the local equations of motion. 2

Our Dirac matrix conventions are omit the wedge for discussions of such “clifform” notation see [12, 13, 14].

We often

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5. Spinor Gauge Invariance of the QSL From the form of the Lagrangian (7), the QSL action for an extended region actually depends on the (normalized) spinor field only through the boundary term, not locally. A change of the spinor field within the interior of the region will leave the action unchanged. Consequently the Dirac spinor field has complete local

gauge invariance subject to the two restrictions (6). This six real parameter spinor gauge freedom can be represented in the form where is a normalized Dirac spinor with constant components and U is the Dirac spinor representation of a Lorentz transformation. Thus the gauge freedom of the normalized spinor field is a kind of local Lorentz gauge freedom. Considering the scalar curvature term in the Lagrangian (7), it can be recognized that the theory also has the usual local Lorentz gauge freedom associated with transformations of the orthonormal frame. Hence there appears to be two Lorentz gauge freedoms here. But are they really independent? The boundary term is

Let us consider a gauge transformed spinor field Then , The gauge transformed boundary term then becomes

The unitary transformations on the gammas induce Lorentz transformations, on the orthonormal frame indices. Thus, the six parameter spinor gauge freedom (with normalization condition) is entirely equivalent to applying the transformation to the orthonormal frame alone. Hence the boundary term really has one physically independent Lorentz gauge freedom. We showed that the QSL is dynamically equivalent to the tetrad (teleparallel) representation [21]. In the QSL we have a spinor field which has a six parameter local gauge freedom which effectively replaces the local Lorentz frame gauge freedom of the tetrad representation. 6. Gravitational Energy-Momentum Density

From Noether’s theorem, with we obtain a canonical energy-momentum 3-form,

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satisfying the conservation of energy-momentum Here the covariant differential D only operates on the spinor but not on the spacetime index. Expression (10) is still another pseudotensor, however, if we consider as a field in Minkowski spacetime, then this is a gravitational energy momentum tensor. Therefore classically, whether we need the concept of an energy-momentum density or not, depends on our viewpoint: (i) When treating the spacetime to be curved, we have no spacetime symmetry in general. According to the equivalence principle, there is no well defined energy-momentum density, (ii) With a Minkowski line element, associated with the spacetime symmetry we have a covariant energy-momentum density. This is useful in microscopic physics, where space and time lose its operational meaning. 7. Discussion

By changing the ten parameter metric variables gµ v to the sixteen parameter tetrad variables eaµ in the variational principle, Møller [9, 10] obtained a local energy-momentum density which is a tensor wrt coordinate transformations but is not a tensor wrt local Lorentz frame rotations. Recently de Andrade, Guillen and Pereira [11] gave a refined version of Møller’s expression, but it still depends on a local Lorentz frame rotation. In this paper, we showed that by adding an auxiliary Dirac spinor field to the action, we can obtain an expression which is a tensor wrt both coordinate and local Lorentz frame rotations. This extra six parameter spinor gauge freedom (eight parameter Dirac spinor with two normalization conditions) was shown to be equivalent to the Lorentz transformation for the associated orthonormal frame. By comparing the formulation with Yang-Mills gauge theory (Table 1), we find that we can define a gravitational field strength FG = DΨ.

Presently it is not so clear here what is the corresponding gauge group in this formulation. Several related approaches may clarify the situation: (1) using a semi-direct sum of the group SL(2,C) and C4 [18], (2) the Teleparallel approach [8, 21, 22, 23], (3) considering the spinor one-form as an anticommuting field

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[24, 25, 26, 27]. We also note that there is an interesting generalization of the quadratic spinor Lagrangian to the Einstein-Maxwell system [28]. We close by noting that the approach discussed here also suggests that we might be able to find an expression for a covariant gravitational “Lorentz force law”. References [1] C. W. Misner, K. S. Thome and J. A. Wheeler, Gravitation, (Freeman, San Francisco, 1973), p 467. [2] R. Penrose, Proc. R. Soc. London A381, 53 (1982). [3] J. D. Brown and J. W. York, Phys. Rev. D 47, 1407 (1993), gr-qc/9209012. [4] C.-M. Chen, J. M. Nester and R. S. Tung, Phys. Lett. A 203, 5 (1995), grqc/9411048. [5] A. J. Dougan and L. J. Mason, Phys. Rev. Lett. 67, 2119 (1991). [6] H. Poincare, La science et l'hypothèse, (Flammarion, Paris, 1904). [7] C. Wiesendanger, Class. Quantum Grav. 13, 681 (1996). [8] F. Gronwald and F. W. Hehl, in Proc. of the 14th Course of the School of Cosmology and Gravitation on Quantum Gravity, P. G. Bergmann et al ed. (World Scientific, Singapore, 1996), gr-qc/9602013. [9] C. Møller, Mat. Fys. Dan. Vid. Selsk. 1, No.10, 1 (1961); Ann. Phys. 12, 118 (1961). [10] J. N. Goldberg, in General Relativity and Gravitation – one Hundred Years After the Birth of Albert Einstein Vol.1, A. Held Ed. (Plenum, New York, 1980), p.478. [11] V. C. de Andrade, L. C. T. Guillen and J. G. Pereira, Phys. Rev. Lett. 84, 4533 (2000), gr-qc/0003100. [12] A. Dimakis and F. Müller-Hoissen, Class. Quantum Grav. 8, 2093 (1991). [13] E. W. Mielke, Geometrodynamics of Gauge Fields, (Akademie, Berlin, 1987). [14] F. Estabrook, Class. Quantum Grav. 8, L151 (1991). [15] J. M. Nester and R. S. Tung, Gen. Rel. Grav. 27 , 115 (1995), gr-qc/9407004 [16] R. S. Tung and T. Jacobson, Class. Quantum Grav. 12, L51 (1995), grqc/9502037.

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[17] D. C. Robinson, Class. Quantum Grav. 12, 307 (1996). [18] D. C. Robinson, Int. J. Theor. Phys. 37, 2067 (1998). [19] R. P. Wallner, J. Math. Phys. 36, 6937 (1995). [20] J. M. Nester, R. S. Tung and V. V. Zhytnikov, Class. Quantum Grav. 11, 983 (1994), gr-qc/9403026. [21] R. S. Tung and J. M. Nester, Phys. Rev. D 60, 021501 (1999), gr-qc/9809030. [22] Y. M. Cho, Phys. Rev. D 14, 2521 (1976). [23] V. C. de Andrade and J. G. Pereira, Phys. Rev. D 56, 4689 (1997), grqc/9703059. [24] I. Bars and S. W. MacDowell, Phys. Lett. B 71, 111 (1977); Gen. Rel Grav. 10, 205 (1979). [25] S. W. MacDowell and F. Mansouri, Phys. Rev. Lett. 38, 739 (1977). [26] F. Mansouri, Phys. Rev. D 16, 2456 (1977). [27] R. S. Tung, Phys. Lett. A 264, 341 (2000), gr-qc/9904008. [28] L. McCulloch and D. C. Robinson, Class. Quantum Grav. 17, 903 (2000).

SPINORS IN AFFINE THEORY OF GRAVITY

HORST-HEINO v. BORZESZKOWSKI Technical University Berlin, Institute for Theoretical Physics Hardenbergstrasse 36 D-10623 Berlin, Germany HANS-JÜRGEN TREDER Rosa-Luxemburg-Str. 17a D-14482 Potsdam, Germany

We consider some mathematical aspects of purely affine theories of gravity. In particular, we show that in affine spaces one can establish a truncated spinor formalism reducing in metric-affine spaces to the standard one. As a consequence, one can formulate gravitational equations with matter sources to which there can exist solutions with a Riemann-Cartan geometry.

1.

Geometrical

fundamentals

1.1 AFFINE GEOMETRY

To begin with, let us make some general remarks on the linear connection field from the point of view of the tangent bundle.1 We consider the connection defined in the tangent bundle over Then to the transformation of the bundle coordinates there corresponds the following transformation of the local components of the Lorentz connection

Here Using in

corresponds to the transformation to the transformation natural bundle coordinates onehas

(with

and (1) reduces to the (Riemann-Christoffel) transformation law of a linear connection under coordinate transformations (it acts only on space-time indices): 1 For the fibre bundle theory see, for example, [1]; for the above discussion, see also [2].

295 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 295-302 © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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On the other hand, for from (1) one obtains the (RicciLevi-Civita) transformation law of the connection under Lorentz transformations (it acts only on the anholonomic indices) Considering now the transformation from anholonomic to holonomic bundle coordinates, then, according to (1) the following (EinsteinCartan) transformation of the connection is associated with it: This relation is equivalent to the so-called Einstein lemma (// denotes here the covariant derivative with respect to the connection and /// the derivative acting simultaneously on holonomic and anholonomic indices) The local Lorentz transformations and the coordinate transformations which are above considered as different transformations of the coordinates in a Lorentz bundle can also be regarded as the homogeneous and inhomogeneous parts of the local Poincaré group2

acting in an affine space. Then, relation (6) can be considered as an implication of the transition from one item of the Poincaré transformation to the other.3 The mathematically equivalent relations (3)-(7) can also be justified by the physical requirement that a physical theory has to enable one to define measurement values. That this requirement is really satisfied by (6), and thus by the related formulas (3), (4), (5), can be seen as follows. The so-called Einstein coefficients are the components of the connection defining the teleparallelism of Einstein and Cartan: The measurement values of vectors and covectors do not change when they are transported by means of this connection:

2 In general, instead of the local Poincaré group (semiproduct of translations and pseudo-orthogonal transformations), one can consider the affine group (semiproduct of translations and general linear transformations) [cf., e.g., [3], [4], [5], [6]]. Here we confine ourselves to the Poincaré group since this group can be interpreted most simply as an requirement following from the principle of general relativity. 3 Following Schouten [7], we here consider the two parts of the Poincaré transformation as assotiated to anholonomic and holonomic transformations. For another understanding, see the Poincaré gauge field theory (Hehl et al. 1995 [6]). In this approach working in the framework of metric-affine space, the homogeneous

(linear) item of the Poincaré group is ascribed to a tensorial connection part which has to be added to the inhomogeneous (translational) part This procedure exploits the fact that the sum of a connection and a tensor is again a connection. To our mind, the generalized connection can conflict with the Weyl lemma establishing the connection between the holonomic and anholonomic representations.

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That the coefficients really provide a teleparallelism can be seen from the fact that the transport is transitive and integrable and thus independent of the way along which it is done. This is reflected by the vanishing of the curvature tensor formed from the Einstein coefficients: With the Ansatz given in (5),this provides the Cartan-Schouten representation of the Riemann tensor in anholonomic coordinates,

In order to couple gravity to matter one has to introduce spinorial quantities in affine spaces. To do so, it is helpful to remind of the fact that spinors are "scalars that satisfy a particular rule of differentiation" [8], [9]. In other words, we employ the one-to-two correspondence of the Lorentz group O(3,l) to the unimodular group SL(2,C) and require that the differentiation has to be covariant with respect to unimodular transformations with and where the

and

are connected by the relations

The coefficients of the spinor connection, variant derivatives [10],

which allow to built the co-

are specified by demanding Einstein's rule of teleparallelism (6) which now takes the form of "Weyl's lemma" demanding the covariant constancy of the spin-vectors4

This provides the following expressions for the coefficients of the connection,

4

This is the notation introduced by Infeld and van der Waerden [10]. In modern literature, sometimes it is also called "soldering form".

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1.2 RIEMANN-CARTAN GEOMETRY AS A SPECIAL CASE OF AFFINE GEOMETRY

In special cases, this purely affine framework can turn into a metric-affine framework, namely when the field equations, have a solution that allows one to introduce a covariantly constant (symmetric) metric tensor satisfying The solution of (17) is given by

where the contorsion is anti-symmetric in the first two indices, Otherwise, the Weyl lemma translating the connection from holonomic (space-time) to anholnomic (internal) coordinates provides for the anholonomic representation of the connection the expression

where

are the Ricci rotation coefficients,

and the last expression

in (19) is defined as both are antisymmetric in the first two indices. As a consequence, the anhlonomic components of the internal connection is antisymmetric in the first two indices, too: Exploiting the validity of the lemma of Ricci together with the relation finally one finds such that and, thus, is satisfied. This recovers the usual spinor formalism:5

5 By means of another line of arguments, this was also shown by Hayashi [11]. Even more, it was demonstrated there that this formalism can only be introduced iff the nonmetricity is of the Weyl form or vanishes.

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2. Affine Theory of Gravity

In the affine framework based on the symmetries given by the Poincaré group, geometry is completely determined by the affine connection Thus, a field theory has to start from a purely affine Lagrangian Its variation with respect to fine field equations

and the matter field provides the following purely af-

A classical example of an affine theory is the Einstein-Schrödinger theory stemming from the Lagrangian density [12], [13] (for its canonical structure, see [14])

According to the intention of their founders, it should represent a unified geometric description of gravitational and matter fields so that an additional matter Lagrangian need and must not be introduced. For we, however, want to regard (28) as the Lagrangian of vacuum gravity, here a matter Lagrangian has to be supplemented. It should be stressed that, as long as we aim at field equations of second order, (28) could only be supplemented by the square root of the determinant of the homothetic curvature. But, for in our framework we shall be interested in special solutions of the field equations which enable us to introduce a metric, this term is equal to zero. Using the Einstein affine tensor

as field coordinates the field equations stemming from (28) read [14]

where

and the Ricci tensor is given as

These equations satisfy the differential identity

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which first time was derived by Einstein and Kaufmann [12]. Eq. (34) holds due to the invariance of (29) under the transformation,

In more detail, it holds: As was shown by Bergmann [15], the principles of general relativity (i.e., of coordinate covariance) and of equivalence (i.e., of minimal coupling), implying a total Lagrangian of the form6 always lead to the following identity

where

the change of a quantity with fixed values of the coordinates,

is the

infinitesimal change of the coordinates coming in via the relation satisfied for a scalar density and and the Euler variations of the Lagrangian with respect to the connection and the matter fields. If the gravitational and matter field equations are satisfied the last three terms on Eq. (38) will vanish. In that case there exists a covariant vector density

satisfying the equation (Noether identity) The divergence (40) is a scalar density, too. In paper [15], it is also shown that, if we consider the case of a rigid displacement along one coordinate direction, const., then the expressions (39) and (40) reduce to the following

The coefficients are no covariant tensors, but only affine tensors and can be considered to be the components of the energy-momentum complex of the total system including gravitational and matter parts.7 In our case of the pure vacuum Lagrangian the relation (38) takes the form

6

For the sake of generality, in this passage, we shall assume that there exists also a metric which is independent of the connection, and we include the the matter term which comes into the game only in Sec. 2 . 7 To be more precice, the points summarized in this passage are in [15] partly formulated for the case of GRT. But they are evidently true for all Lagrangians of the form (37).

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so that, in the case that the field equations (30) are satisfied, Einstein's conservation law results:

Thus, the corresponding Noether identity reads where

Therefore, the Hamiltonian density corresponding to the Lagrangian H is given by the expression

This shows that the Einstein-Schrödinger theory with the field equations (30) can be rewritten in a canonical form, where the field coordinates are given by the components of the Einstein affine tensor and the field momenta by the components of the Ricci tensor Interestingly, as relation (36) shows, the momenta are invariant with respect to the gauge condition (35). As far as the canonical form is concerned, one finds here the same situation as in Maxwell's electrodynamics. Starting in the Maxwell case from the Lagrangian

one obtains the canonical energy-momentum tensor

revealing that the field momenta that are canonically conjugate to the field coordinates are given by the field tensor which are invariants of the gauge group of the electromagnetic theory, This gauge invariance can be preserved in the case of matter coupling considered in [16]. Adding the Dirac Lagrangian to (28) then the total Lagrangian is invariant with respect to gauge transformations of the second kind,

where (50) corresponds to the transformation of the the Dirac spinor In vitue of the Noether theorem, this invariance leads to the vanishung of the covariant divergence of the current density.

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3. References

1. 2. 3. 4. 5. 6.

7. 8. 9. 10.

11. 12. 13. 14. 15. 16.

Kobayshi, L., and Nomizu, K.: Foundations of Differential Geometry, Interscience, London, 1963. Treder, H.-J., and v. Borzeszkowski, H.-H.: Int. J. Theor. Phys. 8, (1973), 219. Hayashi, K., and Nakano, T.: Progr. Theor. Phys. 38 (1967), 49. Treder, H.-J.: Int. J. Theor. Phys. 3 (1970), 23. W., and Trautmann, A.: Spacetime and Gravitation, Wiley, Chisester and PWN Polish Scientific Publishers, Warsaw, 1992. Hehl, F. W., McCrea, D., Mielke, E. W. and Ne'man, Y.: Metric-affine theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance, Phys. Reports 258 (1995), 1-171. Schouten, J. A.: Ricci-Calculus, Springer, Berlin etc., 1954. Einstein, A., and Mayer, W.: Semivektoren und Spinoren, Sitz. Ber. Preuss. Akad. Wiss. Berlin, 522-550 (1932). Weyl, H.: Zs. Physik 56 (1929), 530. Infeld, L., and van der Waerden, B. L.: Die Wellengleichung des Elektrons in der allgemeinen Relativitätstheorie, Sitz. Ber. Preuss. Akad. Wiss. Berlin, 380-401 (1933). Hayashi, K.: Phys. Lett. 65B (1976), 437. Einstein, A.: Zur allgemeinen Relativitätstheorie, Sitz. Ber. Preuss. Akad. Wiss. Berlin, 32 (1923). Schrödinger, E.: Space – Time – Structure, Cambridge University Press, Cambridge, 1950. Treder, H.-J.: Astron. Nachr. 315 (1994), 1. Bergmann, P. G.: General relativity of theory, in Encyclopedia of Physics, vol. II, S. Flügge (ed.), Berlin etc., Springer, 1962. v. Borzeszkowski, H.-H., and Treder, H.-J.: Spinorial matter in affine theory (submitted for publication).

A NEW APPROACH TO QUANTUM GRAVITY An Overview

SARAH B.M. BELL JOHN P. CULLERNE BERNARD M. DIAZ Department of Computer Science I.Q. Group The University of Liverpool Liverpool, L69 7ZF, England

Abstract. We quantize General Relativity for a class of energy-momentum-stress tensors. Key words: quantum gravity, Bohr atom, Sommerfeld atom.

1. The New Method Of Quantization 1.1 THE GENESIS OF A CURVED SPACETIME

It has been generally believed that QED does not allow a classical spin for the particle described. However, it has now been shown that QED permits the spin of the particle to behave like a four-vector (Bell et al. 2000a&b). This was done by mapping the Dirac and photon equation into a new form, the versatile form,

where e is the charge on the electron. This permits the application of QED as a quantum version of General Relativity. We will study one particular sort of curved spacetime here which we will use as the building block for all the others. In this section we discuss its extrinsic description and relationship to the versatile Dirac and photon equations. We define the curvature in terms of co-ordinate systems. We shall call the original flat spacetime that applies to equation (1) the spacetime of the Large observer. We may co-ordinate the and planes of the 303 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 303-312. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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S. B.M. BELL, J. P. CULLERNE AND B. M. DIAZ

Large observer with polar

co-ordinates,

and

where we have and

and

are the arcs corresponding to

and

We introduce two further

observers who we shall call the Medium observer and the Small observer. We give the first the co-ordinate system while the second is given the co-ordinate system The spacetime ofthese observers is curved in such a way that equation (2) is replaced by

where

are constant.

are arcs as before. Both of equations (3) apply for the

Small observer, while the first of equations (3) and the last of equations (2) apply to the Medium observer. The spacetime of the Medium observer is associated with the circular orbits of the Bohr atom while that of the Small observer is associated with the elliptical orbits of the Sommerfeld atom. The Dirac equation (1) and the photon equations are transformed into the curved spacetimes of the Medium and Small observers using variables, for the Medium observer and

for the Small observer. The quaternion matrices are

transformed to lie along the arc and radius. They are then also transformed to the curved spacetime of the Medium and Small observers so that they appear constant for every location in t h e i r s p a c e t i m e s . The r e s u l t for the M e d i u m observer is and

unchanged, while for the Small observer We have, and for the reflector matrices. Similar

expressions apply to M and A. However, while the spacetime of the Medium and Small observers is locally flat, it is globally curved and this will affect any variable that depends on volume. The potential depends on a probability current per unit volume and therefore does not refer to the potential the Medium or Small observer will see. Equations (2) and (3) are used to find the volume seen by the Medium or Small observer and the photon equation to find the corresponding potential. This calculation shows that an inverse distance potential seen by the Large observer becomes a constant potential in the spacetimes of both the Medium and Small observers. More details are provided by Bell et al. (2000f).

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1.2 THE BOHR-SOMMERFELD ATOM We may now solve the Dirac equation amended as described in section 1.1 for the spacetime of the Medium observer to show the derivation of the Bohr atom as a consequence of QED. Our version of Bohr's equations, unlike his original treatment, will be relativistically correct. We suppose that the Bohr electron making up the one-electron atom has a velocity in the direction, that is, the electron has a circular orbit round the proton. We use the rest frame of the proton. We discover that

where we have set electron,

with and we have a plane wave solution

frequency and

the wave number.

the mass of the where

is the

is the potential due to the proton in the Medium

observer's spacetime. We have

where

is the charge on the proton.

is the Bohr radius at which both the Large and

Medium observers see the same potential.

is a constant. We see that the interaction

between the proton and electron has introduced no more than a change in the vacuum potential and swap where co-ordinates are exchanged for co-ordinates We may derive the Bohr equations from the solution to the Dirac equation, equation (4). We see from the third of equations (4) that there is an equivalent free electron with wave number and frequency

This is the Bohr electron. We require that the wave function to be single-valued from the point of view of the Large observer. De Broglie's relation between and the momentum of the Bohr electron then provides

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S. B.M. BELL, J. P. CULLERNE AND B. M. DIAZ

is the velocity of the electron, we have restored

and

is an integer?

This is the first of Bohr's equations. Equations (5) and (6), and de Broglie's relation between and the energy of the Bohr electron provides

However, there is another way of obtaining the total energy and that is to consider the geometry of the path of the Bohr electron. From the Bohr electron's point of view the mass is

and the electron is in a potential field. However, from the point of view of the atom

as whole there is no potential field and atom,

traveling at velocity

is the frequency associated with the mass of the

with respect to the atom's rest frame. From de Broglie's

relation for the atom, we must therefore have

This leads to

and to

which is Bohr's second equation. The total angular momentum of the Bohr atom is made up of an integer contribution from the orbital angular momentum, the half integer spin of the electron and a half integer contribution from the Berry phase (Anandan 1992) the electron gains each orbit because the orbit can be considered as a closed path containing the nucleus. We next derive the Sommerfeld model as a consequence of QED. This time we want to replace co-ordinates and take on the point of view of the Small observer. We adopt s1 as our temporal co-ordinate with

spatial so that the situation

becomes exactly analogous to the our derivation of the Bohr atom above. We can do this because the Dirac equation (1) can be shown to have a suitable tachyon form. We suppose a tachyon particle and anti-particle emerge, if the atom is suitably excited, in the form a bound state. We call the new particles the Green electron and Green positron. We shall call the bound state they form the Green atom. The emergence of the Green atom causes a transition from the spacetime of the Medium observer to that of the Small observer. The wavefunctions of both the Bohr electron and Green electron must be single-valued along the new loop and we may deduce two similar equations to (7). This is sufficient to enable us to calculate the tachyon equivalent of rest mass of the Green electron and positron, where

is an integer called the azimuthal quantum number. We

may also reproduce Bohr's equations for the Green atom assuming

is the temporal

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co-ordinate and the Green positron stationary. The velocity of the Green electron as it appears in Bohr's equations is

We next swap

for

as the temporal axis by

swapping energy and momentum eigenvalues for the Green electron and positron and discover the total contribution the Green atom makes to the frequency of the complete system,

We add the contributions of the Bohr electron and Green atom. This gives us the total frequency and momentum of the whole stucture in the spacetime of the Small observer. Repeated use of analogues to equation (9) allow us to transform these to the spacetime and frame of interest, that of the Large observer and where we retrieve the Sommerfeld expression for the energy levels of the atom. More details are provided by Bell et al. (2000e&f).

1.3 THE NEW QUANTIZATION In section 1.2 we sketched the derivation of the Bohr and Sommerfeld models of the atom from QED. Here we do the inverse, derive QED from the Bohr and Sommerfeld models. We do not need to do more than derive QED from the Bohr model because the extra Sommerfeld energy levels appear when the Bohr model is applied a second time, using the tachyonic Green atom. It is well known that the inverse square of distance law for the strength of the electrical field gives rise to the photon equation in differential form. We extend this using a similar method to the interaction of another particle with the potential field, A, described by the photon equation. We assume the Bohr equations, (7) and (10), apply to an electron interacting with an spherical volume of constant charge density responsible for the potential field, A, at point b in the spacetime of the Medium observer. The Bohr electron orbits round the surface of the spherical volume at the Bohr radius and we may arrange for the electron to make as many orbits as may be required to consider the electron bound whatever the total interaction may describe. We do this by letting the spherical volume decrease in size. As the volume becomes smaller the mass of the electron, will increase along with the charge, e, as in renormalisation. If we add equation (8) describing energy conservation to the Bohr equations we may assemble a wave function that describes the Bohr electron in a manner similar to equation (4). As the spherical volume becomes infinitesimal this wavefunction will satisfy the Dirac equation at point b from the point of view of the Medium observer. However, the steps we took to get from the Dirac equation in the spacetime of the Large observer to the Dirac equation in the spacetime of the Medium observer are reversible. We may return to a Dirac equation and solution with the same frequency term,

and equivalent angular momentum

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at point b in the spacetime of the Large observer. If Bohr's equations are valid for infinitesimal volumes at all points in the field then the Dirac equation must hold everywhere too and we therefore find that the full quantum theory is QED. We note that we have not mentioned the global boundary conditions or constrained the magnitude of the wavefunction in our discussion. The magnitude of the wave function would vary as required by the global boundary conditions. We find that if we negate both charges on our interacting bodies and replace the potential, A by (A–1) QED continues to apply. More details are provided by Bell et al. (2000e).

2. Application Of The New Method To Gravity 2.1 BOHR'S FIRST EQUATION

We now derive Bohr's two equations for quantum gravity. With electromagnetism we are in the position of the Large observer and the spatial loop, provides the initial bound state, with the temporal loop, giving extra fine structure. For gravity we are in the position of the Medium observer, and the first loop we see is temporal. Since we want to distinguish the gravitational hydrogen atom from the electromagnetic equivalent we will call the former the Thalesium atom. We will call the equivalent of the Green electron for a Thalesium atom a Geotron.. We will identify the point of view of the Green positron with that of the Thalesium atom. As we did above, we will assume the point of view of the Thalesium atom initially and assume our temporal co-ordinate is spatial. We therefore have the frame and spacetime ofthe Thalesium atom as being equivalent to that of the Medium observer except that a spatial and temporal co-ordinate are swapped, and the frame and spacetime of the Geotron as being equivalent to that of the Small observer except that a spatial and temporal co-ordinate are swapped and that the Geotron is a moving frame relative to the Thalesium atom. Spacetime appears flat to the Medium observer with his co-ordinates Cartesian since all the curvature produces is a variation in the vacuum potential. We will change our nomenclature and take the metric to be,

where we have assumed the point of view of the Thalesium atom and are treating x1 as temporal. We shall sometimes need the variant

where are polar co-ordinates. The metric we shall use to describe the curved spacetime of the Geotron is

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where we have set the gravitational constant to unity, t is the temporal co-ordinate and the analogue of in section 1. However, we are treating as temporal. a is twice the mass of the source responsible for the gravitational attraction, which takes on the role the Green positron played in the electromagnetic case. We suppose that equation (14) is an intrinsic description of the curvature we now link to QED. We want to consider a temporal loop where the Geotron is bound to the Thalesium atom and in a circular orbit. We therefore use metric (13) and set where is the change in angle according to the Thalesium atom. Equation (14) becomes

We consider a circuit of the Geotron round the centre of attraction in the plane in the metric given by equation (15). From the point of view of the Geotron position four-vector of the Geotron does not change direction, but the co-ordinate frame of the Thalesium atom does. So we can consider the position four-vector parallel transported as for the Berry phase. Let the two-vector giving the position of the Geotron be Following Martin (1995) and assuming the Geotron orbits at a fixed radius r, we then find

is also the angle the position vector of the Geotron makes relative to the Thalesium atom. We require a steady state in which the system returns to start position after a cycle. In that case after n circuits, when from the point ofview ofthe Thalesium atom must be an integer multiple of with both n and m integer. Although the spin is integer from the point of view of the Thalesium atom because the Berry phase is added, from the point ofview ofthe Geotron the spin is half integer as is that ofthe electron which is why we have added an extra factor of 2 for From the Thalesium atom's viewpoint the path length corresponds to an angle We have

Substituting for

and

R is a constant and

in equations (16) we see this imposes a condition on r, r = R where

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R is the equivalent of the Bohr radius and we shall call it that, defining it exactly in section 2.2. We would like to know the interval as the Thalesium atom sees it. Using metric (13) we obtain for the interval

We would also like to know the interval, as the Geotron sees it. Substituting from equation (19) into metric (14) and using equation (17) and (18) we obtain

We see that the Geotron is also entitled to see his spacetime as flat. Equation (18) provides information which will allow us to derive Bohr's first equation.

2.2 BOHR'S SECOND EQUATION

The Lagrangian associated with equation (14) gives us the acceleration of the Geotron for the orbit of the Geotron we discussed in section 2.1. The acceleration is

Next we calculate the acceleration seen by an observer with the frame of the Thalesium atom in his flat spacetime. We apply General Relativity to the problem (Kenyon 1990) and consider the covariant derivative for metric (12) and the equation for the geodesic in flat spacetime. From these we obtain

where

is equivalent to the velocity of the Geotron and we have used equations (19) and

(20) for the last two equations of (22). We see from equations (22) that it does not matter whether we are in frame and spacetime of the Thalesium atom or in the frame and spacetime of the Geotron. We obtain from equations (22) and (21)

From equations (18) and (23) we may formulate the Bohr equations in exactly the same

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form as they took for QED as described by Bell et al. (2000c&d). 2.3

QUANTUM GRAVITATIONAL DYNAMICS

We have been taking the point of view of the Thalesium atom or Geotron for our calculations above. We now wish to assume the point of view of the Small observer for whom the circular loop is temporal. We calculated the energy contribution for such a temporal loop in section 1.2, equation (11). For the current purpose we re-state the equation as We may also calculate the gravitational energy in the standard way (Martin 1995) from the metric (14) and Lagrangian. The two methods agree on the form of the result. On taking up the position where the loop is temporal our metric for the Medium and Small observer become We consider the state of the Thalesium atom. For QED there is no preferred plane for the angular momentum when we are not in an eigenstate. All possible orbits exist as a superposition of states and the atom appears spherically symmetrical. We assume the same for the Thalesium atom. The metric of the Small observer becomes where r labels a sphere of area and and describe colatitude and longitude. The Bohr equations can be derived in an identical manner from this metric as well, provided that the first two terms are set positive. We apply the method of quantization we used in section 1.3. We find that QED is the full quantum theory. Moreover, we can find the equivalent of the electromagnetic potential from the Bohr equations. In the frame of the Geotron it is A = a/(2r). Since we are discussing gravity, while keeping the equation itselfthe same, we will rename the photon equation the graviton equation. If we take the global point of view the potential will be a full four-vector. For the metric, we must set

for the local infinitesimal volume and then Lorentz transform into the global frame by some

Equation (24) covers Schwarzschild metric for the spacetime applying to a spherically symmetric source of gravitation (Martin 1995). We may therefore move to an interpretation of our general metric (24) from the point of view of General Relativity. We impose a change the potential of +1 for each infinitesimal spherical volume in the local rest frame and negate its charge. We negate the charge of the body interacting with the field. QED continues to apply, but each spherical volume has, in the local rest frame, the Schwarzschild metric. This means that, if we have the redefintion of metric (24) applying, at each location and time we have an infinitesimal system composed of two bodies interacting according to the theory of General Relativity. The quantum condition discussed in section 1.3 is still in place. At this

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point it can be shown that there is a one-to-one correspondence between the current density of QED, and the energy-momentum-stress tensor, of the Einstein equation for the permissible metrics of type (24) plus local Lorentz transformation

Acknowledgment One of us (Bell) would like to acknowledge the assistance of E.A.E. Bell.

References Anandan, J. (1992) The geometric phase, Nature 360, 307-313. Bell, S. B. M., Cullerne, J. P. and Diaz, B. M. (2000a) Classical behaviour of the Dirac bispinor, Foundations of Physics 30, 35-57. Bell, S. B. M., Cullerne, J. P. and Diaz, B. M. (2000b) The Dirac bispinor: new properties and location, Presented at the 16th International Conference on Few-Body Problems in Physics, Taipei, Taiwan, March 6-10. Bell, S. B. M., Cullerne, J. P. and Diaz, B. M. (2000c) A new approach to quantum gravity, Presented at the 16th International Conference on Few-Body Problems in Physics, Taipei, Taiwan, March 6-10. Bell, S. B. M., Cullerne, J. P. and Diaz, B. M. (2000d) A new approach to quantum gravity: a summary, to appear in the Proceedings of Physical Interpretations of Relativity Theory VII" at Imperial College, London September 15-18. Bell, S. B. M., Cullerne, J. P. and Diaz, B. M. (2000e) QED and the Bohr model of the atom, To be published. Bell, S. B. M., Cullerne, J. P. and Diaz, B. M. (2000f) QED and the Sommerfeld model of the atom, To be published. Kenyon, I. R. (1990) General Relativity, Oxford University Press, Oxford. Martin, J. L. (1995) General Relativity, a first course for physicists, Revised edition, Prentice Hall, London.

MULTIDIMENSIONAL GRAVITY AND COSMOLOGY AND PROBLEMS OF G

M.A. GREBENIUK Institute of Gravitation and Cosmology, PFUR, MichlukhoMaklaya Str. 6, Moscow 117198, Russia AND V.N. MELNIKOV Center for Gravitation and Fundamental Metrology, VNIIMS, 3-1 M. Ulyanovoy Str., Moscow 117313, Russia Institute of Gravitation and Cosmology, PFUR, MichlukhoMaklaya Str. 6, Moscow 117198, Russia Depto. de Fisica, CINVESTAV-IPN, Apartado Postal 14-740, Mexico 07000, D.F.

Abstract. Main trends in modern gravitation and cosmology and their relation to unified models are analysed. The role of multidimensional gravitational models with different matter sources in solving the basic problems of cosmology is stressed. Exact solutions in multidimensional cosmology with p-branes for some special cases are considered. The Riemann tensor squared is calculated in a general case and investigated for a number of special solutions. Singularities of the Riemann tensor squared are also disscussed. Also the case of static internal spaces is considered. It is shown that external space for this case is de Sitter or anti-de Sitter one. Behaviour of cosmological constant and its generation by p-branes is demonstrated.

1. Introduction The necessity of studying multidimensional models of gravitation and cosmology [1, 2] is motivated by several reasons. First, the main trend of modern physics is the unification of all known fundamental physical interactions: 313 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 313-320 © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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electromagnetic, weak, strong and gravitational ones. During last decades there was a significant progress in unifying weak and electromagnetic interactions, some more modest achievements in GUT, supersymmetric, string and superstring theories. Now theories with membranes, p-branes and more vague M- and Ftheories are being created and studied. Having no any definite successful theory of unification now, it is desirable to study the common features of these theories and their applications to solving basic problems of modern gravity and cosmology. Moreover, if we really believe in unified theories, the early stages of the Universe evolution and black hole physics, as unique superhigh energy regions, are the most proper and natural arena for them. Second, multidimensional gravitational models, as well as scalar-tensor theories of gravity, are the theoretical frameworks for describing possible temporal and range variations of fundamental physical constants [3, 4, 5, 6]. These ideas originated from earlier papers of E.Miln (1935) and P.Dirac (1937) on relations between fenomena of micro and macro worlds and up till now they are under a thorough study both theoretically and experimentally (see review on G in [42]). At last, applying multidimensional gravitational models to basic problems of modern cosmology and black hole physics we hope to find answers to such long standing problems as singular or nonsingular initial states, creation of the Universe, creation of matter and entropy in it, acceleration, cosmological constant, origin of inflation and specific scalar fields which are necessary for their realization, isotropization and graceful exit problems, stability and nature of fundamental constants [4], possible number of extra dimensions, their stable compactification etc. Bearing in mind that multidimensional gravitational models are certain generalizations of general relativity which is tested reliably for weak fields up to 0,001 and partially in strong fields (binary pulsars), it is quite natural to inquire about their possible observational or experimental windows. From what we already know, among these windows are: possible deviations from the Newton and Coulomb laws, possible variations of the effective gravitational constant with a time rate less than the Hubble one, possible existence of monopole modes in gravitational waves, different behaviour of strong field objects, such as multidimensional black holes, wormholes and p-branes, PPN parameters, cosmological models behaviour and standard cosmological tests etc. As modern cosmology already became a unique laboratory for testing standard unified models of physical interactions at energies that are far beyond the level of existing and future man-made accelerators and other

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installations on Earth, there exists a possibility of using cosmological and astrophysical data for discriminating between future unified schemes.

As no accepted unified model exists, in our approach we adopt simple, but general from the point of view of number of dimensions, models based on multidimensional Einstein equations with or without sources of different nature: cosmological constant, perfect and viscous fluids, scalar and electromagnetic fields, plus their interactions, dilaton and moduli fields, fields of antisymmetric forms (related to p-branes) etc. Our main objective was and is to obtain exact solutions (integrable models) for these model self-consistent systems and then to analyze them in cosmological, spherically and axially symmetric cases. In our view this is a natural and most reliable way to study highly nonlinear systems. It is done mainly within the Riemannian geometry. Some simple models in integrable Weyle geometry and with torsion were studied also. 2.

The model

Here we consider the model governed by the action [33, 34]

where

is a metric,

is a dilatonic scalar field,

is a on D-dimensional manifold constant and We consider the manifold

where equation For any

is a time variable and

is the cosmological with the metric

is a metric on we put in (2)

satisfying the

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where is thevolume For dilatonic scalar fields we put

Let

It is not difficult to verify that the field equations for the action (1) with the field configurations from (3), (4) may be obtained as equations of motion corresponding to the action

where function,

is the lapse is the matrix of the minisupermetric of the model and

is the potential. 3.

The case with Ricci-flat internal spaces

Let us consider the harmonic time gauge and also put Here we also assume that the orthogonality conditions

are satisfied for all with the for and We also put Then the exact solutions for the logarithms of scale fields and harmonic gauge in the considered case read

where

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Now let us investigate the obtained solutions for possible configurations. The Riemann tensor squared for the metric (3) takes the following form

We consider the case Due to the orthogonality conditions (8) and the relation (10) we have the following possibilities for the p-branes configuration

if respectively, and Using the rules (12), we can list all possible sets or collections of overlapping 2-brane and 5-brane. Here we study the case with We also assume that In this case the possible set reads

The investigation of this model shows, that the behaviour of the scale factors and thensor Reimann squared is depends on the signatures of the internal spaces. For example in the case with we have the expanding external space with compactified internal spaces. The Riemann tensor squared for the multidimensional metric in this case takes the following form

where and are the Riemann tensor squared and scalar curvature for the external space respectively. Here Thus, if we take the appropriate external space, we can get a nonsingular Riemann tensor squared for the multidimensional metric for this case. 4. The case of static internal spaces

Now let us consider the special case with the In this case we have one differential equation for set of algebraic equations for and

the

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It should be noted that there are two possibilities in this case: i.e. when forms are present in our external space or they are completely defined on internal subspaces. Now let us investigate the solutions in the first case. Let be a synchronous time. Here we put Using the ”fine-tuning” of a cosmological constant

where

Thus, for the new metric

we have As we can see, for we get the de Sitter space with the relations

Example. Now let us consider some more special cases. Here we put and

Thus, resolving the first equation in (18) we get the following formulas for the charges of forms

where

is some arbitrary constant. Here, as we can see, one must put Thus, the effective cosmological term of the space in this case reads

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so, for the case with and we may get the extremely small effective cosmological constant while the multidimensional cosmological bare constant has the Planck scale. The case with If we put we come to the case with Ricci-flat internal spaces. And inflationary solutions are generated in this case only due to the fields of forms (p-branes). 5.

Conclusion

In the paper were investigated some special classical solutions in the framework of the multidimensional cosmological model with additional scalar fields and antisymmetric forms (multidimensional cosmology with p-branes) obtained in [34]. The Riemann tensor squared of the whole multidimensional metric was constructed and investigated for some special solutions with different p-branes configurations. The singularity of the Riemann tensor squared was also considered. Exact solutions for the model were also obtained, when scale factors of internal spaces are constant. It was shown that external space for this case is de Sitter or anti-de Sitter one. Behaviour of cosmological constant and its generation by p-branes was also demonstrated. References 1. V.N. Melnikov, Multidimensional Classical and Quantum Cosmology and Gravitation.Exact Solutions and Variations of Constants, CBPF-NF-051/93, Rio de Janeiro, 1993. V.N. Melnikov. In: Cosmology and Gravitation, ed. M. Novello (Editions Frontieres, Singapore, 1994) p. 147. 2. V.N. Melnikov, Multidimensional Cosmology and Gravitation, CBPF-MO-002/95, Rio de Janeiro, 1995, 210 p. V.N. Melnikov. In Cosmology and Gravitation.II ed. M. Novello (Editions Frontieres, Singapore, 1996) p. 465. 3. K.P. Staniukovich and V.N. Melnikov, ”Hydrodynamics, Fields and Constants in the Theory of Gravitation”, Energoatomizdat, Moscow, 1983, (in Russian). 4. V.N. Melnikov, Int. J. Theor. Phys., 33, No 7, 1569 (1994). 5. V. de Sabbata, V.N. Melnikov and P.I. Pronin, Prog. Theor. Phys., 88, 623 (1992). 6. V.N. Melnikov. In: Gravitational Measurements, Fundamental Metrology and Constants. Eds. V. de Sabbata and V.N. Melnikov (Kluwer Academic Publ.) Dordtrecht, 1988, p.283. 7. V.D. Ivashchuk and V.N. Melnikov, Nuovo Cimento, B102, 131 (1988). 8. K.A. Bronnikov, V.D. Ivashchuk and V.N. Melnikov, Nuovo Cimento, B102, 209 (1988). 9. V.D. Ivashchuk and V.N. Melnikov, Phys. Lett., A135, 465 (1989). 10. V.D. Ivashchuk, V.N. Melnikov and A.I. Zhuk, Nuovo Cimento, B104, 575 (1989). 11. S.B. Fadeev, V.D. Ivashchuk and V.N. Melnikov. In Gravitation and Modern Cosmology (Plenum, N.-Y., 1991) p. 37. 12. V.D. Ivashchuk, Phys. Lett., A170, 16 (1992).

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13. V.D. Ivashchuk and V.N. Melnikov, Teor. Mat. Fiz., 98, 312 (1994) (in Russian). 14. V.D. Ivashchuk and V.N. Melnikov, Int. J. Mod. Phys., D3, 795 (1994), grqc/9403063. 15. V.R. Gavrilov, V.D. Ivashchuk and V.N. Melnikov, J. Math. Phys., 36, 5829 (1995). 16. V.D. Ivashchuk, A.A. Kirillov and V.N. Melnikov, Pis’ma ZhETF, 60 No 4 (1994), 225 (in Russian). 17. V.D. Ivashchuk and V.N. Melnikov, Class. Quantum Grav., 12, 809 (1995). 18. A.A. Kirillov and V.N. Melnikov, Phys. Rev., D52, 723 (1995). 19. K.A. Bronnikov, V.D. Ivashchuk in Abstr. VIII Soviet Grav. Conf., (Erevan, EGU, 1988) p. 156. 20. S.B. Fadeev, V.D. Ivashchuk and V.N. Melnikov, Phys. Lett, A161, 98 (1991). 21. K.A. Bronnikov and V.N. Melnikov, Annals of Physics (N.Y.), 239, 40 (1995). 22. V.D. Ivashchuk and V.N. Melnikov, Class. Quantum Grav., 11, 1793 (1994). 23. V.D. Ivashchuk and V.N. Melnikov, Grav. & Cosmol., 1, No 3, 204 (1995). 24. V.D. Ivashchuk and V.N. Melnikov, Phys. Lett., B384, 58 (1996). 25. U. Bleyer, V.D. Ivashchuk, V.N. Melnikov and A.I. Zhuk, Nucl. Phys., B429 117 (1994). gr-qc/9405020. 26. V.R. Gavrilov, V.D. Ivashchuk, and V.N. Melnikov, Class. Quant. Grav., 13, 3039 (1996). 27. V.R. Gavrilov and V.N. Melnikov, Theor. Math. Phys., 114, No 3, 454 (1998). 28. V.R. Gavrilov, V.N. Melnikov and R. Triay, Class. Quant. Grav., 14, 2203 (1997). V.R. Gavrilov, V.N. Melnikov and M. Novello, Grav. & Cosmol., 1, No 2, 149 (1995). V.R. Gavrilov, V.N. Melnikov and M. Novello, ”Bulk Viscosity and Entropy Production in Multidimensional Integrable Cosmology”, Grav. and Cosmol., 2, No 4, 325 (1996). 29. V.D. Ivashchuk and V.N. Melnikov, ”Multidimensional Gravity with Einstein Internal Spaces”, Grav. & Cosmol., 2, No 3, 177 (1996). hep-th/9612054. 30. V.D. Ivashchuk and V.N. Melnikov, ”Intersecting p-Brane Solutions in Multidimensional Gravity and M-Theory”, Grav. & Cosmol., 2, No 4, 297 (1996). hep-th/9612089. 31. V.D. Ivashchuk and V.N. Melnikov, Phys. Lett. B, 403, 23 (1997). 32. V.D. Ivashchuk and V.N. Melnikov, ”Sigma-Model for Generalized Composite pBranes”, Class. and Quant. Grav., 14, No 11, 3001 (1997). hep-th/9705036. 33. K.A. Bronnikov, M.A. Grebeniuk, V.D. Ivashchuk and V.N. Melnikov, ”Integrable Multidimensional Cosmology for Intersecting p-Branes”, Grav. & Cosmol., 3, No 2, 105 (1997). 34. M.A. Grebeniuk, V.D. Ivashchuk and V.N. Melnikov, ”Integrable Multidimensional Quantum Cosmology for Intersecting p-Branes”, Grav. & Cosmol., 3, No 3, 243 (1997). gr-qc/9708031. 35. M.A. Grebeniuk, V.D. Ivashchuk and V.N. Melnikov, ”Multidimensional Cosmology for Intersecting p-Branes with Static Internal Spaces”, Grav. & Cosmol., 4, No 2 (1998). 36. V.D. Ivashchuk and V.N. Melnikov, Int. J. Mod. Phys. D, 4, 167 (1995). 37. V.D. Ivashchuk and V.N. Melnikov, ”On Singular Solutions in Multidimensional Gravity”, Grav. & Cosmol. 1, No 3, 204 (1996). hep-th/9612089. 38. V.D. Ivashchuk and V.N. Melnikov, ”Multidimensional Classical and Quantum Cosmology with Intersecting p-Branes”, J. Math. Phys., 39, 2866 (1998). hep-th/9708157. 39. V.D. Ivashchuk and V.N. Melnikov, ”Cosmological and Spherically Symmetric Solutions with Intersecting p-Branes”, J. Math. Phys., 40, No 12, 6568 (1999). 40. M.A. Grebeniuk, V.D. Ivashchuk, V.N. Melnikov and R. Triay, ”On Some Cosmological Models with Fields of Forms”, Grav. & Cosmol., 5, No 3, 229 (1999). 41. V.D. Ivashchuk and V.N. Melnikov, Proc. 2nd ICRA Workshop ” The Chaotic Universe” , WS, 2000, p.509. 42. V.N. Melnikov, ”Fundamental Physical Constants and Multidimensional Black Holes”, Grav. & Cosmol., 6, No 2, 81 (2000).

QUANTUM GRAVITY OPERATORS AND NASCENT COSMOLOGIES

LAWRENCE B. CROWELL ALpha Institute of Advanced Study 11 Rutafa Street, H Budapest.. H-11 65, Hungary

1. Introduction The Planck length is a fundamental scale of physics. Suppose that we have a black hole with a whose Scbwarzshild radius is known to be Now let this black hole have a de Broglie wave length

where

the spacetime momentum

of the black hole. Assume this wave length is equal to its diameter defined by the Schwarschild radius, further, the mass of the black hole is The analyst finds that the de Broglie wavelength of this blackhole is,

This length converts into energy units as scale is

2. Operators And Corrections To Gravity Andre Sakharov suggested foundations of gravitation as a metric elasticity [3]. This metric elasticity was based upon a microscopic structure analogous to atomic structure in material elasticity. This lead Wheeler to proposed pregeometry as something more fundamental than geometry. Then the gravitational coupling parameter is a grand sum over many pregeometric modes. So gravitation relies on the quantum physics of the vacuum. A proposed Lagrangian that describes the zero-point energy of the geometrodynamic vacuum advanced by Ruzmaikina and Ruzmaikin is a power series in spacetime curvature [4]

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are coupling coefficients of order unity, and k are the modes of the

Here

pregeometric entities. This corrections to Einstein's general relativity has departures at high energy due to the breakdown of the spacetime picture. The first terni is expected to be removed by renormalization arguments. From the second term we can see that Ashtekar presented the nature ofthe limits as and as they recover classical gravitation and standard quantum mechanics [5]. This illustrates that quantum gravity possesses two limits, or that and are variable or renormalizable. In a similar manner this author has discovered that quantum radiance by horizons in spacetime will generalize the nature of quantum uncertainty [1] Here temperature gravity evaluated by the Killing field normal to the horizon

This demonstrates that the vacuum state is a thermal or mixed state. Now perform these calculations with boson creation and annihilation operators that are thermally distributed,

where these then act on a pure vacuum state

and

are the creation and annihilation operators for a boson field, and equation (8)

is analogous to the Heisenberg picture. The commutator of these generalized operators is then found to be

and uncertainty between position and momentum is The uncertainty of one conjugate variable under a squeeze state operator diverges from the standard Heisenberg result. For large uncertainty of the squeezed variable increases along with the uncertainty ofthe "desqueezed” conjugate variable. II the squeeze state operator

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is applied to the operator ad we find that it is transformed according to

There is then observed to be a gravitational squeezed state operator defined as

that enters into the commutation of the creation and annihilation operators as

The boson field operators examined in this example may be considered to be those for the gravity field. They may be considered as Sen connections written according to Ashtekar variables. Thus quantum gravity involves the renormalization of the Planck scale and a redefinition of quantum uncertainty. 3. Metric Fluctuations And Quantum Fields

Ashtekar [6, 7] provided a demonstration of semiclassical theory according to and which holds here. The quantum expectation then has a maximum that determines the classical limit. Consider a metric fluctuations dual to a term

that is a fluctuation in the

momentum metric conjugate to the metric for a spacial surface of evolution. This fluctuation is then an acceleration that is associated with nongeodesic motion of a test particle on the spacial surface of evolution. Equation 6 shows that quantum fluctuations in the metric involves a renormalization of the Planck constant. We consider momentum metric fluctuations with the ADM Hamiltonian constraint,

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that defines the operator in the Wheeler-DeWitt equation,

We may represent the momentum operator with the fluctuations in the momentum according to

where the momentum metric fluctuation is

where is the amplitude of the cosmological wave function For low energies these fluctuations are small in comparison to the expectation values. From here on we set is the Planck length. These momentum metric fluctuations are related to the metric fluctuations according to

Given that there is an “arrow of time” expressed as

the momentum operator may be

This is computed with the Poisson bracket with the classical ADM hamiltonian as

The corresponding term with fluctuations in the shift parameter vanish. The fluctuation is evaluated to be

The Killing vector is a vector that is constant along the momentum of a particle:

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The fluctuation in momentum of a particle, as it is carried along with a fluctuation in the metric momentum, is then seen to be in components

which means that the fluctuation part of the momentum operator is seen to be

When this momentum fluctuation is projected onto the Killing field this results in the constant vector

N ow the projection of the Killing field onto the momentum fluctuation satisfies

where the first right hand side term vanishes, For a fluctuation mode with frequency

and so the fluctuation propagates on a null direction. We then can construct the Killing vector as

where we have used the Heisenberg uncertainty principle and have included the Planck constant explicitly. Let us consider the fluctuation on a single spacial direction. This Killing vector on null coordinates is

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We have that is the inertial time that is defined for the classical spacetime. The quantum fluctuation determines a Killing time defined on the horizon as

From the Killing vector we have that this equals

or that

We now consider the field expanded into normal modes as

The Killing time is defined by the fluctuations around the classical horizons. As such the phases are restricted to the horizon so that

A Fourier transform of this phase is

For we have that the phase flinction has the generator V. We then perform an analytical extension into the complex plane with to obtain

This means that the Fourier transform is of the form

QUANTUM GRAVITY OPERATORS

It is then possible to examine the same Fourier integral for to

327

which corresponds

and we have that

As this corresponds for we have an extension ofthe wave function across the horizon. We may then demonstrate that,

From this it can be demonstrated that the vacuum state is then defined as

Here it is evident that the modes associated with these fluctuations play the role of a temperature. From this the field assumes the form

The field operators are then generalized according to

which also converge to the zero temperature result for These generalized operators are consistent with the results of Feinstein and Sebastian when the phase is averaged, or equivalently a coarse graining of the phase space. Then the scattering and tunneling of the vacuum has an entropy change S = In cosh (2r) where is the squeezing parameter [2]. The squeezing parameter then is a measure of the tunneling. Here the squeeze parameter may be evaluated as

so the generation of matter is associated with the initial entropy of the cosmology. Here p-,

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is the energy density of the vacuum. The generalized operators in equation 7 leads to the uncertainty spread

which are consistent with the result of Feinstein and Sebastian. Feinstein and Sebastian illustrate that the squeezing of states that are solutions to the Wheeler-DeWitt equation will exhibit tunneling and that the entropy measure of this process is the generation of mass-energy within a cosmology [2]. Above it is illustrated that operators in curved spacetime exhibit a form of gravitationally induced squeezing [1]. These two pieces of the puzzle indicate how the universe spontaneously tunneled out of the vacuum. Further, as the total Gibbs free energy created is zero, the mass-energy of the universe is the temperature times the entropy of the initial state of the universe.

4. Phenomenology With Strings And Membranes

Now it is illustrated how the above theory is related to virtual fluctuations of strings and membranes. What follows is a phenomenological discussion of the problem. Strings and membranes are spaces with fields parameterized on them. The tension of a membrane is the determinant of the cosmological constant. We consider the p-brane as embedded in n dimensions. The stress-energy for a p-brane is then

Here thevielbeins are determined by the Wess -Zumin -Witten action [8,9], and the dynamics as a a model. The membrane in this model is a 2-brane that sweeps out a 3-volnine. We then assume that the membrane is determined by gauge potentials on this volume. The WZW action for this system is then determined by these fields on the boundary of this volume and within it,

This action pertains to the dynamics of a membrane whose boundaries are a two dimensional of spacetime with spacial and temporal directions. The first term can be rewritten with the Stokes theorem

to

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produce the Chern-Simons form as the integrand for the action. If we define the gauge potential in one direction as and apply a gauge covariant Euler-Lagrange equation we find the wave equation

For brevity we consider the gauge potential as being along one spacial direction and restrict it to just Here is the coupling constant for the field. This differential equation requires that the gauge potential contain both Yang-Mills and Lorentz Chern-Simons potentials with different cohomologies [10]. This cubic Klein-Gordon equation has been examined in gauge theories and QCD. This differential equation determines the motion of a soliton wave [11]. We consider this soliton as moving in one direction. The solution to this equation is then

where the coupling constant is related to

We then consider this soliton as

propagating along the membrane, and where the boundaries of this membrane are a virtual fluctuation of a spacetime. We may think of this soliton as propagating on this membrane. The field

then acts upon the vacuum state.

We naively consider the virtual string as a perturbing field that interacts with the membrane. The open ends of the string are tied to the membrane and satisfy Dirichlet Boundary conditions. We then simply consider this interaction as the addition of the field to

If we consider this perturbation to only

we then have that the string field

obeys the differential equation

An analysis may be approached with the raising operators for the scattered field. Let the incident field possess the lowering and raising operators a

and

scattered and reflected wave have the lowering and raising operators b where

and

Then consider the and

We then posit the asymptotic wave functions

The analytic solution to the differential equation is easily seen to be

and and

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which describes a topological soliton. Here A gives the reflection coefficient and B determines the transmission coefficient. The wave ftinction this may be written as

where for comparatively large squeezing parameter

The transmission coefficient

when expressed according to the raising and

lowering operators for the field leads to

which further leads to the result consistent with

These results are consistent with the derivation of a cosmology that tunnels out of the quantum vacuum [2]. This is effectively "creation without creation," and creation from nothing. Thomas Aquinas wrote in his Summa contra Gentiles III "Materia artificialium est a natura, naturalium vero per creationem a Deo." Yet it appears as if nothingness, or the quantum gravity vacuum permits the creation of something from nothing "Creati Ex Nihilo," References [1] L. Crowell, Found. Phys. Left 12, 6 (1999) [2] A. Feinstein, M. Sebastian, Found. Phys. Lett. 13, 2 (2000) [3] A. D. Sakharov, Dokiady Akad. Nank. S. SR. 177, 7({71 (1967),(english) Soviet Phys. Dokiady 12 1040-1041 [4] Ruzinaikina and Ruzmaikin Somet Physics JETP 30, 2, pp.372-374(1970) [5] A. Ashtekar Phys. Rev Lett 77, 24 (1996) [6] A. Ashtekar, Phys. Rev. Lett., 57, 2244-7, (1986). [7] C. Rovelli, Class. Qnant Grav., 8,1613-1675, (1991) [8] E. Witten, Comm. Math. Phw9. 92, 455, (1986). [9] 5. P. Novikov, Usp. Mat Nauk. 37, 3, (1982). [10] M. B. Green, 3. H. Schwartz, E. Witten Superstring Theory vol 2, p 381-383, Cam- bridge, (1987) [11] H. B. Nielson, P. Olesen, Nucl. Phys. B61, 45 (1973)

GRAVITATIONAL MAGNETISM: AN UPDATE

SAUL-PAUL SIRAG International Space Sciences Organization (ISSO) 3220 Sacramento Street, San Francisco, CA 94115

Abstract. Gravitational magnetism (or the Blackett effect) is the generation of a magnetic field by an electrically neutral rotating mass, whose magnitude is determined by analogy with the magnetic field generated by a rotating electric charge. Since 1947, there is increasing evidence for this effect by the measurements of the magnetic fields of the solar planets, the sun, other stars, and even pulsars, as well as the galactic magnetic field. However, the attempt to measure this effect in the laboratory depends on the ability to measure extremely weak magnetic fields and the shielding of extraneous magnetic fields. Early attempts to measure this effect in the laboratory depended on ad hoc extensions of the simple rotational version of gravitational magnetism. Recently there have been more sophisticated laboratory approaches. Also the extended observational evidence has generated a plethora of theoretical attempts to derive the Blackett equation in a larger context. Of particular interest is the work of R.I. Gray, who performed an advanced version of Blackett’s static experiment, and also related the Blackett effect to several other theoretical and empirical relations particularly the Wesson effect--the constancy of the ratio of spin to mass-squared for planetary, stellar, and galactic bodies. Pauli’s anomalous magnetic moment (as a Blackett effect) is also considered as a bridge to the gravitomagnetic field generated by superconductors.

1. The Early Work: 1912 – 1979

In 1912 Arthur Schuster1, in discussing “the possible causes of terrestrial magnetism,” made the very tentative hypothesis: “If magnetisation and rotation go together, the sun and the planets would all be magnetic.” Here presumably mass in rotation would play the role analogous to that of charge in rotation. Schuster’s speculation was tested experimentally by Wilson2 and by Swann and Longacre3. However, these tests were not tests of the straightforward analogy suggested by Schuster. This situation was reviewed by P.M.S. Blackett4 in 1947, when he showed that the ratio of magnetic moment P to angular momentum U for the earth, the sun and the newly measured star 78-Virginis was a close fit to the simple formula:

331 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 331-336. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Where G is Newton’s constant, c is the speed of light, and is a “form factor” which should be close to 1. Note that this equation is usually written in Gaussian electromagnetic units, where the Coulomb constant is But I have rewritten it in SI units where k is the Coulomb constant, to make the parallelism more clear. The consequences of this equation are called the Blackett effect, although for historical reasons it is also called the Schuster law, or the Schuster-Blackett law, or the WilsonBlackett law. [Note well: in my 1979 paper5, I called the consequences of the Blackett equation the Blackett effect and also the gravi-magnetic effect. And other papers have followed this “gravi-magnetic” nomenclature. However, in 1984 a paper was published6, which used the term “gravitomagnetic field” to describe (by analogy with Maxwell’s equations) the purely gravitational field, one of whose consequences is the well known LenseThirring (“frame dragging”) effect. Since then, the term “gravitomagnetism” has become standard. (See especially reference 7: Ciufolini & Wheeler.) Therefore, I will use only the term Blackett effect; and will drop the use of “gravi-magnetism” to avoid confusion with the term “gravitomagnetism.”] It is widely believed8,9 that Blackett did a laboratory experiment which ruled out this equation. However, his huge experimental paper10 described a static test in which a 15-kg gold cylinder at rest was presumed to pick up an induced current (producing a small, but measurable, magnetic field) from the rotation of the earth. In my 1979 paper5 “Gravitational Magnetism” I pointed out that the Blackett effect had yet to be definitively tested in the laboratory. Also I compiled the data from several newly measured astronomical bodies, which was a near fit to the Blackett equation. Thus I proposed that a fresh attempt be made using the latest magnetic field detectors (SQUID-magnetometers) to test the simple rotational version of this equation. The trend line of the data for P/U (magnetic moment versus angular momentum) was fairly systematically offset from the prediction line, where is set to 1, so that the offset is indicated by the average . Of course, there would also be electrical-magnetic effects in many of the astronomical bodies, so that these electrical “dynamo” and other effects serve to dampen somewhat the primary Blackett effect. This idea is supported by the fact that Mercury provides the datum point closest to the prediction line. Because of Mercury’s slow rotation rate and small size, it was presumed that Mercury had no magnetic field. When Mariner 10, which was equipped with a magnetometer to measure the magnetic field of Venus, also measured a magnetic dipole field on Mercury, it was a great surprise11. The only datum point to fall far short of the Blackett prediction line is Mars. This may indicate that we are seeing Mars when the interaction between its primary (Blackett) field is being pumped down by its interaction with the electrical “dynamo” field. Here I should mention that Surdin12 has proposed a formula (based on stochastic electrodynamics, SED) similar to the Blackett effect in which polar flipping is to be expected. He has suggested that Mars is presently being viewed in process of changing polarity. Surdin13 has also done an experiment in which he claims to have measured the rapid changes of polarity of an electrically neutral rotating body. The measurement depends on signal autocorrelation, and various possible “parasitic” effects have to be ruled out. This very intriguing experiment needs to be repeated.

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2. Post 1979 Work

The most straightforward test of the Blackett equation would be to measure directly the magnetic field of a rotating neutral body (which is not also a ferromagnetic substance). Blackett4,10 suggested that a 1-meter bronze sphere spun at 100 Hz would do nicely, except that this is the maximum safe speed, and there are severe problems in nulling out extraneous magnetic fields. With modern SQUIDs and mu-metal shielded rooms, such an experiment can be attempted. Exactly such an experimental design14 was described at the SQUID ‘85 conference in Berlin. However, the results of this experiment have not been published. Another experimental result is not widely known because it is described in the book, Unified Physics, by R.I. Gray15. He found the simple rotational version of the Blackett effect experiment too difficult, so he carried out an improved version of Blackett’s static experiment with positive results. Gray also found an intriguing relationship between the Blackett effect and the somewhat analogous effect described by Wesson16, in which the ratio of angular momentum to the square of the mass of astronomical bodies remains fairly constant over the vast range of planetary bodies to galactic clusters. If “fairly constant” can be idealized to “constant”, these two effects can be compared, in this pristine form, as follows (and note that Blackett’s symbols for magnetic moment P and angular momentum U have been replaced by and J, to make the relationships clear): The Blackett constant can be written as:

The Wesson constant can be written as:

(where Gaussian units have been used; and where G is the gravitational constant, c is the speed of light and 137 is the inverse of the fine structure constant). This suggests various relationships between b and w, such as:

The fundamental nature of such relationships hints at deep connections between the microscopic world and the macroscopic world of astronomical objects, on which both the Blackett effect and the Wesson effect are based. Blackett himself4 was motivated by the possibility of finding a connection between macroscopic physics and microscopic physics. Gray15 has pushed this relationship much further. Jack Sarfatti (in these proceedings) has extended Gray’s ideas to black-hole physics and the Planck-scale world, in which there is a mass-scale duality (as in M-theory) between the Wesson and Blackett effects.

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M-theory is a generalization of superstring theory, which is based on a fundamental Wesson-like parameter. As Abdus Salam17 puts it: “A closed string is a loop which replaces a spacetime point. Its quantum oscillations correspond to particles of higher spins and higher masses, which may be arranged on a linear trajectory in a spin-versus-mass2 (Regge) plot. If the slope parameter of this trajectory – the only parameter in the theory – is adjusted to equal the Newtonian gravitational constant, one can show, quite miraculously, that in the zeroth order of the closed bosonic string there emerges from the string theory Einstein’s gravity in its fullness! (The higher orders give modifications to Einstein’s theory with corrections which have a range ofPlanck

Note that here, Salam is using units in which c is set to 1, so that the Regge plot is being “adjusted” to G/c, which is essentially the Wesson constant 137G/2c for the of astronomical objects. Peter Brosche18 has also analyzed the “mass-angular momentum diagram of astronomical objects” and his work (as well as Wesson’s) was compared to the Blackett effect by V. De Sabbata and M. Gasperini19. This analysis was continued in the book by De Sabbata and C. Sivaram20, Spin and Torsion in Gravitation. Here torsion is described in the standard way of Cartan as “the antisymmetric part of an asymmetric affine connection.” However, in this context, they push torsion beyond the formalism of the Einstein-Cartan theory, where (usually) torsion is minimally coupled to spin and cannot propagate. Nonminimal coupling of electromagnetism and gravitation is implicit in the Blackett effect, as is argued by James F. Woodward21, who devotes most of his paper to an attempt to use 100 pulsars as data points to provide evidence for the Blackett effect. He finds that the Blackett “form factor” must evolve somewhat over the lifetimes of pulsars. This may indicate that there is some peculiar aspect of pulsars not yet understood. Perhaps the most sophisticated approach to the Blackett effect is that of A.O. Barut and Thomas Gornitz.22 Here the concept of magnetic moment is analyzed in the context of 5-dimensional Kaluza-Klein theories of unification of gravity with electromagnetism. In particular, Pauli’s (1933)23 papers on spin in 4-d projective space (with 5 homogenous coordinates) are reviewed. In addition to the usual K-K type minimal coupling of gravity and electromagnetism, Pauli found an anomalous coupling to the electromagnetic field, in which the ratio of the anomalous magnetic moment to the spin of an elementary particle is

Thus Pauli says: “from the extra term it can be concluded that electrically neutral masses with a nonzero spin moment must have a small magnetic moment (which is with respect to the problem of the earth’s magnetism not without interest).” Barut and Gornitz suggest that in a macroscopic body each elementary particle will have an anomalous magnetic moment equal to the spin S multiplied by

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Thus in a rotating macroscopic object, magnetic moment will accumulate and rise to the Blackett magnetic moment:

In this picture, the accumulation of elementary spin magnetic moments is due to the fact that elementary spins are really orbital angular momenta in the 5th dimension. Thus they say: “In conventional gravitational theories it seems to be difficult to understand a relationship of the type [Blackett’s equation] without going to the fifth dimension.” If the idea that spins (and therefore the anomalous Pauli magnetic moments) can accumulate via rotation of macroscopic bodies is correct, then there may be a relationship between the Blackett effect and the gravitomagnetic field. Readers who have heeded the “Note well” caveat near the beginning of this paper, will remember that the gravitomagnetic field is quite different from the Blacketteffect magnetic field, which I called the gravi-magnetic field in 1979. The gravitomagnetic field is not a magnetic field but a gravitational analog by way of writing the equations of general relativity in the form of Maxwell-like vector equations. The bridge between the Blackett-effect magnetic field and the gravitomagnetic field might be built by way of the work of Ning Li and David Torr24,25,26. The key idea seems to be that, although ordinarily, the gravitomagnetic (and electrogravitic) fields are too small to measure, the alignment of spins in the lattice ions of superconductors, make the gravitomagnetic field much larger (by 11 orders of magnitude) than the magnetic field. As we have seen, according to the Barut-Gornitz picture, the rotation of a macroscopic body will accumulate (and thus magnify) the tiny anomalous magnetic moments associated with elementary spins. It would now seem that superconductivity is another way for anomalous magnetic moments to accumulate by the alignment of spins. This accumulation of anomalous magnetic moment would have the same form as the Blackett effect. According to Torr and Li26, “It is shown that the coherent alignment of lattice ion spins will generate a detectable gravitomagnetic field, and in the presence of a timedependent applied magnetic vector potential field, a detectable gravitoelectric field.” It would seem that the Blackett effect may well be entailed in these superconductors via the lattice ion spin analog to the Pauli-Barut-Gornitz rotational mechanism. Some possible technological consequences of work of Li and Torr are described in references 27 & 28. I have listed additional references, 29-35, to the Blackett effect. See especially reference 35, where the Blackett effect is (like the Wesson effect) extended to galactic and intergalactic structures. As Opher and Wichoski say (in Ref. 35 “As far as we know, cosmic magnetic fields pervade the Universe.”

Acknowledgments

It was Hal Puthoff who (at Vigier III) mentioned Paul Wesson’s paper (ref. 16) and later sent it to me. And it was Creon Levit who discovered R.I. Gray’s book (ref. 15), and loaned it to me shortly after I received the Wesson paper.

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References 1. Schuster, A. (1912) A Critical Examination of the Possible Causes of Terrestrial Magnetietism, Proc Lond. Phys. Soc. 24, 121-137(1911-1912). 2. Wilson, H.A. (1923) An Experiment on the Origin of the Earth’s Magnetic Field, Proc. Roy.Soc. A, 104, 451-455 3. Swann, W.F.G, and Longacre, A. (1928) J. Franklin Inst. 206, 421. 4. Blackett, P.M.S. (1947) The Magnetic Field of Massive Rotating Bodies, Nature 159, 658-666. 5. Sirag, S.-P. (1979) Gravitational Magnetism, Nature 278, 535-538. 6. Braginsky, V., Polnarev, A. and Thorne, K. (1984) Foucalt Pendulum at the South Pole, Phys. Rev. Lett. 53, 863. 7. Ciufolini, I. and Wheeler, J.A. (1995) Gravitation and Inertia, Princeton U.P., Princeton. 8. Morrison, P. (1976) The Scientific and Public Life of P.M.S. Blackett, Scientific American 235:4, 138-139. 9. Smith, P.J. (1981) The Earth as a Magnet, in D. Smith (ed.), The Cambridge Encyclopedia Of Earth Sciences, Cambridge U.P., New York. 10. Blackett, P.M.S. (1952) A Negative Experiment Relating to Magnetism and the Earth’s Rotation, Phil. Trans. R. Soc. Lond. Series A, 897, 309-370. 11. Sagan, C. (1975) The Solar System, Scientific American, 233:3, 23-31. 12. Surdin, M, (1979) The Magnetic Field of the Planets, Il Nuovo Cimento,2C:5, 527-53. 13. Surdin, M. (1977) Magnetic Field of the Planets, J. Franklin Inst., 303:6, 493-510. 14. Harasim, A., v. Ludwiger, I., Kroy, W. and Auerbach, H. T. (1988) Laboratory Experiment for Testing Gravi- Magnetic Hypothesis with Squid-Magnetometers, in H.D. Hahlbohm and H. Lubbig (eds.), SQUID ’85, Superconcuctiong Quantum Interference Devices and their Applications, de Gruyter, Berlin. 15. Gray, R.I. (1988) Unified Physics,Naval Surface Warfare Center, Dahlgren, Virginia. 16. Wesson, P.S. (1981) Clue to the Unification of Gravitation and Particle Physics, Phys. Rev. D,23:8, 17301734. 17. Salam, A. (1989) Overview of Particle Physics, in P. Davies (ed.) The New Physics,Camb.U.P., New York. 18. Brosche, P. (1980) The Mass-Angular Momentum Diagram of Astronomical Objects, in P.G. Bergmann and V. De Sabbata (eds.) Cosmology and Gravitation, Plenum, New York. 19. De Sabbata, V. and Gasperini, M. (1983) The Angular Momentum of Celestial Bodies and the Fundamental Dimensionless Constants of Nature, Lett. Al Nuovo Cimento 38, 93-95. 20 De Sabbata, V. and Sivaram, C. (1994) Spin and Torsion in Gravitation, World Scientific, Singapore. 21. Woodward, J.F. (1989) On Nonminimal Coupling of the Electromagnetic and Gravitational Fields, Foundations of Phyhsics 19:11, 1345-1361. 22. Barut, A.O. and Gornitz, T. (1985) On the Gyromagnetic Ratio in the Kaluza-Klein Theories and the Schuster- Blackett Law, Foundations of Physics 15:4, 433-437. 23. Pauli, W. (1933) Annalen der Phyusik 18, 305 & 337. 24. Li, N. and Torr, D. (1991) Effects of a Gravitomagnetic Field on Pure Superconductors, Phys. Rev D 43:2 457. 25. Li, N. and Torr, D. (1992) Gravitational Effects on the Magnetic Attenuation of Superconductors, Phys. Rev. B, 64:9, 5489. 26. Torr, D. and Li, N. (1993) Gravitoelectric-Electric Coupling via Superconductivity, Found.of Phys. Lett 6:4,71. 27. Stirniman, R. (1999) The Wallace Inventions, Spin Aligned Nuclei, the Gravitomagnetic Field, and the Tampere “Gravity-Shielding” Experiment: Is There a Connection? Frontier Perspectives 8:1, 20-25. 28. Wilson, J. (2000) Taming Gravity, Popular Mechanics 177:10, 40-42. 29. Bennett, J.G., Brown, R.L. and Thring, M.W. (1949) Unified Field Theory in a Curvature-Free Five Dimensional Manifold, Proc. R. Soc. London Ser. A 198, 39-61. 30. Ahluwalia, D.V. and Wu, T.-Y. (1978) On the Magnetic Field of Cosmological Bodies, Lett. Nuovo Cimento 23:11, 406-408. 31. McCrea, W.H. (1978) Magnetism and Rotation: Blackett’s Speculation of 1947, Speculations in Science and Technology, 1:4, 329-338. 32. Massa, C. (1989) On the Generalized Gravi-Magnetic Hypothesis, Annalen der Physik 7:46:2, 159-160. 33. Muller-Hoissen, F. (1990) Gravity Actions, Boundary Terms and Second-Order Field Equations, Nucl. Phys. B 337, 709-736. 34. Srivastava, Y. and Widom, A. (1992) Gravitational Diamagentism, Phys. Lett. B 280, 52-54. 35. Opher, R. and Wichoski, U.F. (1997) Origin of Magnetic Fields in the Universe due to Nonminimal Gravitational-Electromagnetic Coupling, Phys. Rev. Letters 78, 787-790.

QUANTUM HALL ENIGMAS

MALCOLM H. MAC GREGOR Lawrence Livermore National Laboratory Livermore, California, USA, 94550

1. Introduction to the enigmas

The quantum Hall effect (QHE) is one of the most striking, and at the same time one of the most enigmatic, physical phenomena to be discovered in the last few decades. Experimentally, the results are simple, straightforward, and of almost unparalleled accuracy. Quantum Hall research facilities have sprung up all over the world, and the “von Klitzing constant” R = 25812.807 has become a metrological standard. Theoretically, however, this same simplicity has not emerged. Although a great many papers have been written on the quantum Hall effect, very few QHE books have appeared, and those that have emphasize the tentative nature of the theories. An introductory text, published in 1994, describes this situation very succinctly [1]: “A somewhat disturbing fact one faces when writing a book on the theory of the QHE is that, unfortunately, for the time being no conclusive theory of this fundamental phenomenon is available – in spite of the numerous attempts that have been undertaken to explain it from the time of its discovery. All that has been achieved so far is the working out of certain aspects which will be part of the ultimate theory that is still to be achieved. ... In general, publishing an introduction to a non-existing ... theory is not considered to be advisable.” In the present paper we list and briefly discuss a series of enigmatic quantum Hall topics. Hopefully these results will lead to further investigations in this important field. 2. The experimental “classical” and “quantum” Hall effects

The Hall effect was discovered by Edwin Hall in 1879. Using a flat metal foil, he measured its longitudinal voltage V and current I, and hence its longitudinal resistance, R=V/I. When he placed the foil horizontally in a vertical magnetic field B, he also measured a transverse voltage proportional to B, which is caused by the displacement of the current slightly to one side of the foil under the action of the Lorentz force. The ratio of to the current I defines the transverse “Hall resistance,” which is linearly proportional to B. We can write the Hall current I as where and v are the electron density and velocity, and e is the charge on the electron. We can similarly write the Hall voltage as (Faraday's law), where the magnetic flux quantum is and the flux density is The Hall resistance thus becomes The Hall conductance is where is denoted as the Hall "filling fraction". This is the “classical” Hall effect. 337 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 337-348. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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The "quantum" Hall effect was discovered by Klaus von Klitzing in 1980. Carrying out a Hall experiment in a two-dimensional (2D) semiconductor (a Hall bar) at a very low temperature and high magnetic field [2], he found that the plot of versus B (or, in his case, gate voltage) exhibits a series of plateaus which are centered around the linear versus B curve. These plateaus occur at integer values of the filling fraction, (k = 1, 2, ...10, ...). They have extremely flat slopes, and the measured integer values have the astonishing accuracy of one part in Furthermore, the longitudinal resistance (magnetoresistance) essentially vanishes on the plateau, which signals the formation of a single coherent quantum state. Thus these integer Hall plateaus are denoted as the “integer quantum Hall effect” (IQHE). Additional quantum Hall results were obtained in 1982, when Horst Stormer and Daniel Tsui extended the quantum Hall measurements to higher B-values [3,4], and unexpectedly discovered Hall plateaus that have fractional filling fractions, (k= 1, 2, 3, ...; m = 3, 5, 7, ...). The k values include all integers, but the m values include only odd integers. These fractional Hall plateaus have the same experimental accuracy as the integer plateaus, and are denoted as the “fractional quantum Hall effect” (FQHE). From a strictly experimental point of view, the integer and fractional quantum Hall effects appear to be identical to one another. In particular, integer and fractional Hall plateaus occur interleaved together in the same set of measurements. Quantum Hall experiments can be carried out either by holding the electron density constant and varying the external magnetic field (semiconductor heterojunctions), or else by holding B constant and varying by means of a “gate” voltage (MOSFETs). In the present paper, we concentrate on the former case. The extreme accuracy of the plateau quantization indicates that the quantum Hall effect is a microscopic phenomenon, in the sense that the Hall plateau basis states all carry the value . 3. The “Fermi sea” (FS) model for the integer quantum Hall plateaus

The theoretical efforts to explain the Hall plateaus have followed the pioneering work of Robert Laughlin [5]. The 2D quantum mechanical motion of electrons in a magnetic field B is described by the Landau equations [1,6]. This motion is essentially the same in a semiconductor as it is for free electrons, except that in the semiconductor the electron mass is replaced by an effective mass which in GaAs is equal to The conduction electrons in the semiconductor, being fermions, each occupy a distinct quantum state, with the continuum of filled states forming the “Fermi sea”. The Landau equations group these electron states into a series of discrete energy bands, which are denoted as “Landau levels”. The electron density of states in each Landau level is Thus for a B-value such that the first Landau level is exactly populated by the available conduction electrons, we have the Hall filling fraction If the field B is now decreased to the value with the electron density remaining constant, then two Landau levels are exactly populated, and we have And so on. The Landau electron density on each Landau level corresponds to the close-packing of single-wavelength Landau cyclotron orbitals. The existence of a Hall plateau, wherein the microscopic Hall conductivity remains constant as B is varied, requires that

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on the plateau. This follows from the Landau level density cited above, and from localization-delocalization (LD) semiconductor theory [1,6], which specifies that only the energy states lying at the center of each Landau level are delocalized, and thus contribute to the Hall current. The other states are pinned by impurities in the semiconductor substrate. Hence, when the surface of the Fermi sea is located between two Landau levels, variations in B affect only the localized electron states, and not the Hall conductance, so that we have plateau formation. As the field B is decreased from its value, the rising Fermi surface sweeps over the Landau levels, successively filling them and generating the spectrum of integer Hall plateaus. This Fermi sea (FS) model is the generally accepted explanation for these integer plateaus, but it cannot account for the fractional plateaus, as we now discuss. 4. The “collective electron excitation” (CEE) model for the fractional plateaus

In a quantum Hall experiment, increasing the magnetic field B above the value that corresponds to the integer Hall plateau means that the conduction electrons all lie in the lower portion of the first Landau energy level. Thus it was not anticipated that Hall plateau formation could occur above This is why the Stormer and Tsui discovery [3,4] of a Hall plateau at the field and with a fractional filling fraction, was such a surprise. This plateau cannot be accounted-for by the FS model with its LD systematics. Hence a new type of quantum Hall theory was devised for the fractional Hall plateaus, wherein the basic quantum states consist of collective electron excitations (CEE) that feature fractional electric charges [5]. Electron coulomb interactions, which are unimportant in the FS model for the integer quantum Hall plateaus, play a dominant role in the CEE models for the fractional Hall plateaus [7-9]. 5. The fundamental quantum Hall enigma: one experiment and two half-theories

The FS model discussed above has played a dominant role in semiconductor physics, and the discrete fermionic Landau energy levels are manifested, for example, as Shubnikov-de Haas oscillations in many experiments [6]. Thus, to associate this model with the IQHE seems a logical procedure. But the FS model cannot accommodate the FQHE. Similarly, the CEE model has led to accurate calculations of many salient features of the FQHE [7-9], but it does not accommodate the IQHE. From an overall physics point of view, we seem to be in the awkward position of having one simple and extremely accurate physics experiment, with two quite different and non-overlapping theories required to account for it. This at least suggests that both of these quantum Hall half-theories may be flawed, and that the search should be continued to find a more unified solution to this problem. In the present paper we discuss a number of enigmas that must be addressed in this search. These include questions about external magnetic flux quantization, trapping, transport, and release; a fermion-to-boson superconducting phase transition and its effect on effective masses; the statistics of 2D multilayer composite particles; and some eigenvalue problems associated with gauge invariance. We also offer an example of a more unified quantum Hall theory, the close-packed composite boson (CPCB) model [10].

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6. The microscopic accuracy of the Hall filling fraction

The Hall plateau filling fraction is the ratio of electrons e to magnetic flux lines in the Hall current I, where k and m are integers. There are two key points to be noted here. (1) The measured accuracy of is one part in (2) The boundaries of the Hall current region are very raggedly defined: there is a Hall “bulk current,” in which the entire plateau quantum state (or at least the part not pinned down by impurities) drifts along; there are Hall “edge currents,” in which the orbiting electrons bounce along the “walls” of the Hall bar; there are sometimes regions in a Hall bar where two different filling fractions co-exist side by side; and there are combinations of these, as can be ascertained by differential voltage measurements across the Hall bar. The only way that points (1) and (2) can be reconciled is if each individual Hall “current element” carries the value , so that is a microscopic quantity. This conclusion is confirmed by the Hall “shot noise” experiments, in which a constriction or bypass is created wherein the Hall current elements pass (tunnel) through one at a time. In the theory of these experiments, the Hall conductance is characterized as a “topological in variant...reflecting the binding of charge and flux” [11]. The task now is to determine the precise nature of these Hall plateau “current elements”. 7. Magnetic flux quantization and the magnetic Reynolds number R M

Classically, a magnetic field B is treated as a continuum. At the quantum level, discrete flux lines can occur, but the quantization of these lines is not (as far as we can detect) innate to the field B, but depends on the quantum states with which the field B interacts. In superconductivity, electrons combine together to form bosonic “Cooper pairs”. When Cooper pairs flow in a superconducting current loop, the magnetic quantization produced is in units of where 2e is the charge of the Cooper pair. In the quantum Hall system, where we have single orbiting electrons, the quantization is in units of The point here is that this quantization is created by the Hall electron orbitals, so that the binding together of the orbitals with the external field lines is a natural result. Measurements made with a high-conductivity fluid such as mercury or sodium demonstrate that for transport times which are short as compared to the diffusion time, magnetic lines are frozen into the fluid and carried along with it [12]. The condition for this to occur is given by the magnetic Reynolds number where is the diffusion time, is the fluid velocity, and L is the flow length: if trapping does not occur, and if it does. Applying these results to a typical Hall bar, we have cm/sec and L ~ 0.1 cm. Off a Hall plateau sec, which gives whereas on the plateau which gives Thus plateau flux trapping is the expected result. When we apply this systematics to the observed flux trapping on the quantum Hall plateaus, we are led to the following result: (7.1) An Landau electron orbital traps m external flux quanta Since a superconducting electron orbital that contains m de Broglie wavelengths generates m induced (diamagnetic) flux quanta we obtain another result: (7.2) In a trapping electron orbital, the number of and (oppositelydirected) magnetic flux lines h / e are equal to one another.

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8. Composite bosons (CBs) and composite fermions (CFs) A “composite particle” (CP) is the bound state of an electron and m magnetic flux quanta We can envision this process as the direct trapping of external flux lines in a Landau electron cyclotron orbital (current loop), as described in (7.1). Alternately, we can picture the external flux lines as creating “vortices” in the 2D “electron charge fluid” of the Hall bar, which then attract the electrons [3]. These vortices act as “fictitious” flux lines that are directed oppositely to the external flux lines, and are thus roughly analogous to the flux lines delineated in (7.2) However, the latter are more localized. In the CEE models, the electron states are not cyclotron orbitals, but rather collective excitations which arise from the mutual coulomb interactions of all of the conduction electrons. In the case of the fractional Hall plateau, for example, each electron occupies three CEE “sites,” so that each site carries an effective fractional charge of 1/3, and each site combines with a magnetic vortex line h/e. A crucial feature of the CPs is that their 2D statistics depends on the number m of trapped flux lines: if m is even, the CP is a composite fermion (CF); and if m is odd, the CP is a composite boson (CB). The Hall plateaus are composed of odd-m CBs, and thus resemble boson condensates. The non-plateau region near contains even-m CFs, which cannot combine bosonically, but which give rise to interesting experimental fluxtrapping results, as we describe in the next section. 9. The

CF Hall non-plateau region and magnetic flux-trapping

Apart from the striking results that occur on the quantum Hall plateaus, important results also occur in the non-plateau region near , where plateau formation does not occur. In the region, there are two available flux quanta for each conduction electron, so that CP formation leads to CFs rather than CBs. These CFs, being fermions, do not combine together as a bosonic quantum macrostate [13], but instead operate as individual entities. This gives us an opportunity to investigate two essential CF features: (1) the flux-trapping process; (2) the nature of the CF bound states. The concept of CFs was introduced by Jain [14], and CFs play a prominent role in CEE theories, where they account for many systematic Hall results [9,15]. Formally, the CF is defined by a Slater-determinant wave function that contains no mass parameter [14]. Thus the experimental determination of the CF properties is an important task. The trajectories of individual CFs in a magnetic field B have been measured in twoaperture experiments in the region [14]. For B-values below the value there are more electrons than pairs of magnetic field lines, and the field lines are all captured, so that no CF trajectories through the two apertures are observed. Slightly above the value, there are more flux lines than available electrons to capture them, and CF trajectories are observed that reflect the residual magnetic field When this experiment is repeated with electrons at magnetic field values the peaks in the electron two-aperture spectra match the CF peaks [14]. These experiments show two important things: (a) the CF composite state, which was originally introduced as a CEE collective state involving many electrons [14], exists as a localizable entity that has a particle-like trajectory in a magnetic field; (b) the trapping process that binds

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two magnetic field lines to each CF electron actually removes these field lines from the field experienced by the CFs. Thus this flux-trapping mechanism is more that just a theoretical artifact: the reality of these even-m CF states indicates that the odd-m CB states on the Hall plateaus, and their associated flux-trappings, are just as real. The final information about CFs comes from the experimental measurements of their properties [14]. Experimentally, the CF has a negative electric charge of -e, a spin equal to that of the electron, and an electron-like gyromagnetic ratio. Furthermore, whereas GaAs semiconductor electrons have effective masses the CF state has a mass roughly equal to the electron mass There is an old vaudeville saying that if something looks like a duck and quacks like a duck, it probably is a duck. Application of this show-business wisdom to the CF states suggests that the electron in the CF composite state is a real orbiting electron, and not a CEE many-electron collective state. 10. The fermion-to-boson phase transition

The conduction electrons in a semiconductor live in a Fermi sea, and their FS motion in a magnetic field is defined by the Landau equations. Thus, when the integer quantum Hall plateaus were first discovered, it seemed natural to associate them with the Landau energy bands. This belief was not shaken when the fractional quantum Hall plateaus were discovered, which have nothing to do with the Landau FS band structure. The quantum Hall theorists went on to invoke a different (CEE) theory of interacting fermions that applies just to the fractional plateaus. However, experiment seems to be telling us a different story. The main facts that we know about the quantum Hall plateaus suggest that these plateaus actually operate in a "boson sea," where the FS limitations no longer apply. These facts can be summarized as follows: (a) The quantum Hall plateaus, with one exception [13], involve odd numbers of flux quanta and hence have bosonic (CB) current-element states (b) The longitudinal resistance on a quantum Hall plateau decreases by up to 14 orders of magnitude, which indicates superconducting (and hence bosonic) behavior. (c) The extreme accuracy of the Hall filling fraction means that each plateau CB state carries this filling fraction, so that the uniform distribution of the external flux quanta is matched by an equally uniform distribution of CB states. This in turn requires that the CBs all have the same "size" (same quantum state), which happens only for bosons. This evidence indicates that a fermion-to-boson statistical phase transition occurs during the formation of a quantum Hall plateau, which has an important consequence: (d) The transformation of the electron wave functions from fermions to bosons decouples the plateau electrons from their mutual semiconductor "landscape potential" and couples them instead to the magnetic field lines, which logically leads to an increase in their effective mass values as is observed in the CF experiments near . An increase in lowers the energy state of the electron, thus providing an energy gap and stabilizing the fermion-to-boson phase transition. Once we have freed the Hall plateau solutions from the confines of the Fermi sea, we can select, for each Hall plateau, an appropriate (single) Landau bosonic quantum state that accurately tiles the plateau, and thereby obtain a unified approach to both the integer and fractional Hall plateaus, as we describe in the next section.

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11. A close-packed composite boson (CPCB) integer and fractional plateau model

The idea of representing a quantum Hall plateau as a boson condensate was first introduced by Girvin and MacDonald [16], and has subsequently been discussed [17]. In this context, we can picture the integer Hall plateau as a close-packed hexagonal array of identical single-wavelength CB Landau cyclotron orbitals [18], each of which contains one trapped flux line Since the close-packed CB orbitals have no freedom of movement, their coulomb interactions are unimportant [3]. This CPCB picture [10] is in agreement with the usual way of viewing the IQHE plateau. In order to apply this same close-packed CPCB model to the fractional Hall plateaus we simply replace the Landau orbitals with suitable orbitals [10]. Experimentally, the fractional Hall plateaus are successively reached by increasing the magnetic field by factors of 3, 5, ... . The Landau cyclotron orbitals are specified by the quantum numbers m and n, where m is the number of orbital de Broglie wavelengths and n is the radial quantum number (Sec. 16, with It turns out that the (n,m) = (0,1), (1,3), (2,5), ... Landau orbitals in the magnetic fields all have the same area (same value for and hence each correspond to one conduction electron. Since the orbitals on the plateau each trap external flux quanta (see 7.1 above), the fractional filling factors are reproduced in the same close-packed manner as the integer filling factor, which is the result that is suggested by experiment. 12. Multilayer quantum Hall plateaus

If we start with the IQHE plateau and its magnetic field, and then successively lower the field to values of we generate the IQHE plateaus. As we do this, the Landau orbitals grow in size so that they each still contain one flux line but they now each contain 2, 3, ... conduction electrons. The question then arises as to the nature of these multi-electron Hall current elements. The solution we offer [10] is to treat them as 2, 3, ... layers of identical quantum Hall sheets, with each flux line penetrating corresponding orbitals on each of the 2, 3, ... sheets. This same situation applies to the FQHE plateaus. Starting with the plateau, which is a sheet of close-packed orbitals, we generate the FQHE plateaus as 2, 4, 5, ... layers of orbitals, with a "flux bundle" penetrating corresponding orbitals on each of the sheets. And so on for the m = 5, 7,... FQHE plateaus. It should be noted that this same multi-sheet formalism also occurs in the conventional FS picture of the IQHE. In the FS approach, the plateau consists of one fully filled Landau level, with one flux line per orbital, The plateau consists of two fully filled Landau levels, with one flux line per two orbitals. In order to maintain a microscopic filling fraction, it seems clear that in the FS model, each flux line must be associated with one orbital on each of the two Landau levels, just as for the CPCB model described above. The difference in this FS model is that the orbitals on the second Landau level are larger in size than on the first level. When we consider IQHE plateaus with filling fractions or greater, this FS orbital size increase gets very substantial. In the boson models, we can make all of the orbitals on a plateau the same size.

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13. The 2D statistics of composite particles with more than one electron

According to the Ehrenfest-Oppenheimer-Bethe rule, a CP composite particle is bosonic (fermionic) if it contains an even (odd) number of elementary fermions. If we picture the magnetic flux lines as elementary fermions, then we immediately account for the CF and CB composite particles described in Sec. 8. These are single-electron states. But if we now associate more than one electron with a CP, as we do for the Hall plateaus that have k > 1, we then have a problem with the EOB rule. On the IQHE plateaus, we have 2, 3, 4 electrons bound together with a single flux quantum These plateaus are experimentally similar, and it seems clear that they cannot represent alternations between CF and CB Hall current elements. Hence in the 2D Hall geometry, the CP statistics must be determined by exchanges or rotations within the 2D geometry of each individual layer on a Hall plateau, plateau, the two electron orbitals (one on each layer) that are so that (e.g.) on the combined with a single flux line represent two associated CBs, and not one CF. 14. Quantum Hall plateau formation

The existence of quantum Hall plateaus means that the Hall conductance is independent of B over a range of B values. In the FS model, the IQHE plateaus are explained by the LD theory, which restricts electron movement to the central region of each Landau level. However, this LD mechanism does not apply to the FQHE plateaus. In the CPCB model [10], both the IQHE and FQHE plateaus are attributed to a different mechanism: electron density trapping (EDT) in the plateau CBs. EDT follows from the fact that both the radius and the mean spacing of the plateau CBs vary as Thus the density of CBs on a plateau varies as B. If the electrons are locked into the CB orbitals, then the electron density varies as B (instead of remaining constant, as it does off the plateau). Since the flux density also varies as B, the Hall filling fraction is invariant. Hence the Hall plateaus have very accurate zero slopes. 15. External magnetic flux capture, transport, and release

In the classical Hall effect, an external magnetic field forces an electron current to one side of a 2D Hall bar, where it creates a transverse Hall voltage and electric field which in turn create a longitudinal Hall current I under the action of the E×B drift [12]. In the drift frame of reference, which moves at the velocity the transverse electric field is canceled out by the moving flux lines which create an opposing electric field via the Faraday effect. In the quantum Hall effect, as it is envisaged in the CPCB model [10], the external magnetic field lines are captured and carried along by the CBs in the drifting Hall plateau current. These transported field lines create a transverse Faraday-effect voltage as they cross the region where the Hall voltage is measured. There are two different scenarios possible as to how this process takes place: Scenario I. The moving CBs in the drifting Hall current have captured all of the external field B, so they move along in a zero-field environment (similar to that which

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occurs in the CF region at . Thus there is no lateral displacement of the CBs, and the Hall voltage is provided solely by the captured and transported flux lines Scenario II. When the captured flux lines move along the Hall bar, they create a Hall voltage But when they reach the outlet region of the Hall bar and are released, they "spring back", thus creating an opposing voltage that cancels out the voltage produced by the forward-moving flux lines. The "spring-back" field lines interact with the Hall-current CBs and force them to one side, thus recreating the voltage When these two scenarios are traced through relativistically, they give the same Hall voltage There is one effect that may distinguish between these two possibilities. When the Hall current reaches a certain critical drift velocity, the Hall plateau abruptly breaks down [2]. This breakdown occurs at the point where the current enters the 2D electron sheet, and it is intrinsically a microscopic phenomenon [19]. It could be a "magnetic viscosity" effect [12] caused by a pileup of flux lines in the inlet region, which would suggest Scenario II. 16. The enigmatic magnetic-moment term in the Landau energy equation

The Landau Schrödinger wave function for a 2D electron in a magnetic field B separates into azimuthal and radial components:

The canonical angular momentum operator has eigenvalues The radial wave equation in the symmetric gauge A = 1/2 B×r is [6]

where is the cyclotron angular velocity and is the effective mass of the electron in the semiconductor. The corresponding eigenvalue equation is

The term on the right-hand side of Eq. (16.2) is the orbital magnetic moment interaction [6]. In order to obtain the standard Landau energy levels in Eq. (16.3), it is necessary to select negative values for However, these are not the values that are singled out experimentally. When an electron moves in a Landau cyclotron orbital, the magnetic moment that is generated is negative, due to the negative charge on the electron. Thus the interaction term is positive, so that is positive. It is possible to formally obtain negative by choosing the radial coordinate r as r=x–iy instead of the customary r=x+iy [1,7], but it is not clear that this choice is in agreement with experiment [20]. When the Landau equations are written out using the Landau gauge instead of the symmetric gauge, the energy ---dependence in Eq. (16.3) vanishes. This apparently solves the energy eigenvalue problem, but it raises questions about gauge invariance, as we discuss in the next section.

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17. The enigma of gauge invariance in quantum Hall theories

In both classical [12] and quantum [6] electrodynamics, a magnetic field B can be formally introduced into the Hamiltonian by replacing the electron kinetic momentum with the momenta where is the canonical momentum, A is the vector potential, and eA is what we can denote as the gauge momentum. The magnetic field B is given by the equation B = curl A. It turns out mathematically that different choices for A can lead to the same value for B, so this procedure has a certain amount of "gauge freedom". This freedom leads to different ways of dividing up the kinetic momentum (the total particle momentum) into its canonical and gauge components, and eA. Since the different choices for A all reflect the same physical system, the correct solutions should be independent of the choice for A; that is, the solutions should be gauge-invariant, both classically and quantum mechanically. A classical example of gauge freedom is provided by 2D non-relativistic electron motion in a quantized cyclotron orbit in a magnetic field The Newtonian energy for the orbital is When this problem is calculated in the Hamiltonian formalism, using the symmetric-gauge vector potential A = (0, 1/2 Br, 0) in polar coordinates, the electron (instantaneous) kinetic orbital velocity v divides equally into canonical and gauge orbital velocity components. When this same problem is calculated using the Landau-gauge vector potential A = (0, Bx, 0) in cartesian coordinates, the kinetic orbital velocity v now divides into a linear x-axis canonical velocity with components and a correlated linear y-axis gauge velocity w i t h components The symmetric-gauge solution gives circular cyclotron motion for both the and velocities, whereas the Landau-gauge solution gives harmonic oscillator motion along the x-axis and correlated harmonic oscillator motion along the y-axis. These two different gauges both add up to the same cyclotron motion and same energy spectrum. The classical solution is gauge invariant. The situation with respect to gauge invariance is different when we examine the quantum mechanical Landau cyclotron equations. The symmetric-gauge formulation of these equations was given in Sec. 16. In this gauge, the kinetic momentum divides equally into canonical and gauge eA momenta, just as it did in the classical case. The energy equation has two solutions, depending on whether we select positive or negative values for the angular momentum quantum number The observed Landau levels are obtained by choosing negative but the magnetic moment interaction indicates positive When we solve these equations in the Landau gauge A = (0, Bx, 0), the wave equation is expressed in terms of plane waves in the y direction and a harmonic oscillator solution in the x direction [6]. The electron velocity in the y direction contains both canonical and gauge components, which combine together to give harmonic oscillator motion. The velocity in the x direction contains only a canonical component, which also oscillates harmonically. However, the Landau-gauge eigenenergies contain an enigma. These energies are given by the equation The energy that was shown in Eq. (16.3) has disappeared. Thus the Landau equations as customarily formulated [1,6,7] are not gauge-invariant.

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18. Final comments about the quantum Hall Landau solutions

It should be emphasized that the Landau equations for the quantum mechanical motion of an electron in a magnetic field apply only to the possible eigenstates of a single electron. They do not apply directly to a semiconductor ensemble of Fermi sea conduction electrons. The symmetric-gauge Landau equations, for example, are for a set of electron orbitals that all have the same center. Although the energies of the symmetric-gauge Landau electron states can be made degenerate in by choosing negative in Eq. (16.3), the sizes of these states are not degenerate, and the orbitals increase in size with increasing The Landau-gauge electron states are for a series of electron wave functions that are closely-spaced in one direction, and extend the length of the Hall bar in the other direction. These Landau-gauge wave functions are not related to specific angular momentum states , and each Landau-gauge electron state is in fact a linear combination over all angular momentum values. This makes it difficult to single out the magnetic moment contributions to the Landau-gauge energy eigenvalues. The electron orbitals in a "normal" semiconductor state are quite different from the electron orbitals on a quantum Hall plateau. In the "normal" state, the interactions of the electrons with the substrate are very strong, as manifested by their anomalous effective masses These electrons are trying to establish quantized cyclotron orbitals, but the orbits are constantly being disrupted by inelastic scattering with the substrate, and stable orbits with stable magnetic moments do not occur. The electrons in a Landau level are synchronized coherently into the Fermi sea by their common averaged interactions with the hills and valleys (the landscape) of the substrate, and they spread out so as to occupy the phase space predicted for them by the Landau equations. Thus they sequentially fill the Landau levels and produce the observed Shubnikov-de Haas oscillations. But they are not neatly occupying the various orbital states that are predicted by the Landau equations for non-perturbed electron motion. They are in what might be termed a condition of "coherent quantum chaos". When a quantum Hall plateau is created, this chaotic condition abruptly changes. The linkage between the electrons and the substrate landscape vanishes, and a new linkage is established with the external magnetic field lines Stable orbitals are now formed, and these orbitals capture flux lines in a ratio that is rigidly defined by the Hall plateau filling fraction . This requires the resulting composite particles to spread out evenly over the 2D surface of the Hall bar, which in turn requires them to all have the same "size", and hence to be in the same Landau quantum state. This is only possible in a boson condensate, so that the transition from fermions to bosons, when viewed globally, seems to be an inevitable requirement for Hall plateau formation. The coherence between the individual CBs in a Hall plateau boson condensate cannot be attributed to quantum states that are linked together by a single "master equation" (such as occurs for example with the electron states in an atom). Rather, this coherence follows from the close packing of the CBs, in a manner that has been directly observed in experiments with individual quantum dots [21]. Hence the systematics of the quantum Hall plateaus leads not just to CB condensates, but directly to CPCB condensates [10].

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References. 1. Janssen, M., Viehweger, O., Fastenrath, U., and Hajdu, J.: Introduction to the Theory of the Integer Quantum Hall Effect, VCH, Weinheim, 1994. 2. Von Klitzing, K.: Nobel Lecture: The quantized Hall effect, Rev. Mod. Phys. 58, 519-531 (1986). 3. Stormer, H.L.: Nobel Lecture: The fractional quantum Hall effect, Rev. Mod Phys. 71, 875-889 (1999). Fig. 5 of this paper has the wrong sign for the Lorentz force. 4. Tsui, D.C.: Nobel Lecture: Interplay of disorder and interaction in two-dimensional electron gas in intense magnetic fields, Rev. Mod. Phys. 71, 891-895 (1999). 5. Laughlin, R.B.: Nobel Lecture: Fractional quantization, Rev. Mod. Phys. 71, 863-874 (1999). 6. Davies, J.H.: The Physics of Low-Dimensional Semiconductors, Cambridge University Press, New York, 1998. 7. Prange, R.E. and Girvin, S.M., eds.: The Quantum Hall Effect, Second Edition, Springer-Verlag, New York, 1990. 8. Chakraborty, T. and Pietiläinen, P.: The Quantum Hall Effects; Fractional and Integral, Second Edition Springer, Berlin, 1995. 9. Das Sarma, S. and Pinczuk, A., eds.: Perspectives in Quantum Hall Effects, Wiley, New York, 1997. 10. Mac Gregor, M.H.: A Unified Quantum Hall Close-Packed Composite Boson (CPCB) Model, Found. Phys. Lett. 13, 443-460 (2000). 11. Kane, C.L. and Fisher, M.P.A.: Nonequilibrium Noise and Fractional Charge in the Quantum Hall Effect, Phys. Rev. Lett. 72, 724 (1994). 12. Jackson, J.D.: Classical Electrodynamics, Second Edition, Wiley, New York, 1975. 13. The v = 5/2 Hall plateau is the only well-established even-denominator Hall plateau [see Ref. 4]. 14. Jain, J.K.: The Composite Fermion: A Quantum Particle and Its Quantum Fluids, Physics Today, April 2000, 39-45. 15. Heinenon, O., ed.: Composite Fermions: A Unified View of the Quantum Hall Regime, World Scientific, Singapore, 1998. 16. Girvin, S.M. and MacDonald, A.H.: Off-Diagonal Long-Range Order, Oblique Confinement, and the Fractional Quantum Hall Effect, Phys. Rev. Lett. 58, 1252 (1987). 17. Kivelson, S., Lee, D.-H., and Zhang, S.-C.: Electrons in Flatland, Sci. Amer. 274 (3), 86-91 (1996). The composite boson model illustrated in the bottom right-hand figure on page 89 may be flawed 18. See, for example, Aoki, H.: Quantized Hall Effect, Rep. Prog. Phys. 50, 655-730 (1987), 665, who does not, however, employ a minimum-energy hexagonal (Abrikosov) array. 19. Van Son, P. C., Kruithof, G. H., and Klapwijk, T. M.: Intrinsic sequence in the breakdown of the quantum Hall effect, Surface Science 229, 57-59 (1990). 20. One possible solution to this impasse is to note that the W = –µ • B term is not the total interaction energy of a magnetic moment in a magnetic field [12], and that if nonadiabatic interactions with a constant-current magnet power supply are taken into account, the sign of this term is reversed [12]. 21. Collier, C. P. et al., Science 277, 1978 (1997); Physics Today, Dec. 1997, 9.

Note This work was performed under the auspices of the U. S. Department of Energy by the Lawrence Livermore National Laboratory under contract number W-7405-ENG-48.

ON THE POSSIBLE EXISTENCE OF TIGHT BOUND STATES IN QUANTUM MECHANICS

A. DRAGIC, Z. MARIC Institute of Physics Zemun Pregrevica 118, 11080 Belgrade, Yugoslavia J.P. VIGIER Gravitation et Cosmologie Relativistes Tour 22-12, 4 eme etage, Boite 142, 4 Place Jussieu, 75005 Paris, France

Abstract. The possibility of tight bound states in quantum mechanics has been often emphasized [1]. Two different mechanisms are proposed for their occurrence: strong magnetic interactions at small distances and creation of the "anti-Born-Oppenheimer" states, corresponding to rapid motion of the heavy particles around almost fixed electrons. The observed "excess heat" in cold fusion experiments could be explained by these new tightly bound states [1]. In the present paper both ideas are analyzed.

1. The Barut-Vigier Two-Body Model And Positronium Magnetic Resonances

The usual practice in atomic physics treats electromagnetic interactions other than Coulomb (spin-orbit, spin-spin etc.) as perturbations, giving only small corrections to the energy levels. Although these terms in Hamiltonian with distance behavior and are indeed small compared to atomic scale Coulomb terms, they are comparable or even much higher than later at shorter distances. In principle, a possibility that magnetic interactions at short distances give rise to new phenomena, not explained by perturbative treatment. The first exploration is made by Corben in an unpublished paper. He noticed that motion of a point charge in the field of magnetic dipole at rest, is highly relativistic and the orbits are of nuclear dimensions. The further investigation has been undertaken by Schild [2], but the most systematic treatment of this problem is given by Barut (see for example [3]) A two-body system where magnetic interactions play the most significant role is positronium. Both electron and positron have large magnetic moments which contribute to the second potential well in effective potential, at distances much smaller then Bohr radius. Barut and his coworkers predicted that this second potential well can support resonances [4, 5]. A twobody model, suitable for non-perturbative treatment of magnetic interactions is presented by Barut[3]and Vigier[l]. 349 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 349-356. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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The model essentially represents an extension of the Pauli equation to a twobody system and is defined by Hamiltonian:

where: is the mass, the momentum, the charge, particles is electromagnetic vector potential and interaction term:

the position of the is dipole-dipole

In the center of mass frame and with a normal magnetic moment: Hamiltonian (2) becomes:

where are quantities related to the relative motion of bodies, and is a reduced mass. The standard Pauli approximation leading to eq.(3) can be improved by keeping the energy term in the Hamiltonian. This correction is essential in the positronium problem, since the reduces mass is of the order of resonance energies we are interested in. The new Hamiltonian depends upon energy only through the effective mass given by:

In terms of total spin and angular momenta Hamiltonian for this self-consistent Barut-Vigier model can be written as:

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where operator Q is:

This most important spin channel for the resonance phenomena is S=1, L=1, J=0, because the attractive spin interactions are strongest and effective potential which appears in the radial Schrödinger equation:

has second potential well at distances of order The schematic plot of the effective potential in that channel, obtained using the effective mass corresponding to the energy E = 600 KeV, is shown in Fig. 1. Before attacking the problem of magnetic resonances, the method is first tested in the low energy limit (the usual bound states of positronium) and results are compared with perturbative QED. Energy eigenvalues are determined by an iterative procedure. The approximate energy at each step was obtained by numerical integration of the Schrödinger equation with implementation of the ”shooting” method [6]. This energy was put back in the effective mass (eq. (4)) and the procedure was repeated until convergence was achieved. The known bound state energies are reproduced with five digits accuracy. Resonant states are defined as solutions of the radial Schrödinger equation which satisfy ”the outgoing wave only” boundary condition at large distances. As a consequence of a complex boundary condition, the energy is treated as complex quantity: The real part E is resonance energy, while imaginary part is resonance width. Resonances are investigated by means of complex coordinate rotation technique [7]. In contrast to bound state problem, matching of logarithmic derivatives of inward and outward wave functions now amounts to a two-dimensional problem which is treated by Newton-Raphson method. The other difference concerns the iterative procedure, since both the resonance energies and widths must be selfconsistently determined. However, it should be noted that only the real part E of a complex energy enters the effective mass To start the iterative procedure the lattice of initial guesses of energies and widths is formed, covering energy range of interest and iterative procedure is started at each of this points.

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We have not found any resonances in the energy range 1-2 MeV in the S=1,L=1,J=0 channel. Stability analysis shows that our iterative procedure is convergent and we can conclude that negative result indeed means that resonant solutions do not exist. In addition, phase shifts are calculated. Two different methods of calculation are used: direct numerical integration of Schrödinger equation (NI) and variable phase method (VPM). Methods are consistent with each other (Table 1) and phase shift does not show any resonant behavior. The details of calculations will be published [8]. 2. Anti-Born-Oppenheimer States

The three body problem has been well understood in the framework of the Born-Oppenheimer approximation. If there is a situation where the motion of the lighter particles (electrons) is suppressed, the energy spectrum would look quite different. A method for estimating energy levels in this ”anti-Born-Oppenheimer” approximation (corresponding to rapid motion of the nuclei around almost fixed electron) has been presented by Barut[9].

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2.1 MOTIONLESS ELECTRON

The general idea can be most easily seen from configuration with static central particle (electron). For configuration of the three particles shown on Fig. 1 and by taking into account only Coulomb interactions, in the c. m. frame Hamiltonian is given by:

(7)The Bohr quantization of symmetric circular orbits of the two particles M around the center of mass:

gives

The minimization of the energy

can give us prediction of energy levels.

This method of quantization gives for and H e ground state energies values very close to the correct ones. In the case of the anti-Born-Oppenheimer situation, the above method gives for :

The ground state energy for

is ~ 28KeV and for

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2.2 TRIANGULAR CONFIGURATION

In the more general treatment electron can perform small oscillations between the nuclei(Figure 2) As in the Born-Oppenheimer approximation, the motion of the electron is quantized first.

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The electron Hamiltonian is approximately an oscillator around: with frequency: The quantization of H by means of

gives

Next, the nuclear Hamiltonian is quantized.

In the radial wave equation for appears:

the effective potential

The effective potential has two minima. First, for L = 0, in the region of the Bohr radius of the electron. The second for f o r in the region of the Bohr radius of the proton. Thus, this method predicts the existence of both and Following described procedure one arrives at an approximate expression for energy levels of normal states of and

In the case of

and

energy levels are:

2.3 SPIN INTERACTIONS

Spin is encountered for by ”minimal substitution”:

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For static configuration (Figure 2.) with spin Hamiltonian for

is:

For triangular configuration energy levels of are calculated by keeping terms linear with respect to the nuclear spin. They are on the same scale as in the static case. Although theory here presented is proposed as a mechanism for occurrence of ”excess heat” in the cold fusion experiments, available experimental data can not conclusively be related to it. Also, it is not clear at which experimental situation systems would exhibit behavior predicted by this model.

References [1] Vigier J. P. paper, presented at ICCF 4, Hawai 1993. [2] Schild A. Phys. Rev. 131 (1963) 2762. [3] Barut A. O. Surv. High Energy Phys. 1 (1980) 113. [4] Barut A.O. and Kraus J. J. Math. Phys. 17 (1976) 506. [5] Barut A.O., Berrondo M. and Garc'ia-Calderon G., J. Math. Phys. 21 (1980) 1851. [6] Koonin S.E. & Meredith D.C. Computational Physics, Fortran Version. Addison-Wesley, Reading,MA (1990). [7] Connor J.N.L. and Smith A.D. J. Chem. Phys. 78 (1983) 6161. [8] DragiA., BeliA. and MariZ. Facta Universitatis (in press). [9] Barut A.O. - Inter. Journal of Hydrogen Energy, 15 (1990), 907.

A CHAOTIC-STOCHASTIC MODEL OF AN ATOM CORNELIU CIUBOTARIU*, VIOREL STANCU Technical University Gh. Asachi of Iasi, Department of Physics, Bv. D. Mangeron No 67, RO-6600 Iasi, Romania CIPRIAN CIUBOTARIU Al. I. Cuza University of Iasi, Faculty of Computer Science, RO-6600 Iasi, Romania, email: [email protected]

Abstract. The idea that a 'magnetized' charged particle in interaction with ‘resonant’ photons operates from an energy level to another higher one by a stochastic acceleration effect suggests that such effects may represent a phenomenological physical mechanism which explains how an electron jumps to higher atomic orbits when it absorbs resonant photons. If we increase the number of iterations of the corresponding nonlinear system of equations, we obtain a Bohr image of an atom. Such (quantum-transition) jumps, their duration and physical mechanism have never been explained by the quantum theory of atoms. We thus offer through such a cascade of chaotic kicked (stochastic acceleration) effects a physical explanation of the quantum model of absorption of energy by an atom. The proposed equations can model a circuit biased with a traveling electromagnetic wave. Such a circuit can also simulate a stochastic acceleration and a chaotic atom.

1. Introduction

It is often claimed that the recent (re-)discovery and applications of chaos theory (or ergodic theory if we remind the old name of this subject) constitute the third great revolution in physics in the twentieth century, the first two being the invention of relativity and of quantum theory. Many systems in nature exhibit classical chaos (i.e. sensitive and exponential dependence on initial conditions) but, for the time being, there is no universally accepted definition of quantum chaos (see, for example, [l]-[3]). However, there are different approaches to this concept, such as Quantum Chaology (QC)1 and the Random Matrix Theory (RMT)2. In a way, quantum chaos refers to the * 1

Corresponding author, email: [email protected] Quantum Chaology represents the study of semiclassical behaviour of systems whose classical motion

exhibits chaos [4]. 2 It has been shown that the statistics of eigenvalues, and eigenfunctions of classically chaotic systems are similar to infinite symmetric matrices whose elements are random numbers.

357 R.L. Amoroso et al (eds.), Gravitation & Cosmology: From the Hubble Radius to the Planck Scale, 357-366 © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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quantum mechanics of classically chaotic systems and concerns itself with the quantumclassical correspondence, specifically, deducing quantum properties of a system using only classical quantities (primarily trajectories and periodic orbits). For classically regular (integrable) systems (which form only a subset of measure zero of all Hamiltonian systems3), methods for doing this have been known since the early days of the quantum ideas, before wave mechanics. However, for classically chaotic systems, the methods are still being developed [5]. The purpose of the present paper is essentially threefold: (i) to draw attention that the quantum chaos with all the characteristics of a classical chaos (e.g., the butterfly effect) manifests itself at any quantum level of atomic systems in a specific quantum way (e.g. the quantum butterfly effect); (ii) for a charged particle-electromagnetic field system, the counterpart of the quantum resonance is represented, for example, by the classical resonance between the frequency of photons and cyclotron frequency ; and (iii) a quantum resonant transition (e.g., in a Bohr's semiclassical theory of an atom) corresponds to a Fermi (stochastic) acceleration mechanism. 2. Poincare Sections and Bogomolny Sections It is well known that the transition from order to classical chaos is studied by the examination of the phase space properties by the way of Poincaré-section plots (classical surface of phase-space sections, CSOS). A quantum (or Bogomolny's semiclassical) surface of section (QSOS) is similarly drawn through the (configuration) space of the classical Hamiltonian but now, instead of only marking the points where the different (periodic, quasi-periodic, or chaotic) trajectories cross the surface, one also calculates the semiclassical phase which has accumulated since the previous crossing. In other words, one can use the behaviour of classical trajectories from one crossing of a QSOS to the next to compute the energy eigenvalues of the system. We remind that is the action accumulated along the trajectory (closed loop)

If a

Hamiltonian system is classically integrable4 then its trajectories are constrained to invariant tori, and EBK-quantization5 can be applied. In terms of action-angle variables, the Hamiltonian is a function of the actions only, Then the EBK (semiclassical) quantization rules are (k = 1, 2,...,l)

3

Generally, in a Hamiltonian system the motion is governed, for example, by Newton-like equations without dissipation.

4

i.e. if it has as many constants of motions in involution (i.e., their Poisson brackets with each other vanish) as degrees of freedom, where one constant of motion is the total energy E which is equal to the classical Hamiltonian H(p, q) =E. For 1-D Hamiltonians (all of which are integrable), the tori are just the periodic orbits, and the analogous WKB approximation method can be applied. 5 Einstein-Brillouin-Keller quantization

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where the are integer quantum numbers, and the integers are the Maslov indices that count the numbers of caustics along the trajectories6. The semiclassical (EBK) quantization rules (1) (based on action-angle variables) are not applicable to the chaotic systems since the definition of is meaningless for ergodic systems which possess no invariant tori7. The distance between two neighbouring trajectories increases exponentially along a unstable manifold, and decreases exponentially along a stable manifold. The problem is to find a semiclassical quantization approach to the chaotic systems. In other words, the question is to find a classical approach to the quantum-mechanical energy levels of (nonintegrable) ergodic systems. Martin Gutzwiller[6], opened this field by his semiclassical trace formula (for the quantization of chaos)8 which sums purely classical information about periodic orbits (i.e., classical orbits which are closed in phase space9) into an expression for the quantum mechanical density of states. The poles of the expression lead to quantum eigenenergies. Thus, semiclassical mechanics is now liberated from the torus and allowed to wander free. In a way, quantum chaos is in fact a semiclassical mechanics off the torus. If we want to explore all phase space, we do not need to use only (Gutzwiller) periodic orbits, but we can use any set of (spray) trajectories with their contributions on a QSOS. We mention that all these methods (Gutzwiller's periodic-orbit theory, Bogomolny's QSOS method, dynamical zeta functions etc.) to generate approximations to the quantum energy levels of chaotic systems may be considered as a substitute for the WKB-method for the case of integrable systems and its multidimensional generalization, the EBK-quantization conditions.

3. Nonlinear Equations for Unstable Orbits In the framework of quantum chaology, the basic idea is to relate the behaviour of the eigenvalues and eigenfunctions (of a quantum Hamiltonian H) of a quantum system to the physical and formal structure of the phase space of the corresponding classical system in the regular and chaotic region. For example, the spectral statistic P(s) of the energy levels is a quantity which can be calculated only from an observed sequence of levels. P(s) is the distribution level (nearest-neighbour) spacings of levels It has been found that the spectral statistics (or spectral fluctuations) of quantum systems with

6

The motion takes place on a Lagrangian manifold, and (the number of conjugate points or the Morse index of a trajectory) is determined by the topology of the Lagrangian manifold in phase space with respect to

configuration space. 7 Individual trajectories in a chaotic dynamical system are ergodic in the sense that they are wandering through phase space in a similar manner to a random (stochastic) variable. 8 The Gutzwiller's trace formula represents the trace of the energy-dependent Green's function which is the Fourier transform of the time-evolution operator. In the semiclassical limit when the leading contribution to the Feynman path integral [9], [10] comes

9

from the classical orbits. However, the contributions to the trace of the time-evolution operator come from those classical orbits which are closed in coordinate space.

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corresponding classical (strongly) chaotic behaviour (Κ-systems) are in agreement with the Gaussian distribution of a Unitary Ensemble (GUE)10,

whereas quantum analogs of classically integrable systems are described by non-Gaussian (e.g., Poissonian) statistics,

In other words, systems which are classically chaotic, and those which are integrable belong to different Universality classes. So far we referred only to the signatures of quantum chaos in spectra. However, the same question can be asserted for the eigenfunctions of H. The distributions of the values of individual eigenfunctions of H have a Gaussian density in the semiclassical limit when the classical system is chaotic, whereas for classically integrable systems the density is non-Gaussian. Furthermore, a class of chaotic systems was found for which P(s) nearly behaves as it is expected for classically integrable systems. This is the case of the so-called arithmetical quantum chaos and shows us that there do not exist universal signatures of classical chaos that manifest themselves in P(s). It thus seems desirable to introduce a visualization technique that distinguishes quantum systems with chaotic classical limits from those with integrable ones, and which in a more intuitive way displays the random character of P(s) in the former case. It is the aim of this paper to put forward a numerical approach to this subject. We obtain a chaotic generalization of the De Broglie-Bohr quantization of an atom. Instead of quantizing by suspending standing-wave configurations on stable Keplerian orbits, we suspends the standing-wave configurations on the infinity of unstable orbits. Such unstable periodic orbits are observed experimentally in some atoms (helium, hydrogen and other systems [11]). The starting point for our approach is represented by the physical conditions which define the applicability of the WKB semiclassical approximation. We remind that the Schrödinger equation for a wave function is equivalent to the equation for the function S:

Comparing this equation with the classical Hamilton-Jacobi equation for the action function S,

10

If the chaotic system has a time-reversal symmetry, the distribution corresponds to the Gaussian Orthogonal Ensemble (GOE) of RMT:

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we notice that the last term in equation (4), which is proportional to Planck's constant gives minor corrections to the classical equation if the following condition (semiclassical approximation) is satisfied:

Considering that

this condition can be written in the form

In particular, for the 1-D case, we get

As

we can rewrite (8) in the form

Thus, the semiclassical approximation is not applicable for small values of the particle momentum, especially at points where the particle must come to rest Such a situation arises, for example, when a particle in a potential well is reflected at the potential barrier and starts moving in the opposite direction (turning point). We consider that, locally, in a semiclassical approximation, an electron of an atom interacting with photons may be modelled by the motion of a charged particle in a field of a transverse monochromatic electromagnetic wave (photons) and within a constant external magnetic field which simulates the curvature of spacetime inside an atom. The nonlinear dynamics of this model is described by the following equations (Maxwell equations, and relativistic equations of motion):

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obtained with a non-dimensionalization scheme (see Ref. [12] for notations). 4.

Numerical Results and Bohr Orbits

Since there exist no efficient analytic methods for describing stochastic components of a motion, it is advisable to apply a numerical integration of equations (10)-(12). In this section we propose to analyse numerically the process of the stochastization (chaotization or randomization) of the wave-particle interaction. We initiate the calculus for the following values of parameters: cyclotron frequency = 0.5, order of resonance n = 4, initial energy = =2, initial momentum = = = The dynamic system of equations (10)-(12) generates a flow in a 3-dimensional (3D) phase space Numerical solutions have been obtained by applying a fifth order Runge-Kutta algorithm with an adaptive stepsize control [13]-[15]. The initial values for are [0.001, 0.001]. As may be expected, the particle motion for small dimensionless amplitudes of the external electromagnetic wave (e.g. H= 0.01) remains regular and no acceleration arises. If the amplitude is increased to H = 0.05 the particle motion becomes more complex but retains a regular character. The onset of a stochastization is observed once H exceeds 0.5 but the energy gain remains initially still small. When H = 0.6 there emerges a strong acceleration and a `gun effect' is initiated, whereby a sudden expulsion of a particle from the system takes place in a certain direction. The gain in energy becomes significant rising from to and thus a state of a resonance overlap is obtained.

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A major chaotic gun erupts for H = 0.9 (see Fig. 1). In such a case we distinguish three situations: first, a localized chaotic regime (which we name a chaoson) emerges, subsequently, a high frequency oscillation with chaotic modulation of the amplitude appears (see also the time series in Fig. 2) and finally we observe the sharp rectilinear part of the trajectory which in fact displays the gun effect. It is easy to show that the electromagnetic gun is indeed chaotic because a small change in the initial value of for instance of leads to a dramatic change of the phase portrait and the velocity of the particle. At the same time the chaotic motion of a charged particle inside a chaoson will certainly lead to a localised stochastic domain of the electromagnetic field which we may denote as an electromagnetic stochaston. In a stochaston the field has a stochastic or braided structure. A fundamental question arises now: Are these two physical concepts (chaoson and stochaston) related by a reciprocal stochastic induction in the sense that “a chaoson (a localised chaotic motion of a charged particle) induces an electromagnetic stochaston (localised domain of the electromagnetic field) and a stochaston induces a chaoson". If stochastic induction were true we could accelerate particles in chaotic fields. In a way the answer appears affirmative since it is well known that any sufficiently turbulent magnetized plasma may accelerate particles through resonant interactions with chaotic plasma waves. However, the question refers also to the acceleration of particles as from chaotic static fields. We hope to study this problem in a future paper. A very interesting result arises if we increase the number of iterations and the duration of the operation (see Fig. 3). We notice that in the process of acceleration the energy is increased by `quantum jumps' so that a charged particle gains energy within a sequence of a cascade of gun effects. The jump from a `spiral Larmor orbit' to another one is performed by a gun effect. The radii of the Larmor circles and the corresponding energy of the particle increase. The existence of a multi-gun cascade via different Larmor orbits is illustrated clearly by the time series for (Fig. 4), and and also through the Poincaré sections (Fig. 5). In Fig. 4 the horizontal lines display the chaotic gun effects between different Larmor orbits represented here by vertical oscillations. From the time series (Fig. 4) we

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deduce that, during a gun regime the `longitudinal momentum' (Px) is constant and represents the highest value which may be achieved in the current Larmor spiral. This highest value is transferred to the following Larmor spiral in which the particle absorbs again energy from the electromagnetic field until a new gun effect erupts. In fact, on a specific Larmor spiral the particle will go around repeatedly, receiving a kick of energy each time it completes an orbit. We observe a classical Fermi acceleration combined with the gun effect which generates the jumps.

In fact a gun effect hides a very localized (in space and time) phenomenon (a chaoson) which leads to a very ordered motion of a charged particle. The long duration of the chaotic gun effect (with respect to the short duration of motion on the Larmor orbits) permits the extraction of an accelerated particle with a certain energy. Moreover, the gun timescale, for an escape varies considerably with energy and this variation may not be of the same order as that of other timescales (e.g. the timescale for diffusion, the timescale for advection etc). This situation may have some implications on the steady state and the time evolution of particle distributions in physically realistic situations.

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It is also worth noting that the gun effect has a stabilizing effect on the component which after each stop of the multi-gun cascade is reduced to the same initial value. Thus, the particle is accelerated and guided preferentially into the direction of the electromagnetic wave vector. In a phase portrait (Fig. 5) the motion seems apparently regular but a sharp sight can distinguish one jump from a Larmor circle to the next one as generated by a chaotic gun effect. All vertical lines display in fact a gun effect. The central vertical segment represents the chaotic gun illustrated in Fig. 1. 5. Conclusion And Comment

The situation described in this paper is similar to that of the semiclassical Bohr orbits in an atom whereby an electron jumps to orbits of larger radii by absorption of electromagnetic energy (photons). Such jumps, their duration and physical mechanism have never been explained by the quantum theory of atoms. We offer through a cascade of chaotic gun effects a phenomenological physical explanation of the semiclassical model of absorption of energy by an atom which can be interpreted as a chaoson immersed in a stochaston. In other words, an atom (or other quantum mechanical systems) represents a semiclassical chaotic (Hamiltonian) system (chaoson) coupled with a stochastic (quantum) field (stochaston). Finally, we notice that, in a very schematic (but general, for a wide range of physical phenomena) way, the nonlinear equations (10)-(12) are equivalent with the equation which describes the motion of a non-linear oscillator perturbed by a plane wave. This situation proves that indeed the proposed stochastic model (a K-system) for an atom appertains to the Gaussian class of distribution11.

11

We remind that, for example, the coordinate distribution in the ground state of a quantum oscillator is

Gaussian:

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Thus, a quantum system is equivalent with a specialized sensor that works under resonance between frequencies and in the same way as any circuit based on an analog computer version of the Lorenz or Rössler, or Chua butterfly equations for a classical chaos, or a Josephson junction for a quantum chaos works. In the framework of the present paper, the Pound-Rebka-Snider redshift experiments [16]-[18] as a test of geodesic motion are just an application of the high sensitivity of chaotic quantum systems at the very small variation of specific parameters (frequency of electromagnetic waves).

Acknowledgments The authors are thankful to Professor Richard L. Amoroso (Noetic Advanced Studies Institute, Orinda, CA, USA) for encouragement, and to Professor John Argyris (Institute for Computer Applications, ICA 1, University of Stuttgart) for discussions.

References [1] Steiner, F.: Quantum Chaos, Invited contribution to the Fetschrift Universität Hamburg 1994: Schlaglichter der Forschung zum 75. Jahrestag (Ed.R. Ansorge) published on the occasion of the 75th anniversary of the University of Hamburg, Dietrich Reimer Verlag, Hamburg 1994, pp. 542-564. [2] Manfredi, V. R. and Salasnich, L.: Different Facets of Chaos in Quantum Mechanics, Int. J. Mod. Phys. B, 13 (18) (1999), 2343-2360. [3] Aurich, R., Bolte, J. and Steiner, F.: Universal Signatures of Quantum Chaos, DESY Report 94-024, February 1994. [4] Berry, M. V.: in Dynamical Chaos, The Royal Society, London, 1987. [5] Haggerty, M. R.: Semiclassical quantization in a smooth potential using Bogomolny's quantum surface of section, PhD Thesis, Massachusetts Institute of Technology, 1994. [6] Gutzwiller, M. C.: Energy Spectrum According to Classical Mechanics, J. Math. Phys. 11 1970, 1791-1806. [7] Gutzwiller, M. C.: Periodic Orbits & Classical Quantization Conditions, J. Math. Phys. 12 (1971), 343-358. [8] Gutzwiller, M. C.: Chaos in Classical and Quantum Mechanics, Springer, New York, 1990. [9] Feynman, R. P.: Space-Time Approach to Non-Relativistic Quantum Mechanics, Rev. Mod. Phys. 20 (1948), 367-387. [10] Feynman, R. P. and Hibbs, A. R. Quantum Mechanics and Path Integrals. McGraw-Hill, New York, 1965. [11] P.: Classical and Quantum Chaos, version 6.0.2, Feb 2, 2000, printed April 7, 2000 (www.nbi.dk/ChaosBook/). [12] Argyris, J. Ciubotariu, C.: A new physical effect modelled by an Ikeda map depending on a monotonically time-varying parameter, Int. J. Bif. Chaos 9 (1999), 1111-1120. [13] Butcher, J. C., The Numerical Solution of Ordinary Differential Equations.Wiley, Chichester, 1987. [14] Koonin, S. E., Computational Physics. Benjamin Cummings, Menlo Park, 1986, Chap. 4. [15] Press, W. H., Flannery, B. P., Teukolski, S. A. and Vetterling, W. T. Numerical Recipes. Cambridge University Press, Cambridge, 1986. [16] Pound, R. V. and Rebka, G. A.: Apparent weight of photons, Phys. Rev. Lett. 4 (1960), 337-341. [17] Pound, R. V. and Snider, J. L.: Effect of gravity on nuclear resonance, Phys. Rev. Lett. 13 (1964), 539-540. [18] Pound, R. V. and Snider, J. L.: Effect of gravity on gamma radiation, Phys. Rev. B 40 (1965), 783- 803.

SYNCRONIZATION VERSUS SIMULTANEITY RELATIONS, WITH IMPLICATIONS FOR INTERPRETATIONS OF QUANTUM MEASUREMENTS

JOSE G. VARGAS AND DOUGLAS G. TORR Department of Physics University of South Carolina Columbia, SC

Abstract

The incorrect extrapolation into the realm of dynamics of the thesis of conventionality of one-way speeds (i.e. that, unlike two-way speeds, they depend on the choice of synchronization) has been detrimental for developments of physical theories based on an absolute relation of simultaneity. In response to the perceived need for a better understanding of issues such as interpretation of the collapse of the wave function, quantum teleportation, etc., we explore the quantum mechanics (QM) sector of a theory canonically implied by an absolute relation of simultaneity, thus by a preferred frame field (PFF). It constitutes a formidable platform containing a geometric Dirac equation from which to develop a deeper interpretation of the usual, dualistic QM. 1. The Issue of the Relation of Simultaneity of the World.

Participants in this workshop try to harness the potential of teleportation to synchronize clocks at spatially separated points. Looming in the background is the thesis of conventionality of synchronizations. The two camps on the issue remain hopelessly opposed. Is the con-ventionality issue relevant for applications of teleportation to synchronizations? Not in the short term, as one tries to find out whether QM allows phase teleportation using a two level transition. In the longer term, one has to contend with the related issue of conventionality of simultaneity. If simultaneity were a matter of convention, why would quantum teleportation make any sense at all? For the wave function’s collapse to teleport from one photon to another, the concept of “at which photon (in an entangled pair) the wave function was made to collapse” must be nonconventional. Workers at the foundations of QM do not share this conventionalist view. They view nature’s relation of simultaneity as objective or non-conventional. They believe that it is the one prescribed by standard special relativity (SR) but, unfortunately, they also believe that the experimental evidence confirms this. 367 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 367-376. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Our resolution of this issue is that whereas synchronizations are for most purposes about our conventional setting of clocks at a distance, the relation of simultaneity is a dynamical, non-conventional issue that only nature decides. For most purposes coordinates are simply markers. We can change from one set of them to another, in particular from the Einstein markers to the (primed) markers for absolute synchronization:

where (v,0,0) is the velocity of these frames with respect to some ab initio PFF. But, if the world abides by an absolute relation of simultaneity, using Einstein’s synchronization when finding out at which of two entangled photons did we cause their common wave function to collapse will be a wrong purpose. The right synchronization of clocks can, for instance, be achieved through the use of the candidate PFF, where the slow clock transport and Einstein’s synchronizations (implicit in the use of Lorentz transformations) coincide with the absolute one. The objective significance of the relation of simultaneity is embodied in the bundles of frames dual to the coordinate systems, if not in the coordinates themselves. The standard and “absolute” bundles are related by:

(velocity components The unprimed frames represent the pseudoorthonormal frames of SR and the primed ones correspond to absolute simultaneity. Caban and Rembielinski have shown that one can adapt QM to absolute simultaneity (primed frames), with advantage over the standard QM (unprimed frames)1. In view of this, we shall consider as proven that the relation of simultaneity is not a matter of convention. Assuming then that the actual relation of simultaneity of the world is the absolute one, one can still choose for most purposes a set of markers (coordinates) which are not dual to the primed frames, but rather correspond to SR, given that the slow clock transport and Einstein synchronizations are the only expedient synchronizations and that both give the same result. The coordinate transformations would then be the Lorentz transformations for most purposes even if the actual frames chosen by nature’s relation of simultaneity are not the pseudo-orthonormal frames. What experimental evidence determines the actual relation of simultaneity of nature? It must be one which conventionalists would say that it cannot exist, or that they claim to be circular. They rarely have to contend with specific experimental proposals, which, in any case, are not of a patently meaningful and easily recognizable nature. If they were, conventionalists would have conceded that there are meaningful (non circular) experiments that show that nature has a specific relation of simultaneity. Given limitations of space to deal with this issue here, this persistent attitude of conventionalists, intelligent scientists after all, will be viewed as “proof of the statement that the relation of simultaneity of SR has not been demonstrated by experiment in an incontrovertible way.

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2. The Physics Canonically Determined by an Absolute Relation of Simultaneity The post-relativistic postulate of absolute simultaneity implies a PFF, with respect to which the dual coordinate transformations are:

X,Y,Z,T representing the PFF. The latter canonically determines a rule to compare vectors at different points or affine connection, the one according to which the frames of the PFF are all equal, like the connection on the earth where the rhumb lines are taken to be lines of constant direction. We then have zero affine curvature, to be denoted as teleparallelism (TP). The roundness of the earth still shows in the non-zero metric curvature. But, as we now show, the geometry in PFF theories must be Finslerian, even if the metric is Riemannian. The group property is another “non-conventional purpose” for coordinates. Let A denote transformations like (3), taking us from the PFF to “fiducial frames” with velocity These frames, and those obtained from them through O(3) rotations, will be called para-Lorentzian (PL). Let R1, and R2 be rotations. transforms between fiducial frames of velocities and the transformation T given by

transforms between general PL frames, which can be put in a one to one correspondence with the Lorentz frames. This correspondence speaks of the fact that transformations 4 constitute a non linear representation of the Lorentz group2. The group property extends similarly to large families of flat spacetime structures which differ from SR in not complying with the independent experiments by Michelson-Morley, Kennedy-Thorndike and Ives-Stilwell3. The transformations represented in (4) act on a seven dimensional space, the three additional coordinates being the components of the velocity3. This space is the base space of Finsler connections and of Finsler geometry. This geometry is about manifolds S(M) which have been constructed from other manifolds M like we have constructed our sevendimensional space from spacetime4,5. Remarks: (a) there is affine Finsler geometry not involving metrics, (b) there are Finsler connections on Riemannian metrics and (c) Finsler distances may be irrelevant to physics. As with Riemannian geometry, the metric (nonaffine) origin of Finsler geometry is a historical accident. The process followed to obtain the coordinate transformations 4 does not work for their dual frames, as becomes clear by simply trying it. This is a consequence of how the frames sit over S(M), M now being spacetime. Since this S(M) is seven-dimensional and the set of orthonormal frames is 10-dimensional, there is a three-fold of PL frames related by the O(3) rotations sitting at each point s of S(M), rather than a six-fold of frames related by the Lorentz group. This O(3) group is a remnant of the vanished Lorentz group. It becomes SU(2) in the QM sector of the theory. The Lorentz group survives as a transformation between coordinates in the aforementioned non-linear realization. The Finslerian structure thus is the structure of PFF theories. Furthermore, it can be shown that the Euclidean signature is the canonical signature of Finsler geometry5. This Finsler

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bundle must, however, be reformulated as a 5-d Kaluza-Klein space where it becomes evident that U(1) also is a remnant of the Lorentz boosts on the frames6. It is in this space that the deepest QM must take place. 3. The Unified Theory Canonically Determined by Preferred Frame Fields

Unlike Caban-Rembielinski1, we let the canonical mathematics of a PFF, or Finslerian TP, guide us. A superseding, combined Maxwell-Einstein-Dirac theory results7 (in contrast, Riemannian geometry does not take us beyond general relativity (GR)). Let be the affine connection generated by the PFF and the metric. Let be 1forms whose components are the Christoffel symbols. Let be the contorsion. The statement that the affine curvature is zero becomes:

The left hand side of (5) is the metric curvature. The parenthesis and the last term yield gravitational and non-gravitational energy-momentum by standard contraction, after identification of the different non-gravitational interactions in the different components of the Finslerian torsion. This one is of the form:

where d is the exterior covariant derivative and dP is the translation form. . contribute to the classical equations of the motion, but and do not8. These are not, however, terms to be compared with the standard model, because this torsion is a concept from a theory of frames. QM does not deal with these macroscopic objects. Instead of Maxwell’s equations, we now have:

is the interior derivative, which, applied to tensor valued forms, becomes The derivative the interior covariant derivative. It is defined as in the TP version9 of the Kähler calculus10. It replaces the Dirac equation with:

where a and u constitute input and output differential forms (both scalar-valued for electrodynamics). is the sum of the interior and exterior derivatives. A conserved current is implied by equation (9), for any a. Kähler developed his theory for the LeviCivita connection. One then has that, if

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This means that u and are solutions for the same a. In particular, the product of solutions for a=0 is another solution for a=0. In TP, this is no longer the case, but Eq. 10 remains valid in a weak field approximation9. The reformulation of Finslerian TP over a canonical Kaluza-Klein space yields a “generalized Dirac equation”

where

is the element of proper time, now viewed as a fifth coordinate, where where d P is given by and where dv is equivalent to belongs to a very rich algebraic structure. The unification of micro and macrophysics results. When J is written in terms of as prescribed in the Kähler theory, the system of Eqs. 6, 7 and 11 is closed ( given in terms of the metric and in terms of the torsion). The complexity of the mathematics has impeded the recognition of this program for unification among theoretical and even mathematical physicists in general, except for the Clifford algebra community (see Preface to Proceedings in Ref. 6), where there is a tradition and some success in relating internal symmetries as spacetime symmetries (See examples in Ref. 6). For null J, Equations 7 can be written as which thus is a Kähler-Dirac equation for a=0. The expansion of these equations in the PFF and for S=0, i.e. for electromagnetic (EM) vacuum, yields:

We have not tried to find solutions for such sophisticated system. It contains non-linear terms similar to those that cause the beautiful solitons of the Muraskin system11

These equations 12 happen to state (not intentionally) that, in TP with Lorentzian metric, where the are the components of the contorsion. Muraskin obtained different types of numerical solutions of these equations over a 20 year period, solutions which very much depend on the initial values. For a particular set of initial values, the plot of as a function of x for given values of the other coordinates is shown in the figure, to be used in the next section (no special significance should be attached to this set of indices) We now show how this theory relates to the PFF of absolute simultaneity. In physical TP, the PFF is attached to autoparallel curves. The distribution of matter and charge determines these curves. The PFF at the basis of absolute simultaneity is the one in which

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we neglect “local motion” of matter and consider only the average over very large scales, matter which is neutral. The PFF field of TP in specific regions of the spacetime manifold becomes the PFF of cosmology by disregarding the motion of matter except on the very large scales (comoving matter frame). In view of the unexpected emergence of such formidable theory, we have sidestepped the issue of whether the frames are PL or orthonormal-cum-PFF3. Based on the experience gained, we expect that the correct flat spacetime structure may still come in the wash. 4. Consequences of the Canonical Theory of PFF for Quantum Measurement and for Teleportation.

In this section, we shall overlook that not all the equations have been worked out to the point where precise statements can be made. This need not impede a view of the big picture. We shall proceed in stages of increasing structure. (A) The canonical development of TP leads to the Kähler equation, (9), not the Dirac equation. A major difference is the fact that this equation, based in a calculus of differential forms, has neither gamma matrices nor Hamiltonians. It is geometric. For EM coupling, the geometric character of the Kähler equation

is not apparent because is scalar-valued, and affine geometry proper involves vector-valued forms. It is far fetched to say that the Hamiltonian is built into this equation, given the presence of the factor i in the mass term. There are other differences, like the number of components of the wave function, and yet, Eq. 14 solves the hydrogen atom. The energy operator, (now with standard is simply the Lie operator relating the solutions of (14) under time translation symmetry. The differences become even more obvious for systems of n particles. In the paradigm, these systems are typically handled with the Schrödinger equation in 3n+1 dimensions. The Kähler equation is an equation in 4 dimensions regardless of the value of n. It is, therefore, of the densityfunctional type, ab initio. In the standard, Hamiltonian-based QM, one feels the need to justify why density functional theory works. Kohn, who does not use Kähler theory, objects to the concept of multiparticle wave function, as he did in his Nobel acceptance lecture12, with statements such as “In general the many-electron wave function for a system of N electrons is not a legitimate scientific concept, when where (emphasis in original). In Kohn, the concept of multiparticle Hamiltonian goes down the drain with the concept of many-electron wave function. Of course, a QM based on the Kähler equation will not be complete without a derivation from it of the multiparticle equation for numbers of particles which are not too large.

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(B) The implications of (A) are even richer for spacetimes with torsion. The equation remains valid, now as a weak-torsion approximation, with v in the role of boson and u as matter, fermions in particular. The wave function for a pair of a measuring instrument (to be called fermion), u, and a boson (say a photon) v, would be given by when apart. When they are together, they constitute a fermion, for the same a. By the same argument, now forward in time, Eq. 10 admits the interpretation that evolves back into the original fermion and boson. Concepts like confinement, absorption and wave function collapse appear to be related (a) among themselves, (b) to the compliance or not compliance of Eq. 10 and (c) to the fact that the wave function lives in spacetime (complemented by velocity in one way or another), but not in a 3n+1 space, as we now argue. Consider the following statements in Dirac’s classic book on “If a system is small, we cannot observe it without producing a serious disturbance...” (p. 4), “...beam of light passing through a crystal of tourmaline... each photon polarized perpendicular to the axis passes unhindered and unchanged through the crystal, while each photon polarized parallel...” (p. 5) and “When we make the photon meet a tourmaline crystal, we are subjecting it to an observation... The effect of making this observation is to force the photon entirely into the state of parallel or entirely ...” (p. 7). Suppose we have two parallel tourmaline crystals with their optical axes also in parallel arrangement. Light going through the first crystal will become polarized and, consistently with will pass undisturbed. But according to this is a measurement. So, it is a measurement which does not produce disturbance, contrary to The morals of the

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story is not to try to invalidate the uncertainty principle, which is implicit here. But one should perhaps be more careful in its statement, or perhaps Dirac could have been more careful. For instance, could be modified with the remark that the observation or the measurement finishes when the intensity of the passing beam is measured by absorption on a screen beyond the second tourmaline crystal. That is where we wanted to be. It is the absorption that counts. The through photons are in a regime where equation (10) works. Photons that are absorbed are in the regime where the torsion is not negligible and does not satisfy equation 9. The wave function now remains a solution of (9) but is no longer The extreme robustness of the photons in modern technology would be a manifestation of the fact that, except perhaps at very high intensities, the regime of Eq. (10) is at work in the form if Quark-gluon confinement would be the statement that, like in the case and unlike in a molecular system (EM interactions), the Kähler equation does not admit a reformulation in multi-particle format. This is not surprising since the S terms are of a completely different and unusual nature R terms (EM is of the R type). The question is: how can the polarization of photons (relative to the tourmaline’s orientation) be a determining factor of whether one or the other regime applies. At this point, we need to go one step further and assume TP. (C) Given the structural sophistication of the canonical PFF theory and the still primitive state of development of its details, we are not in a position to describe even qualitatively what causes a photon’s wave function to be absorbed or get through. We may, however, resort to the proxy Muraskin equations to “guesstimate” some form of mechanism at work. In the PFF theory, the EM vacuum equations are i.e. of the type (when using specific u’s and v’s, we start to use boldface if applicable). In other words, the vacuum EM field is a spinor -in the sense of solution of the Kähler equation- of the type called harmonic. Except for the fact that we have to use the torsion appropriate for the K-K space14, which limitations of space forbid, the equation may be viewed as being of the type with i.e. equal to zero. This is a constraint not unlike requiring that an EM field satisfied . Hence, the boson and fermion equations are unified in TP! Assume for the sake of the argument, that the Muraskin equations were the equations of the EM vacuum, its solitons being the photons (these are not packets or superposition of plane waves of any sort). Since the linearized Muraskin theory yields the addition of the quadratic terms to the left hand side of generates the solitons and modulated background of Eqs. (13). The high dependence on initial of thetype of solution that one obtains is to be contrasted with the irregular behavior (though bound in magnitude) in the zones between the solitons. Considering that the values of the at any point Q in such zone can be viewed as a set of original conditions for the second soliton, one is led to conclude that the behavior of the solution to the right of Q (i.e. the emergence of the virtually identical second soliton, or other behavior as in the book in Ref. 11) depends on the existence of conserved algebraic relations between the and thus into classes of solutions. Given the argument of the previous paragraph, it is natural to assume similar behavior for matter. Polarizations of incident photons that are parallel and orthogonal to the axis of the tourmaline are interpreted as being such that the wave

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function (equi-valently, the set of gammas) of the pair crystal-photon when the two come close evolves into different classes. Different states of polarization would then give rise to wave functions belonging to two different classes of classes, respectively corresponding to whether or not Eq. 10 is satisfied. No collapse is needed, just bifurcation points. The question remains of whether this deterministic picture is affected in any way by Bell’s inequalities. Let us start with the observation that Bohm’s hidden variable theory satisfies the same inequalities as standard QM. Hence, there are limits to the applicability of these inequalities, certainly when the hidden variable theory is governed by equations which are quantum mechanical. But, in addition, the “loophole” type limitations are overwhelming in this case. depends on the torsion, through the connectiondependent operator and also depends on The dependence on the torsion constitutes a non-local way of depending on the potential. We may then say that the dependence on constitutes a non-local way of depending on the torsion. Furthermore, this equation is not to be viewed as one to be solved for when and are given, but rather as making part of the system of equations where one has to solve for the (structuredependent) and at the same time as one solves for Finally, the system of Eqs. 6, 7 and 11 is a closed system, but the physical system to which it applies cannot be considered as closed. We send photons into a region of space. They carry “their background” as in the figure, but there is already a background there (This is an issue that has to do with the Cauchy problem of which is radically different from the Cauchy problem of GR because the field equations involve the curvature and not just the Einstein tensor). It should be clear that present-day Bell theorems cannot cope with this sophistication. They would amount to collections of potential loopholes. 5. Concluding Remarks

In canonical Finslerian TP, which is also the canonical theory of PFF, teleportation may not be needed since the information as to the future behavior of a photon when encountering a tourmaline crystal is contained in its wave function (See (c) above). The measurement (absorption of photons) is regulated by deterministic laws, though this determinism is possibly unfathomable in practical circumstances. But even if Copenhagen-type indeterminism is more than a historical accident of twentieth century physics, the great richness of evolutions of solutions of the Muraskin equations speak of the possibility of defining other velocities associated with the propagation of the solitons, in addition to the velocity of the solitons themselves. Thus, an absolute, superluminal velocity might perhaps be possible, though apparently unnecessary, in this theory. It might be associated in one way or another with the zones among the solitons. But we lack the imagination to go any further in this direction without violating causality. The examples just discussed show that the richness and wealth of information contained in the QM sector of this theory goes way beyond present day QM and the Copenhagen interpretation, not to mention the consequences of replacing gauge geometry (i.e. geometry in secondary bundles) and concomitant internal symmetries with geometry in the principal bundle. Hence there is no need in canonical PFF theory for extreme views

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(at least retrospectively) like “there is no wave function, just probability amplitudes”15. The arguments put forward cannot be considered anything but qualitative, and even tentative. The reason is that the principle of the theory, Finslerian TP cum Kähler calculus, is extremely removed from any physical concepts. It will take an enormous amount of effort to bring this theory to where its physical consequences can be derived with a great degree of reliability. There are, however, windows of opportunity. One of them is, for instance, the fact that non homogeneous electric fields produce gravitational fields by virtue of source terms that contain the derivatives of this field, rather than by virtue of the energy-momentum contents. An experiment recently performed by T. Datta and M. Yin at this institution has already shown the existence of gravitational fields induced by a non-homogeneous electric field, although we have not yet tried to match it with exact predictions of the theory. Time may soon tell.

Acknowledgments One of us (J.G.V.) deeply acknowledges generous funding from the Office of the Dean of the School of Science and Mathematics of the University of South Carolina at Columbia, and the invitation and funding by JPL to participate in the NASA-DoD on Quantum Information and Clock Synchronization for Space Applications (September, 2000). We thank Professor J. Anandan for reference 15.

References 1.

P. Caban and J. Rembielinski, Phys. Rev. 59A, 4187 (1999). J. Rembielinski, Physics Letters 78, 33 (1980). 3. J. G. Vargas, Foundations of Physics 16, 1231 (1986). 4. J. G. Vargas and D. G. Torr, J. Math. Phys. 34, 4898 (1993). 5. J. G. Vargas and D. G. Torr, The Cartan-Clifton Method of the Moving Frame: Finslerian Bundles on Riemannian Distances, to be published in Algebras, Groups and Geometry (September 2000), as part of the th Proceedings of the 11 Romanian Conference on Finsler and Lagrange Geometry. 6. J. G. Vargas and D. G. Torr, Clifford- Valued Clifforms: A Geometric Language for Dirac Equations in R. Ablamowicz and B. Fauser (eds.), Clifford Algebras and their Applications in Mathematical Physics", Birkhauser, Boston (2000), 135-154. 2.

7.

J. G. Vargas and D. G. Torr, From the Cosmological Term to the Planck Constant, to be published in a

Volume of The Fundamental Series of Physics, Kluwer Academic Publishers, devoted to the Proceedings of the Third Vigier Symposium, Berkeley, 2000. 8. J. G. Vargas and D. G. Torr, Found. Phys. 29, 1543 (1999). 9. J. G. Vargas and D. G. Torr, Found. Phys. 28, 931 (1998). 10. E. Kähler, Rendiconti di Matematica 21, 425 (1962). 11. M. Muraskin, Physics Essays 8, 99 (1995); Mathematical Aesthetic Principles/Nonintegrable Systems" (World Scientific, Singapore, 1995). 12. W. Kohn, Rev. Mod. Phys. 71, 1253 (1999) 13. P.A.M. Dirac, The Principles of Quantum Mechanics (Oxford U. P., London, 1958) 14. J. G. Vargas and D. G. Torr, Found. Phys. 27, 533 (1997). 15. B. Kayser and L. Stodolsky, Phys. Letters B 359, 343 (1995).

CAN NON-LOCAL INTERFEROMETRY EXPERIMENTS REVEAL A LOCAL MODEL OF MATTER?

*J. MARTO, **J. R. CROCA *Departamento de Física da Universidade da Beira Interior 6200 Covilhã, Portugal Email: [email protected] **Departamento de Física Faculdade de Ciências, Universidade de Lisboa Campo Grande, Ed. C1, 1700 Lisboa, Portugal Email: [email protected]

Abstract: In the following we consider the possibility of interpretating non-local interferometric experiments according to the De Broglie causal model. With the help of a simplified mathematical model based on wavelet analysis it is indeed possible to explain it in a causal way. Furthermore we show the distinction between the two formalisms and discuss some experimental conditions that may make these differences evident. 1. Introduction

Quantum mechanics has a formalism exhibiting non-separability [1], in that context interferometric experiments with neutrons carried by Rauch [2] [3] seem to show this peculiar feature. The authors assert that it is possible to get an increase in the visibility of the interference pattern using a proper postselection state process. The interference is observed without superposition, at the mixing region, of the wave packet that represents the neutron. An alternative causal interpretation is possible following the ideas of De Broglie. An early attempt was made [4], in the framework of this theory maintaining however Fourier non-local analysis in the particle description. In it a physical reality was assigned to the plane waves. We feel some uneasiness in understading how is possible to ascribe some character of physical reality to plane harmonic waves that are infinite in time and space. This assumption implies that a wave packet built up by an infinite sum of plane harmonic waves is not separable; therefore it is not possible to individualize real physical systems at a microscopic level. 377 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 377-384 © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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With those dificulties in mind here we present a causal local particle model and discuss experimental conditions that could settle the problem of non-separability. Whether is it a result of non-local analysis or, due to technical limits in carrying out the experiments that lead to an inability in distinguishing the kind of waves composing the wave packet? 2. Finite pulse superposition

In order to built a wave packet we follow a process similar to Fourier analysis, but instead of using non-local infinite plane harmonic waves as the building bricks we use finite waves that represent a quantum particle. Consider, for instance, a point like source emitting monochromatic finite pulses described analytically, for the sake of simplification, by the Morlet wavelet [5]:

Now suppose also, for simplicity reasons, that the emitted pulses have gaussian wave number distribution. Then the resulting wave packet is obtained superimposing the finite waves:

In non-dispersive medium the relation between the wave number and the frequency of the wave is that is what to be expect in a non-dispersive medium, so the wave train (2) is after some calculations:

where,

This last relation and its physical meaning, as a more general form for the uncertainty relations, was derived and extensively discussed some time ago by one of us [6], It appears that the possibility of further testing this expression is now possible if some modifications in the experimental apparatus could be made [2][3][7][8][9]. 3. Non-local interferometry

Let us consider in figure 1 a Mach-Zehnder interferometer made assymmetric by a modification in the arm lengths.

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When the two pulses arrive at the first beamsplitter they are split into two different independent paths. The presence of a phase shifting device in one arm of the interferometer induces a relative shift between the two pulses. In such conditions one is allowed to write for the first arm of the interferometer

And for the second arm:

Therefore, at the interferometer output port we will have the sum of the two waves To obtain the intensity collected at the detector must be averaged over time,

in such conditions and after some easy calculations we get,

which represents the single pulse at the output port before reaching the optical filter. Recalling the De Broglie postulates [10], the probability that the particle-singularity cross the interferometer and the filter is given by,

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3.1. MONOCHROMATORS The monochromator plays a key role in this kind of so-called non-local interferometric experiments. There are monochromators of many types, for photonics we have the optical filters, Fabry-Perot etalons, and crystal analysers for neutrons. In any case the inclusion of this filter at the interferometer exit leads to an increasing of the coherence length of the output wave train. In the present work our attention is focused on experimental situations where the input sources produces a beam with gaussian wave number distribution. Therefore the practical effect of the mentioned filter is of narrowing the bandwidth of those distributions on the output beam. In some works [2][11], on those matters, we found that the typical bandwidth acceptance on the analysed beam where approximately of the order of Angström. The effect of this kind of filter is described by a narrow bandwidth limiting function:

3.2. VISIBILITY OF THE INTERFERENCE PATTERN Assuming a gaussian distribution in the wave number for the pulses and summing over the entire spectrum of the particles emitted by the source, we find the counting intensity at the detector,

The interference pattern visibility for those experiments is by definition,

which by substitution of the intensities leads to the causal visibility:

where

We must point out that this result is formally similar to the one obtained if we had used non-local Fourier analysis [2], which would give:

The question now is how to establish what is the right formula for describing the experimental visibility obtained. Till present these results have been interpreted in the usual way, that is using non-local Fourier analysis. Now, we have shown that there is an

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alternative way to derive formally the same expression for the visibility. This new process is based in the local wavelet analysis and assumes that the quantum particle is a real physical entity, with a well-defined energy localized in space and time. The obvious way to find out what is the correct model for describing a quantum particle is using a filter which bandwidth goes to zero, In such case the usual visibility (15) goes to one, for any arbitrary large value in the optical path difference between the two arms of the interferometer. This fact is easily understandable in the light of the non-local Fourier analysis. When the bandwidth of the filter goes to zero this means that in such case the output wave is practically a monochromatic plane wave with an infinite coherence length. Therefore, no matter how long is the optical path difference between the two arms of the interferometer there is always a complete overlapping of the two infinite plane waves and therefore the visibility is always one. Predictions change drastically if instead of considering infinite plane waves one assumes that quantum entities ought to be represented by finite wavelets. In this case when the bandwidth of the filter goes to zero the causal visibility assumes the form,

which means that the causal visibility depends on the optical path difference. Therefore if the optical path difference is much larger than the size of the wavelet no interference is to be expected and the visibility goes to zero. For shorter path differences the visibility starts growing up till it reaches the value one for no optical delay between the two arms of the interferometer. This problem was already analyzed [12], and it was pointed out that what has been interpreted as sound evidence in favor of "postponed compensation", that is time compensation after the actual physical effect has taken place, could as well be explained simply as a classical problem of recovering an interference pattern hidden by noise with the help of an appropriate filter. 4. Spectral modifications

In order to test the possibility of finite waves description for the quantum particles, we should look for a discrepancy in the experimental visibility. In figure 2 the mentioned discrepancy manifests through the fact that causal visibility no longer tends to 1 for spatial delays larger than the wavelet size

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From figure 2 we may infer that the causal visibility fall to zero for the shorter wavelet size. What happens now is that in order to reproduce the same result, as with plane waves, the wavelet size must be comparable or greater the coherence length of f the plane wave packet. That is for a plane wave packet with a bandwidth the wavelet size must satisfy:

There is an inconclusive problem with this kind of experiment because we can always say that wavelets explain the visibility obtained if the wavelet size satisfy (17). However there is another way to settle the problem if we consider the study of spectral modifications [7][8][9] on similar non-local interferometry experiments performed with neutrons and photons. In fact it was reported that when the spatial delay in the Mach-Zehnder (figure 1) is greater than the coherence length of the wave packet, we could observe a modulation in the spectral distribution. Recalling equation (8) but with plane waves instead of the Morlet wavelet we find for the spectral modulation:

All the experiments mentioned confirm the presence of modulation in the spectral distribution as predicted by (18), but the same experiments reveal an increase of the spectral modulation depth with the path delay Consequently the expected minima of (18) do not go to zero.

The modulation depth was explained [9] by studying the effect of vibrations on the table that sustained the experimental apparatus. We can assume that there is a fluctuation in the spatial delay and so equation (18) is modified:

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It is easy to guess that if we consider those fluctuations in the finite wave point of view, we obtain the following result:

and we will predict that the modulation minima don’t go to zero. However the most interesting aspect is that expression (20) predicts also a different behaviour for the modulation. In fact the computation of (19) and (20) can be summarized in the following expression:

where A is the attenuation term, function of the vibration parameter v (both (19) and (20)) and the wavelet size (only for (20)). However in (21) assumes a very different behaviour for plane waves and finite ones, and if we graphically represent it the difference become evident:

It is possible to conceive a conceptual experiment where induced vibrations could open a way to test the idea of finite versus infinite waves in quantum physics.

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5. Conclusion Non-locality and non-separability are concepts rising from the quantum formalism, and have been the core of a debate old as quantum theory. Some experiments done and mentioned here tried to prove a non-local behaviour for the quantum particle, and seem suitable to test the idea if the undulatory aspect of it is of infinite range. We have presented an examination of those ideas and described an experiment that could get us farther in the elucidation on the nature of non-separability that these interferometric experiments pretend to reveal. 6. References [1] N. Bohr, (1928), in Como Lectures, Collected Works, Vol. 6, (North-Holland, Amsterdam, 1985). [2] H. Kaiser, R. Clothier, S. A. Werner, H. Rauch and H.Wölwitsch, "Coherence and spectral filtering in neutron interferometry," Phys. Rev. A 45, (1992) 31. [3] H. Rauch, "Superposition experiments in neutron interferometry," Il Nuovo Cimento B110, (1995) 557. [4] V. L. Lepore, in Waves and Particles in Light and Matter, pp. 387-394, (Edited by A. van der Merwe and A. Garrucio) (Plenum Press, New York, 1994). [5] P. Kumar and E. Foufoula-Georgiou, "Wavelet analysis for geophysical applications," Rev. of Geophysics 35, (1997)385. [6] J. R. Croca, in Fundamental Problems in Quantum Physics, pp. 73-82, (Kluwer Academic Publishers, 1995); J. R. Croca, A. J. Rica da Silva, "Dispersion relations for the superposition of monochromatic finite pulses," submit. for publication; J. R. Croca, "The Limits of Heisenberg's Uncertainty Relations," in Causality and Locality in Modern Physics, (Edited by S. Jeffers et al., in print) (Kluwer Academic Publishers, 1997). [7] D. L. Jacobson, S. A. Werner, H. Rauch, "Spectral modulation and squeezing at high-order neutron interferences," Phys. Rev. A 49, (1994) 3196. [8] X. Y. Zou, T. P. Grayson and L. Mandel, "Observation of quantum interference effects in the frequency domain," Phys. Rev. Lett. 69, (1992) 3041. [9] D. Narayana Rao and V. Nirmal Kumal, "Experimental demonstration of spectral modification in a MachZehnder interfrometer, " J. mod. Optics 41, (1994) 1757. [10] L. De Broglie, in The current interpretation of wave mechanics (a critical study) (Elsevier Publishing Company, 1964), pp. 43-46. [11] W. Tiller, "Bandwidth measurement of high-resolution Fabry-Perot etalons," Appl. Opt. 23, (1984) 3931. [12] J. R. Croca, R. N. Moreira, A. J. Rica da Silva,"Recovery of an interference pattern hidden by noise," in Causality and locality in modern physics, (Edited by S. Jeffers et al., in print) (Kluwer Academic Publishers, 1997).

BEYOND HEISENBERG’S UNCERTAINTY LIMITS

JOSEE R. CROCA Departamento de Física Faculdade de Ciências Universidade de Lisboa Campo Grande Ed. C1 1749-016 Lisboa Portugal Email: [email protected]

1. Introduction Heisenberg’s uncertainty relations state that it is impossible to predict the result of a measurement with a precision lying beyond its limits. In practice this formula defines, in standard quantum mechanics, the limits of the previsional measurement space accessible to any measurement. Beyond it is impossible to perform any concrete measurement. In the last years a strong effort was made to develop causal quantum physics in the spirit of de Broglie and Einstein. These efforts were carried out mostly with the help of a recent mathematical tool, the wavelet analysis, developed precisely to overcome some inconveniencies of the Fourier non-local analysis. As a consequence of this work, a more general expression for the uncertainty relations, containing the usual ones as a particular case, were derived. These more general uncertainty relations define a predictive measurement error space including the usual Heisenberg space. In this new space the prevision of the precision of a result of a measurement depends only of the basic measuring tool used in a concrete experiment. If one wishes to predict the result of a measurement with greater precision it is necessary to change the basic measuring tool. In the language of local wavelet analysis this corresponds to say that one has to use a mother wavelet of a smaller scale. By changing the scale of the basic wavelet it is possible, in principle, to scan all the predictive measurement space. For some time those ideas had no experimental support, even if the new approach explained the experimental evidence, which is not a surprise, since the new relations contain the usual ones as a particular case. Now, thanks to the recent development of a new generation of microscopes that have a resolving power that goes far beyond Abbe’s limit of half wave length, there are measurements, done everyday, that are outside the Heisenberg’s space. 2. Wavelet Analysis And The New Uncertainty Relations The wavelet local analysis [1] was developed in the early eighties by Jean Morlet, a French geophysicist working in oil prospection. He devised this mathematical tool for 385 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 385-392. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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denoising the very sharp seismic signals. These early attempts were developed and formalized by Grossmann, Meyer, and many others till the wavelet local analysis was made into a powerful mathematical tool, competing side by side with the Fourier nonlocal analysis. Today, there is a whole universe of scientific literature dealing with the all-different aspects of wavelets. This field is developing so fast that even the very definition of wavelet that at the beginning seemed fairly settled, is today an uncertain ground that has lead some to authors to say that the precise definition of wavelet is a kind of scholastic question. Here, only the basic property of finitude of the wavelet is retained. 2.1. WAVELET LOCAL ANALYSIS VERSUS NON-LOCAL FOURIER ANALYSIS In order to understand the conceptual and practical importance and gasp the deep significance of this new local analysis by wavelets it will useful to make a short comparison with the non-local Fourier analysis. The Fourier analysis must be considered non-local or global because its basic elements, its constituting bricks, are monochromatic harmonic plane waves, infinite both in space and time. The non-local or global character of the Fourier analysis can be easily understood with the following example.

Suppose that it is necessary to record, line by line, the first image shown in Figure 2.1, for instance, in a CD. In order to simplify the problem, let us consider only the line indicated in the picture. The plot of the intensity of the picture along that line is also represented below in the same picture. In order to Fourier represent this variation in intensity it is necessary to find the amplitudes, frequencies and phases of the infinite monochromatic harmonic plane waves that sum up reproducing the initial function. This immensity of plane waves interferes negatively in all points of space except in those two regions where the interference is constructive. Let us now suppose that, for instance, the first ship moves to the right, while the other remains in the same position. This situation is represented in the second image of Figure 2.1. Next, we ask how this motion of the first ship is to be Fourier represented. In this type of analysis, both the first and the second ship are made of the same infinite harmonic plane waves. In these circumstances, a modification in the position of the first ship implies naturally a modification in the amplitude frequency and phase of all the waves, so that their interference is constructive now only in the new position. This means that, even though the two ships may seem separate entities, they are in fact, treated as a whole entity

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in this process of non-local Fourier analysis and reconstruction. Hence, a modification in the position of one of the ships implies a simultaneous modification of the constituents of the other, no matter how distant they are from one another. This mathematical consequence of dealing with infinite harmonic plane waves, determines the very deep roots of the non-locality of the usual quantum mechanics. The non-local or global character of Fourier analysis has also serious technical drawbacks, namely when, as is the case, one wants to register video images. In general, as is very well known, from video image to video image there are only slight variations. Nevertheless, in this type of nonlocal or global analysis, a minor modification on one region of the image implies the necessity of analyzing and rebuilding the whole image. If, instead of the non-local Fourier analysis, one uses local wavelet analysis, the modification of the position of one ship as nothing to do with the other, if the two objects are sufficiently far one from another. In this case, the group of wavelets that sums up to recreate one object is completely independent from the wavelets corresponding to the reconstruction of the other ship. This is precisely the reason why this type of analysis is called local analysis by finite waves. The difference between the two types of analysis is shown graphically in Figure 2.2.

Another important advantage of the local wavelet analysis is related to the fact that, in order to represent a given signal, one is free to choose different types of basic wavelets. In these circumstances, a wise choice of the basic wavelet can further increase the rate of compression and the quality of the reconstruction, or the denoising capability. 2.2. THE NEW UNCERTAINTY RELATIONS

Analytically, the usual uncertainty relations are a mathematical consequence of the nonlocal Fourier analysis [2]; if, instead of infinite harmonic plane waves, one uses the local wavelet analysis to represent a quantum particle, the form of uncertainty relations may consequently change [3]. In order to make the derivation of the new uncertainty relations more comprehensible, it is convenient to do it step-by-step, in parallel and with the usual derivation. This process is shown in the following sketch, Figure 2.3

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From the table, Figure 2.3, it is seen that the new uncertainty relations derived with the local wavelet analysis have the form

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The same analysis could, in principle, be done with other mother wavelets. The Gaussian Morlet wavelet [4] was chosen among the other possibilities because of its interesting properties: - From the mathematical point of view, it has a very simple form. Therefore, the necessary calculations can fully be carried out without approximations. - Another very interesting property of this wavelet comes from the fact that, when its size increases indefinitely, it transforms itself in the kernel of the Fourier transform. In this sense, this local analysis contains the non-local Fourier analysis as a particular case. When the size of the basic wavelet is sufficient large, the new relation transforms itself into the old usual Heisenberg relations, which is a very satisfactory result - It allows a reasonable representation of fair localized particle with a well-defined velocity. - Written in terms of space and time, it is a solution of a Schrödinger non-linear master equation [5]. The plot of the uncertainty is shown in Figure 3.4. The new uncertainty relations are represented by the solid line for three finite values of the size of the basic wavelet. The usual Heisenberg uncertainty relations, which correspond to the case are shown as a dashed line.

2.3. GENERALIZED MEASUREMENT SPACE The new, more general uncertainty relations were derived in a causal framework, assuming that the physical properties of a quantum system are observer independent, and, even more, that they exist even before the measurement. Naturally due to the unavoidable physical interaction during the measurement process, when the other conjugated observable is to be measured, the quantum system does not remain in the same state. In last the instance, the precision of the measurement, for a non-prepared system, depends

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on the relative size between the measurement basic apparatus and the system upon which the measurement is being done. Until now, the most basic interacting quantum device is the photon. Nevertheless if the photon has an inner structure, as is assumed in the Broglie model, it will imply the ability to perform measurements beyond the photon limit. Since in the derivation of the new uncertainty relations, the quantum systems were assumed to be described by local finite wavelets, the measurement space resulting from those general relations must depend on the size of the basic wavelet used. As the width of the analyzing wavelet changes, the measurement scale also changes. This can be seen in the following plot Figure 2.4.

From this picture it is seen that, as the width of the basic wavelet changes, all the measurement space is scanned. This space is only limited by Heisenberg’s space. The smaller the greater is the precision of the position measurement. That is: the smaller is the uncertainty for any value of the error in the momentum. Given that the new relation contains the usual as a particular case, it implies that the measurement space available to the general uncertainty relations is the whole space, as shown in Figure 2.5.

These results are rather satisfactory, because in this causal paradigm the quantum measurement process depends, in last instance, on a standard used. We are, in principle, free to choose the size of the mother wavelet more suitable for the measurement precision we want to attain.

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3. Beyond Heisenberg’s Measurement Space

The middle of the eighties saw the development of super-resolution microscopes. The first of this new type of microscopes was developed for electrons by Bennig and Roher [6], who received the Nobel Prize for the discovery. The principal characteristic of these microscopes is that their resolution goes far beyond those given by Abbe’s rule of half wavelength. Abbe derived his theoretical resolution rule based on the Rayleigh diffraction criteria for the separation of two circles in a far field. Super-resolution microscopes were also developed in 1984 by Pohl et al. [7] for the optical domain. Initially, these microscopes had a spatial resolution of ten years later [8] they attained resolutions of or even better. There are many types of super-resolution optical microscopes, as can be seen in pertinent literature. In most straightforward of these microscopes the light emitted by the sample is simply collected by the sensing probe. In order to see if there are some very special experimental situations were the predicted errors of the two conjugated observables do not lie in Heisenberg’s measurement space, it is convenient to consider the well-known Heisenberg microscope in parallel with the super-resolution microscope optical microscope.

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From the table, one sees that there is a very significative difference of 1/25 in the products of the uncertainties related to the measurement of the two conjugated observables, position and momentum. For the common Fourier microscope, this product lies well within Heisenberg’s measurement error space For the superresolution microscope, we have that lies outside Heisenberg’s measurement space, and is therefore contained in the more general wavelet precision measurement space.

Acknowledgements

I want to thank Fundação Calouste Gulbenkian, for a grant, allowing me to be present at the Vigier III Conference.

References 1. A. Grossmann and J. Morlet, SIAM J. Math. Anal. 3 (1989),723; C.K Chui, An Introduction to Wavelets, Academic Press, N.Y. 1992. 2. N. Bohr, (1928) – Como Lectures, Collected Works, Vol. 6, North-Holand, Amsterdam, 1985. 3. J.R. Croca, Apeiron, 4 (1997), 41. 4. P. Kumar, Rev. Geophysics, 35 (1997), 385. 5. Ph. Gueret and J.P. Vigier, Lett. Nuovo Cimento, 38 (1983), 125; J.P. Vigier, Phys. Lett. A, 135 (1989), 99. 6. G. Benning and H. Roher, Reviews of Modern Physics, 59 (1987), 615. 7. D.W. Pohl, W. Denk and M. Lanz, Appl. Phys. Lett. 44 (1984), 651. 8. H. Heiselmann, D.W. Pohl, Appl. Phys. A, 59 (1994), 89.

TOWARDS A CLASSICAL RE-INTERPRETATION OF THE SCHRODINGER EQUATION ACCORDING TO STOCHASTIC ELECTRODYNAMICS

K. DECHOUM Departamento de Fisica Moderna Universidad de Cantabria 39005 Santander, Spain Email: [email protected] H.M. FRANCA Instituto de Fisica, Universidade de Sao Paulo CP 66318, 05315-970 Sao Paulo, Brasil Email: [email protected] C.P. MALTA Instituto de Fisica, Universidade de Sao Paulo CP 66318, 05315-970 Sao Paulo, Brasil Email: [email protected]

Keywords: Zero-point fluctuations; Stochastic Electrodynamics.

Abstract

We study the statistical evolution of a charged particle moving in phase space under the action of vacuum fluctuations of the zero-point electromagnetic field. Our starting point is the Liouville equation, from which we derive a classical stochastic Schrodinger like equation for the probability amplitude in configuration space. It should be stressed that we are not deriving the Schrodinger wave equation. An equation formally identical to the Schrodinger equation used in Quantum Mechanics is obtained as a particular case of the classical stochastic Schrodinger like equation. An inconsistency appearing in the standard Schrodinger equation, when vacuum electromagnetic fluctuations and radiation reaction are taken into account, is clearly identified and explained. The classical stochastic Schrodinger like equation, however, is consistently interpreted within the realm of Stochastic Electrodynamics. 393 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 393-400. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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1. Introduction

The classical electromagnetic theory has been largely extended by the program of the Stochastic Electrodynamics (SED) [1, 2, 3], due to the inclusion of the effects of the real electromagnetic zero-point radiation. According to the SED picture, there is a clear similarity between the nonrelativistic Heisenberg equations of motion, for a spinless charged particle interacting with the quantized electromagnetic field of Quantum Electrodynamics (QED), and the classical (Langevin type) Abraham-Lorentz equation with real vacuum fluctuation forces. At the same time, it is well known [4] that for a special class of potentials the dynamical evolution of any Schrödinger wave packet is entirely determined by the laws of Classical Mechanics. Therefore we shall develop a method that is aplicable to any deterministic potential, in order to better clarify the relation between the classical and the quantum theories of the microscopic world. We shall derive a classical Schrödinger like equation from the Liouville equation, using a procedure similar to that introduced by Wigner [5] for describing Quantum Mechanics in phase space. Our approach introduces a free parameter in the Wigner type transform [6]. We shall show that this procedure enables us to make a clear distinction between the free parameter and the Planck’s constant Only the vacuum electromagnetic fluctuations will depend on the numerical value of the Planck’s constant

2. Connecting The Stochastic Liouville Equation To A Schrodinger Like Classical Styochastic Equation

The description of classical phenomena by classical statistical mechanics is based on the concept of phase space. The probability density distribution, evolves in time according to the Liouville equation

where and are obtained from the classical Hamilton’s equations of motion. Consider an ensemble of systems which consist of a nonrelativistic spinless charged particle interacting only with the electromagnetic field. The Hamiltonian which describes the time evolution of the whole system

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(particle plus field) is

where

and are the charge and mass of the particle, respectively, is the scalar potential, and

is the vector potential. The term is an external deterministic disturbance. The term is the vector potential associated with the real vacuum fluctuations, and can be written as [3]

where is the volume containing the particle and the radiation field, is the wave vector, is the polarization index, and are the polarization vectors. The amplitudes are taken to be random variables. The random character of the field is contained in these variables which are such that and denotes the ensemble average). The term is the vector potential that describes the radiation reaction [1, 3] and is the Hamiltonian of the background radiation field (contains only variables of the field). In the case of zero temperature, can be written as [3]

where so that The extension to a non-zero temperature T is obtained by introducing the factor coth Each particle of the ensemble evolves in time according to the Hamilton’s equations

and

Substituting the equations (1.6), (1.7) into (1.1) we get the Liouvillian form of the equation governing the time evolution of the ensemble of

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particles for each realization of the stochastic field

namely

Prom the solution of (1.8) the ensemble average over all possible realizations of the field (in order to obtain the average distribution is performed by considering the average over the random Gaussian amplitudes in (1.4). Notice that in (1.8) and are explicit functions of the variables and Consider the Fourier transform defined by [5, 6]

where is a point in the configuration space and is a free parameter having dimension of action. The meaning of the free parameter will be discussed further below. Notice that (1.9) corresponds to the well known Wigner transform [5] if The use of the conventional Wigner transform within classical SED was discussed in great detail by França and Marshall in 1988 [7]. In what follows we shall concentrate our attention in the particular case of very small In this case the right hand side of (1.9) is different from zero only if is small, and the following approximations will be used and

Using (1.9) in (1.8), plus the approximations (1.10), (1.11), we obtain the following Schrödinger type equation for

and the corresponding equation for with the vector potential A as given in (1.3) and (1.4) (see [8] for details). Therefore, the Schrödinger type equation depends on the Planck’s constant only due to its presence in defined in (1.4). In other words, equation (1.12) has terms which are proportional to and Moreover, the solutions of (1.12) must be interpreted by considering that the limit must be taken in the end of the calculations.

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3. Incompatibility Of The Standard Schrodinger Equation With The Zero-Point Field The above derivation shows a clear correspondence between the quantum Schrödinger equation for spinless particles, and the classical stochastic Schrödinger like equation given by (1.12). The case of neutral spinning particle has been already discussed by Dechoum, França and Malta [9]. The limit of the solution of the classical stochastic Schrödinger like equation corresponds, physically, to classical (non-Heisenberg) states of motion as shown by Dechoum and França [6]. Nevertheless, we shall observe several effects, arising from the vacuum fluctuations, which depend non-trivially on the Planck’s constant This is better understood by means of very simple examples. One interesting example, discussed in reference [9], is the derivation of the Pauli-Schrödinger equation in the spinorial form, starting from the Liouville equation. The experimental results of the Stern-Gerlach experiment, and also the Rabi type molecular beam experiments, were appropriately described and interpreted classically, in the limit that is, in the classical limit where the particles have well-defined trajectory, and also continuous orientation of the spin vector. The zero-point field has not been included. The best example, however, is the one-dimensional harmonic oscillator discussed in many details in previous works [6, 7]. In order to apply equation (1.12) to the charged harmonic oscillator, we shall assume that the scalar potential is the simple static function satisfying being the natural frequency of the oscillator. We have shown in ref. [6] that by introducing the function we obtain for (1.12) the equivalent equation

where equation

and

depend only on

(dipole approximation). In this

is the harmonic force,

is the ra-

diation reaction force, and is the random force. The exact solution of equation (1.13) is a Gaussian coherent state, Here is the classical stochastic oscillator trajectory [6] such that Using this exact

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This gives the correct value at zero temperature, namely in the limit This is an important result that is very easy to understand within the realm of SED. The inevitable conclusion is that the standard Schrödinger equation, namely equation (1.13) with does not give consistent results if the zero-point electromagnetic field is fully considered. 4. Discussion

Dalibard, Dupont-Roc and Cohen Tannoudji [10], and França, Franco and Malta [11] provided an identification of the contribution of the radiation reaction and the vacuum fluctuation forces to the processes of radiation emission and atomic stability. Using the Heisenberg picture and perturbative QED calculations it is possible to show that [10]

This equation is the quantum generalization of the Larmor formula for the rate of radiation emission, including the zero-point field effects, of an electron in the quantum state (Dirac notation) with energy We see that the inclusion of the zero-point electromagnetic field simply doubles the rate of the radiation emission, being thus very important for obtaining agreement with experiment [11, 12]. Our last remark is concerned with the necessity of the vacuum electromagnetic field within the Heisenberg picture of the quantum theory. A good discussion of this point is given in the book by P. Milonni [13]. The zero-point electromagnetic field, is necessary for the formal consistency of the quantum theory, as only its inclusion guarantees the equal-time commutation relation This result indicates that the operator already contains some effects of the zero-point electromagnetic field when the standard Schrödinger equation (equation (1.13) with is used. Acknowledgments This work was partially supported by FAPESP and CNPq.

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References

[1]

Marshall, T.W. (1963): “Random Electrodynamics", Proc. Royal Soc. (London) A, 276, 475-491.

[2]

Santos, E. (1968): "Is there an electromagnetic background radiation underlying the quantum phenomena ?’. Anales de la Real Sociedad Espanola de Fsica y Qumica, Tomo LXIV, 317-320.

[3]

de la Pena, L. and Cetto, A.M. (1996): “The Quantum Dice: An Introduction to Stochastic Electrodynamics" ( Kluwer Academic Publishers).

[4]

Manfredi, G., Mola, S. and Feix, M.R. (1993): “Quantum systems that follow classical dynamics", Eur. J. Phys.14, 101-107.

[5]

Wigner, E.P. (1932): “On the quantum correction for thermodynamic equilibrium", Phys. Rev., 40}, 749-759. See also O'Connell, R.F. and Wigner, E.P. (1981): “Quantum-mechanical distribution functions: conditions for uniqueness", Phys. Lett. A 83}, 145-158; Lee, H.-W. and Scully, M.O. (1983): “The Wigner phase-space description of collision processes", Found. Phys. 13, 61-72.

[6]

Dechoum, K. and Frana, H.M. (1995): “Non-Heisenberg states of the harmonic oscillator", Found. Phys., 25, 1599-1620. (1996): “Non-Heisenberg states of the harmonic oscillator: Erratum", Found. Phys., 26, 1573-1573.

[7]

Franca, H.M. and Marshall, T.W. (1988): “Excited states in stochastic electrodynamics", Phys. Rev. A, 38, 3258--3263. This paper shows that the set of Wigner functions associated with the excited states of the harmonic oscillator constitutes a complete set of functions. An arbitrary positive probability distribution can be expanded in terms of these Wigner functions.

[8]

Dechoum, K., Franca, H.M. and Malta, C.P. (in press): “The role of the zeropoint fluctuations in the classical Schrodinger like equation", Il Nuovo Cimento B.

[9]

Dechoum,K., Fran\ca, H.M. and Malta, C.P. (1998): “Classical aspects of the Pauli-Schrodinger equation", Phys. Lett. A, 24, 93-102. This paper discusses various classical aspects of the Pauli-Schrodinger equation, for a neutral spinning particle, including the classical interpretation of the Stern-Gerlach experiments.

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[10]

Dalibard, J., Dupont-Roc, J. and Cohen-Tannoudji, C. (1982): “Vacuum fluctuations and radiation reaction: identification of their respective contributions", J. Physique, 43, 1617-1638. According to Dalibard et al. “If self reaction was alone, the ground state would colapse and the atomic commutation relation would not hold".

[11]

Franca, H.M., Franco, H. and Malta, C.P. (1997): “A stochastic electrodynamics interpretation of spontaneous transitions in the hydrogen atom", Eur. J. Phys., 18, 343–349.

[12]

Franca, H.M., Marshall, T.W. and Santos, E. (1992): “Spontaneous emission in confined space according to stochastic electrodynamics", Phys. Rev. A, 45, 6436–6442.

[13]

P.W. Milonni (1994): “The quantum vacuum: An introduction to quantum electrodynamics, Academic Press Inc., Boston. According to Milonni (see chapter 2, section 2.6). The origin of the Planck’s constant in this expression can be traced back to the zero-point radiation with spectral distribution (see also our equations (1.4) and (1.5)

THE PHILOSOPHY OF THE TRAJECTORY REPRESENTATION OF QUANTUM MECHANICS

EDWARD R. FLOYD 10 Jamaica Village Rd. Coronado, CA 92118-3208 USA

Abstract. The philosophy of the trajectory representation is contrasted with the Copenhagen and Bohmian philosophies. 1. Introduction

The seminal work on the trajectory representation was published in 1982 [1]. The equations of motion for the trajectories are developed from a quantum HamiltonJacobi formulation. These trajectories are deterministic and continuous. Ergo, there is no need for any collapse of the wave function during observation. Recently, Faraggi and Matone [2] have independently generated the same quantum Hamilton-Jacobi formulation from an equivalence principle free from any other axioms. Faraggi and Matone have shown that although all quantum systems can be connected by an equivalence coordinate transformation (trivializing map), all systems in classical mechanics are not so connected. The development of the equivalence principle is beyond the scope of this exposition. We present the philosophical aspects of the trajectory representation of quantum mechanics that distinguish this representation. We exhibit its philosophy, which we contrast to the Copenhagen philosophy and the Bohmian philosophy. Our findings are presented in closed form mostly, but not exclusively, in one dimension for the time-independent case whenever one dimension suffices. The work in one dimension for the time-independent case is sufficient to refute many of the tenets of the Copenhagen and Bohmian theories. The trajectory representation is not just another interpretation of quantum mechanics because it also predicts results that differ with Copenhagen. Tests have been proposed consistent with Copenhagen epistemology [3,4]. In Section 2, we briefly present the fundamentals of the trajectory representation sufficient to support a philosophical comparison with Copenhagen and Bohm. In Section 3, we present different predictions rendered by trajectories and Copenhagen. We continue in Section 4 to contrast the trajectory representation to the Copenhagen interpretation. In Section 5, we compare the trajectory representation with the Bohmian stochastic representation. A gentler, extended version of this opus with more details is available at the LANL electronic archives [5]. 401 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 401-408 © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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2. The Trajectory Representation

The trajectory representation is based upon a phenomenological, nonlocal quantum stationary Hamilton-Jacobi equation (QSHJE) in one dimension It is a third-order non-linear differential equation given by [6,7]

where W is the reduced action (also known as the generalized Hamilton’s characteristic function), W' is the momentum conjugate to V is the potential, E is energy, is the mass of the particle, and where in turn is Planck’s constant. Faraggi and Matone have independently derived the QSHJE from an equivalence principle [2]. We note that W and W' are real even in classically forbidden regions. The general solution for W' is given by [8]

where is a set of real coefficients such that and is a set of normalized independent solutions of the associated stationary Schrödinger equation The independent solutions are normalized so that their Wronskian, is scaled to give This ensures that and that W' is real in the classically forbidden regions (V > E). The motion in phase space is specified by Eq. (2) and is a function of the set of coefficients In general, the conjugate momentum expressed by Eq. (2) is not the mechanical momentum, i.e., Actually, The solution for the generalized reduced action, W, is given by

where K is an integration constant that we may set to zero herein. The reduced action is a generator of motion. The equation of motion in the domain is rendered by Jacobi’s theorem (often called the Hamilton-Jacobi transformation equation for constant coordinates). The procedure simplifies for coordinates whose conjugate momenta are separation constants. Carroll has shown that for stationarity Jacobi’s theorem applies for W' is a Legendre transform [9]. For stationarity, E is a separation constant for time. Thus, the equation of motion for time, relative to its constant coordinate is given as a function of by

where the trajectory is a function of a set of coefficients epoch.

and

specifies the

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The set can only be a set of independent solutions of the SSE. Direct substitution of Eq. (2) for W' into Eq. (1) gives

For the general solution for W', the real coefficients are arbitrary within the limitations that and from the Wronskian that Hence, for generality the expressions within each of the three square brackets on the left side of Eq. (5) must vanish identically. The expressions within the first two of these square brackets manifest the SSE, so the expressions within these two square brackets are identically zero if and only if and are solutions of the SSE. The expression within third bracket vanishes identically if and only if the normalization of the Wronskian is such that For and must be independent solutions of the SSE. Hence, and must form a set of independent solutions of the SSE. Equation (5) is independent of any particular choice of ansatz. When comparing trajectories to Copenhagen and Bohm, we have broad selection for choosing a convenient ansatz to generate the equivalent wave picture (nothing herein implies that the trajectories need waves for completeness; only convenience). By Eq. (2), W' is real in the classically forbidden zone. Inside barriers, W' still manifests a trajectory. For bound states of finite the trajectories go to turning points at where regardless of the because at least one of becomes unbound as This is a nodal point singularity. For a given energy eigenvalue, E, of the SSE, there exist infinitely many microstates (trajectories or orbits with turning points at specified by These microstates are not distinguishable from the SSE for bound states [2,7].

3. Different Predictions between Trajectories and Copenhagen

First, we examine impulsive perturbations. Trajectories and Copenhagen render different predictions for the first-order change in energy, due to a small, spatially symmetric perturbing impulse, acting on the ground state of a infinitely deep, symmetric square well [3]. In the trajectory representation, is dependent upon the particular microstate This has been investigated under a Copenhagen epistemology even for the trajectory theory, where complete knowledge of the initial conditions for the trajectory as well as knowledge of the particular microstate are not necessary to show differences for an ensemble suffi-

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ciently large so that all microstates are individually well represented. In the trajectory representation, the first-order change in energy, is due to the location of the particle in its trajectory when the impulse occurs. The trajectory representation finds that the perturbing impulse, to first order, is as likely to do work on the particle as the particle is to do work perturbing system. Hence, the trajectory representation evaluates On the other hand, Copenhagen predicts to be finite as Copenhagen evaluates by the trace ground-state matrix element at the instant of impulse. Due to spatial symmetry of the ground state and In an actual test, we do not need perturbing impulses, which were used for mathematical tractability. A rapid perturbation whose duration is much shorter than the period of the unperturbed system would suffice [3]. We also consider a redundant set of constants of the motion. For a square well duct, we have proposed a test where consistent overdetermination of the trajectory by a redundant set of observed constants of the motion would be beyond Copenhagen [4]. The overdetermined set of constants of the motion should have a redundancy that is consistent with the particular trajectory. On the other hand, Copenhagen would predict a complete lack of consistency among these observed constants of the motion as Copenhagen denies the existence of trajectories. Such a test could be designed to be consistent with Copenhagen epistemology [4].

4. Other Differences between Trajectories and Copenhagen

As the trajectory exists by precept in the trajectory representation, there is no need for Copenhagen’s collapse of the wave function. The trajectory representation can describe an individual particle. On the other hand, Copenhagen describes an ensemble of particles while only rendering probabilities for individual particles. The trajectory representation renders microstates of the Schrödinger wave function for the bound state problem. Each microstate is sufficient by itself to determine the Schrödinger wave function [7]. Thus, the existence of microstates is a counter example refuting the Copenhagen assertion that be an exhaustive description of nonrelativistic quantum phenomenon. The trajectory representation is deterministic. We can now identify a trajectory and construct, for convenience, its corresponding witb sub-barrier energy that tunnels through the barrier with certainty [10]. Tunneling with certainty is a counter example refuting Bern’s postulate of the Copenhagen interpretation that attributes a probability amplitude to As the trajectory representation is deterministic and does not assign a probability amplitude to it does not need a wave packet to describe or localize a particle. The equation of motion, Eq. (4), for a particle (monochromatic wave) has been shown to be consistent with the group velocity of the wave packet [11]. Though probability is not needed for tunneling through a barrier [10], the trajectory representation for tunneling is still consistent with the Schrödinger representation

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without the Copenhagen interpretation [10]. Trajectories differ with Feynman’s path integrals in three ways. First, trajectories employ a quantum reduced action while a path integral is based upon a classical reduced action. Second, the quantum reduced action is determined uniquely by the initial values of the QSHJE while path integrals are democratic summing over all possible classical paths to determine Feynman’s amplitude. While path integrals need an infinite number of constants of the motion even for a single particle in one dimension, motion in the trajectory representation for a finite number of particles in finite dimensions is always determined by only a finite number of constants of the motion. Third, trajectories are well defined in classically forbidden regions where path integrals are not defined by precept. By the QSHJE, knowledge of a set of initial conditions [W', W", W'"] at some point is necessary and sufficient to specify E and the quantum motion while for the classical stationary Hamilton-Jacobi equation the set of initial conditions is reduced to [W'] at to specify E and classical motion [6]. Thus, the Heisenberg uncertainty principle assumes a subset of initial conditions that is insufficient to specify E and quantum motion [the SSE operates in the QSHJE in by a canonical transform]. The Heisenberg uncertainty principle is premature since Copenhagen uses an insufficient subset of initial conditions to try to describe quantum phenomena [12]. Bohr’s complementarity postulates that the wave-particle duality be resolved consistent with the measuring instrument’s specific properties. But Faraggi and Matone [2] have derived the QSHJE from an equivalence principle without evoking any axiomatic interpretation of the wave function. Furthermore, Floyd [7] and Faraggi and Matone [2] have shown that the QSHJE renders additional information beyond what can be gleaned from the Schrödinger wave function alone.

5. Trajectories vis-a-vis Bohmian mechanics

The trajectory representation differs with Bohmian representation [13,14] in many ways despite both representations being based on equivalent QSHJEs. We describe the various differences between the two representations in this section. These differences may not necessarily be independent of each other. The two representations have different equations of motion. Jacobi’s theorem, Eq. (4), gives the equations of motion for the trajectory representation. Meanwhile, Bohmian mechanics eschews solving the QSHJE for a generator of the motion, but instead assumes that the conjugate momentum be the mechanical momentum, which could be integrated to render the trajectory. But the conjugate momentum, as already shown herein, is not the mechanical momentum [1,2,9,11]. Bohmian mechanics considers to form a field that fundamentally effects the quantum particle. The trajectory representation considers the SSE to be only a phenomenological equation where does not represent a field. To date, no one has ever measured such a Bohm postulates a quantum potential, Q, in addition to the standard potential,

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that renders a quantum force proportional to [The negative of Bohm’s in one dimension appears on the right side of the QSHJE, Eq. (1).] But by the QSHJE, is dependent upon E and the microstate of a given eigenvalue energy E because

Therefore, is path dependent and cannot be a conservative potential. Consequently, does not generally render a force. While Bohmian mechanics postulates pilot waves to guide the particle, the trajectory representation does not need any such waves. Bohmian mechanics uses an ansatz that contains an exponential with imaginary arguments. The Bohmian ansatz in one dimension is As Eq. (5) is valid for any set other ansätze including trigometric forms are acceptable [1,2,5]. Bohmian mechanics asserts that particles can never reach a point where vanishes. On the other hand, trajectories have been shown to pass through nulls of with finite conjugate momentum, W' [1,2]. Bohmian mechanics asserts that bound-state particles have zero velocity because the spatial part of the bound-state wave function can be expressed by a real function. On the other hand, the QSHJE, Eq. (1) is still applicable for bound states in the trajectory representation. For bound states, the trajectories form orbits whose action variables have the Milne quantization independent of the microstate Bohmian mechanics asserts that a particle will follow a path normal to the surfaces of constant W. Meanwhile, trajectories in higher dimensions are not generally normal to the surfaces of constant W [4,11]. In higher dimensions, trajectories are determined by Jacobi’s theorem, Eq. (4) rather than by Bohmian mechanics asserts that the possible Bohmian trajectories for a particular particle do not cross. Rather, Bohmian trajectories are channeled and follow hydrodynamic-like flow lines. On the other hand, the trajectory representation describes trajectories that not only can cross but can also form caustics as shown elsewhere in an analogous, but applicable acoustic environment [16]. The two representations differ epistemologically whether probability is needed. The trajectory representation is deterministic. Bohmian mechanics purports to be stochastic and consistent with Born’s probability amplitude [14]. Let us consider three dimensions in this paragraph to examine the familiar stationary auxiliary equation

to the three-dimensional QSHJE. Bohm and Hiley [14] identify R as a probability amplitude and Eq. (6) as the continuity equation conserving probability. Hence, must be divergenceless. The trajectory representation can now show a non-probabilistic interpretation of Let us consider a case for which the stationary Bohm’s ansatz, is applicable. Bohm [13] used

TRAJECTORY REPRESENTATION OF QM

407

and

where and Hence, by the superposition principle, and are solutions to the SSE. Upon substituting and into Eq. (6), we get the intermediate step which is a “three-dimensional Wronskian”. Whether or not this “three-dimensional Wronskian” renders a constant, it is divergenceless [5]. Therefore, the trajectory representation finds that the auxiliary equation contains a “three-dimensional Wronskian” that satisfies Eq. (6) without any need for evoking a probability amplitude. Bohm had expressed concerns regarding the initial distributions of particles. Bohm [13] had alleged that in the duration that nonequilibrium probability densities exist in his stochastic representation, the usual formulation of quantum mechanics would have insoluble difficulties. The trajectory representation has shown that the set of initial conditions may be arbitrary and still be consistent with the Schrödinger representation [6]. Stochastic Bohmian mechanics, like the Copenhagen interpretation, uses a wave packet to describe the motion of the of the associated As previously described herein, the deterministic trajectory does not need wave packets to describe or localize particles. Holland [17] reports that the Bohm’s equation for particle motion could be deduced from the SSE but the process could not be reversed. On the other hand, the development of Eq. (5) is reversible. In application, the two representations differ regarding tunneling. Dewdney and Hiley [18] have used Bohmian mechanics to investigate tunneling through a rectangular barrier by Gaussian pulses. While Dewdney and Hiley assert consistency with the Schrödinger representation, they do not present any results in closed form. Rather, they present graphically an ensemble of numerically computed trajectories for eye-ball integration. On the other hand, the trajectory representation exhibits in closed form consistency with the Schrödinger representation. Also, every Bohmian trajectory that successfully tunnels slows down while tunneling even though Steinberg et al [19] have shown that the peak of the associated wave packet speeds up while tunneling. Our trajectories that successfully tunnel speed up [4,10] consistent with the findings of others [20–23].

Acknowledgement

I am pleased to thank M. Matone for many discussions. I also thank D. M. Appleby, G. Bertoldi, R. Carroll, and A. E. Faraggi.

1. Floyd, E. R.: “Modified Potential and Bohm’s Quantum Potential”, Phys. Rev. D 26 (1982), 1339–1347. 2. Faraggi, A. E. and Matone, M.: “The equivalence postulate of quantum mechanics”, Int. J. Mod. Phys. A 15 (2000), 1869–2017, hep-th/9809127.

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3. Floyd, E. R.: “Which causality? Differences between tajectory and Copenhagen analyses of an impulsive perturbation”, Int. J. Mod. Phys. A 14 (1999), 1111–1124, quant-ph/9708026. 4. Floyd, E. R.: “Reflection time and the Goos-Hänchen effect for reflection by a semi-infinite rectangular barrier”, Found. Phys. Lett. 13 (2000), 235–251, quantph/9708007. 5. Floyd, E. R.: (2000) “Extended Version of ‘The Philosophy of the Trajectory Representation of Quantum Mechanics’ ”, quant-ph/0009070. 6. Floyd, E. R.: “Arbitrary initial conditions of hidden variables”, Phys. Rev. D 29 (1984), 1842–1844. 7. Floyd, E. R.: “Where and why the generalized Hamilton-Jacobi representation describes microstates of the Schrödinger wave function”, Found. Phys. Lett. 9 (1996), 489–497, quant-ph/9707051. 8. Floyd, E. R.: “Closed-form solutions for the modified potential”, Phys. Rev. D 34 (1986), 3246–3249. 9. Carroll, R.: “Some remarks on time, uncertainty and spin”, J. Can. Phys. 77 (1999), 319–325, quant-ph/9903081. 10. Floyd, E. R.: “A trajectory interpretation of tunneling”, An. Fond. Louis de Broglie 20 (1995), 263–279. 11. Floyd, E. R.: “A trajectory interpretation of transmission and reflection”, Phys. Essays 7, (1994) 135–145. 12. Floyd, E. R.: “Classical limit of the trajectory representation of quantum mechanics, loss of information and residual indeterminacy”, Int. J. Mod. Phys. A 15 (2000), 1363–1378, quant-ph/9907092. 13. Bohm, D.: “A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. I”, Phys. Rev. 85 (1952), 166–179. 14. Bohm D. and Hiley, B. J.: “An ontological basis for the quantum theory”, Phys. Rep. 144 (1987), 323–348. 15. Milne, W. E.: “The numerical determination of characteristic numbers”, Phys. Rev. 35 (1930), 863–867. 16. Floyd, E. R.: “The existence of caustics and cusps in a rigorous ray tracing representation”, J. Acous. Soc. Am. 80 (1986), 1741–1747. 17. Holland, P. R.: The Quantum Theory of Motion, Cambridge U. Press, Cambridge, UK, 1993, p. 79. 18. Dewdney, C. and Hiley, B. J.: “A quantum potential description of the onedimensional time-dependent scattering from square barriers and square wells”, Found. Phys. (1982), 12, 27–48. 19. Steinberg, A. M., Kwiat, P. G. and Chiao, R. Y.: “Hidden and unhidden information in quantum tunneling”, Found. Phys. Lett. 7 (1994), 223–237. 20. Olkhovsky, V. S. and Racami, E.: “Recent developments in the time analysis of tunnelling processes”, Phys. Rep. 214 (1992), 339–356. 21. Barton, G.: “Quantum mechanics of the inverted oscillator potential”, An. Phys. (New York) 166, (1986), 322–363. 22. Hartmann, T. E.: “Tunneling of a wave packet”, J. Appl. Phys. 33 (1962), 3427– 3433. 23. Fletcher, J. R.: “Time delay in tunnelling through a potential barrier”, J. Phys. C 18 (1985), L55–L59.

SOME PHYSICAL AND PHILOSOPHICAL PROBLEMS OF CAUSALITY IN THE INTERPRETATION OF QUANTUM MECHANICS

B. LANGE Departament for Logic, Methodology and Philosophy of Science, University of ul. 5, 80-951, Poland

1. Introduction The International Conference of Physicists in Warsaw took place from 30th May to 3rd June 1938. The Conference was organised by the International Institute of Intellectual Cooperation, and the Polish Commission as its associate body. The talks were chiefly devoted to philosophical issues that emerged along the development of quantum theory and theory of relativity, and other basic issues related to interpretation of new discoveries and theories. The need for organising such meetings resulted from the necessity of discussing and confronting the ideas of most prominent scholars in relation to the most basic problems emerging in the period of rapid development of contemporary physics. It was the first of the whole pre-planned series of conferences, and it should be pointed out that the very first conference took place in Poland, which is a proof of a high world rank of Polish physics. The conference was initiated and then chaired by an eminent Polish physicist, Professor Czeslaw Bialobrzeski. The conference was also attended by other eminent scholars, like Niels Bohr, Leon Brillouin, Charles Darwin, Arthur Eddington, George Gamow, Hendrik Anthony Kramers, Paul Langevin, John von Neumann, Eugen Wigner, and others. Poland was represented by Professors Wojciech Rubinowicz, Szczepan Szczeniowski, Jan Weyssenhoff, Ludwik Wertenstein, and F. J. Wisniewski. Complicated political situation of that time, however, was the reason why some invited scholars failed to arrive. During the conference eight papers were delivered, and they were afterwards published together with the abstracted discussion in the volume titled Les Nouvelles Theories de la Physique [1]. The outbreak of the War was the cause why the information and the conference materials did not reach most of the world centres of physics. Therefore the conclusions drawn from the papers and discussion did not become a source of further debates. Thus the Conference did not have some major impact on further development of the foundations and interpretation of quantum physics. It seems, however, that there is a need to present chief ideas discussed during the conference, particularly since some of them – like the proposition of Bialobrzeski, for instance - were attempts to overcome basic problems in interpreting formalism of quantum theory. Those issues have not been successfully solved so far, and they still are a point of interest to physicists and philosophers. The following paper aims at presenting the discussion during the Conference, which was joined by Bohr, von Neumann, Bialobrzeski, and Kramers. The discussion developed over the paper of Bohr, whose intervention seemed to have stirred the highest interest among the participants. The proposal of interpretation of the reduction process of the wave function presented during the discussion by C. Bialobrzeski [1], seems to be of particular interest here. 409 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 409-412. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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2. A Dispute Between N. Bohr, J. Von Neumann, Cz. Bialobrzeski, And H. A. Kramers On Causality In Atomic Physics While analysing a problem of the use of the laws of mechanics and thermodynamics in explaining atom's stability Bohr stated that we should make additional assumptions, i.e. that any well defined change in the status of electron within the atom depended of the shift of the atom from one stationary position into another, similar one. According to Bohr, quantum postulates are not only completely alien to the notions of classical mechanics, but they also assume divergence from any casual description of such processes. This particularly refers to possible shift of the atom from a stationary position into another one. Bohr claimed that "the atom faces a choice with no determining circumstances. Situation like that, any forecasts may refer only to the probability of various possible directions of atomic processes, which are susceptible to direct observation" [1]. Further on Bohr claimed that the "utilisation of purely statistical consideration is the only instrument which allows for generalisation of usual description, necessary for explaining individual character expressed by quantum postulates..., (and) this is how we limit classical theory to an extreme case, where the changes of action are large as compared to the quantum of action. The only reason for formulating such a generalisation was the need for utilisation of classical notions in the widest possible scope which could be reconciled with quantum postulates" [1]. Basic assumptions of the Bohr's intervention can be presented as follows: measurement cannot mean anything else but unbiased comparison of a given property of the object under research, with a corresponding property of another system used as a measuring instrument. Properties of the measuring instrument directly result from the definition taken from classical physics. The above mentioned comparison of the properties must be convergent with the definition expressed in a common language. Whereas in the field of classical physics such a comparison may be performed without substantial intervention into the object itself, in the quantum theory, however, the interaction between the object and the measuring instrument will basically influence the phenomenon itself. We have to be particularly aware of the fact that the interaction cannot be exactly separated from the behaviour of the undisturbed object. Thus the necessity of basing our description on classical notions implies disregarding all the quantum effects in the very description, particularly disregarding more precise control over the object's reaction than it might result from the uncertainty principle. Adopting the complementarity point of view we may avoid unnecessary discussion on determinism. There is no need to discuss indeterminism when we state a general scheme of causative idea which may aim at the synthesis of the phenomena which can be described within the function of the behaviour of the objects irrespective of the mode of their observation. Bialobrzeski, in turn, thought that there were doubts concerning logical coherence of quantum theory. The interpretation of Bohr and Heisenberg presented two, complementary sides of elementary creations of the material world: the corpuscular and the wave sides. It is rightly said that the formalism of quantum mechanics gives synthesis which unifies the two images of the single reality. We must refer to these images to describe natural phenomena within space-time continuum. Bialobrzeski thinks, however, that it is possible to find dualism of a different nature within the notions of quantum mechanics. This dualism questions uniformity of a doctrine, unless it can be explained within the chief postulates of the doctrine. The dualism means - as Bialobrzeski stated - that on the one hand we have a casual sequence of phenomena governed by the Schrödinger equation, on the other hand, however, when we measure a certain volume of "A", the status represented by function is subject to rapid non-casual change. The sudden change in function results from the fact, that we can obtain a result of a measurement in the form of any eigenstate which would correspond to measured value. "We assume," the scholar says, "that we are dealing with a discontinuous system of these states. States are

PROBLEMS OF CAUSALITY IN QUANTUM MECHANICS

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independent from the initial state, decisive only for probability of their realisation expressed by the formula If we consider a large number of the systems in the same state of then the measurement of the volume "A" performed on the set of all these identical systems turns the set into a mixture, which is a proof of non-casual character of the process. The shift from the state into one of the states must be either instant or brief, which on the one hand leads to indeterminacy of energy related to time by one of the Heisenberg's uncertainty relations, on the other hand it clearly separates time from the co-ordinates defining position, which is in contradiction to the postulate of relativity" [1]. Bialobrzeski asked a basic question: "What is the mechanism of realisation of the state from the state naturally if there exists describable mechanism of the change?" [1]. Bialobrzeski’s proposal looks as follows: after accepting the thesis, that von Neumann's analysis defined impossibility (within quantum mechanics) of eliminating indeterminism by introducing hidden parameters for preserving the description of the measuring process with the use of Schrödinger equation, we must assume that coupling of the measuring instrument with the object under research is the cause of distortion. The distortion is discontinuous and changeable in such a way, that a probability of realisation of state is strictly defined by the formula Discontinuity and indeterminacy, which were eliminated from the measurement, reappeared in the distorted operator of energy, which ought to change in a discontinuous and indeterminable way. As a result, we obtain nothing which could explain the mechanism under discussion. Bialobrzeski concludes his comments with the assumption that the action of taking measurement performed upon the system in which its natural alternation of state is governed by Schrödinger equation causes another type of alteration of indeterminate character, as it is solely governed by statistical laws. Anyway, it seems to be necessary - as Bialobrzeski believes - to assume the existence of a basic postulate which refers to the very act of taking measurement: "quantized physical quantity is basically indeterminate, and we can only recognise the probability of realisation of each of eigenvalues of the quantity. It is just determined by the formula J. von Neumann noticed that we must have an observer placed somewhere within a given system, therefore it is necessary to draw the borderline between the observer and the observed. It is not necessary, however, that the border overlapped with physical body limits of the observing person. He believes that it is possible to "shrink" or "blow up" the observer. We may include, for instance, everything that is happening in the observer's eye and include it all into the part under observation, which is described in a quantum manner. The observer would begin then behind the retina of the eye. We could also include some of the equipment used for physical observation, e.g. a microscope, into the observer. The principle of psycho physical parallelism defines a situation, where the abovementioned border can be shifted as deeply into the body of the observer as we wish. As far as the issue discussed by Bialobrzeski was concerned, Bohr thought that the dualism mentioned by him was simply a problem of selecting most adequate description of an experiment. As for von Neumann's comment on differentiation between the phenomenon and the observer, Bohr thought that the differentiation was a natural process with the phenomena we are dealing with within quantum theory, however we try to explain a given phenomenon in a classical way. Kramers in turn briefly referred to Bialobrzeski’s comments, and expressed his own opinion on them. He thought that if we adopted calculation scheme utilised, as a matter of fact, in many real physical problems, it would be dubious if we could really talk about independent postulate referring to measurements. Kramers believed that "demanding such a postulate is a consequence of too a classical approach to the problem. We cannot speak about Schrödinger equation and about the ways we build up Hamiltonian unless we think about experiments at the same time. In other words, we cannot reject results of measurements which allowed us to make use of wave functions, and we cannot neglect the way in which the value of mechanical quantity can be linked with the wave image. To measure is nothing but confirming details in the instrument's state. These details carry the sense

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directly derived from the notions of classical physics. According to the notions, the condition of a given system may be either like this or like that. There is neither indefiniteness nor distribution of probability among various possible states. Natural laws, however, say that the result frequently cannot be predicted even if the starting stage of an experiment is defined in the most precise way." [1] Kramers saw in this the role for the quantum of action in stabilising Nature. On the other hand, the role of the quantum of action is of the type that does not allow us to talk about some 'mechanism' regulating the selection performed by an instrument when we want to measure something. To put forward a question like this means - according to Kramers - a shift into the scheme of classical determinism. This determinism, however, had been rejected the moment we accepted wave function. In the further part of discussion Bialobrzeski stood by his opinion by saying, that he had not properly understood Kramers's explanation. He noticed that the way of thinking in the field of quantum theory was the very way we are used to in natural sciences. The situation could always be presented as an interaction between an object and an instrument. The system could be studied - the scholar thought - with the use of additional expression in the hamiltonian formula. If the expression exists, the condition changes. The emergence of a spectral line on a photographic plate may be an example here. We have phenomenon which can be described with the expression defining disturbance. Then the whole process of the phenomenon can be defined. Thus we can use this particular postulate, because - as von Neumann said - a division line within a measuring system can be drawn anywhere. In his reply Kramers stated that we could adopt this procedure for more precise analysis of a measuring instrument, we could introduce to hamiltonian words that would define interaction of the instrument with variables characteristic for the state of the instrument. We could prove by this that we understood the function of measuring instruments well. According to Kramers, however, this did not constitute the 'mechanism' of making choice. In his summary, Bohr said that there were a few ways of expressing relations between classical and quantum physics. In the theory of classical electromagnetism, for instance, theoretical description is inseparable from certain experimental methods used to define certain basic notions. In quantum mechanics the situation is different in the sense that interpretation of various measurements which could be performed, whatever their precision be, do not allow for classical definition of the system. According to Bohr it would be very difficult to describe such a situation by saying that it would take additional mathematical postulate to link a result of a measurement with the other principles of the theory.

3. Conclusions Bialobrzeski's idea that interpretation difficulties forced us to recognise a choice during measurement of eigenvalue of the measured quantity, as an independent postulate which was mathematically expressed by the formula defining probability of the choice of any of eigenvalues, did not receive recognition. Bialobrzeski did not give up, however, the development of his idea, and made it more comprehensible in his book entitled Cognitive Foundations of the Physics of the Atomic World, [2].

References 1. 2.

Les Nouvelles Theories de la Physique, (1939) International Institute of Intellectual Cooperation, Paris. Bialobrzeski C., (1956) Podstawy poznawcze fizyki swiata atomowego, (Cognitive Foundations of the Physics of the Atomic World,), PWN Warsaw.

THE FORCE THE POWER QUANTUM MECHANICS

AND THE BASIC EQUATIONS OF

LUDWIK KOSTRO Department for Logic, Methodology and Philosophy of Science, University of Gda1sk, ul. Biela1ska 5, 80-951 Gda1sk, Poland E-mail: [email protected]

1. Introduction

In two recent papers [1-2] the quantity was interpretated as the greatest possible force in Nature. In a third paper [3] following I.R. Kenyon [4] the quantity was interpretated as the greatest possible power. In the three mentioned papers I have limited myself to classical considerations. I have shown e.g. that the classical Newton law and the classical Coulomb law can be rewritten in the following way: Newton force when

when

Coulomb force

It was also indicated that the quantities and and their inverses appear in the equations of General Relativity [1,2,3] and Kenyon’s interpretation [4] of this fact was presented. In my considerations I use the following constants and constant coefficients: c velocity of light in vacuum; G - the gravtational constant; ...- Planck’s constant; e - the elementary electrical charge; m - the mass of an elementary particle; I take into account also the units of lenght, time and mass determined by the following set of constants (c,G,m) (c,G,e) (c,G, ..) Where m, e, are the respective charges of four fundamental interactions. Using the dimensional analyse we obtain the fallowing units: 413 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 413–418. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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L. KOSTRO

(a) gravitational lenght

time

and mass

(b) J.G. Stoney’s lenght

, time

, and mass

(c) M. Planck’s lenght

time

and mass

(d) Lenght

time

(e) Lenght

time

and mass

and mass

introduced by him in 1874 [5–6]

introduced by him in 1899 [7]

connected withe the strong interactions

connected withe the weak interactions

It is interesting to note that forces F and powers P connected with these units all are equal:

These forces and powers appear especially when the density of the matter is the greatest possible one. The following formulae present the limiting density of the gravitationally, electrically, strongly and weakly charged matter and the Planck’s density.

As we can see the respective limiting density depends upon the inverse of the square of the respective charge multiplied by the factor In other words, it constitutes the product of the square of the respective charge and the factor

BASIC EQUATIONS OF QUANTUM MECHANICS

2. The Quantities Equivalence

And

415

And Einstein’s Principle Of Mass And Energy

It is interesting to note that the Einstein’s Principle of mass and energy equivalence can be rewritten in the following way:

This fact shows, may be once again, the dynamical nature of the matter. If an elementary particle could deliver its total energy acting on the path equal to and during the time equal to then it could show its greatest force and power If it could happen then the extremal force and the extremal power would be hidden in every particle. Perhaps in the future mankind will find out the circumstances in which it will be possible. At the present day, however, we can ose only to interpretate the two quanties as exremal ones.

3. The Quantities

And

And Schrödinger Equation

As is well known Schrödinger equation is the basic equation of the non relativistic Quantum Mechanics. In textbooks it is written in the following way

where V can be the Coulomb potential In these equations we find the constants .., m, e and the coefficient K. Since the constants used in physics and the units determined by them are correlated and interconnected therefore it is not difficult to rewrite the Schrödinger equation in such a way that the quantities and and the considered units appear in it. The Coulomb potential can be rewritten as follows

and the Schrödinger equation in the following way

Since

(where

is the fine structure constant) we obtain also

As we can see, in the Schrödinger equation written in such a way, threre appear not only the quantities and but also the Planck’s lenght and time and Stoney’s

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L. KOSTRO

lenght and time.We see also that Planck’s constants ...is related to the quantities as follows:

When we divide the both sides of the eq. (8) by

and

we obtain

We must be aware, however, that such a division changes the numerical value and the dimensions of the both sides of the equation. The threedimensional Hamilton operator

can be rewritten introducing

and

as follows

When

4. The Quantities

then we can write

And

And The Klein-Gordon Equation

Let’s consider the Klein-Gorden equatiom written e.g. for the mesons

Taking into consideration the quantities be rewritten:

and

the Klein-Gordon equation can

As we can see the Planck’s charge raised to the second power as follows

Since

.

is related to

the eq. (15) can be also written as follows

BASIC EQUATIONS OF QUANTUM MECHANICS

417

Since (where is the coupling constant of gravitational interactions between two particles of the same mass i our case the coupling constant of grawitational interactions between two mesons the eq. (16) can be rewritten as follows

Dividing the both sides of eq. (17) by

we obtain

We must be aware, however, that such a division changes the numerical value and the dimensions of the both sides of the equation.

5. The Quantities

And

And The Dirac Equation

The Dirac equation can be written as follows:

where the matrix

have the following properties

(where † means the hermitonian coupling) Taking into consideration the quantities rewritten:

where

and

is the Planck’s time and

the Klein-Gordon equation can be

is the gravitational time

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L. KOSTRO

6. Conclusion It was very easy to introduce the quantities and into the basic equations of Quantum Mechanics (and we can even say that such an introduction constitutes a very trivial operation) but it is very dufficult to interpretate their part played in these equations. When we introduce the quantities and e.g. into the classical equations of Newton and Coulomb we see immediately their meaning as limiting quantities i.e. the greatest force and the greatest power but when we introduce them into the basic equations of Quantum Mechanics we do not see clearly their physical meaning. They work, however in these equations and therefore we can think that also here they play not only a role of constants but also as limiting quantities, as the greates force and the greatest power.

References 1.

Kostro L. and Lange, B. (1999) Is No 1, 182

2.

Kostro L. and Lange B., (1998) The force and Relativity Theory, in M.C. Duffy (ed), Physical Interpretations of Relativity Theory. (Proceedings) British Society for Philosophy of Science, Imperial College, London, pp. 183-193

3.

Kostro L. and Lange B., (1998) The Power and Relativity Theory, in M.C. Duffy (ed), Physical Interpretations of Relativity Theory. (Later Papers) British Society for Philosophy of Science, Imperial College, London, pp. 150–155 Kenyon I.R. (1990) General Relativity, Oxford University Press Barrow J.D. (1983) Q.Jl R. Astr. Soc. 24, 24 Stoney G.J. (1881). Phil. Mag., 5, 381 Planck, M. (1899) Sitzungsberichte d. Preus. Akad. Wiss. /Mitteilung/5, 440 and Planck, M. (1900) Ann. Phys. 1, 69

4. 5. 6. 7.

the greatest possible force in nature? Physics Essays, 12,

PROGRESS IN POST-QUANTUM PHYSICS AND UNIFIED FIELD THEORY

JACK SARFATTI Internet Science Education Project & International Space Sciences Organization [email protected]

Abstract

Progress in extending the de Broglie- Bohm-Vigier (AKA dBBV) quantum ontology to the experimental mind-matter problem is reported in Part I. Progress in extending Einstein’s classical orthodox holonomic topology-conserving general relativity of 1915 to the unified field theory including topology-changing anholonomic torsion fields from the “hyperspace” of M-theory is reported in Part II. I also make a conjecture that the empirical duality in the Wesson compared to the Sirag data plots, noted by Gray in 1988, is actually showing the M-theory T-duality The Wesson “Regge trajectory” (Kaluza-Klein excitation number m) mass scale The “dual” Blackett-Sirag magneto-gyro mass scale

where

and

What is clear is that we now have a new "telescope" directly into the quantum gravity scale showing strong anholonomic unified field effects beyond Einstein’s 1915 theory. This is as important as the Hubble flow, the cosmic microwave background, missing mass, gravity waves, and the anomalous acceleration of the universe. A completely new conception of COSMOS is now emerging from the actual data. Part I: Progress In Post-Quantum Physics

Orthodox quantum theory has many “degenerate”2 informal interpretations that appear to have no crucial experimental tests to “lift the degeneracy”. The experimental situation is now changing dramatically and quickly with my recognition of the real meaning of data lying around in journals unread for twenty years. All of the contemporary competing

1 2

11.6 eq. (11.6.2) p. 477 “Intro to Superstrings and M-Theory”, M. Kaku, Springer-Verlag, 1999. In the sense of atomic spectroscopy with “degenerate energy eigenfunctions” of the Hamiltonian operator.

419 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 419-430. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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interpretations3, save dBBV, only have the quantum wave and not the particle in their models of quantum reality. The “particle” or “Bohm point” moving on the landscape formed by the quantum potential and other forces is “eschewed”.4 This immediately causes confused thinking, e.g., in the recent articles claiming to divide the charge e of the electron into two equal pieces.5 This confusion comes from falsely assuming that the quantum wave “is the essence of an electron” carrying the charge because there is no particle at the micro level. The experimental result of increased current is easily understood intuitively in the dBBV interpretation in which the complete indivisible electron particle, of screened charge e, is completely localized in only one of the two bubbles that the physically real pilot wave divides into. One of the paired bubbles has a real but empty branch. The smaller bubbles move faster through the liquid helium explaining the observed increase of electric current without having to cut the electron in half. Therefore, this experiment seems to lift the degeneracy.6 Bohm’s ontology works better for this experiment. In another recent development, Henry Stapp7 has proposed a model of ontological collapse8 of the quantum wave with infinite speed in the preferred cosmological rest frame of the Hubble flow9 in the standard cosmological model. This is quantum theory on a classical curved space-time geometry not full blown quantum gravity. Stapp’s conjecture is similar to Bohm’s and Hiley’s in which the quantum potential Q acts instantaneously in this same preferred frame. There is no reason to suppose that classical Diff(4) local gauge symmetry10 of the 1915 general theory of relativity should be valid when quantum nonlocality is important. However, experiments by Gisin11 et-al in Geneva seem to rule out this idea, although Stapp and I are in serious disagreement12 on how to think about this problem. I am only giving my biased perspective here. Special relativity still works locally. According to the Einstein 3

E.g., Stapp’s “ontological collapse”, Penrose’s “R” and “OR”, “many worlds” in all of its variations such as David Deutsch’s “multiverse”, Gell-Mann/Hartle decohering histories, John Cramer’s “transactional” with weak backward causation consistent with “signal locality” AKA “passion at a distance” (Abner Shimony). 4 “Bohmian Mechanics and Quantum Theory: An Appraisal” Ed. J.T. Gushing, A. Fine, S. Goldstein (Kluwer, 1996) 5 New Scientist magazine, 14 October 2000 “This sounds harmless enough, but the implications are staggering. If the bubble split, half of the electron's wave function would be trapped in each of the two daughter bubbles .... As the wave function is the essence of an electron, the electron would be split into two. The indivisible would have been divided. ... ‘There were more bubbles, and being smaller they were more mobile,’ says Maris. Although the total charge in the system remained the same, the smaller bubbles felt less drag in the helium, and thus moved faster. Consequently, the current went up,” 6 My solution here is reminiscent of King Solomon’s when asked to divide the baby in half. The electron, like the baby, has been thrown out with the bathwater in all the alternative interpretations save dBBV. 7 Discussion by e-mail among Stapp, Stan Klein and myself. 8 “speed of quantum information” (Gisin et-al, ref 11) 9 In which the cosmic microwave black body background radiation is isotropic to about one part in 105. 10 Integrable holonomic general coordinate transformations that are global 1-1 conserving topology of the 4d spacetime manifold. 11 Quant-ph/0007008 4 July 2000 “The Speed of Quantum Information and the Preferred Frame: Analysis of Experimental Data” V. Scarani, W. Tittel, H. Zbinden, N. Gisin 12 There is no preferred frame in global special relativity, consequently whether or not nonlocal EPR correlations are observed cannot depend upon the common state of uniform motion of the two detectors relative to any other frame. However, global special relativity breaks down in general relativity where it can only be used locally. Therefore, this issue is an experimental one. Stapp inconsistently tries to maintain global special relativity and the preferred frame of the Hubble flow together in my opinion.

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addition of velocities, an infinite speed in one frame is a finite superluminal speed where is the subluminal speed of the moving “G-frame” relative to the allegedly preferred global rest frame of the Hubble flow. For example, the rotating Earth’s motion has a 24 hr periodic projection u, of its CM motion on to the flight line between the G-detectors, of amplitude Therefore, except for ~ 5 seconds every 43,200 seconds, the G-frame qubit speed is less than the minimum required dipping down to reaching a peak less than

for most of the data run. In contrast, Gisin cites “a

lower bound for the speed of quantum information in this “G-frame” at “ for 15 the parameters of the actual experiment in Geneva of EPR photon pair correlations over a distance 10.6 km. Therefore, G-frame detectors, each with speed u relative to the Hubble flow, separated by a distance 10.6 km requiring a qubit speed of at least to travel between them in the 5 picosecond time uncertainty16 in the detections, should not show the actually observed EPR correlations most of the time. In fact, the EPR correlations are seen all of the time. This argument assumes that the qubit speed in the Hubble frame is infinite, so that objective collapse with a real qubit speed is a useful way to picture how the nonlocal EPR correlations are maintained. Therefore, Stapp’s conjecture here is falsified by the actual experiment. Does this also shoot down the Bohm-Vigier conjecture that the quantum potential Q acts instantly in the preferred Hubble flow frame? Yes, the model that does survive is that of “backward causation” 17 originating in the Wheeler-Feynman 1940 program for classical electrodynamics eliminating independent dynamical degrees of freedom for the electromagnetic field with “Everything is particles.”18 However, it appears that can be 19 redefined in terms of backward causation. Note that the term “ret” for “retarded” means “from the past”, or what Aharonov calls the “history” state vector. Similarly, the term “adv” for “advanced” means “from the future”, or what Aharonov calls the “destiny” state vector. Feynman already used this idea in his original paper on the path integral in nonrelativistic quantum theory.20

13

Same as de Broglie’s relation of wave to particle speed at the beginning of the wave mechanics of matter. “Introduction to Cosmology”, J.V. Narlikar. p.301 (Cambridge, 1993) 15 Time resolution of 5 picoseconds in footnote [19] of ref 11. 16 Photon pulse temporal width 17 “Feynman zig zag” (O. Costa de Beauregard), “transactional interpretation” (J. Cramer), “history and destiny state vectors” (Y. Aharonov et-al), Hoyle-Narlikar “advanced response of the universe” in “Cosmology and Action at a Distance Electrodynamics” World Scientific, 1996. “Time’s Arrow and Archimedes Point”, Huw Price, Oxford, 1996. 18 “Geons, Blackholes & Quantum Foam”, Wheeler’s autobiography with Ken Ford. 19 There is no collapse in the Bohm ontology, hence no “qubit speed” for collapse. We have seen that the standard backward causation models without the particle cannot explain the apparent splitting of the charge on the electron as naturally as the Bohm ontology can. 20 Rev. Mod. Phys. 20,267, “Space-Time Approach to Non-Relativistic Quantum Mechanics” (1948) 14

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Bohm showed that the pilot wave is a physical field of qubits in the configuration space of the piloted material. The ideas of quantum probability21 are not fundamental. God does not play dice with post-quantum reality. Therefore, unlike the orthodox statistical interpretations relying fundamentally on ensembles of identically prepared simple atomic systems such as particle beams in scattering experiments, Bohm’s ontology is ideally suited to explain unique complex highly entangled systems such as the living human brain. There was only one Shakespeare22 and to invoke shadow Shakepeares in a multiverse of parallel worlds is “excess metaphysical baggage”.23 This is not to deny the possibility of “other worlds” close by in material hyperspace, less than a millimeter away, as in M-theory with “3D membranes” folded by anholonomic torsion fields.24 One can even imagine traversable wormholes connecting these worlds to each other. Bohm and Hiley also emphasize that the quantum pilot field is “nonmechanical” and “organic” with no “preassigned interactions between the parts”. In this sense, the pilot field is not at all like a classical machine. The pilot field in configuration space for entangled subsystems is form-dependent and intensity-independent totally opposite to classical fields in ordinary space. It is intimate25, immediate, and undiminished with increasing separation unlike the classical dynamical force fields of electromagnetism, gravity, and torsion confined to ordinary space. These are all desiderata for the “mental field” out of which our thoughts, feelings, and perceptions arise in consciousness. Indeed, the pilot field idea immediately explains how thought can move matter. What Bohm’s and Vigier’s “causal theory” cannot qualitatively explain in principle is how matter influences thought to create the inner conscious experience. This is because of an argument Bohm and Hiley give26 that the standard statistical predictions of quantum theory for ensembles of identically prepared simple independent unentangled systems27 require that there can be no direct reaction or “back-action” of matter on its pilot field. Such a compensating postquantum reaction to quantum pilot action would result in “signal nonlocality” violating the Stapp-Eberhard no-go theorem. The latter forbids the use of quantum nonlocality as a direct communication channel for what Einstein called “spooky telepathic action at a distance” in violation of the retarded causality postulate of the classical theory of special relativity. The modern theory of quantum computing, cryptography and teleportation would fall apart if the signal-locality of orthodox quantum theory could be violated. Yet, 21

Feynman’s phenomenological rules: add complex amplitudes before squaring for indistinguishable alternatives. Square amplitudes before adding for distinguishable alternatives. 22 Whether or not the real Shakespeare was really the brothers Francis and Tony Bacon at the Scriptorium in London with Ben Johnson (Sirag “Shakespeare’s doublet reversed in First Folio picture”) not the issue. 23 Wheeler refuting “many worlds” he once endorsed. 24 August 2000, Scientific American “The Universe’s Unseen Dimensions” p.62, N. Arkani-Hamed et-al. 25 Literally attached to a material configuration like the private mind of an individual living brain. 26 P.30 and Ch. 14 in “The Undivided Universe” (1993) 27 We can call this the “actuary’s limit” of sub-quantal equilibrium of the nonlocal hidden variables, AKA “subquantal heat death” in Antony Valentini’s Ph.D. dissertation under Dennis Sciama at Cambridge.

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this is precisely what the human mind does when it experiences and knows. The quantum potential Q characterized by action without reaction is “fragile”.28 It is this “fragility” that maintains irreducible uncontrollable local quantum randomness29 even in nonlocally entangled systems. It is not possible to control a quantum probability at a distance in orthodox quantum theory. Yet, this is precisely what happens in the brain, indeed, in the entire living body beyond the neural transmission of electrical signals and the transport of chemical messenger molecules as important as they are. The pilot field of the Bohm-Vigier “causal theory”, with deterministic particle trajectories, is an “absolute physical object”30. Therefore, this pilot wave is just like Newton’s absolute space and absolute time before special relativity and just like the absolutely flat spacetime of special relativity before Einstein introduced the direct back-reaction of matter on spacetime geometry to bend it into gravity. The unified field theory goes further twisting the spacetime geometry and changing the topology of the 3D membranes to create and destroy traversable wormholes as one example. Why does quantum theory work so well? I propose the following model. Imagine two barriers. One barrier is for the action of the quantum pilot field on the matter it is piloting of height per qubit31. The other barrier is for the reaction or “backaction” of the matter on its pilot field. When the quantum action is balanced in strength by this new post-quantum reaction, one forms a self-organizing feedback control loop between pilot field and its matter suppressing quantum randomness with “signal nonlocality”. Orthodox quantum theory is strongly violated in this situation. The BohmVigier “causal theory” breaks down completely in this new regime. The particle paths are no longer deterministic, rather, they are self-determining . The entangled individual particle paths, inseparable in configuration space, exhibit strangely synchronized motions beyond the local contact forces from classical signal connections when viewed in ordinary space. This is the essence of the “self in biological life. Indeed, the particle paths co-evolve with the changing shape of the landscape formed by the now postquantum potential . Both the particle paths and the shape of the landscape they ride on in configuration space are tweaked by external perturbations from the non-self environment. There is a further mathematical generalization in that the co-evolutionary flow of the nonlocally entangled particle paths on the landscape of their common pilot field is no longer an integrable holonomic path-independent flow characterized by exact differentials. One now has a more complex dynamically changing self-determining topology of closed inexact differential nonintegrable anholonomic flows on the landscape. Indeed one must use the Pfaffian theory of nonintegrability of R. Kiehn32 in which one has a nonstatistical 28

Bohm and Hiley’s term in “The Undivided Universe”. The late Heinz Pagels, killed in a way he foresaw in a dream, discusses how quantum randomness prohibits the use of nonlocality as a direct communication channel in “The Cosmic Code”. This book closes with Pagels’ strange precognitive dream of his then future death. 30 “On the Ether” Albert Einstein, 1924 31 A “qubit” is the basic unit of quantum information analogous to the Shannon “c-bit” of classical information associated with negative thermodynamic entropy. The qubit is a relative phase coherent two-state quantum system. The spin of a single electron forms a qubit. A single hydrophobically-caged electron inside the protein dimer molecules tiled around subneuronal microtubule (e.g. Stuart Hameroff’s papers) forms the qubit we are specifically interested in. 32 See Kiehn’s paper in these proceedings. 29

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topological irreversibility33 constituting a self-organizing process of an open system far from thermodynamic equilibrium with memory and learning. I postulate the Ansatz that the height of the reaction barrier of matter back on its pilot qubit information field is

where H is the Hubble cosmological parameter whose dependence on global cosmic time differs for different solutions of the Einstein field equations.34 This Ansatz is interim since the Einstein field equations need to be changed to include the anholonomic unified field from the nonsymmetric connection for parallel transport of vectors around closed loops in the deformed spacetime manifold. These closed loops in the manifold project into broken loops with translational gaps in the flat tangent spacetime fiber erected at the common beginning and end of the closed loop in the manifold.35 is the number of coherently entangled qubits forming a nonlocally connected network of qubits, each of rest mass m and electric charge e, of N nodes. We have post-quantum signal-nonlocality so that the local motion of a single node in the intelligent net is not random, but is synchronized with the simultaneous motion of the other nodes in the approximation of their common rest frame. Galilean relativity works fine here since the dimensionless velocity parameter for each node obeys The height of the backaction barrier is normalized per qubit node. The in the denominator is the coherence complexity factor lowering the height of the reaction barrier36. The fine structure constant is the strength of the lowest order Feynman diagram connecting two electron nodes with the exchange of a virtual longitudinally polarized near field photon. One needs virtual photons in the same field oscillator mode to phase coherently connect all N nodes together to lowest order Feynman diagram expansion in the N node many-body problem. I make the second Ansatz that this quantum coherent non-radiating near field is the 40 Hz Crick brain field in the specific application of this model to the actual living human brain. Since the virtual photons are “off the mass shell”,37 the equation does not apply to them. The Crick near electric field38 “enslaves”39 the N electron qubit nodes into a coherently-phased interferometric array across the whole cortex forming the brain hologram of Bohm and Pribram. Impose the action-reaction principle, in this case the heights of the two barriers are equal forming a resonant two way feedback control loop in contrast to the one way action of orthodox quantum theory 33 34

Pfaffians of degree 3 and 4 in the Cartan theory of differential forms. When the backaction barrier is much higher than the action barrier. So quantum theory works. 35 “Spin and Torsion in Gravitation” pp. 10, 11 Figs. 1,2,3 V. de Sabbata & C. Sivaram, World Scientific (1994) 36 This coherent lowering of the barrier also applies to Schwinger’s theory of cold fusion. 37 For photons, the mass shell Fourier transformed to spacetime is the classical light cone. Virtual photons are off the light cone either inside it or outside it. The superluminal virtual longitudinal photons outside the classical light cone dominate the static near field Coulomb force between pairs of charges. Far field radiation of real transverse polarization plays a small role here. The near electric and magnetic induction fields are what are biophysically most significant. 38 “The Astonishing Hypothesis”, F. Crick, Scribner’s , 1994. 39 “Principles of Brain Functioning”, 4.2.2, 4.2.3, 4.2.4 H. Haken, Springer-Verlag, 1996.

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with uncontrollable randomness from a “fragile” integrable quantum potential field without memory. The nonintegrable post-quantum field, with both memory from the past and “presponse”40 from the future, is robust stabilized by the action-reaction loop.

for the critical nonlocal entanglement complexity needed to obey the action-reaction principle. Stuart Hameroff http://www.consciousness.arizona.edu/hameroff/ informed me that this is the correct order of magnitude41 for the total number of hydrophobically-caged electrons in the human brain. Furthermore, from the simple post-quantum model of Bohm and Hiley42 the duration of a single undivided moment of self-organization43 is

One must erase the configuration of a billion billion qubits every 0.16 seconds in order to form the Jamesian “stream of consciousness”. The required power dissipation to accomplish this is

The resting adult human body metabolizes at about 100 Watts, so this is a small power consumption to generate consciousness in this model. What is unique about my model 40

AKA “presentiment” “The Conscious Universe”, Dean Radin, Harper Edge, 1997 replicated by Dick Bierman. 41 A billion billion electron qubits. 42 “The Undivided Universe” 14.3, 14.6, Bohm & Hiley op-cit 43 I interpret this as Whitehead’s “occasion of experience” i.e. an undivided conscious moment in the intrinsically mental quantum information pilot field.

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here44 is that human consciousness has a cosmological origin45 in a kind of mental version of Mach’s Principle compatible with the Hoyle-Narlikar “future response of the universe” 46with backward causation based upon a generalization of the WheelerFeynman conjecture.

Part II: Progress In Classical Unified Field Theory This is the “particle” part of the dBBV ontological “wave-particle duality”. As I show below in some detail, Vigier’s idea that elementary particles have extended spatial structure in which the center of charge is displaced from the center of mass with a finite rest mass of the photon in a superconducting Dirac “aether” of correlated virtual electronpositron pairs is getting increasing experimental confirmation. Bo Lehnert47 has shown several electromagnetic anomalies such as an effective electric charge density in the classical vacuum needed to explain observed data. I suspect that all of Bo Lehnert’s observations can be adequately explained by Corum’s anholonomic

Where

is the Maxwell electromagnetic field tensor and

field48 equation

is the electromagnetic 4-

potential now a local classical observable. The equivalence principle used by Einstein in 1915 only applied to the nonuniform holonomic translational motion with nonrotating local noninertial frames. The use of a nonsymmetric connection comes from extending the equivalence principle to nonuniform anholonomic rotations of local noninertial frames.49 Corum’s equation is not U(l) gauge invariant. One can make it gauge invariant with a minimal coupling using Dirac’s50 string quantization of electric charge from the magnetic monopole.

44

Compared to Stapp’s or Penrose’s. The connection of the Hubble parameter to the mass of the spatially extended electron has also been noted by R. I. Gray op-cit who derives where 46 “The Intelligent Universe”, Fred Hoyle, Holt, Rinehart&Winston (1983). 47 See contribution to this conference. Lehnert is at the Royal Institute of Technology in Stockholm. 48 This third rank tensor field under holonomic Diff(4) gauge symmetry of 1915 general relativity is the antisymmetric part of the connection field for nonintegrable parallel transport of vectors along paths in the manifold. J.Math Physics, 18, 4, pp. 770-776, 1977 “Relativistic rotation and the anholonomic object” James F. Corum 49 e.g. Gennady Shipov “A Theory of Physical Vacuum” Moscow, 1998 and Vladimir Poponin’s paper in these proceedings. 50 “Geometry, Particles and Fields” pp 18, 391, 491, Bjorn Felsager, Springer-Verlag, 1998. 45

POST-QUANTUM PHYSICS AND UNIFIED FIELD THEORY

Where

427

is the mechanical 4-momentum and n is a topological winding number. This

means that one must violate holonomic Diff(4) gauge symmetry when generating the anholonomic

field in order to preserve the internal U(l) gauge symmetry of the

electromagnetic force. Kleinert’s theory (http://www.physik.fu-berlin.de/~kleinert/) of “super-tetrads” for the unified field beyond Einstein’s 1915 theory is required. Eq. (7) should also explain the Blackett-Sirag Effect51 for rotating neutral astronomical objects that show an anomalous magnetic moment (http://stardrive.org/Jack/blackett1.pdf). Sirag’s empirical data plot for the magneto-gyroscopic coefficient obeys

The Kerr-Newman solution for a rotating charged black hole has a non-radiating solution with zero Hawking radiation when the Pythagorean theorem is obeyed in control parameter space52, i.e.,

51

“Gravitational Magnetism” Nature, Vol. 278, 535, April, 1979. “The gravi-magnetic hypothesis is that a rotating mass,” [electrically neutral] “measured in gravitational units has the same magnetic effect as that of an electrostatic charge, measured in esus moving at the same angular velocity at the same distance.” 4-1 “Unified Physics”, R. I. Gray, O.B.E., in-house report, 1988 from Naval Surface Warfare Center, Dahlgren, VA. This also implies EM radiation from accelerating neutral matter. Also p. 459 eq. (11.2.4) ref 1 on Schwinger’s classical “dyon”. 52 In the sense of Rene Thom’s “catastrophe theory” and V.I. Arnold’s “singularity theory”. Folds and other kinds of controllable catastrophes in the dynamic 3D membrane embedded in 10d hyperspace are expected.

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T is the string tension [energy/length] and of the dilaton field. Changing string tension

is the dimensionless vacuum expectation

changes the effective gravity parameter G and the

. I conjecture that this right triangle constraint must be generalized to

the general triangle constraint

with torsion “hair”53 in the cross-term for a unified field generalization of the orthodox black hole and wormhole solutions beyond Einstein’s symmetric 1915 general relativity. Model the bare electron as a non-radiating wormhole mouth with charge and bare mass The magneto-gyroscopic coefficient is then

Therefore the bare mass that fits the data in (8) is

In other

words, the magneto-gyroscopic measurements of astronomical objects act as a “telescope” directly into the quantum gravity era! This is consistent with Vigier’s notion of the spatially extended bare elementary particle in a superconducting virtual dressed electron-positron screening plasma in the quantum vacuum that is not dragged along with the rotating bare core due to zero superfluid viscosity. We have additional empirical astronomical data of P.S. Wesson54 53

The black hole is not completely bald when we add torsion. Wheeler had in 1955 “Geon” paper. Phys. Rev. D, 23, 8, 1730, April 1981 “Clue to the Unification of Gravitation and Particle Physics”. P.S. Wesson.

54

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that fits my 197355 association of Regge trajectories to tiny non-radiating rotating black holes with strong short-range gravity. The hadronic resonances of the strong quantum chromodynamic SU(3) color force have

In contrast (12) is ~ 36

powers of 10 flatter, i.e. Note that the new parameter p, not found in the 1915 Einstein general relativity, has the dimensions of vorticity flux per unit mass for the circulation of the aether flows. Therefore, this shows a strong presence of the anholonomic unified field at all scales of the universe from planets to pulsars to galaxies to clusters and superclusters. The Blackett-Sirag data together with the Wesson data rank in equal importance to the data on Hubble’s law for the cosmological redshift and the isotropic cosmic microwave blackbody background radiation. Wesson points out the strong dimensionless anholonomic unified field coupling strength parameter and conjectures from his data that

This corresponds to the mass scale

closer to the GUT unification compared to the Sirag-Blackett mass scale of 11.7.56 The Blackett-Sirag mass scale is roughly dual, in the sense of superstring M-theory, to the Wesson mass scale using the Planck mass as the standard. Therefore, I conclude that the empirical evidence from both Blackett-Sirag and Wesson are effective “telescopes” down to Wheeler’s “quantum foam” at the Planck scale confirming M-theory qualitatively. This shows that bare matter is made from non-radiating spatially-extended rotating charged wormholes whose gyromagnetic properties decouple from the zero viscosity superconducting virtual dressed electron-positron Dirac-Vigier quantum vacuum.57 55

“The Eightfold Way as a Consequence of the General Theory of Relativity” Collective Phenomena, 1, 1974 (edited by H. Frohlich and F.W. Cummings), & “Speculations on Gravitation and Cosmology in Hadron Physics”, Collective Phenomena, pp 163-167 (1973); “Quantum Mechanics as a Consequence of General Relativity” IC/74/9 International Centre Theoretical Physics, Trieste, Italy; “Gravitation, Strong Interactions and the Creation of the Universe”, Nature-Physical Science (December 4, 1974) “The Primordial Proton”, Physics Today letter (May, 1974) 69 also Andrew Salthouse “Is Symmetry Breaking in SU(3) a Consequence of General Relativity”, UM HE 73-29 cites my work and fits nuclear data to it. “Space-Time and Beyond” p. 168 (Dutton, 1975) also strong finite range gravity to scale 1 micron p. 129, 137. 56 57

Not quite superstring dual but the data is not accurate enough yet. “Dirac’s Aether in Relativistic Quantum Mechanics”, J.P.Vigier, N.C. Petroni, Fdn. Physics, 13, 2, 29 (1983).

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Acknowledgements

Saul-Paul Sirag, Creon Levit, Vladimir Poponin, Axel Pelster, Bo Lehnert, James Corum, Tony Smith, R. Kiehn, Henry Stapp, Stan Klein, and Hal Puthoff have made valuable contributions to this work in private conversations and e-mail. All errors and confusions are solely mine.

POLARIZABLE-VACUUM APPROACH TO GENERAL RELATIVITY

H. E. PUTHOFF Institute for Advanced Studies at Austin 4030 W. Braker Lane, Suite 300, Austin, Texas 78759

Abstract

Topics in general relativity (GR) are routinely treated in terms of tensor formulations in curved spacetime. An alternative approach is presented here, based on treating the vacuum as a polarizable medium. Beyond simply reproducing the standard weak-field predictions of GR, the polarizable vacuum (PV) approach provides additional insight into what is meant by a curved metric. For the strong field case, a divergence of predictions in the two formalisms (GR vs. PV) provides fertile ground for both laboratory and astrophysical tests. 1. Introduction

The principles of General Relativity (GR) are generally formulated in terms of tensor formulations in curved spacetime. Such an approach captures in a concise and elegant way the interaction between masses, and their consequent motion. "Matter tells space how to curve, and space tells matter how to move [1]." During the course of development of GR over the years, however, alternative approaches have emerged that provide convenient methodologies for investigating metric changes in other formalisms, and which yield heuristic insight into what is meant by a curved metric. One approach that has intuitive appeal is the polarizable-vacuum (PV) approach [2-3]. The PV approach treats metric changes in terms of equivalent changes in the permittivity and permeability constants of the vacuum, and , essentially along the lines of the so-called methodology used in comparative studies of gravitational theories [4-6]. In brief, Maxwell's equations in curved space are treated in the isomorphism of a polarizable medium of variable refractive index in flat space [7]; the bending of a light ray near a massive body is modeled as due to an induced spatial variation in the refractive index of the vacuum near the body; the reduction in the velocity of light in a gravitational potential is represented by an effective increase in the refractive index of the vacuum, and so forth. As elaborated in Refs. [3-7], PV modeling can be carried out in a self-consistent way so as to reproduce to appropriate order both the equations of GR, and the match to the classical experimental tests of those equations. Under conditions of extreme metric perturbation, however, the PV approach predicts certain results at 431 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 431-446. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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variance with the standard GR approach. We discuss these variances in terms of testable implications, both in the laboratory and with regard to astrophysical consequences.

2. The Polarizable Vacuum The electric flux vector

in a linear, homogeneous medium can be written

where and are the permittivities of the medium and of the vacuum, respectively, and the polarization corresponds to the induced dipole moment per unit volume in the medium whose polarizability per unit volume is . The identical form of the last two terms leads naturally to the interpretation of as the polarizability per unit volume of the vacuum, treated as a medium in its own right. This interpretation is explicitly corroborated in detail by the quantum picture of the vacuum where it is shown that the vacuum acts as a polarizable medium by virtue of induced dipole moments resulting from the excitation of virtual electron-positron pairs [8]. To represent curved-space conditions, the basic postulate of the PV approach is that the polarizability of the vacuum in the vicinity of a mass (or other mass-energy concentrations) differs from its asymptotic far-field value by virtue of vacuum polarization effects induced by the presence of the mass. That is, we postulate for the vacuum itself

where K is the (altered) dielectric constant of the vacuum (typically a function of position) due to (GR-induced) vacuum polarizability changes under consideration. Throughout the rest of our study the vacuum dielectric constant K constitutes the key variable of interest. 2.1. VELOCITY OF LIGHT IN A VACUUM OF VARIABLE POLARIZABILITY In this section we examine quantitatively the effects of a polarizable vacuum on the various measurement processes that form the basis of the PV approach to general relativity. We begin by examining a constraint imposed by observation. An appropriate starting point is the expression for the fine structure constant,

By the conservation of charge for elementary particles, and the conservation of angular momentum for a circularly polarized photon propagating through the vacuum (even with variable polarizability), and can be taken as constants. Given that can be expected with a variable vacuum polarizability to change to and the vacuum

POLARIZABLE-VACUUM permeability may be a function of constant therefore takes the form

433 , The fine structure

which is potentially a function of K. Studies that consider the possibility of the variability of fundamental constants under varying cosmological conditions, however, require that the fine structure constant remain constant in order to satisfy the constraints of observational data [9-11]. Under this constraint we obtain from Eq. (4) thus the permittivity and permeability constants of the vacuum must change together with vacuum polarizability as1

As a result the velocity of light changes inversely with K in accordance with

Thus, the dielectric constant of the vacuum plays the role of a variable refractive index under conditions in which vacuum polarizability is assumed to change in response to GRtype influences. As will be shown in detail later, the PV treatment of GR effects is based on the use of equations that hold in special relativity, but with the modification that the velocity of light in the Lorentz transformations and elsewhere is replaced by the velocity of light in a medium of variable refractive index, Expressions such as are still valid, but now take into account that and and may be functions of and so forth. 2.2. ENERGY IN A VACUUM OF VARIABLE POLARIZABILITY Dicke has shown by application of a limited principle of equivalence that the energy of a system whose value is in flat space takes on the value

1

This transformation, which maintains constant the ratio the impedance of free space) is just what is required to maintain electric-to-magnetic energy ratios constant during adiabatic movement of atoms from one point to another of differing vacuum polarizability [3]. Detailed analysis shows that it is also a necessary condition in the formalism for an electromagnetic test body to fall in a gravitational field with a composition-independent acceleration WEP, or weak equivalence principle, verified by Eötvös-type experiments) [4-6]. Finally, this condition must be satisfied by any metric theory of gravity, which constitutes the class of viable gravity theories.

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in a region where This is due to that fact that the self-energy of a system changes in response to changes in the local vacuum polarizability, analogous to the change in the stored energy of a charged air capacitor during transport to a region of differing dielectric constant. The energy relationship given by Eq. (7) also implies, via which becomes a corollary change in mass

again a consequence of the change in self-energy. 2.3. ROD AND CLOCK (METRIC) CHANGES IN A VACUUM OF VARIABLE POLARIZABILITY Another consequence of the change in energy as a function of vacuum polarizability is a and change in associated frequency processes which, by the quantum condition Eq. (7), takes the form

This, as we shall see, is responsible for the red shift in light emitted from an atom located in a gravitational potential. From the reciprocal of Eq. (9) we find that time intervals marked by such processes are related by

Therefore, in a gravitational potential (where it will be shown that the time interval between clock ticks is increased (that is, the clock runs slower) relative to a reference clock at infinity. With regard to effects on measuring rods, we note that, for example, the radius of the ground-state Bohr orbit of a hydrogen atom

becomes (with

and

constant as discussed earlier)

Other measures of length such as the classical electron radius or the Compton wavelength of a particle lead to the relationship Eq. (12) as well, so this relationship is general. This dependence of fundamental length measures on the variable K indicates that the dimensions of material objects adjust in accordance with local changes in vacuum

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polarizability - thus there is no such thing as a perfectly rigid rod. From the standpoint of the PV approach this is the genesis of the variable metric that is of such significance in GR studies. We are now in a position to define precisely what is meant by the label "curved space." In the vicinity of, say, a planet or star, where if one were to take a ruler and measure along a radius vector R to some circular orbit, and then measure the circumference C of that orbit, one would obtain (as for a concave curved surface). This is a consequence of the ruler being relatively shorter during the radial measuring process (see Eq. (12)) when closer to the body where K is relatively greater, as compared to its length during the circumferential measuring process when further from the body. Such an influence on the measuring process due to induced polarizability changes in the vacuum near the body leads to the GR concept that the presence of the body "influences the metric," and correctly so. Of special interest is the measurement of the velocity of light with "natural" (i.e., physical) rods and clocks in a gravitational potential which have become "distorted" in accordance with Eqns. (10) and (11). It is a simple exercise to show that the measured velocity of light obtained by the use of physical rods and clocks renormalizes from its "true" (PV) value to the value The PV formalism therefore maintains the universal constancy of the locally measured velocity of light. 2.4. THE METRIC TENSOR

At this point we can make a crossover connection to the standard metric tensor concept that characterizes the conventional GR formulation. In flat space a (4-dimensional) infinitesimal interval is given by the expression

If rods were rigid and clocks non-varying in their performance in regions of differing vacuum polarizability, then the above expression would hold universally. However, a -length measuring rod placed in a region where for example, shrinks according to Eq. (12) to . Therefore, the infinitesimal length which would measure were the rod to remain rigid is now expressed in terms of the -length rod as (Such constitutes a transformation between "proper" and "coordinate" values.) With a similar argument based on Eq. (10) holding for clock rate, Eq. (13) can be written

Therefore, the infinitesimal interval takes on the form

where

in the above expression defines the metric tensor, and

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The metric tensor in this form defines an isotropic coordinate system, familiar in GR studies. 3. Classical Experimental Tests of General Relativity in the PV Model

In the previous sections we have established the concept of the polarizable vacuum and the effects of polarizability changes on metric (rods and clocks) behavior. In particular, we found that metric changes can be specified in terms of a single parameter the dielectric constant of the vacuum. This is the basis of the PV approach to GR. In this section we note, with the aid of expressions to be derived in detail in Section 5, how changes in the presence of mass, and the effects generated thereby. The effects of major interest at this point comprise such classical tests of GR as the gravitational redshift, the bending of light and the advance of the perihelion of Mercury. These examples constitute a good testbed for demonstrating the techniques of the PV alternative to the conventional GR curved-space approach. For the spherically symmetric mass distribution of a star or planet it will be shown later from the basic postulates of the PV approach that the appropriate PV expression for the vacuum dielectric constant K is given by the exponential form

where G is the gravitational constant, M is the mass, and is the distance from the origin located at the center of the mass M. For comparison with expressions derived by conventional GR techniques, it is sufficient to restrict consideration to a weak-field approximation expressed by expansion of the exponential to second order as shown. As an example of application of the PV approach to a standard experimental test of GR, we consider the case of gravitational redshift. In a gravitation-free part of space, photon emission from an excited atom takes place at some frequency , uninfluenced by vacuum polarizability changes. That same emission process taking place in a gravitational field, however, will, according to Eq. (9), have its emission frequency altered (redshifted) to . With the first-order correction to given by the first two terms in Eq. (17), emission by an atom located on the surface of a body of mass and radius will therefore experience a redshift by an amount

where we take . Once emitted, the frequency of the photon remains constant during its propagation to a relatively gravitation-free part of space where its frequency can then be compared against that of local emission, and the spectral shift given by Eq.

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(18) observed. Measurement of the redshift of the sodium line emitted on the surface of the sun and received on earth has verified Eq. (18) to a precision of 5% [12]. Experiments carried out on the surface of the earth involving the comparison of photon frequencies at different heights have improved the accuracy of verification still further to a precision of 1% [13-14]. With the two ends of the experiment separated by a vertical height h, the first-order frequency shift is calculated with the aid of Eqns. (9) and (18) as

where M and R are the mass and radius of the earth. This experiment required a measurement accuracy of for a height meters. It was accomplished by the use of Mössbauer-effect measurement of the difference between emission and absorption frequencies at the two ends of the experiment. In similar fashion, we could consider in detail other canonical examples such as the bending of light rays or perihelion advance of planetary orbits near a mass. However, rather than treating these cases individually, we can take a more general approach. In standard textbook treatments of the classical tests of GR one begins with the Schwarzschild metric, which in isotropic coordinates is written [15]

Expanding the metric tensor for small departures from flatness as a Maclaurin series in we obtain

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Similarly, in the PV approach one begins with the exponential metric defined by Eqns. (14) - (17),

This, when expanded to the same order as the Schwarzschild metric tensor above for small departures from unity vacuum dielectric constant, yields

Comparison of Eqns. (24) - (25) with Eqns. (21) - (22) reveals that, to the order of expansion shown, the two metric tensors are identical. Since the classical tests of GR do not require terms beyond these explicitly displayed, the agreement between theory and experiment is accounted for equally in both the conventional GR and in the alternative PV formalisms. For a charged mass the Schwarzschild metric is replaced by the Reissnermetric in the GR approach, while in the PV approach the exponential metric is replaced by a metric involving hyperbolic functions (see Section 6.2). Again, for the weak-field case, it can be shown that the two approaches are in precise agreement to the order shown in the charge-free case. 4. Coupled Matter-Field Equations

In the preceding section we have seen that the classical tests of GR theory can be accounted for in the PV formalism on the basis of a variable vacuum dielectric constant, K. To carry that out we stated without proof that the appropriate mathematical form for the variation in K induced by the presence of mass is an exponential form. In this section we show how the exponential form is derived from first principles, and, in the process, establish the general approach to the derivation of field equations as well as the equations for particle motion. The approach consists of following standard Lagrangian techniques as outlined, for example, in Ref. 16, but with the proviso that in our case the dielectric constant of the vacuum is treated as a variable function of time and space. 4.1. LAGRANGIAN APPROACH

The Lagrangian for a free particle is given by

POLARIZABLE-VACUUM

which, in the presence of a variable vacuum dielectric constant aid of Eqns. (6) and (8) to read

439

, is modified with the

This implies a Lagrangian density for the particle of

Following standard procedure, the particle Lagrangian density can be extended to the case of interaction with electromagnetic fields by inclusion of the potentials

The Lagrangian density for the electromagnetic fields themselves, as in the case of the particle Lagrangian, is given by the standard expression (see, e.g., Ref. 16), except that again is treated as a variable,

We now need a Lagrangian density for the dielectric constant variable , which, being treated as a scalar variable, must take on the standard Lorentz-invariant form for propagational disturbances of a scalar,

where is an arbitrary function of . As indicated by Dicke in the second citation of Ref. 3, a correct match to experiment requires that we take thus,

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We can now write down the total Lagrangian density for matter-field interactions in a vacuum of variable dielectric constant,

4.2. GENERAL MATTER-FIELD EQUATIONS Variation of the Lagrangian density

with regard to the particle variables

leads to the equation for particle motion in a variable dielectric vacuum,

We see that accompanying the usual Lorentz force is an additional dielectric force proportional to the gradient of the vacuum dielectric constant. This term is equally effective with regard to both charged and neutral particles and accounts for the familiar gravitational potential, whether Newtonian in form or taken to higher order to account for GR effects.2 Variation of the Lagrangian density with regard to the K variable leads to an equation for the generation of GR vacuum polarization effects due to the presence of matter and fields. (In the final expression we use to obtain a form convenient for the following discussion.)

2

Of passing interest is the fact that as

but

the deflection of a zero-mass particle

(e.g., a photon) in a gravitational field is twice that of a slow-moving particle important result in GR dynamics.

an

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Thus we see that changes in the vacuum dielectric constant K are driven by mass density (first term), EM energy density (second term), and the vacuum polarization energy density itself (third term). The fact that the latter term corresponds to the energy density of the K variable can be seen by the following argument. We start with the Lagrangian density Eq. (32), define the momentum density by and form the Hamiltonian energy density to obtain

Eqns. (34) and (35), together with Maxwell's equations for propagation in a medium with variable dielectric constant, thus constitute the master equations to be used in discussing matter-field interactions in a vacuum of variable dielectric constant as required in the PV formulation of GR. 5. Static Field Solutions

We demonstrate application of field Eq. (35) to two static field cases with spherical symmetry: derivation of the expression introduced earlier for the gravitational field alone, and derivation of the corresponding expression for charged masses. 5.1. STATIC FIELDS (GRAVITATIONAL)

In space surrounding an uncharged spherical mass distribution the static solution to Eq. (35) is found by solving

or

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where we have used The solution that satisfies the Newtonian limit is given by

or

which can be verified by substitution into the equation for particle motion, Eq. (34). We have thus derived from first principles the exponential form of the variable dielectric constant in the vicinity of a mass as used in earlier sections. As indicated in Section 4, this solution reproduces to appropriate order the standard GR Schwarzschild metric predictions as they apply to the weak-field conditions prevailing in the solar system. 5.2. STATIC FIELDS (GRAVITATIONAL PLUS ELECTRICAL)

For the case of a mass with charge we first write the electric field appropriate to a charged mass imbedded in a variable-dielectric-constant medium,

Substitution into Eq. (35) yields (for spherical symmetry)

where The solution to Eq. (39) as a function of charge (represented by b) and mass (represented by is given below. Substitution into Eq. (34) verifies that as this expression asymptotically approaches the standard flat-space equations for particle motion about a body of charge and mass .

(For

the solution is trigonometric.) As noted earlier (Section 4), for the weak-field case the above reproduces the familiar metric [17].

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6. Strong-Field Tests As noted in the Abstract, both the conventional and PV approaches to GR problems lead to the same results for small departures from flatness. For increasingly larger departures from flatness, however, the two approaches, although initially following similar trends, begin to diverge with regard to specific magnitudes of effects. In the PV approach the solution for the static gravitational case yields a metric tensor that is exponential in form, in the conventional GR approach the somewhat more complex Schwarzschild solution. This discrepancy has shown up previously in other general curved-space approaches to GR as well.3 6.1. ASTROPHYSICAL TESTS A major difference between the Schwarzschild (GR) and exponential (PV) metrics is that the former contains an event horizon at which prevents radially-directed photons from escaping ("black holes"), whereas the latter has no such discontinuity (only increasingly "dark gray holes"). One consequence is that whereas the Schwarzschild solution limits neutron stars (or neutron star mergers) to ~2.8 solar masses because of black hole formation, no such constraint exists for the exponential metric. This raises the possibility that such anomalous observations as the enormous radiative output if isotropic) of the gamma ray burster GRB990123 [20] might be interpreted as being associated with collapse of a very massive star (hypernova), or the collision of two high-density neutron stars [21]. The collection of additional astrophysical evidence of this and related genres would be useful in the search for discriminants between the standard GR and alternative PV approaches. 6.2. LABORATORY TESTS For small departures from flatness it is useful to express the generalized metric in terms of the PPN (parametrized post-Newtonian) form

where and and comprise the PPN parameters. For the case of a central mass, both the conventional Schwarzschild and PV-derived exponential solutions

3

Of special interest is the so-called Einstein-Yilmaz tensor form, in which Einstein's equations are modified by inclusion of the stress-energy tensor of the gravitational field itself on the R.H.S. of the equations, in addition to the usual matter/field stress-energy [18]. The Yilmaz modification yields exponential solutions in the form derived here by means of the PV approach. The EinsteinYilmaz equations satisfy the standard experimental tests of GR, as well as addressing a number of mathematical issues of concern to general relativists, and are thus under study as a potentially viable modification to the original Einstein form [19].

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require by virtue of the classical tests of GR that however, a predicted discrepancy with regard to the fourth parameter,

There exists, , which is

As detailed in Ref. [18], an argument has been put forward that the isotropy-of-mass experiments of Hughes et al. [22] and by Drever [23], and the neutron phase-shift measurements of Collela et al. [24] yield a value . The data, analysis, and interpretation of such experiments provide yet further opportunities for discriminants between the standard GR and the alternative PV approaches. 7. Discussion

In overview, we have shown that a convenient methodology for investigating general relativistic (GR) effects in a non-abstract formalism is provided by the so-called polarizable-vacuum (PV) representation of GR. The PV approach treats metric perturbation in terms of a vacuum dielectric function K that tracks changes in the effective permittivity and permeability constants of the vacuum, a metric engineering approach, so to speak [25]. The structure of the approach is along the lines of the formalism used in comparative studies of gravitational theories. The PV-derived matter-field Eqns. (34)-(35) are in principle applicable to a wide variety of problems. This short exposition, covering but the Schwarzschild and metrics and experimental tests of GR, is therefore clearly not exhaustive. Consideration was confined to cases of spherical symmetry and static sources,4 and important topics such as gravitational radiation and frame-dragging effects were not addressed. Therefore, further exploration and extension of the PV approach to specific problems of interest is encouraged, again with cross-referencing of PV-derived results to those obtained by conventional GR techniques to ensure that the PV approach does not generate incomplete or spurious results. With regard to the epistemology underlying the polarizable-vacuum (PV) approach as compared with the standard GR approach, one rather unconventional viewpoint is that expressed by Atkinson who carried out a study comparing the two [26]. "It is possible, on the one hand, to postulate that the velocity of light is a universal constant, to define 'natural' clocks and measuring rods as the standards by which space and time are to be judged, and then to discover from measurement that space-time, and space itself, are 'really' non-Euclidean; alternatively, one can define space as Euclidean and time as the same everywhere, and discover (from exactly the same measurements) how the velocity of light, and natural clocks, rods, and particle inertias 'really' behave in the neighborhood of large masses. There is just as much (or as little) content for the word 'really' in the one approach as in the other; provided that each is self-consistent, the 4 However, it is known that the PV-related approach is sufficiently general that results obtained for spherically symmetric gravitational fields can be generalized to hold for nonsymmetric conditions as well.

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ultimate appeal is only to convenience and fruitfulness, and even 'convenience' may be largely a matter of personal taste..." On the other hand, from the standpoint of what is actually measured with physical rods and clocks, the conventional tensor approach captures such measurements in a concise, mathematically self-consistent formalism (the tensor approach). Therefore, the standard approach is more closely aligned with the positivist viewpoint that underlies modern scientific thought. Nonetheless, the PV model, with its intuitive, physical appeal, can be useful in bridging the gap between flat-space Newtonian physics and the curvedspacetime formalisms of general relativity.

Acknowledgements

I wish to express my appreciation to G. W. Church, Jr., for encouragement and useful suggestions in the development of this effort. I also wish to thank H. Yilmaz, E. Davis, M. Ibison and S. R. Little for stimulating discussions of the concepts presented herein. References 1. Misner, C.W., Thorne, K.S., and Wheeler, J.A. (1973) Gravitation, Freeman, San Francisco, p. 5. 2. Wilson, H.A. (1921) An electromagnetic theory of gravitation, Phys. Rev. 17, 54-59. 3. Dicke, R.H. (1957) Gravitation without a principle of equivalence, Rev. Mod. Phys. 29, 363-376. See also Dicke, R.H. (1961) Mach's principle and equivalence, in C. Mø11er (ed.), Proc. of the Intern'l School of Physics "Enrico Fermi" Course XX, Evidence for Gravitational Theories, Academic Press, New York, pp.l49. 4. Lightman, A.P., and Lee, D.L. (1973) Restricted proof that the weak equivalence principle implies the Einstein equivalence principle, Phys. Rev. D 8, 364-376. 5. Will, C.M. (1974) Gravitational red-shift measurements as tests of nonmetric theories of gravity, Phys. Rev. D 10, 2330-2337. 6. Haugan, M.P., and Will, C.M. (1977) Principles of equivalence, Eötvös experiments, and gravitational redshift experiments: The free fall of electromagnetic systems to post-post-Coulombian order, Phys. Rev. D 15, 2711-2720. 7. Volkov, A.M., Izmest'ev, A.A., and Skrotskii, G.V. (1971) The propagation of electromagnetic waves in a Riemannian space, Sov. Phys. JETP 32, 686-689. 8. Heitler, W. (1954) The Quantum Theory of Radiation, 3rd ed., Oxford University Press, London, p. 113. 9. Alpher, R.A. (Jan.-Feb. 1973) Large numbers, cosmology, and Gamow, Am. Sci. 61, 52-58. 10. Harrison, E.R. (Dec. 1972) The cosmic numbers, Phys. Today 25, 30-34. 11. Webb, J.K., Flambaum, V.V., Churchill, C.W., Drinkwater, M.J., and Barrow, J.D. (1999) Search for time variation of the fine structure constant, Phys. Rev. Lett. 82, 884-887. 12. Brault, J.W. (1963) Gravitational red shift of solar lines, Bull. Amer. Phys. Soc. 8, 28. 13. Pound, R.V., and Rebka, G.A. (1960) Apparent weight of photons, Phys. Rev. Lett. 4, 337-341. 14. Pound, R.V., and Snider, J.L. (1965) Effect of gravity on nuclear resonance, Phys. Rev. Lett. 13, 539-540. 15. Ref. l,p. 840. 16. Goldstein, H. (1957) Classical Mechanics, Addison-Wesley, Reading MA, pp. 206-207. 17. Ref. l,p. 841. 18. Mizobuchi, Y. (1985) New theory of space-time and gravitation - Yilmaz's approach, Hadronic Jour. 8, 193-219. 19. Alley, C.O. (1995) The Yilmaz theory of gravity and its compatibility with quantum theory, in D.M. Greenberger and A. Zeilinger (eds.), Fundamental Problems in Quantum Theory: A Conference Held in Honor of Professor John A. Wheeler, Vol. 755 of the Annals of the New York Academy of Sciences, New York, pp. 464-475. 20. Schilling, G. (1999) Watching the universe's second biggest bang, Science 283, 2003-2004. 21. Robertson, S.L. (1999) Bigger bursts from merging neutron stars, Astrophys. Jour. 517, L117-L119.

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22. Hughes, V.W., Robinson, H.G. and Beltran-Lopez, V. (1960) Upper limit for the anisotropy of inertial mass from nuclear resonance experiments, Phys. Rev. Lett. 4, 342-344. 23. Drever, R.W.P. (1961) A search for anisotropy of inertial mass using a free precession technique, Phil. Mag. 6, 683-687. 24. Collela, R. Overhauser, A.W., and Werner, S.A. (1975) Observation of gravitationally induced quantum mechanics, Phys. Rev Lett. 34, 1472-1474. 25. Puthoff, H.E. (1996) SETI, the velocity-of-light limitation, and the Alcubierre warp drive: an integrating overview, Physics Essays 9, 156-158. 26. Atkinson, R. d'E. (1962) General relativity in Euclidean terms, Proc. Roy. Soc. 272, 60-78.

THE INERTIA REACTION FORCE AND ITS VACUUM ORIGIN

ALFONSO RUEDA Department of Electrical Engineering, ECS Building, California State University 1250 Bellflower Blvd. Long Beach, CA 90840, USA. E-mail: [email protected] BERNARD HAISCH California Institute for Physics and Astrophysics 366 Cambridge Ave. Palo Alto, CA 94306. E-mail:[email protected]

Abstract. By means of a covariant approach we show that there must be a contribution to the inertial mass and to the inertial reaction force on an accelerated massive object by the zero-point electromagnetic field. This development does not require any detailed model of the accelerated object other than the knowledge that it interacts electromagnetically. It is shown that inertia can indeed be construed as an opposition of the vacuum fields to any change to the uniform state of motion of an object. Interesting insights originating from this result are discussed. It is argued why the proposed existence of a Higgs field in no way contradicts or is at odds with the above statements. The Higgs field is responsible for assigning mass to elementary particles. It is argued that still the underlying reason for the opposition to acceleration that massive objects present requires an explanation. The explanation proposed here fulfills that requirement. Keywords: Quantum-vacuum, inertia-reaction-force 1. Foreword Among the several proposed explanations for the origin of inertia [1], we review in this article a recent one [2,3,4] that attributes the inertia reaction force to an opposition by the vacuum fields to accelerated motion of any real object that possesses mass. In his recent book, “Concepts of Mass in Contemporary Physics and Philosophy”, Max Jammer [1] examines in detail many aspects of the mass concept including the origin of inertia. In particular, in reference to our recent work [2,3,4] he states [5]: 447 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale,447-458. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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A. RUEDA & B. HAISCH “However, debatable as their theory still is, it is from the philosophical point of view a thought-provoking attempt to renounce the traditional priority of the notion of mass in the hierarchy of our conceptions of physical reality and to dispense with the concept of mass in favor of the concept of field. In this respect their theory does to the Newtonian concept of mass what modern physics has done to the notion of absolute space: As Einstein [6] once wrote, ‘the victory over the concept of absolute space or over that of the inertial systems became possible only because the concept of the material object was gradually replaced as the fundamental concept of physics by that of the field.’ ”

Here we outline an attempt to show that the inertia reaction force has a contribution from the vacuum electromagnetic field. It strongly suggests that other vacuum fields (weak, strong interactions) do also contribute. A recent proposal by Vigier [7] that there is also a contribution from the Dirac vacuum very much goes along this line. Most of the work outlined here appears in Refs. [2] and [3] (Those two papers we will refer to as RH, and specifically in most cases RH will directly refer to Ref. [2]). Another related paper prior to these is Ref. [4] (This one we denote by HRP). 2. General Comments

We highlight here the main aspects of the RH approach [2]. In general the RH analysis is carried out in two complementary but totally independent and comprehensive ways (in addition to the completely different original HRP approach [4] that we shall not discuss in this paper). In the first way (section 3), which is the more intuitive, one calculates the radiation pressure resulting from a non-zero Poynting vector of the electromagnetic zeropoint field (ZPF) as viewed by an accelerated object. This radiation pressure is exactly opposite to the direction of the imposed acceleration, and in the subrelativistic case turns out to be directly proportional to the magnitude of the acceleration. The second way (section 4) leads to this same results by showing why and how an accelerating object acquires its four-momentum. This turns out to be directly related to the amount of ZPF energy and momentum contained within the object: it is that fraction of the contained energy that interacts with the fundamental particles comprising the object. From any change in this four-momentum one can straightforwardly calculate the resulting inertia reaction force. Not surprisingly this proves to be exactly the same force as in the first case. In both representations one obtains an (electromagnetic) expression for the corresponding inertial mass that is essentially the amount of ZPF energy, divided by c2, instantaneously contained within an object and which actually interacts with the object. This mass is a factor of 4/3 too large in the first two (noncovariant) versions (sections 3 and 4) we present below. This requires a correction. The correct form then comes about from a fully covariant derivation that we briefly outline at the end (section 5). There we introduce an important contribution from the ZPF electromagnetic Maxwell stress tensor that was previously neglected.

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3. The Inertia Reaction Force And The Electromagnetic Vacuum Radiation Pressure On An Accelerated Object We consider a small material object to be undergoing hyperbolic motion, i.e. uniformly accelerated motion, or motion with constant proper acceleration constant. Consider the object, that for simplicity we may identify with a particle, at the point 0,0) of a frame S that is rigid and noninertial because it comoves with the particle. At particle proper time the particle point of S exactly coincides and instantaneously comoves with the corresponding point of an inertial frame that we denote by and call the “inertial laboratory frame.” We take the direction of the particle acceleration vector to coincide with the positive x-direction in both S and , and in general in all subsequent frames that we will introduce. We consider also an infinitely continuous family of inertial frames, each denoted by , and such that each one of them has its point instantaneously coinciding and comoving with the particle at the point of S at the particle proper time . Clearly then The acceleration of the particle at

of S as seen from

is

The frame S is

called the Rindler noninertial frame and as it is rigid, its acceleration is not the same for all its points but we will only be interested in points in the neighborhood of the point. The particle undergoes well-known hyperbolic motion, in which the velocity of the particle point with respect to is then or

and then

The position of the particle in

and the time in

as function of particle proper time is then given by

when the proper time of the particle is is

Observe that we select

at

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We refer the reader to RH [2] for a detailed classically exact representation of the stochastic form of the electromagnetic field as well as for the details on the stochastic averaging. In what follows we will also omit a number of subtle points, fully discussed in App. C of RH [2], in which the reader may also find some useful intuitive analogies and a more thorough discussion of essential aspects of the arguments sketched here. According to an observer fixed to (say at the point of ), the object moves through the ZPF as viewed in with the hyperbolic motion described above. At proper time the object instantaneously comoves with the corresponding point of the inertial frame and thus at that point in time is found at rest in the inertial frame . We calculate the frame Poynting vector, but evaluated at the point o f . This allows us to obtain the net ZPF rate of momentum density that is accumulating in the object due to the uniformly accelerated motion. Recall that this is the ZPF of . This Poynting vector we denote by

, and

where the angular brackets represent stochastic averaging. That only the x-direction is relevant follows from symmetry. There are several subtle points that we must sweep under the rug here, details of which may be found in RH [2], especially App. C. We refer particularly to the so-called k-spheres of integration. Each inertial frame has its own spheres and even though the ZPF is Lorentz-invariant and it has the same form of energydensity spectrum and is homogeneous and isotropic in every inertial frame, the ZPF of does not appear to be that way to an observer of

and vice versa.

In concise form we present the calculations as follows:

Observe that in the last equality of eqn. (6) the term proportional to the projection of the ordinary ZPF Poynting vector of vanishes as it should. The integrals

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are taken with respect to the ZPF background (using then the k-sphere of of. App. C of RH [2]) as that is the background that the observer considers the accelerated object to be sweeping through. The net amount of momentum of the background that the object has swept through after a time as judged again from the viewpoint is then

the momentum density, is introduced and represents the volume of the object as seen in Clearly then, because of Lorentz contraction, where is the proper volume. In obtaining eqn. (7) from eqn. (6) and for the following step we use the fact that

With the last two equalities of eqn. (8), eqn. (6) becomes

where we used the fact that

and we introduced the frequency function

where for all This represents the fraction of ZPF radiation that actually interacts with the object at a given frequency. Clearly we expect sufficiently rapidly. The force applied by the ZPF to the uniformly accelerated particle or physical object can now be easily calculated:

where is the particle proper acceleration and where use was made of eqns. and (2). We have thus obtained what can be called the ZPF inertia reaction force

with an “inertial mass” of the form

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This is an invariant scalar with the dimension of mass. It represents the amount of ZPF radiation enclosed within the object (or particle) of proper volume that actually interacts with it. In eqn. (12) we have omitted a factor of 4/3 that appears in eqn. (A10). One of the revenues of a fully covariant analysis (of. App. D of RH [2]), that we sketch below in 5, is to show that the 4/3 factor disappears when proper use is made of the electromagnetic stress tensor which has been omitted so far. 4. The Inertia Reaction Force And The Electromagnetic Vacuum Momentum Content Of An Accelerated Object This approach is totally independent from the one above. However, it is strongly complementary. It is “the other side of the coin.” One approach requires the other. From Newton's Third Law applied to the accelerated object, when an external agent applies a motive force, and thus uniformly accelerates the object, according to the view proposed here the vacuum applies an equal and opposite force,

in the opposing direction, i.e.

where the star subscripts just means that we refer here to the laboratory inertial frame Eqn. (13) implies that the corresponding impulses, at time

and

and -

also obey after a short lapse of time

taken to be zero, say, and correspondingly a

short proper time lapse

Integrating over longer times in

from zero to some final time

we can write

This allows us to introduce an equation between the corresponding momentum densities where and thus

INERTIA REACTION FORCE where we have already confronted

in eqn. (7), and

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as carefully argued in App. B

of RH [2], corresponds to the fraction of the momentum density of the ZPF radiation within the object that interacts with the object. Expressing this momentum density in terms of the corresponding Poynting vector we write

where

is the Poynting vector due to the ZPF as measured in

at the object's

point, at proper time of eqns. (3) and (4), in the laboratory frame. Because of symmetry again, only the x-component appears. Recall however that we are calculating the ZPF momentum associated with the object. At proper time the object is instantaneously at rest in the inertial frame This means (cf. App. C of RH [2]) that we must perform the integrals over the k-sphere of the frame. This becomes more revealing when we Lorentz-transform the field in eqn. (17) from to

where again we have a term that vanishes, namely the one proportional to the xcomponent of the ZPF statistically-averaged Poynting vector in Recall that integrals are performed with respect to the k-sphere of (App. C of RH [2]). We then have

where as in

Al we have used the fact that

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The prime in indicates just that frequencies and fields refer now to those in In eqn. (19) we again used the fact that and we introduced the factor again because only the fraction of the ZPF radiation that actually interacts with the particles in the object is relevant. If we differentiate the impulse with respect to time we get the corresponding force, and with eqn. (13) we obtain the inertia reaction force due to the ZPF as

which reproduces eqn. (10) as should be expected. Now again, the formulae (11) and (12) follow accordingly. 5. A Covariant Approach

Here we briefly sketch the argument and calculation that lead to the fully relativistic form of the inertia reaction force and that as a byproduct eliminates the bothersome 4/3 factor. In the previous sections only the contribution due to the momentum density g was included (or equivalently the Poynting vector There is however an additional contribution to the momentum p that was neglected. In order to obtain it, one has to perform a covariant generalization of the previous analysis. We cannot perform here a detailed account of it (see App. D of RH [2] for details). The analysis is a bit more easily grasped by the intuition if the momentum-content approach of 4 is used. For this reason we select in this section the momentum-content approach. The covariant extension of momentum in the radiation field is the four-vector.

Where

We see that p now carries an additional term of the form Maxwell stress tensor,

where T is the

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With i, j = x, y, z. the angular brackets as usual signify stochastic averaging. In

the

zero component we find again the energy density U (as in eqn. (8)), but there is now an additional contribution to the energy in the zero component, namely a term of the form –v.g. the

Since we are now considering a relatively small proper volume, frame we have

A detailed calculation of

written in

yields

and then

From this we may obtain the inertia reaction force of eqn. (21)

But now mi is precisely

and the factor 4/3 does not need to be artificially removed as in eqn. (10) to (12) and (21) where it was de facto removed. Of course in eqn. (30) we again introduced the factor for the fraction of the ZPF that interacts with the particles in an object. On the other hand the zero-component of the momentum, for sufficiently small proper volume can be written as

Detailed calculation (App. D of RH[2]) yields then

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A. RUEDA & B. HAISCH

is the mass found in eqn. (30).

Putting together eqns. (28), (30) and (32) we recover the conventional form of the four-momentum in relativistic mechanics, namely,

With the “inertial mass” of eqn. (30) that indeed represents the amount of electromagnetic ZPF energy inside the object volume that actually interacts with the object. From eqn. (33) we can then obtain the relativistic form of Newton’s Second Law

where the star indices have been suppressed for generality. And where p is the relativistic momentum

The origin of inertia in this picture becomes remarkably intuitive. Any material object resists acceleration because the acceleration produces a perceived and instantaneous flux of radiation in the opposite direction that scatters within the object and thereby pushes against the accelerating agent. Inertia in the present model appears as a kind of acceleration-dependent electromagnetic vacuum-fields drag force acting upon electromagnetically-interacting particles.

6. Discussion

In the Standard Model of particle physics it is postulated that there exists a scalar field pervasive throughout the Universe and whose main function is to assign mass by transferring mass to the elementary particles. This is the so-called Higgs field or more specifically, the Higgs boson, and it originated from a proposal by the British physicist Peter Higgs who introduced that kind of field as an idea for assigning masses in the Landau-Ginzburg theory of superconductivity. Recent predictions of the mass that the Higgs boson itself may have, indicate a rather large mass (more than 60 GeV) and this may be one of the reasons why, up to the present, the Higgs boson has not been observed. There are alternative theories that give mass to elementary particles without the need to postulate a Higgs field, as e.g., dynamical symmetry breaking where the Higgs boson is not elementary but composite. But the fact that the Higgs boson has not been detected is by no means an indication that it does not exist. Recall the 26 years which passed

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between the proposal by Pauli in 1930 of the existence of the neutrino and its first detection when the Reines experiment was performed. It should be clearly stated that the existence (or non-existence) of the hypothetical Higgs boson does not affect our proposal for the origin of inertia. In the standard Model attempt to obtain, in John Wheeler's quote, “ mass without mass” the issue of inertia itself does not appear. As Wilsczek [8] states concerning protons and neutrons: “Most of the mass of ordinary matter, for sure, is the pure energy of moving quarks and gluons. The remainder, a quantitatively small but qualitatively crucial remainder – it includes the mass of electrons – is all ascribed to the confounding influence of a pervasive medium, the Higgs field condensate.” An explanation of proton and neutron masses in terms of the energies of quarks motions and gluon fields falls short of offering any insight on inertia itself. One is no closer to an understanding of how this energy somehow acquires the property of resistance to acceleration known as inertia. Put another way, a quantitative equivalence between energy and mass does not address the origin of inertial reaction forces. And the manner in which, say the rest mass of the neutrino, is taken from the Higgs field, does not at all explain the inertia reaction force on accelerated neutrinos. Many physicists apparently believe that our conjecture of inertia originating in the vacuum fields is at odds with the Higgs hypothesis for the origin of mass. This happens because of the pervasive assumption that inertia can only be intrinsic to mass and thus, if the Higgs mechanism creates mass, one automatically has an explanation for inertia. If inertia is intrinsic to mass as postulated by Newton, then inertia could indeed be considered to be a direct result of the Higgs field because presumably the Higgs field is the entity that generates the corresponding mass, and inertia simply comes along with mass automatically. However, if one accepts that there is indeed an extrinsic origin for the inertia reaction force, be it the gravity field of the surrounding matter of the universe (Mach's Principle in senso stricto) or be it the electromagnetic quantum vacuum (or more generally the quantum vacua) that we propose, then the question of how mass originates – possibly by a Higgs mechanism – is a separate issue from the property of inertia. This is a point that is often not properly understood. The modern Standard Model explanation of mass is satisfied if it can balance the calculated energies with the measured masses (as in the proton) but obviously this does not explain the origin of the inertia reaction force. It is the inertia reaction force associated with acceleration that is measurable and fundamental, not mass itself. We are proposing a specific mechanism for generation of the inertia reaction force resulting from distortions of the quantum vacua as perceived by accelerating elementary particles. We do not enter into the problems associated with attempts to explain inertia via Mach's Principle, since we have discussed this at length in a recent paper [9]: A detailed discussion on intrinsic vs. extrinsic inertia and on the inability of the geometrodynamics of general relativity to generate inertia reaction forces may be found therein. It had already been shown by Rindler [10] and others that Mach's Principle is inconsistent with general relativity, and Dobyns et al [9] further elaborate on a crucial point in general relativity that is not widely understood: Geometrodynamics merely defines the geodesic that a freely moving object will follow. But if an object is constrained to follow some different path, Geometrodynamics has no mechanism for creating a reaction force. Geometrodynamics has nothing more to say about inertia than does classical Newtonian

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physics. Geometrodynamics leaves it to whatever processes generates inertia, to generate such a force upon deviation from a geodesic path, but this becomes an obvious tautology if an explanation of inertia is sought in Geometrodynamics. We would like to point out that Mach’s Principle in senso stricto is, as described above, the hypothesis that inertia here is due to the overall matter in the distant Universe that produces a net gravitational effect so that the inertia reaction force is generated on an accelerated object by the gravitational field of all the Universe. This Mach’s Principle in a strict sense is not compatible with our proposal that inertia is generated by the vacuum fields [9]. However, a more broad interpretation of some of Mach’s ideas is the view that inertia is not just inherent to mass but due to an external agent that acts on the accelerated massive object. Such agent is different from the accelerated massive object itself and should reside in the external Universe. This view is then perfectly compatible with the view that we propose, namely that the vacuum fields are the entities responsible for producing, on the accelerated massive object, the inertia reaction force. We finally, acknowledge that Newton’s proposal that inertia is intrinsic to mass looks, superficially at least, more economical (Occam’s razor) but it is also oversimplistic as one may always continue asking for a deeper reason for the operation of physical processes or for more fundamental bases for physical laws. The question of why the mass associated with either matter or energy should display a resistance to acceleration is a valid question that needs to be addressed even if the Higgs boson is experimentally found and confirmed as the origin of mass.

Acknowledgement We acknowledge NASA contract NASW 5050 for support of this research. AR acknowledges additional support from the California Institute for Physics and Astrophysics (CIPA).

References 1. M. Jammer, “ Concepts of Mass in Contemporary Physics and Philosophy” , Princeton University Press (2000). 2. A. Rueda and B. Haisch, Foundations of Physics 28, 1057 (1998). 3. A. Rueda and B. Haisch, Physics Lett. A 240, 115 (1998). 4. B. Haisch, A. Rueda and H. E. Puthoff, Phys. Rev. A 49, 678 (1994). 5. See pg. 166 of Ref. [1]. For a whole discussion see pp. 163-167 of Ref. [1]. 6. A. Einstein, Foreword in M. Jammer “Concepts of Space” (Harvard Univ. Press, 1954 or Dover, New York, 1993) p xvii. 7. J.-P. Vigier, Found. Phys. 25, 1461 (1995). 8. F. Wilsczek, Physics Today, Nov. 1999 p 11 and Jan. 2000 p.13. 9. Y. Dobyns, A. Rueda and B. Haisch, Found. Phys. (2000), in press. 10. W. Rindler, Phys. Lett. A 187, 236 (1994) and Phys. Lett. A 233, 25 (1997).

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TREVOR W. MARSHALL Dept. of Mathematics, Manchester University, Manchester M13 9PL

Abstract. As a continuation of the programme reported by me to the first Vigier conference, I report that a theory of nonlinear optical phenomena, based on a real zeropoint or “vacuum” electromagnetic field, has successfully explained a wide range of allegedly nonlocal experimental data. The same theory is seen to be capable of making quite accurate predictions of new phenomena not foreseen by the prevailing photon theory of Quantum Optics, as well as having some important engineering applications.

1. The vacuum is full of light waves

”Vacuum” is the name commonly given to a region of space from which all atoms and all ”photons” have been extracted, the latter by cooling to zero Kelvin. It is a bad name, because we now know that this ”vacuum” actually contains a lot of radiation, known as the Zero Point Field (ZPF). The ZPF [1, 2] was a hypothesis put forward by Max Planck in 1911, and developed, by him and Walther Nernst, between 1911 and 1916, because they did not believe the phenomena of light emission and absorption could be adequately explained by Einstein's hypothesis of ”light quanta”, which subsequently became photons. In 1947 and 1948 the effect of the ZPF was directly demonstrated in the Lamb shift and the Casimir effect. After more than 50 years these very clear demonstrations of the ZPF go unacknowledged. Meanwhile the very objections, made by Planck to the photon hypothesis, have been transformed, by its enthusiasts, into what they consider to be great achievements. They claim to have proved that our world is ”nonlocal”, when all they have actually done is demonstrate that the photon description is nonlocal. For a discussion of the experiments purporting to have proved nonlocality, notably the atomic-cascade experi459 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 459-468 © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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ments of Clauser and Freedman and of Aspect, see my article for the first Vigier congress [3] and also my web page[4]. Thanks to the techniques of nonlinear optics, the last 50 years have witnessed great technical achievements which rely on the ZPF, but, as with heat engines two hundred years ago, our advance in engineering has oustripped our scientific understanding. The first, and most spectacular achievement is the laser. Then, with the intervention of nonlinear crystals, we have seen that certain modes of the ZPF may be squeezed below their normal (that is Uncertainty-Principle) amplitude. Also, certain pairs of ZPF modes may be amplified in a correlated manner, when they are nonlinearly coupled to a pumping laser. This phenomenon is known as Parametric Down Conversion (PDC). The description of PDC correlations offered by photon theorists hardly qualifies as an explanation, and, I shall argue, has deservedly given the physics of the latter half of the last century a reputation for mysticism. On the other hand, the ZPF description is now sufficiently advanced, not only to explain in detail the experimental data of PDC, but also to predict a new phenomenon, called Parametric Depletion of the Vacuum (PDV). Willis Lamb[5] said, in 1996, In 1947 I proved the vacuum does not exist. He was thereby associating himself with the point of view I have just expressed. The ZPF is real and fills the “vacuum”, which we should now really call a plenum. Schematically his experiment of 1947 established that

According to the ZPF description each mode of the vacuum electromagnetic field has an average energy of At first glance this is not too different from the description of Quantum Electrodynamics (QED), but there is a crucial difference. In the ZPF description this mean energy is subject to fluctuations, so that it has a standard deviation, also of Thus this “ground state” of the radiation field is not to be interpreted as an energy eigenstate. More radically we must recognize that such a continuum of energies is incompatible with the concept of a photon, since there is no way in which a quantum of superposed on such a ZPF background can give anything other than a continuum of energies in all modes of the field. The simplistic, that is corpuscular, notion of the photon underlies all of the magical “nonlocal” interpretations of optical phenomena, from the Clauser-Freedman experiment mentioned above to such exotic recent arrivals as teleportation[6]. The incompatibility I speak of has been long recognized by exponents of the ZPF. For example Planck said, in a letter to Einstein in 1907[7]

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I am not seeking the meaning of the quantum of action (light quantum) in the vacuum but rather in places where emission and absorption occur, and I assume that what happens in the vacuum is rigorously described by Maxwell’s equations. More recently Willis Lamb[8], in an article entitled Antiphoton, made an essentially similar point. 2. Parametric down conversion (PDC)

PDC is a process, depicted in Fig.l, in which a (very small) percentage of

photons from a laser are converted, by a nonlinear crystal (NLC), into photon pairs Different photons emerge in different directions and the relationship between and is determined by the requirements of energy and momentum conservation, after taking account of the different refractive indices of the various photons within the crystal. Where any photon comes from is already a mystery. We need only consult[9] Feynman's discussion with his own father Father Atoms emit photons. Right? Son Yes. Father So do atoms contain photons?

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Son No. Not exactly. Father So where do photons come from? Son We don't know. One could pose the same questions about the photons that come out of a nonlinear crystal. The difference is that in this case, we are speaking of a macroscopic object, whose diameter is about 1cm, instead of an atom. The macroscopic nature is further emphasized by our ability, literally, to cook the PDC rainbow by changing the temperature of the crystal, thereby changing and That in turn changes Every “photon” is produced by the entire crystal. Furthermore, I emphasize that the process of photon cookery, as well as being macroscopic, is very much a disordering one. As such it is not at all like what occurs in a superconductor. 3. The mind boggling experiment

In 1991 Wang, Zou and Mandel (WZM) performed an experiment[10] which they dubbed Induced coherence without induced emission. My colleagues and I [11, 12, 13, 14, 15, 16, 17] have published a series of articles showing that all of the mystifying results claimed by photon theorists have a ready explanation once one recognizes the reality of the ZPF. I believe the WZM experiment deserves singling out for two reasons; it has been so singled out by the photon theorists themselves, and also the apparently paradoxical result, once it has been demystified, is the basis of a new prediction which, once verified, will give a convincing new demonstration of the ZPF. Greenberger, Home and Zeilinger[18] call the WZM experiment “mind boggling”, which may be translated as “mind frightening”. Reading their article one cannot help wondering whether they believe it is the actual experimental data that should frighten us, or whether perhaps, after the manner of Uri Geller, they think we should be frightened by the magical powers possessed by WZM. The title of the article is Two photon interference, and in view of Dirac’s[19] famous statement about photons never interfering with each other, it indicates something of the boggling situation in which they find themselves. The WZM experiment is depicted in Fig.2. Photons from a laser v are separated so that they go either into channel or into Some of those going into undergo PDC (see Fig.l) at the nonlinear crystal NLC1 and down convert into while some of those going into down convert at NLC2 into WZM observe interference fringes between and by varying the two optical paths from the crystals to the beam splitter BS2, where these two signals are combined. This may be done, for example, by varying the orientation of the phase shifter PS thereby producing a

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variation in the counting rate at the detector D. When WZM replaced PS by an absorber, the fringes disappeared. The “explanation” of these results offered by WZM is that and interfere because they are indistinguishable. We cannot tell which path a given photon detected at D has travelled in order to arrive there. Indeed, it is an important condition of the experimental arrangement that the unobserved (so called “idler”) photons, and are made indistinguishable by very careful alignment of NLC2 with NLC1. But, so the “explanation” continues, replacement of PS by an absorber renders and distinguishable. We could, if we wished, put a second detector in channel if it clicked we would know that PDC had occurred in NLC2, and if it did not we would know it had occurred in NLC1. This is an example of a kind of observer subjectivism inevitable in any purely quantum mechanical (that is ignoring the presence of fields) analysis. Not only does measurement cause a “collapse” of an information wave function; the possibility of making such a measurement also causes collapse. Incidentally, another feature of this type of argumentation is a kind of schizophrenia which finds difficulty in distinguishing real from thought experiments. Only in the purest fantasy do detectors of the above either/or type actually “exist”. 4. Parametric Amplification of the Vacuum

To begin the process of demystification we have to return to Fig.1, and recognize the active role played in the phenomenon known as PDC by the zero point field. I claim that the failure to understand PDC is so deep, among the school of photon theorists, that the phenomenon should really be

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given a new name. I call it Parametric Amplification of the Vacuum (PAV) in recognition of the fact that, not only would there be no phenomenon at all if the “vacuum” were truly vacuous, but also it is not correct to say that a laser photon “down converts”; rather we should say that a laser mode (since there are no photons) interacts with a ZPF mode to give a signal mode at the difference frequency. Of course, it is precisely at this point that the nonlinearity of the crystal plays a crucial role. The PAV process is depicted in Fig.3. It is really two independent

processes. In one of them a mode of frequency together with the laser mode of frequency polarizes the crystal so that it vibrates at frequency The polarization current radiates the signal at this latter frequency, and the inducing mode has its amplitude modified to In the second process the roles of signal (s) and idler (i) are reversed. The amplitudes of the outgoing modes may be calculated by the techniques of classical nonlinear optics[20]. The modes which are substantially modified are those for which the partial waves from the various parts of the crystal interfere constructively. This gives rise to Phase Matching Conditions, which, apart from a cancelling factor of are identical with the energy and momentum conservation conditions given with Fig.1. We have shown that and have intensities greater than and It may be assumed, as explained in Ref.[3], that only intensities above the zeropoint level give rise to detection events, which is why we see strongly correlated events in the two outgoing channels. We have here a causal de-

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scription of the process known as PDC; the incoming laser field, together with the ZPF, polarizes the crystal, and the resulting polarization current radiates into a mode whose frequency is the difference of the frequencies of the exciting modes. 5.

The mind unboggled

We now apply the realist analysis of the previous section to the Mind Boggling Experiment of WZM. The vital missing link required to progress from the nonlocal “explanation” offered previously is the component of the ZPF added in Fig.4. The field v is split into mutually coherent

components which go into both channels. These interact with the same component of the ZPF at the two crystals NLC1 and NLC2 to give coherent outputs When PS is replaced by an absorber, then NLC1 and NLC2 still have coherent and incident on them, but the ZPF components, and are independent. So and are then mutually incoherent. Again we must change the name given by photon theory to a phenomenon; instead of the name given to the Mind Boggler by its discoverers we should call it Induced emission with induced coherence. The alignment of NLC1 and NLC2, may, as they said, ensure indistinguishability of the photons and but that has nothing to do with the explanation of the phenomenon. Physics is a science, and it requires causality, not teleology. The real achievement of ZWM’s alignment of the crystals was in ensuring that the same component of the ZPF interacted with both and inducing thereby both the emission and the coherence.

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6. New Science

In the sequence of articles I cited above[3, 11, 12, 13, 14, 15, 16, 17] we have shown that the whole family of allegedly nonlocal optical phenomena, from atomic-cascade coincidences to teleportation, has a natural explanation in terms of a wave theory of light which is simply the one bequeathed to us by Fresnel, Faraday and Maxwell, the only novel feature being the recognition of Planck’s zeropoint field. I claim that we have thereby achieved the only convincing explanation of all these phenomena, and it is only because, in the twentieth century, physicists abandoned their duty, as scientists, to find explanations that our results have received so little recognition. It was the received wisdom of that century (thankfully we are now in a new one!) that science should concern itself only with prediction, and that explanation belonged to a bygone immature phase. Fortunately biologists took no notice of this message from the “senior” science. Nevertheless, prediction has its place, so I will now describe a fairly straightforward application of our theory leading to a striking new prediction, which I think could not have been made from the current photonbased version of Quantum Optics. We just saw that the process, within the crystal, which produces the PDC signal is the polarization current at the difference frequency Now another polarization current at the frequency also exists within the crystal. Indeed, in the case that the mode has an intensity comparable with that of the pump this sum mode produces an “up-conversion” signal which has already been observed [20]. Now the calculation of the expected intensity, when this second mode is reduced to its zeropoint level, is a straightforward extension, and the question of whether a detectable signal emerges is settled by calculating whether the mode intensity is or is not amplified above the zeropoint level. I have done this calculation, and reported its results[21]. As in the case of PAV, described above, we have to study two processes together, namely

We find that the ZPF mode with the higher frequency, that is actually has its intensity reduced below the vacuum level. I propose to call this Parametric Depletion of the Vacuum (PDV); it is somewhat different from squeezing, since in this case both quadrature modes are reduced in intensity, while in squeezing one of them is reduced at the expense of the other. Such a phenomenon will not easily be demonstrated experimentally, but the other mode, that is is amplified. So we need to know where to look for and what intensity to expect. I have published sufficient details of this on

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the Los Alamos archive, as cited above, so all you have to do is look at the predicted angles and you will see the phenomenon. 7. New technology?

I think that, once the ZPF has been fully accepted as a real physical object, it will have an engineering, as well as a scientific impact. A prime candidate for this is the field of laser design. To emphasize the relevance of the ZPF in a laser, I point out that a laser is a device which selectively amplifies certain modes of the ZPF. Indeed the acronym “laser” stands for light amplification by stimulated emission of radiation, and it is precisely the ZPF which does the stimulating! It is no accident of history that effectively the first working laser (called, at that time a maser) was the apparatus used by W. E. Lamb and colleagues in 1947 to measure the Lamb shift. As we have seen, Willis Lamb is outstanding, among the pioneers of Quantum Electrodynamics, in his recognition both of the importance of the ZPF and of the inadequacy of the Photon concept. Lasers are extremely inefficient heat engines. They convert incoherent (that is thermal) light energy from the pump into a coherent output (“work”), but typically, in a narrow-band continuous-wave laser, about 10 kilowatts of pump power is required to produce 500 milliwatts of coherent output. Good coupling of the pumped atoms to the ZPF, both outside and inside the laser, is achieved by good cavity design, and I think the standard Fabry-Perot cavity can be immensely improved, once we have understood how the ZPF may be engineered. There is a historical parallel here. Heat engines played a central role in the Industrial Revolution between 1770 and 1850, but scientists at that time accepted a deeply incorrect theory of heat; it was thought to be a material substance called “caloric”. Then Thermodynamics was correctly formulated and heat engines became even better, culminating in the automobile. We have made a lot of progress with lasers since 1947, even though our theory of Optics has become dominated by mythical objects called “photons”. The time has come, when we should recognize, in the zeropoint field of Max Planck, a new branch of Thermodynamics, that of zero degrees Kelvin. 8.

Theses on Magic

No competent magician believes in magic. A competent magician is really a scientist pretending to be a magician. A competent magician is one who knows how the rabbit got into the hat ...

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. . . and to distinguish between a scientist and a competent magician is a problem in ethics rather than science. References 1. L. de la Peña and A. M. Cetto, The Quantum Dice, (Kluwer, Dordrecht, 1996) 2. P. W. Milonni, The Quantum Vacuum, (Academic, San Diego, 1993) 3. T. W. Marshall and E. Santos, The myth of the photon in The Present Status of the Quantum Theory of Light, eds. S. Jeffers et al, (Kluwer, Dordrecht, 1997) pages 67–77. 4. T. W. Marshall, www.demon.co.uk and homepages.tesco.net/~trevor.marshall 5. W. E. Lamb, interview in La Nueva España, Oviedo, Spain (July, 23, 1996) 6. D. Bouwmeester and A. Zeilinger, Nature 388, 827-828 (1997) G. Weihs, T. Jenneswein, C. Simon, H. Weinfurter and A. Zeilinger, Phys. Rev. Lett, 81, 5039 (1998) 7. A. Pais, Subtle is the Lord, page 384 (Clarendon, Oxford, 1982) 8. W. E. Lamb, Appl. Phys. B, 60, 77-82 (1995) 9. R. P. Feynman, You must be joking Mr. Feynman 10. L. J. Wang, X. Y. Zou and L. Mandel, Phys. Rev. A, 44, 4614 (1991) 11. A.Casado, T.W.Marshall, and E.Santos, J. Opt. Soc. Am. B, 14, 494–502 (1997). 12. A.Casado, A.Fernández Rueda, T.W.Marshall, R.Risco Delgado, and E.Santos, Phys.Rev.A, 55, 3879–3890 (1997). 13. A.Casado, A.Fernández Rueda, T.W.Marshall, R.Risco Delgado, and E.Santos, Phys.Rev.A, 56, R2477-2480 (1997) 14. A. Casado, T. W. Marshall and E. Santos, J. Opt. Soc Am. B, 15, 1572-1577 (1998) 15. A.Casado, A.Fernández Rueda, T.W.Marshall, J. Martinez, R.Risco Delgado, and E.Santos, Eur. Phys. J., D11,465 (2000) 16. K.Dechoum, T. W. Marshall and E. Santos, J. Mod. Optics, 47, 1273 (2000) 17. K. Dechoum, L. de la Peña and E. Santos, Found. Phys. Lett., 13, 253 (2000) 18. D. M. Greenberger, M. A. Home and A. Zeilinger, Phys. Today, 46 No.8, 22 (1993) 19. P. A. M. Dirac, Principles of Quantum Mechanics, page 9 (Clarendon, Oxford, 1958) 20. B.E.A.Saleh and M.C.Teich, Fundamentals of Photonics, (John Wiley, New York, 1991) 21. T. W. Marshall, http://xxx.lanl.gov/abs/quant-ph/9803054.

THE PHOTON AS A CHARGE-NEUTRAL AND MASS-NEUTRAL COMOSITE PARTICLE Part I. The Qualitative Model

HECTOR A. MUNERA Department of Physics Universidad National de Columbia A.A. 84893, Bogota, Columbia

Keywords: Photon, Aether, Composite photon model, Maxwell equations, Advanced fields, Charge neutral particles, Mass neutral particles. Abstract. In the context of a 4D aether model, where rest mass is associated with a flow of primordial mass (preons), the photon was described as an electron-positron pair. Such a composite particle is then a charge-neutral and mass-neutral entity; thus accounting for photon standard properties: zero charge and null restmass. The electromagnetic field of such photons contain both advanced and retarded components, without any causality breach. The model obeys conventional Maxwell equations.

1. Introduction

The idea that the photon may be a composite particle is not new. Long ago De Broglie (1932) suggested that the photon was a composite state of a neutrino-antineutrino pair; such pair, however, did not obey Bose statistics. To avoid this difficulty Jordan (1935) introduced neutrinos with different momenta. Over the years, additional adjustments were made by Kronig (1936), Pryce (1938), Barbour et al. (1963), Ferreti (1964), Perkins (1965), Bandyopadhyay and Chaudhuri (1971). Since the photon rest-mass is zero, or very small (Vigier, 1997), neutrinos are chosen as its components. However, 469 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 469-476 © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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one would naïvely expect that if the photon is a composite particle it may under some conditions decay or be separated into its components. Indeed, photon pair production leads to an electron-positron pair, but not to a neutrino-antineutrino one. However, from the viewpoint of total rest mass, an electron plus a positron can not be the components of a low energy photon. There is a clear difficulty for composite models along this line of thought. On the other hand, the idea that the modern “vacuum” (= aether in this paper) is of hydrodynamic nature is a recurrent one. Recent examples: a superfluid of particle-antiparticle pairs (Sinha et al., 1976), a fluid of “stuff” particles (Di Marzio, 1977), and a variety of fluids (Shekhawat, 1976; Widom and Srivastava, 1990; Winterberg, 1997; Ribaric and Sustersik, 1997). From such fluids, some authors derive electrodynamic and particle models: Thomson (1931), Hofer (1998), Marmanis (1998), Dmitriyev (1999). This author recently proposed a four dimensional (4D) hydrodynamic model that allows for a variable component of the 4-velocity along the time axis (Munera, 1999), which leads to a 4D-force as the gradient of the 4pressure; the 3D-electromagnetic force is a particular case (Múnera, 2000). Also, we have argued elsewhere (Chubykalo, Múnera and Smirnov-Rueda, 1998) that, in the context of Maxwell’s equations, the concept of zero charge in vacuum may be interpreted as neutrality of charge almost everywhere, rather than as complete absence of charge. In this note we take one step further. The photon is modelled as a source-sink pair (into and out of our 3D world), having a zero net mass flow into our 3D world, thus accounting for the photon zero rest mass. Next section 2 summarizes the 4D aether model, and section 3 sketches the photon model. A final section 4 closes the paper.

2. A Four Dimensional Fluid

Let us assume the existence of a four-dimensional (4D) flat Euclidean space space where the time dimension behaves exactly the same as the 3 spatial dimensions (Múnera, 1999; 2000). Further, let be filled with a fluid of preons (= tiny particles of mass m and and Planck length dimensions). These particles are in continual motion with speed No a priori limits are set on the speed of preons along the w-axis. (Notation: 4D-concepts and vectors are represented either by calligraphic or by Greek uppercase letters, while 3D-vectors are in the usual bold face). Note that the limitations of the special theory of relativity (STR), if applicable, refer to (= the speed of particles in 3D) not to = the projection of the 4D-velocity on the w-axis. Here, we extend the

PHOTON AS COMPOSITE PARTICLE

notion of absolute space to

471

whereas the spacetime of STR is

(ct,x,y,z), i.e.

Motion of individual preons in is governed by a 4D-equation of motion, given by the matrix expression (Múnera, 1999):

where is the preonic fluid mass density, n is the number of preons per unit 3D-volume, the column vector is the 4D-velocity of individual preons, refers to the time-arrow, the vector operator is a 4D gradient, the 4×4 matrix is the 4Dstress tensor, and is the pressure generated by the preonic fluid; the Greek index Finally, the energy-momentum tensor 4×4 matrix) results from the dyadic product Consider now a 3D-hypersurface formed by a projection of the 4Duniverse onto the w-axis, say (Fig. 1). The plane w-r may be interpreted in two complementary ways: Interpretation 1 (Fig. 1a). At a fixed time (say the present), the line divides the plane into three classes of particles: preons moving with (upper region), preons moving with (lower region), preons moving with (on the horizontal line). Interpretation 2 (Fig. 1b). For the class of preons moving with the line divides the plane into three periods of time: the future for (upper half-plane), the past for (lower half-plane), the present (on the line). The conventional worldlines of STR and the space underlying Feynman diagrams belong to interpretation 2 with unspecified. If we postulate that we live in a 3D-hypersurface where then all preons in our world move with constant speed c along the time axis. This brings in a novel intrpretation for constant c: the speed at which our hypersurface slides from the past to the future (Interpretation 2 above). The meaning of the w-r plane under Interpretation 1 can now be rephrased as: at a given (say the present) our 3D-world separates superluminal from subluminal preons. Furthermore, as seen below, there is a continuous exchange of preons between our hypersurface and the two half-spaces above and below. For events inside our hypersurface, eq. (1) reduces to

472

where the elements of given by the conventional the w-dimension are

HECTOR A. MUNERA

associated with the 3D spatial dimensions are viscosity matrix. The elements associated with

where is a (displacement) energy flux along axes x,y,z (dimensions: energy per unit time per unit area), and the source/sink is a concentrated energy flow along the w-axis (dimensions: energy per unit time), and represents the position of the energy source/sink (positive/negative respectively), and is a 3D-Dirac's delta function (dimensions: Eq. (3) may be interpreted as a transfer of energy by displacement from the waxis into the spatial axes (or the other way around), whereas eq. (4) is a transfer of energy along the w-axis. Therefore, the 4D-source simply represents a "convective" transfer of preons from one region of the 4Dfluid into another, i.e. there is conservation of energy in the whole 4Duniverse. Note that other fluid theories contain expressions similar to our eq. (2) (for instance, eq. 3 in Ribaric and Sustersic, 1998). However, our approach is fundamentally different because we allow for interaction between our world and other regions of with (described by the more general eq. 1). This interaction gives rise to the 4D-source 5 described by eqs. (3) and (4). By analogy with the standard 3D-case, the 4D-preonic fluid exerts force, and performs work along the four dimensions (w,x,y,z), via its hydrodynamic pressure P (in this sense, P is interpreted as potential energy per unit volume). This immediately leads to a component of force along the w-dimension, which is responsible for the appeareance of sources and sinks in our hypersurface, via the following mechanisms: Sources are produced by the fourth component of force, which acts upon preons outside our hypersurface, via two mechanisms (Fig. 2a): preons moving with are decelerated to enter our world at with preons moving with are accelerated to enter our world at with

PHOTON AS COMPOSITE PARTICLE

473

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HECTOR A. MUNERA

Sinks are produced by the fourth component of force, which acts upon preons in our hypersurface that move with for Two mechanisms (Fig. 2b): acceleration of preons which leave our world at with and deceleration of preons which leave our world at with For additional details see Múnera (1999; 2000). 3.

The Photon As A Source-Sink Dipole

It is widely known that Maxwell equations (and also the homogeneous wave equation) have two sets of solutions: retarded (= outgoing) and advanced (= incoming) ones. The latter are typically neglected on the grounds of causality violations (for instance, Panofsky and Phillips, 1962, p. 244). To account for the advanced solutions in the presence of charges, Wheeler and Feynman (1945, 1949) located an absorber of radiation at some distance from a charge (see also Panofsky and Phillips, 1962, ch. 21). In the model for the photon described next, we also allow for advanced and retarded solutions without causality violations. In the 4D aether described in previous section, a particle (antiparticle) is a source (sink), whose rest mass is proportional to Energy is transported into (or out of) our 3D world by a flow of preons. Such representation immediately explains away the difficulties associated with infinities in potential energy (gravitational, electrical, or otherwise) Let the photon be a pair electron-positron, both of them inside a small region of diameter Since the rest masses of the constituent particles are identical, then which implies that the effective rest mass of the composite particle is zero. Evidently, the net charge is also null, except inside a neighbourhood A simple 3D-analogy is a water filled vessel, with a source and a sink allowing equal flows of water in and out of the vessel. Consider a differential volume located at an arbitrary point P connected to by a line of flow. Water particles emitted from at reach P at a later time according to the speed of propagation of the particle, thus giving rise to a retarded field of pressure. Likewise, consider a particle that is absorbed by the sink at a time along a line of flow coming from P. Of course, such a particle was at P at an earlier time, and gives rise to an advanced pressure field. However, there is no causality breach. Simply put, some of the particles located at a given time inside the small volume at P may come from the source, some may reach the sink at a later time, and many other particles may have quite a different fate.

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475

The photon is then a rotating dipole in the normal 3D Euclidean space. Let the time-dependent retarded (and advanced) electric field associated with the electron (positron) be N (P). It can be shown that fields N and P obey the symmetric Maxwell’s equations that we described some time ago (Múnera, 1997). Such symmetric system is tautologically equivalent to the conventional Maxwell equations, provided that the standard electric and magnetic fields be defined as:

4.

Concluding Remarks

In the context of a 4D aether model, where rest mass is associated with a flow of primordial mass (preons), the photon was described as an electronpositron pair. Such composite particle is then a charge-neutral and massneutral entity, thus accounting for the photon standard properties: zero charge and null rest mass. The electromagnetic field of such photon contains both advanced and retarded components, without any causality breach. The model leads to a symmetric system of Maxwell’s equations (Múnera, 1997), containing two sources (electron and positron). Such system immediately leads to the conventional Maxwell equations (the details will be published elsewhere). Falaco solitons were recently reported (Kiehn, 2000) as pairs of solitons that exist on the surface of a fluid (water), and are interconnected through the third spatial dimension. Our model for the photon is a pair of 3D solitons interconnected through the fourth dimension. An open question to be pursued at a later stage is the connection between the equation of motion describing the 4D aether proposed here and the BohmVigier (1958) relativistic hydrodynamics.

References Bandyopadhyay, P. and P. R. Chaudhuri, “The photon as a composite state of a neutrinoantineutrino pair”, Phys. Rev. D 3, No. 6 (1971) 1378-1381. Barbour, I. M., A. Bietti, and B.F. Toushek, Nuovo Cimento 28 (1963) 453. Bohm, D. and J.-P. Vigier, “Relativistic hydrodynamcis of rotating fluid masses”, Phys. Rev. 109, No. 6(1958) 1882-1891.

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Chubykalo, A. E., H, A. Múnera, and R. Smirnov-Rueda, “Is the free electromagnetic field a consequence of Maxwell’s equations or a postulate?”, Found. Physics Lett. 11, No. 6 (1998) 573-584. De Broglie, L, Compt. Rend. 195 (1932) 862; Compt. Rend. 199 (1934) 813. Di Marzio, E. A., “A unified theory of matter. I. The fundamental idea”, Found. Physics 7 (1977) 511-528. “II. Derivation of the fundamental physical law”, ”, Found. Physics 7 (1977) 885-905. Dmitriyev, V. P., “Turbulent advection of a fluid discontinuity and Schrödinger mechanics”, Galilean Electrodynamics 10, No. 5 (1999) 95-99. Ferreti, B., Nuovo Cimento 28 (1964) 265. Hofer, W. A., “Internal structures of electrons and photons: the concept of extended particles revisited”, PhysicaA 256 (1998) 178-196. Kiehn, R. M., paper at Vigier 2000 Symposium, University of California, Berkeley, USA (August 2000). Kronig, P. Physica 3 (1936) 1120. Marmanis, H., “Analogy between the Navier-Stokes equations and Maxwell's equations: Application to turbulence”, Phys.Fluids 10, No. 6 (1998) 1428-37. “Erratum”, Phys.Fluids 10, No. 11 (1998) 3031. Múnera, H. A. “A symmeric formulation of Maxwell’s equations”, Mod. Phys. Lett. A 12, No. 28 (1997)2089-2101. Múnera, H. A. “A realistic four-dimensional hydrodynamic aether interpreted as a unified field equation”, presented at the International Workshop Lorentz Group, CPT and Neutrinos, Universidad Autónoma de Zacatecas, Zacatecas, Mexico (June 1999). Published in the Proceedings edited by A. Chubykalo, V. Dvoeglazov, D. Ernst, V. Kadyshevsky and Y.S. Kim. Múnera, H. A. “An electromagnetic force containing two new terms: derivation from a 4D aether”, Apeiron 7, No. 1-2 (2000) 67-75. Panofsky, W. K. H., and M. Phillips, Classical Electricity and Magnetism , 2nd edition, Addison-Wesley Publishing Co. (1962) 494 pp. Perkins, W. A., Phys. Rev. 137 (1965) B1291. Pryce, M.H.L., Proc. Roy. Soc. (London) 165 (1938) 247. Ribaric, M. and L. Sustersik, “Transport theoretic extensions of quantum field theories”, Eprint archive: hep-th/9710220 (Oct. 97) 36 pp; “Framework for a theory that underlies the standard model”, LANL electronic file hep-th/9810138 (Oct. 1998). Shekhawat, V., “Some preliminary formulations toward a new theory of matter”, Found. Physics 6 (1976) 221-235. Sinha, K. P., C. Sivaram, and E. C. G. Sudarshan, Found. Physics 6, No. 1 (1976) 65-70. Thomson, J. J., “On the analogy between electromagnetic field and a fluid containing a large number of vortex filaments”, Phil. Mag.. Ser. 7 12 (1931) 1057-1063. Vigier, J.-P., “Relativistic interpretation (with non-zero photon mass) of the small ether drift velocity detected by Michelson, Morley and Miller”, Apeiron 4, No. 2-3 (1997) 71-76. Widom, A. and Y. N. Srivastava, “Quantum fluid mechanics and quantum electrodynamics”, Mod. Phys. Lett. B 4 (1990) 1-8. Wheeler, J. A. and R. P. Feynman, Revs. Modern Phys. 17 (1945) 157, and Revs. Modern Phys. 21 (1949) 425. Winterberg, F. “Planck aether”, Zeistch. für Naturforsch. 52a (1997) 185.

PREGEOMETRY VIA UNIFORM SPACES

W.M. STUCKEY and WYETH RAWS Department of Physics & Engineering Elizabethtown College Elizabethtown, PA 17022

Abstract. We begin by motivating a pregeometric approach to quantum gravity. A pregeometry is then introduced over denumberable sets which employs the discrete uniform space and a uniformity base induced by a topological group. The entourages of provide a non-metric notion of locality consistent with the open balls of a differentiable manifold, thereby supplying a pregeometric basis for macroscopic spacetime neighborhoods. Per the underlying group structure, entourages of provide a pregeometric model of quantum non-locality/non-separability, thereby supplying a pregeometric basis for microscopic spacetime neighborhoods. In this context, a robust pregeometric correspondance between microscopic and macroscopic spacetime structures is enumerated. Finally, we show how the pseudometric generated by a uniform space may be used to provide the M4 geodetic structure. This affine structure may produce a metric structure by requiring the covariant derivative annihilate the spacetime metric per standard Riemannian geometry. An example on is provided.

1. Introduction There is as yet no definitive course of action in the search for a theory of quantum gravity (QG). Since QG must satisfy the correspondence principle with general relativity (GR) and quantum mechanics (QM), it must provide a basis whence the locality of GR and the non-locality of QM. Demaret, Heller, and Lambert write [1], In this Section we analyse fundamental concepts of quantum mechanics. We show that they lead to some problems which force us to modify the usual notion of spacetime. ... The second problem is related to the famous E.P.R. paradox which introduces the idea of non-locality or more precisely of non-separability with respect to space. In fact, in quantum mechanics space cannot be viewed as a set of isolated points. These problems lead to a deep modification of our representation of “quantum” space-time. Stuckey has argued [2] that a non-local, reductive model of reality precludes the fundamental use of trans-temporal objects. Since trans-temporal objects are fundamental to kinematics and kinematics is fundamental to dynamics, concepts such as mass, momentum, and energy are excluded from the foundation of a rational reductionist theory of QG. It is difficult to imagine where to begin modeling reality without reference to the 477 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 477-482. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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concepts of dynamics. Weinberg writes [3], “How can we get the ideas we need to formulate a truly fundamental theory, when this theory is to describe a realm where all intuitions derived from life in space-time become inapplicable?” Accordingly, QG may require an approach a la Wheeler’s pregeometry. That is [4], “the features of the conventional space-time, such as its continuity, dimensionality, and even causality and topology, should not be present from the beginning, but should emerge naturally in the transition process from pregeometry to the usual space-time dynamics of our conventional physical theories.” Further, it may behove us to base the program in discrete mathematics. Butterfield and Isham write [5], “For these reasons, a good case can be made that a complete theory of quantum gravity may require a revision of quantum theory itself in a way that removes the a priori use of continuum numbers in its mathematical formalism.” Au writes [6], “One can see how a discrete theory could reduce to a continuum one in the large scale limit, but to shed light on a discrete theory while working from the perspective of a continuum one seems difficult to achieve.” And Sorkin writes [7], “The dynamical principles learned from quantum mechanics just seem to be incompatible with the idea that gravity is described by a metric field on a continuous manifold.” Thus, we are motivated to search for QG using pregeometry based in denumerable sets. 2. The Model

We are guaranteed that a uniform space U may be constructed over any denumerable set X by introducing a group structure and the discrete topology over X. And, the discrete uniform space induces the discrete topology over X while its entourages provide a conventional, but non-metric, definition of a ball centered on Thus, induces the topology required for U while providing a pregeometric definition of macroscopic spacetime neighborhoods. Given that the introduction of a group structure over X underlying provides a uniformity base for U, we have the means to define microscopic spacetime neighborhoods independently of, but consistently with, macroscopic spacetime neighborhoods. Specifically [8], for x and y elements of X, a symmetric entourage V is a subset of X x X such that for each is also an element of V. is the collection of all symmetric entourages. For the distance between x and y is said to be less than V. The ball with center x and radius V is

and is denoted

B(x, V). A neighborhood of x in the topology induced by is Int B(x,V), so all possible balls about each are established. This is precisely in accord with the conventional notion of locality, i.e., open balls about elements of the spacetime manifold. Therefore, B(x, V) is a perfect pregeometric definition of a macroscopic spacetime neighborhood of x for denumerable X (cf. Sorkin's finitary topological spaces [9]). In section 4, we will show that this definition of macroscopic spacetime neighborhoods accommodates the topological priority of causal chains over metric balls per Finkelstein [10]. To show that the introduction of a group structure G over X underlying allows for the construct of U, we construct its uniformity base via neighborhoods of the identity e of G in the following fashion [11]. The entourage of U is

PREGEOMETRY VIA UNIFORM SPACES

479

where 2 is a neighborhood of e in the topology over X. When X is

denumerable of order N,

is partitioned equally into the

entourages

for the N - 1, order-two neighborhoods of e, i.e.,

is generated by {e, x}. The entourages

and

constitute a base

for U. Entourages generated by larger neighborhoods of e are given by members of i.e., {e, x, y} generates etc. While for some group structures all members of are elements of e.g. the such that Klein 4-group [12], this is not true in general. In fact, This, since for such that and therefore, For the base members and such that we have where This, since for

such that

and

therefore, We may now construct the largest element of via multiplication of the members of With a subset of any entourage (uniquely and axiomatically), we have in general for entourages A and B that

and

where

Next, consider and with and and In addition to these account exhaustively for the elements of and For any such pair and by definition and since The N pairs (x,z) with account exhaustively for the elements of and, excepting the impact of on the N pairs (x, z) account exhaustively for the elements of Again, the impact of on is to render and Therefore, So, if G is cyclic with generator x, where and is of course the largest element of [This is of particular interest, since the cyclic group structure exists for all and is the unique group structure for N prime.] If G is not cyclic, one may produce via for since when We are also guaranteed to produce via some variation of where {x, y, ..., z} = X, according to G. It should also be noted that, as implied supra, the entourage of U is generated by the entire set X.

3. Consequences Should we define microscopic spacetime neighborhoods with the members of analogously to macroscopic spacetime neighborhoods per the symmetric entourages of Dx, we note the following interesting consequences.

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W.M. STUCKEY AND WEYTH RAWS 1.

for so when the distance between elements of is non-separable from that of lest we compromise the symmetry of our pregeometric notion of distance. For s = w, i.e., and our microscopic spacetime structure accommodates locality. Thus, the degree to which our spacetime is to accommodate quantum non-separability is determined by the choice of G. 2. The choice of G over X underlying is all that is needed to produce the microscopic spacetime structure embedded in the macroscopic spacetime structure. 3. The members of the base of the microscopic spacetime structure U may be combined via entourage multiplication to yield the largest element of the macroscopic spacetime structure Complementing this, is equivalent to the entourage of U generated by the entire set X. Thus, a robust pregeometric correspondance between the microscopic spacetime structure and the macroscopic spacetime structure is provided. 4.

Nexus to Physics

First, we show how a pseudometric induced by yields the geodetic structure of M4. We borrow from a proof of the following theorem [13]: For every sequence = X x X and X such that for every

of members of a uniformity on a set X, where for i = 1,2,..., there exists a pseudometric > on the set

To find consider all sequences of elements of X beginning with x and ending with y. For each adjacent pair in any given sequence, find the smallest member of containing that pair. [The smallest will have the largest i, since Suppose is that smallest member and let the distance between and be Summing for all adjacent pairs in a given sequence yields a distance between x and y for that particular sequence. According to the theorem, is the smallest distance obtained via the sequences. While this pseudometric is Euclidean rather than Minkowskian, it may be used to define geodesics for either space, since their affine structures are equivalent. Thus, we define a geodesic between x and y to be that sequence yielding Since some sequences might contain ‘distant’ adjacent pairs, our definition is suitable only for [To consider curved spacetimes, we would have to restrict our attention to sequences harboring only ‘local’ adjacent pairs.] The finest resolution would result when considering sequences of maximal length. The construct of maximal sequences is possible with X denumerable and finite. Of course, this immediately suggests a pregeometric foundation for the path integral formulation of QM. And, should one consider various sequences of entourages satisfying so that the combinatorics of both element and entourage sequences are considered, then a pregeometric foundation for quantum field theory is also

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481

intimated. Since each sequence of entourages produces a pseudometric, a combinatoric formalism over entourage sequences is analogous to the path integral formulation of quantum cosmology per Hartle and Hawking [14]. The transition from pregeometry to classical spacetime dynamics might be initiated by constructing an affine definition of 4-momentum Let a worldline be defined by a sequence of elements of X. Then, 4-velocity is defined by adjacent pairs in the sequence with direction specified via ascending order of the sequence. Thus, for a particle of mass m we have To accommodate curved spacetimes the definition of >(x,y) would have to be restricted to sequences constructed of ‘local’ adjacent pairings. Speculatively, an equivalence relation might be used to partition X into cells providing this restriction. These local, affinely M4 frames would then be pieced together so that the spacetime metric is consistent with the 4-momentum distribution a la Einstein's equations. Since this demands a relationship between affine and metric structures, a reasonable axiom is that of Riemannian geometry as noted by Bergliaffa et al. [15], i.e., the covariant derivative annihilates the spacetime metric. We have which yields

As an example, consider

with the standard polar

and

coordinates. Let

and azimuthal

The affine structure yields

= 0 and the other Christoffel symbols can be computed by using parallel transport and where

and

We know

= 0. Combined with the annihilation condition, we have is a constant. The annihilation condition also gives thus completing our example.

so and

Thus, or

Acknowledgement This work was funded in part by a grant from AmerGen Energy Company/Three Mile Island Unit 1.

References 1. J. Demaret, M. Heller, and D. Lambert, Found. Sci. 2, 137 (1997). 2. W.M. Stuckey, “Pregeometry and the Trans-Temporal Object,” to appear in Studies on the structure of time: From physics to psycho(patho)logy, R. Buccheri, V. Di Gesù and M. Saniga, eds., (Kluwer, Dordrecht, 2001). 3. S. Weinberg, Sci. Amer. 281, 72 (1999). 4. Ibid 1. 5. J. Butterfield and C.J. Isham, “Spacetime and the Philosophical Challenge of

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Quantum Gravity,” gr-qc/9903072 (1999). 6. G.K. Au, “The Quest for Quantum Gravity,” gr-qc/9506001 (1995). 7. R. Sorkin in The Creation of Ideas in Physics, J. Leplin, ed, (Kluwer, Dordrecht, 1995) 167. 8. R. Engelking, General Topology (Heldermann Verlag, Berlin, 1989). 9. R. Sorkin, Int. J. Theor. Phys. 30, 923 (1991). 10. D. Finkelstein, Phys. Rev. 184, 1261 (1969). 11. R. Geroch, Mathematical Physics (Univ of Chicago Press, Chicago, 1985). 12. W.M. Stuckey, Phys. Essays 12, 414 (1999). 13. Ibid 8. 14. S.W. Hawking in 300 Years of Gravitation, S.W. Hawking and W. Israel, eds., (Cambridge University Press, Cambridge, 1987) 631. 15. S.E.P. Bergliaffa, G.E. Romero, and H. Vucetich, Int. J. Theor. Phys. 37, 2281 (1998).

A ZPF-MEDIATED COSMOLOGICAL ORIGIN OF ELECTRON INERTIA

M. IBISON Institute for Advanced Studies at Austin 4030 Braker Lane West, Suite 300 Austin, TX 78759, USA

Abstract Support is found for a fundamental role for the electromagnetic zero-point-field (ZPF) in the origin of inertia. Simply by requiring that that a universal noise field be selfconsistent in the presence of the lightest charge, it is shown that this field must be the ZPF, and that the mass of that charge must be close to The ZPF functions as homeostatic regulator, with the electron mass decided by cosmological quantities. The calculation validates Dirac’s second Large Number hypothesis.

1.

Introduction

Several speakers at this conference have been pioneers championing the cause of an electromagnetic zero-point-field (EM ZPF) origin for inertia. Notable amongst these have been Haisch, Rueda and Puthoff [1-11]. Currently the implementations are classical, with a ‘classicized’ ZPF as conceived within the program of Stochastic Electrodynamics (SED), (see Kalitsin [12], Braffort [13] and Marshall [14] for the original works, and Boyer [15] for a review of this field). Epistemologically, a common theme of their work is that the ZPF is the cause of resistance to acceleration. In mathematical practice though, the end result is an inertial mass-energy that attributable to the ZPF. Broadly, the ZPF is seen as an external, energizing influence for a local degree of freedom, which, classically, is the co-ordinate of the particle whose mass we wish to explain. Thus the program has – in part – some of the flavor of Mach, because the ZPF provides a ‘background’ against which the acceleration can be measured. The particle, once energized, is conceived as having attributable energy, and therefore inertia. Although Haisch et al [11] in particular have made a distinction between the inertial and energetic aspects of matter, this distinction appears to be largely epistemological; since any ‘localized’ packet of energy is found to resist acceleration, it is sufficient to explain, within the context of this program, how EM ZPF energy can become localized. Even so, within this program, there are two quite different possible implementations distinguished by different degrees of non-locality for the origin of mass. To date, 483 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 483-490. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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despite the active role of the ZPF, current implementations result in a value for the inertial mass that is intrinsic to the particle in question. In contrast, the purpose of this paper is to argue for an alternative implementation, wherein the value of the inertial mass is determined entirely by external – cosmological – factors. For this reason the approach described here is much closer to the original conception of Mach (ca. 1883) than previous ZPF-as-background approach. In the following is given contrasting descriptions of the intrinsic and extrinsic approaches to ZPF-originated inertia. These are followed by a calculation supporting the latter, wherein one of Dirac’s large number hypotheses [16,17] is derived and interpreted as evidence of a cosmological ZPF-origin of the inertial mass of the electron.

2.

Role Of The ZPF In A Model Of Locally Determined Inertial Mass

In a locally determined, ZPF-originated, model of inertia, there exists a local dynamical degree of freedom, such as an oscillator amplitude [1], or a resonator excitation level. This co-ordinate is conceived initially as quiescent, and having no intrinsic energy. Subsequently, if the ZPF is switched on, then the oscillator or resonator is energized, and the ZPF-originated energy that is now associated with the oscillator or resonator can be regarded as the ‘rest’ mass. In some work, the local properties ultimately deciding the mass of the charged particle enter as a Fourier form factor governing the spectral response to the ZPF [3,4,5]. This has the advantage of leaving open whether the response is due to the energetic resonance of an oscillator or geometric structure. But in either case it is taken to be a local property. The end result is the same in that not only is there resistance to acceleration, but there is also a localized energy density that can be associated with the particle in question. Clearly, in this approach, the object has zero true rest mass, whilst the ZPF-energized mass may nonetheless be statistically at rest due to the homogeneity and isotropy of the ZPF. Also, thanks to the peculiar k-space distribution of the ZPF - the ZPF retains the same homogeneous energy density in every inertial frame – it follows that with suitably chosen dynamics it should be possible to make the rest mass a fully invariant scalar. These original ideas have stimulated new thinking about the origin of inertia and brought forth some encouraging responses [18,19,20]. However, it is premature to claim that the origin of inertia has been found in the ZPF, because there are some unresolved and unsatisfactory aspects of the current approach: 1. The particle oscillator or resonator must contain electrical charges capable of interaction with the ZPF, so the model cannot describe a neutral elementary particle such as a massive neutrino, for example. 2. The electron also poses a problem unless it is admitted that it has some – as yet unobserved – structure. 3. The values of the intrinsic mass (of the electron, muon, and tau say), are not predicted, but must be inserted by hand. The final mass – the energy stored in the oscillator or resonator – is decided by intrinsic qualities i.e.: locally, wherein combinations of charge–field coupling, geometric form factor, and spectral form factor, must be chosen to give the desired final mass. (The existence of these energy-storing

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485

‘degrees of freedom’ rests upon the presumed existence of a structure for the particle in question – points 1 and 2 - and neither this structure nor these coupling and form factors are explained.) 4. There is no clear path of development for the theory by which it can unite the inertial and gravitational aspects of mass. There is the hope that it also has something to do with the EM ZPF, as first suggested by Sakharov [21,22]. But to date there have been no successful implementations of a ZPF origin for gravity. Perhaps the most attractive feature of the current thinking along these lines is that the proposed energetic source by which means both gravity and inertia may perhaps be united - the ZPF - is a ‘ready-made’, omnipresent, influence. In the next section is investigated a different implementation which retains this foundational feature, but which overcomes some of the above enumerated difficulties.

3.

A Model For Non-Locally Determined Inertial Mass

By non-local model is here meant that inertia is conceived not as an intrinsic, unitary property, but as arising out of a non-local mutual interaction. Just as the mutual interaction energy of charges, current elements, and (gravitating) masses cannot be assigned to either partner in the interaction, so - it is suggested - inertial mass-energy cannot be ascribed to a single particular particle, but results from the multiple mutual (pair-wise) interactions with distant partners. Like the foregoing examples, the Casimir and van-der-Waals energies are also mutual, yet these are different in that they exist only by virtue of the ZPF. Though an apparently intrinsic Casimir energy does exist for a conducting curved surface embedded in the ZPF [23], one may regard this energy as arising out of the mutual interaction of local elements of the curved structure, just as in a Casimir cavity. To date, no one has identified a mutual yet distant interaction energy of electromagnetic origin that can explain inertia. And this is the reason why the ZPFinertia advocates have concentrated on local, ‘unitary’ qualities that might cooperate to localize ZPF energy. A detailed description of the distant interaction believed to be responsible will be given in a future document. Briefly, a consequence of that work is that, like the local models, the positional / motional particle degree of freedom may be regarded as ‘energized’ by the ZPF. But unlike local models, the energy of interaction turns out to be mutual, involving all distant particles. A good metaphor is provided by van-der-Waals binding energy, except that the rate of radial fall-off precludes it from candidature. For now, the following calculation is presented as evidence to support the claim that inertial mass is a non-local energy - with the ZPF as its means. 4.

Derivation Of The Electron Mass

In the following calculation it will be assumed that associated in some way with a charged particle is a resistance to acceleration equal to a final renormalized (not bare) inertial mass It will be assumed that underlying this mass is a particle in micromotion (commonly, but not very accurately termed zitterbewegung). We also need to

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assume that an EM noise field and the micro motion of the charge are consonant in that the ‘in’ fields impacting the source and the ‘out’ fields leaving the source have the same statistics. In other words, it will be assumed that the EM noise field has attained a selfconsistent state in the presence of the particle micro-motion. (A causal flow is not implied here: the particle’s motion does not cause the field nor does the field cause the particle’s motion. Rather, the field and motion are to be viewed as mutually consistent.) So far this sounds rather like the SED program that results in a ZPF-energized, but nonetheless intrinsic-valued, inertia. However, the particle employed here is deemed to have no intrinsic structure, and therefore cannot have an intrinsic-valued inertia - with or without the ZPF. Very broadly, this can be concluded simply from the absence of any length scale that could conceivably be associated with a mass. More specifically, it can be shown that a massless classical point charge dropped into the ZPF leaves the electromagnetic spectrum and energy density unchanged from that of the charge and ZPF considered apart from each other - unless the charge is permitted to interact with other charges. It must be admitted at the outset that the particle that will be singled out by this calculation is the electron. This is because it is the lightest charged particle. As a consequence it gives the largest acceleration per unit field, and therefore the largest out field per unit in field. It follows that, provided the in and out fields are universally selfconsistent, the fields must be maintained predominantly by the electron. Therefore, in the following calculation, it will be assumed that electrons, sprinkled approximately uniformly throughout the visible universe, cooperate in the maintenance of an EM noise field. Further, it can reasonably be assumed that the micro-motion will have a coherence length somewhere between the classical electron radius and the Compton wavelength. From this it follows that, although widely varying, the local environment of electrons is to a good approximation of no consequence to the presumed micro-motion, since the coherence length of the latter is relatively so short. In this paper the self-consistency calculation will be simplified by assuming that the micro-motion is non-relativistic. For this to be true for all electrons from the perspective of our own earthly reference frame, we must necessarily consider only a static universe – i.e. without expansion. This is because the electrons near the Hubble radius will turn out to dominate the self-consistent field calculation. (A more complicated calculation admitting expansion gives a very similar result, as discussed below.) With this restriction the electromagnetic noise-induced acceleration is approximately

for which the outgoing radiation, in the far-field, has electric field

The corresponding 3D orientation averaged energy density from both the magnetic and electric fields, as viewed in the frame in which the particle’s expected position is always at the origin, is

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In the presumed static cosmology there are N sources approximately uniformly distributed throughout the universe of static radius R, i.e., The expected energy density from all the sources is therefore

By contrast, the energy density of the in field at the particle in question can be expressed in terms of the acceleration using Eq. (1):

For self-consistency the energy density of the in field must, at all locations, equal the energy density due to all the out fields:

Consequently, one obtains the Dirac large number hypothesis [16]

With

and R set to the Hubble radius of cm, this computes to kg, i.e. 40% of the observed value of the electron mass – well within the tolerance set by the uncertainty in (which expressed as a factor is between about 0.5 and 2).

5.

Discussion

The above calculation establishes a linear relationship between the in and out fields. It follows directly that it does not matter how strong or weak is the noise field; the electron mass given by Eq. (7) would have the same value whatever. Another consequence of the linearity is that the computed electron mass is also insensitive to the energy spectrum of the ZPF. (A consequence of the fact that the charge-field scattering is elastic.) A relativistically correct version of this calculation performed in a flat expanding universe turns out to give, apart from a numerical coefficient of order unity, the same Dirac relation, and therefore the same electron mass as a function of cosmological constants, with the former remaining independent of time. Specifically, taking into account the special role played by the ZPF in maintaining homeostasis, the second Dirac hypothesis [16] - that is constant - can be validated. Dirac’s suggestions seem to have been rejected mostly on the basis of his first relation, which predicts a

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time-dependent gravitational constant, and is considered to be incompatible with observation; see [24] and [25] for reviews. Although not directly impacting the validity of the second relation under investigation here, it is nonetheless interesting that similar ZPF-mediated arguments have led Puthoff [25,27] to claim a time-independence for the first relation. It is hoped to reproduce elsewhere the detailed calculations and qualifying cosmologies discovered to maintain constancy of the second relation. In those calculations it turns out that appeal must be made to velocity-invariant statistics of the EM noise field. That is, the self-consistent field must be, at least at the level of expectations of quadratic field operators, the electromagnetic zero-point field as furnished by second-quantized Maxwell (henceforth the ZPF). The reader may be alert to the fact that a proper relativistic treatment necessitates the use of the Lorentz-Dirac equation with non-linear radiation reaction terms, or – in the quantum domain - the corresponding Heisenberg equation of motion [26]. This, and related issues concerning the bandwidth of the self-consistent field and the origin of time asymmetry, require a much more detailed treatment, and will also be addressed elsewhere. A concern expressed by some is that the cosmological distribution of matter is such that any alleged derivation of particle constants from cosmology will suffer from an unacceptable level of frame dependent, or perhaps time dependent, variability. A rough estimate of the variability in the predicted value of mass is to entertain fluctuations in N, which are likely to be of order This gives rise to corresponding fluctuations in of order unity, i.e. one part in Therefore, at least by this mechanism, cosmological variability does not lead to a detectable variability in the mass. It must be emphasized that nowhere in the above was inertial mass ‘explained’. Rather, this calculation tells us only that if the fields are to be self-consistent, the electron mass could not have any value other than the one it is observed to have, given the cosmological numbers. The calculation does not explain the mechanism of the mass. Despite these caveats, the success of the calculation provides support for the novel emphasis placed on the ZPF by Haisch and others in their work on inertial mass. This work continues that effort, but with a different role for the ZPF. Here, the ZPF is the means by which homeostasis is maintained; it is the means by which the electrons throughout the universe come into electromagnetic equilibrium with each other, whereby the electron mass attains universal consistency.

6.

Cosmological Origin Of Length-Scale

In natural units where e = 1, mass has units of the classical length corresponding to the electron mass (the classical electron radius) is, in S.I. units, which is about m. In the previous calculation based upon cosmological self-consistency this length is found from Eq. (7) to be where both R and N are cosmological constants. Despite all the talk of electromagnetism and ZPF-induced micro-motion, the final result constructs a very small length out of cosmological constants. If the cosmic somehow determines this length, as implied in this work, then there should be a direct cosmological interpretation for this very small length, without any reference to electrons. This is the focus of the following discussion.

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Notice that the integral in Eq. (4) may be regarded as computing the expectation of over the Hubble volume. For a homogeneous distribution, for any n, so nearly the whole contribution to the integral comes from matter at the Hubble radius. In other words, for the purposes of computing a self-consistent field, and to a very good approximation, all the matter in the universe appears to be at the Hubble radius. It is as if all matter is projected onto the Hubble sphere, creating the appearance of a surface density at the Hubble radius of whereupon is approximately the mean nearest neighbor distance between the points. That is, is the mean nearest neighbor distance on the Hubble sphere between the points that are the radial projections of all the electrons in the universe. To within a factor of order unity this is the previously computed mass-length of the electron, and therefore this distance must be the corresponding cosmological entity, and, allegedly, the origin of that length. It is clear from the above that a sufficiently large telescope could, in principle, be used to resolve the individual electrons in the universe if its probing radiation had a wavelength shorter than the mass-length. This means that the universe of electrons must be at least partly transparent to ZPF ‘radiation’ at this and shorter wavelengths. Ignoring for now the possibility of future collapse, it follows that the electrons cannot maintain a universally self-consistent noise field beyond the mass-frequency. (In a more realistic cosmology it is to be hoped that this quantity will look like a frame independent cutoff.) In other words, the mass-length is also the critical wavelength at which the universe of electrons starts to become transparent. Therefore, based upon the considerations of this and the previous section combined, one may conclude that the locally observed mass-length is authored cosmically and broadcast by the ZPF (as a cutoff at that wavelength). No attempt has been made to investigate, from this cosmological perspective, the relationship between the Compton wavelength and the mass length. Since their ratio is the fine structure constant, a search for a cosmic relation is therefore equivalent to a search, in this context, for a geometric interpretation of It is interesting that Wyler [29] (see [30] for a review in English) found an expression for involving the ratio of projections of volume elements – especially since the mass-length calculation above also involves a projection. Obviously, the dimensionality here is wrong because we have ignored universal expansion; if the two paths do converge there remains much more work to be done.

Acknowledgements

The author gratefully acknowledges the kind encouragement and the many productive conversations with H. Puthoff and S. Little. Thanks also to the organizers of the conference for putting together an intense and inspiring program of interesting physics.

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References 1. 2. 3.

4. 5. 6. 7. 8.

9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Haisch, B., Rueda, A., and Puthoff, H. E. (1994) Inertia as a zero-point-field Lorentz force, Phys. Rev. A 49, 678-694. Haisch, B., Rueda, A., and Puthoff, H. E. (1997) Physics of the zero-point-field: Implications for inertia, gravitation and mass, Speculations in Science & Technology 20, 99–114. Haisch, B., Rueda, A., and Puthoff, H. E. (1998) Advances in the proposed electromagnetic zero-point field theory of inertia, proc. 34th AIAA/ASME/SAE/ASEE AIAA Joint Propulsion Conference, AIAA paper 98-3143. Rueda, A., and Haisch, B., (1998) Contribution to inertial mass by reaction of the vacuum to accelerated motion. Found. Phys. 28, 1057-1108. Rueda, A., and Haisch, B., (1998) Inertia as reaction of the vacuum to accelerated motion, Phys. Letters A 240, 115-126. Haisch B. and Rueda, A. (1998) The zero-point field and inertia, in G. Hunter, S. Jeffers & J.-P. Vigier (eds.) Causality and Locality in Modern Physics, Kluwer Academic Publishers, 171-178. Rueda, A. and Haisch B. (1998) Electromagnetic vacuum and inertial mass, in G. Hunter, S. Jeffers & J.-P. Vigier (eds.) Causality and Locality in Modern Physics, Kluwer Academic Publishers, 179-186. Haisch, B., and Rueda, A., (1999) Progress in establishing a connection between the electromagnetic zero-point field and inertia, in M. S. El-Genk (ed.) Proc. Space Technology and Applications International Forum (STAIF-1999), AIP Conf. Publication 458, 988-994. Haisch, B., and Rueda, A. (1999) Inertial mass viewed as reaction of the vacuum to accelerated motion, Proc. NASA Breakthrough Propulsion Physics Workshop, NASA/CP-1999-208694, pp. 65. Haisch, B., and Rueda, A., (2000) Toward an interstellar mission: zeroing in on the zero-point-field inertia resonance, Proc. Space Technology and Applications International Forum (STAIF-2000), AIP Conf. Publication 504, 1047-1054. Haisch, B., Rueda, A., and Dobyns, Y. (2000) Inertial mass and the quantum vacuum fields, Annalen der Physik, in press. Kalitisin, N. S. (1953) JETP 25, pp. 407. Braffort, P., Spighel, M., and Tzara, C. (1954) Acad. Sci. Paris, Comptes Rendus 239, 157. Marshall, T. W. (1963) Proc. R. Soc. London, Ser. A 275, pp. 475. Boyer, T. H. (1980) A brief survey of Stochastic Electrodynamics, in A. O. Barut (ed.), Foundations of Radiation Theory and Quantum Electrodynamics, Plenum Press, New York, 49-63. Dirac, P. A. M. (1979) The Large numbers hypothesis and the Einstein theory of gravitation, Proc. R. Soc. London, Ser. A 365, 19-30. Dirac, P. A. M. (1938) Proc. R. Soc. London, Ser. A 165, pp. 199. Davies, P. C. W. (1992) Mach’s Principle, Guardian Newspaper, 22nd September, “http://www.physics.adelaide.edu.au/itp/staff/pcwd/Guardian/1994/940922Mach.html”. Jammer, M. (1999) Concepts of mass in Contemporary Physics and Philosophy, Princeton University Press, Princeton. Matthews, R. (1994) Inertia: Does empty space put up the resistance? Science 263, 612-613. Sakharov, A. D. (1968) Vacuum fluctuations in curved space and the theory of gravitation, Sov. Phys. Doklady 12, 1040-1041. Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973) Gravitation, Freeman, San Francisco. Candelas, P. (1982) Vacuum energy in the presence of dielectric and conducting surfaces, Annals of Physics 143, 241-295. Alpher, R. A. (1973) Large numbers, Cosmology, and Gamow, American Scientist 61, 51-58. Harrison, E. R. (1972) The cosmic numbers, Physics Today 25, 30-34. Puthoff, H.E. (1989) Source of vacuum electromagnetic zero-point energy, Phys. Rev. A 40, 48574862. Puthoff, H.E. (1991) Reply to “Comment on ‘Source of vacuum electromagnetic zero-point energy”’, Phys. Rev. A 44, 3385-3386. Sharp, D. H. (1980) Radiation reaction in non-relativistic quantum theory, in A. O. Barut (ed.), Foundations of Radiation Theory and Quantum Electrodynamics, Plenum Press, New York, 127-141. Wyler, A. (1969) Theorie de la Relativité – L’espace symétrique du groupe des équations de Maxwell, Acad. Sci. Paris, Comptes Rendus 269A, 743-745. GBL (1971) A mathematician’s version of the fine-structure constant. Physics Today 24, 17-19.

VACUUM RADIATION, ENTROPY AND THE ARROW OF TIME

JEAN E. BURNS Consciousness Research 1525 - 153rd Avenue San Leandro, CA 94578

Abstract The root mean square perturbations on particles produced by vacuum radiation must be limited by the uncertainty principle, i.e., where and are the root mean square values of drift in spatial and momentum coordinates. The value where m is the mass of the particle, can be obtained both from classical SED calculation and the stochastic interpretation of quantum mechanics. Substituting the latter result into the uncertainty principle yields a fractional change in momentum coordinate, where p is the total momentum, equal to where E is the kinetic energy. It is shown that when an initial change is amplified by the lever arm of a molecular interaction, in only a few collision times. Therefore the momentum distribution of a collection of interacting particles is randomized in that time, and the action of vacuum radiation on matter can account for entropy increase in thermodynamic systems. The interaction of vacuum radiation with matter is time-reversible. Therefore whether entropy increase in thermodynamic systems is ultimately associated with an arrow of time depends on whether vacuum photons are created in a time-reversible or irreversible process. Either scenario appears to be consistent with quantum mechanics.

1.

Introduction

In this paper we will see that entropy increase in thermodynamic systems can be accounted for by vacuum radiation, and then discuss the relationship between vacuum radiation and the arrow of time. The problem in accounting for entropy increase has always been that dynamical interactions which occur at the molecular level are time-reversible, but thermodynamic processes associated with entropy increase, such as diffusion and heat flow, only proceed in one direction as time increases. In the past it was often held that entropy increase is only a 491 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 491-498. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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macroscopic phenomenon, which somehow appears when a coarse-grain average is taken of microscopic processes. But no averaging of time-reversible processes has ever been shown to account for phenomena which are not time-reversible.[l] Nowadays entropy increase is often viewed as coming from effects of the environment, such as walls of a container or thermal radiation, not taken into account in the description of a system. Unruh and Zurek [2] have given examples in which entropy increase is produced in this way. However, the second law of thermodynamics specifies that entropy increase must also occur in an isolated system. So if we are to hold that entropy increase is produced by a physical process at the microscopic level, we must also understand how it can be produced in this way in an isolated system. Any explanation must satisfy the basic assumptions of statistical mechanics. Classical statistical mechanics has only one assumption: At equilibrium it is equally probable that the system will be in any (classical) state which satisfies the thermodynamic constraints. Quantum statistical mechanics has two basic assumptions. The first is essentially the same as for classical, except that states are now counted quantum mechanically. Thus:

At equilibrium it is equally probable that the system will be in any (quantum) state which satisfies the thermodynamic constraints. The second assumption of quantum statistical mechanics is: At equilibrium the relative phases of the eigenvectors describing the system are random. Once these fundamental assumptions are made, one can then define entropy as klog(number of states), where k is Boltzmann's constant. It is always also assumed that the number of molecules, and therefore the number of states, is extremely large. One can then develop the physics of the microcanonical ensemble in the usual way, by requiring that different parts of an isolated system be in equilibrium with each other at temperature T. By placing the system in equilibrium with a heat bath one can then derive the physics of the canonical ensemble, and so forth. [3] In order to talk about entropy, we must specify the context in which we refer to the ensemble of all possible states. In the coarse-grain view we would use an ensemble of states with all possible initial conditions, and then argue that because the number of states is very large, the only states we are apt to see are the most probable ones (and not ones in which all molecules are clustered in a corner of a box, for instance). Thus equilibrium merely refers to the most probable state in a large collection of systems. In the view in which entropy is produced at the microscopic level, we start with a single system which has

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specified initial conditions (classical or quantum mechanical) and look for a process which produces many random perturbations and by this means places the individual system into its most probable state. In order to inquire about an isolated system, let us consider the system to be comprised of not only the interacting molecules under consideration, but also the walls of their container, any heat bath surrounding them, and all the thermal radiation which might affect them. It would seem that we have taken into account all interactions which could possibly affect the system. What then could serve as an "environment" which would account for entropy increase? Let us ask if an interaction could take place within the limits of the uncertainty principle which would affect molecules randomly? If this interaction could randomize the momentum of each molecule and (when quantum mechanical description is needed) randomize the quantum phases of the eigenvectors describing the system, this process would then account for entropy increase. Yet the interaction itself could not be detected in measurements of the system. Vacuum radiation acts at the limits of the uncertainty principle, and clearly it would perturb molecules in a random way. But are these effects large enough? A mermodynamic system goes to equilibrium in a few molecular collision times. [3] So in order to account for entropy increase, vacuum radiation would have to randomize the momentum of a system and the quantum phases of its eigenvectors in that short time. Let us first take up the question of momentum.

2.

Randomization Of Momentum By Vacuum Radiation

2.1. DRIFT IN SPATIAL COORDINATE It has been shown by Rueda [4] in a classical stochastic electrodynamics (SED) calculation that the coordinate drift produced on a free particle by vacuum radiation can be described by diffusion constant where m is the mass of the particle. A quantum mechanical calculation of this effect of vacuum radiation has not been done. However, when only energy and momentum transfer are involved and not anything specifically quantum about the nature of the radiation involved, it is reasonable that an SED calculation will give the same result as a quantum mechanical one.[5,6] Rueda showed that vacuum radiation moves electrons in a random walk at relativistic speeds and that this motion accounts for nearly all of their mass, with step length varying from the Compton wavelength to the de Broglie wavelength. The radiation acts on hadrons at the quark level and moves the hadrons at sub-relativistic velocity. [4] We note that the stochastic interpretation [7] of the Schrödinger equation, which has no direct connection to vacuum radiation, but attributes a quantum brownian motion to particles, yields the same diffusion constant. In a similar vein, the stochastic action of particles, with the same range of step lengths as above, can be derived directly from the

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uncertainty principle in the following way. Suppose that we have an ensemble of particles, labeled 1, 2,.... Each is subject to a series of position measurements at equal time intervals. Particle 1 is measured with resolution particle 2 with resolution and so forth, with According to the uncertainty principle, as measurement resolution becomes increasingly fine, particle momentum is increasingly more uncertain, and the path is more erratic. Using this point of view, a particle can be described as following a continuous, non-differentiable path of fractal dimension two, which corresponds to brownian motion. [8] Further analysis shows that the step lengths vary from the Compton wavelength to the de Broglie wavelength. [9] The above diffusion constant yields a root mean square spatial drift so

The above result can be confirmed experimentally using a tightly collimated beam of low energy electrons. For instance, if a beam of 100 ev electrons has (where x is the forward direction of travel), the spread in beam width due to the above process will be larger than the spread due to diffraction in the first 19.5 cm of travel.[11] This experiment has not presently been done, however. 2.2. RANDOMIZATION OF MOMENTUM

Vacuum radiation acts at the limits of the uncertainty principle, so we write where is the root mean square shift in momentum component of the particle produced by vacuum radiation. It is then easily found that

where p is the total momentum of the particle and is the energy. We see that is proportional to so momentum is conserved as time becomes large. Perturbations in momentum of a particle will change its original value, and when momentum has been completely randomized. We wish to know how long this will take. In order to have a concrete example, let us start with air at standard conditions. At the end of one collision time (i.e., the time to travel a mean free path), However, any change in momentum is multiplied by a lever arm is the mean free path and r the molecular radius, during the

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next collision.[11] In air at standard conditions Therefore, the momentum distribution of the molecules has been randomized in two collision times. The product is proportional to .[11] Therefore, momentum is randomized in a few collision times for all gases except those at very high pressures (> 100 atm, or higher if the temperature is substantially more than 300 K). In solids and liquids many particles interact simultaneously, so it is reasonable to suppose that momentum will randomize within a few collision times in these also.[11]

3. Randomization Of The Phases Of The Eigenvectors In order to fulfill the second fundamental assumption of quantum statistical mechanics, it is necessary to show that vacuum radiation can randomize the relative phases of the eigenvectors describing the system within a few collision times. We make no calculation here, but simply show that this is likely to be the case. First, we note that perturbation theory tells us that components of eigenvectors added to a system because of a perturbation are out of phase with the original state vector. [12] Furthermore, because vacuum radiation will produce many small, independent effects, we can see by considering either a coordinate or a momentum representation of the eigenvectors that these effects would affect different eigenvectors differently. So we would expect the relative phases of the eigenvectors to be randomized. The above does not tell us how quickly this randomization would occur. However, Unruh and Zurek [2] have shown in various examples that when an environment perturbs a system, the off-diagonal elements of the density matrix go to zero in a much shorter time scale than effects involving spatial and momentum distributions. Thus it seems likely that vacuum radiation can diagonalize the density matrix in a shorter tune than it takes to randomize momentum.

4. The Arrow Of Time The dynamical laws of physics are time reversible, i.e., for any given trajectory described by them, the time reversed trajectory is also a solution of the equations. And in nearly all cases, both the process described by these equations as time moves forward and the process described when time is reversed can be observed to occur. But curiously, there are a few exceptions to this rule. The decay of K-mesons violates CP and therefore (assuming CPT holds) is not time symmetric. Electromagnetic waves emanate from a source out to infinity, but do not converge from infinity to a source. Collapse of the wave function is a one-way process.[13,14] And as Prigogine and co-workers have shown, in systems which are so unstable that they cannot be described analytically in an ordinary dynamical framework, process can go in only one directional.[15] Such processes can be called irreversible, and they are accounted for by saying they are governed by an arrow of time. It is not known

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what an arrow of time is, what it has to do with the rest of physics, or whether any of the above arrows of time have anything to do with each other. It has been shown herein that entropy increase in thermodynamic systems is produced by the interaction of vacuum radiation with matter. This interaction is time reversible. However, we can go back a step and ask how vacuum radiation is produced. Whether an arrow oftime is ultimately involved in entropy increase depends on the answer to this question, as we will see. In examining this issue, let us start with a classical (SED) analysis. Puthoff [16] has shown that if vacuum radiation with its frequency-cubed spectrum once exists, then random interactions with matter in which radiation is absorbed and matter accelerates and reradiates maintain this frequency-cubed spectrum indefinitely. From this perspective, the random nature of the interaction of vacuum radiation with any given particle is caused by the random distribution in position and momentum of other particles the radiation previously interacted with. All interactions are time-reversible, and it is not necessary to invoke an "arrow of time" to explain entropy increase in thermodynamic systems. In quantum mechanics photons exist in quantized units of energy However, the average energy per photon of vacuum radiation is For that reason it is commonplace to explain the average energy by supposing that photons spontaneously and causelessly arise out of the vacuum, exist for the time allotted by the uncertainty principle, and then annihilate themselves back into the vacuum. In this scenario information describing the state of the newly created vacuum photon arises from nothing, the photon interacts with matter and modifies the information describing its state according to this interaction, and this modified information is then destroyed when the photon annihilates itself. The dynamical information which is introduced in the creation of virtual photons is purely random. However, the information which is removed is no longer random (or potentially is not because the virtual photons could have interacted with an ordered system). Thus the beginning and end points are inherently different, and an arrow of time is defined. According to this view, entropy increase is therefore ultimately associated with an arrow of time.[11] On the other hand, it would seem that quite different views of the arising and disappearance of photons are possible. The basic equations of QED and quantum field theory do not tell us how vacuum photons (or other virtual particles) arise. And creation and annihilation operators, although they have evocative names, simply describe mappings from one state to another in Hilbert space, the same as any other operators. The idea that vacuum photons arise spontaneously out of the vacuum is basically a pictorial device to account for the average energy per photon of Alternatively, one can conceive that, comparably to the classical picture, vacuum photons arise and disappear through constructive and destructive phase interference of a large number of photons traveling in different directions. To be consistent, one would have to view all other virtual particles as also arising and disappearing through constructive and destructive interference of quantum phase, perhaps through interaction with negative energy particles. But the appearance and

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disappearance of virtual particles could perhaps occur in this way. Another possibility is that the seemingly random appearance and disappearance of virtual particles comes about through interactions in the extra dimensions provided by string theory. In each of these cases processes would be entirely time-reversible, and no arrow of time would be involved. We can put this issue another way by asking: Is the universe a continuous source of random dynamical information, creating virtual particles which can interact with matter and then return some of the previous dynamical information describing this matter to the vacuum? Or does the universe merely transform dynamical information, with virtual particles arising and disappearing through a process such as the above? At present there is no answer to these questions and, given quantum indeterminacy within the limits of the uncertainty principle, there may never be any conclusive answer.

5. Conclusion As vacuum radiation interacts with particles, it exchanges momentum with them. The fractional change in momentum of a particle after one collision time, when multiplied by the lever arm of succeeding molecular interactions, becomes greater than one in only a few collision times. Therefore, particle momentum is randomized during that time, and vacuum radiation can account for entropy increase in thermodynamic systems. Vacuum radiation interacts with matter in a time-reversible process. Therefore, whether entropy increase in thermodynamic systems should be viewed as ultimately connected with an arrow of time depends on whether the arising and disappearance of vacuum photons should be considered as a time-reversible or irreversible process. Either possibility appears to be consistent with quantum mechanics.

References 1. 2. 3. 4.

5. 6. 7.

Zeh, H.-D. (1989) The Physical Basis of the Direction of Time, Springer-Verlag, New York. Unruh, W.G. and Zurek, W.H. (1989) Reduction of a wave packet in quantum Brownian motion, Phys. Rev. D 40(4), 1071-1094. Huang, K. (1963) Statistical Mechanics, Wiley, New York. Rueda, A. (1993) Stochastic electrodynamics with particle structure, Part I: Zero-point induced brownian behavior, Found. Phys. Lett. 6(1), 75-108; (1993) Stochastic electrodynamics with particle structure, Part II: Towards a zero-point induced wave behavior, Found. Phys. Lett. 6(2), 139-166. Milonni, P.W. (1994) The Quantum Vacuum: An Introduction to Quantum Electrodynamics, Academic, New York. SED calculations are known to give the same result as quantum mechanical ones for the Casimir effect, van der Waals forces, the shape of the blackbody spectrum, and the Unruh-Davies effect. See Ref. 5. Chebotarev, L.V. (2000) The de Broglie-Bohm-Vigier approach in quantum mechanics, in S. Jeffers, B. Lehnert, N. Abramson, and L. Chebotarev (eds.), Jean-Pierre Vigier and the Stochastic Interpretation of Quantum Mechanics, Apeiron, Montreal, pp. 1-17.

498 8.

9. 10. 11. 12. 13. 14. 15.

16.

J. E. BURNS Abbott, L.F. and Wise, MB. (1981) Dimension of a quantum-mechanical path, Am. J. Phys. 49(1), 37-39; Cannata, F. and Ferrari, L. (1988) Dimensions of relativistic quantum mechanical paths, Am. J. Phys. 56(8), 721-725. Sornette, D. (1990) Brownian representation of fractal quantum paths, Eur. J. Phys. 11, 334-337. Haken, H. (1983) Synergetics, Springer-Verlag, New York. Burns, J.E. (1998) Entropy and vacuum radiation, Found. Phys. 28(7), 1191-1207. Peebles, P.J.E. (1992) Quantum Mechanics, Princeton University Press, Princeton, NJ. Penrose, R. (1994). Shadows of the Mind. New York: Oxford University Press, pp. 354-359. It should be noted that not all interpretations of quantum mechanics assume there is such a thing as collapse of the wave function. See, e.g., Ref. 7. Prigogine, I. (1997) From Poincaré's divergences to quantum mechanics with broken time symmetry, Zeitschrift für Naturforschung 52a, 37-47; Petrosky, T. and Rosenberg, M. (1997) Microscopic nonequilibrium structure and dynamical model of entropy flow, Foundations of Physics 27(2), 239-259. Puthoff, H.E. (1989) Source of vacuum electromagnetic zero-point energy, Phys. Rev. A 40(9), 4857-4862; (1991) Reply to "Comment on 'Source of vacuum electromagnetic zero-point energy'", Phys. Rev. A 44(5), 3385-3386.

QUATERNIONS, TORSION AND THE PHYSICAL VACUUM: THEORIES OF M. SACHS AND G. SHIPOV COMPARED

DAVID CYGANSKI Worcester Polytechnic Institute Worcester, MA [email protected] WILLIAM S. PAGE Daneliuk & Page, Kingston, Ontario [email protected]

Abstract. Of several developments of unified field theories in the spirit of Einstein’s original objective of a fully geometric description of all classical fields as well as quantum mechanics, two are particularly noteworthy. The works of Mendel Sachs and Gennady Shipov stand apart as major life works comprising tens of papers, several monographs and decades of effort. Direct comparison of these theories is hampered however by differences in notation and conceptual view-point. Despite these differences, there are many parallels between the fundamental mathematical structures appearing in each. In this paper we discuss the main tenets of the two approaches and demonstrate that they both give rise to a factorization of the invariant interval of general relativity.

1. Introduction The theories reviewed in this paper represent a return to the ideas initiated by Einstein after the development of general relativity. After brie y introducing both theories we develop the representations used by each factorization of the general invariant space-time line element. In his book, General Relativity and Matter [1] Mendel Sachs presents a unified field theory incorporating gravitation, electromagnetism, nuclear interactions and the inertial properties of matter. In a later book, Quantum Mechanics from General Relativity [2] Sachs extends the formalism of general relativity in the manner originally envisioned by Einstein to obtain a general theory of matter including those properties of matter that are now usually described by quantum mechanics. To achieve this unification Sachs writes the field equations of general relativity in a factored form having a similar relationship to the usual field equations of general relativity that the Dirac equation has to the Klein-Gordon equation in relativistic quantum mechanics. Factoring the field equations involves introducing a generalization of Riemann geometry that admits coordinate transformations involving all 16 parameters of the Einstein group rather than the usual 10 parameters of the Poincare group. These extra parameters represent spin degrees of freedom. 499 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 499-506. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Sachs expresses this in terms of the algebra of spinors and quaternions. Applying this same factorization to Maxwell’s equations leads to an explanation the of Lamb shift without involving quantum field theory. Exact solution of the field equations corresponding to a ground state of bound particleantiparticle pairs suggests a picture of the physical vacuum quite different than the virtual particle sea of contemporary relativistic quantum mechanics. Instead of annihilating, ground state particleantiparticle pairs constitute a ubiquitous very weakly interacting background which provides an alternate physical interpretation of phenomena such as anomalous scattering and magnetic moments that are well described numerically by contemporary relativistic quantum mechanics but lack an intuitive physical interpretation in that formalism. Gennady Shipov, in his book “A Theory of Physical Vacuum" [3] also presents a unified field theory with conclusions very similar to those of Mendel Sachs. Shipov's program involves a completely geometric representation of the field equations of general relativity as equivalent to the structural equations of

geometry [8]. Gravitation and the inertial properties of matter in non-inertial frames

of reference are described in terms of the contorsion part of the general affine connection of

while

a generalization of electromagnetism is derived from the Christoffel part of this connection. Solutions to the structural equations for the situation corresponding to anti-particle particle pairs bound by the generalized electromagnetic interaction yield the same picture of the physical vacuum as proposed by Sachs. Shipov's theory achieves William Clifford's vision[4] that preceded Einstein's general relativity by more than 30 years, of a representation of the material world entirely in terms of the curved and twisted geometry of space itself. It is remarkable that in addition to the description of gravitational, electromagnetic and nuclear interactions that are well known in physics, Shipov’s theory also admits solutions involving only the torsion of space. Shipov proposes some novel interpretations and potential applications of this fact that are very controversial. From the surface resemblance seen in the above comparison one is led to consider the possibility that the two theories may be linked at a fundamental level However differences not only in notation but in the choice of affine connections and geometry act as barriers to direct comparison. We have undertaken a research program directed towards construction of a bridge between the formalisms and determination of their relationship to each other. The objectives of this paper which contains some early results from this effort include: familiarizing readers with the existence of the two theories; making available a readable derivation of the spinor affine connection used by Sachs and a parallel development for Shipov1s connection' identifying the number of spin degrees of freedom retained in each theory's metric factorization and as a result demonstrating another parallel between the approaches.

2. Spinors Fundamental to the development by Sachs is the application of the spinor representation of spacetime. Cartan[5] introduced the spinor as an irreducible representation of the proper Lorentz group of special relativity. The splitting of the four dimensional Riemannian space into a direct product of two spinor spaces was first introduced by Van Der Waerden and Infield [6] with the introduction of spinor analysis, This application of spinors was further developed by Bergmann[7] and many others, eventually taking a form that is today recognized as a theory of connections on a complex valued fibrebundle. Recall that the covariant derivative of a vector is given by

QUATERNIONS, TORSION AND THE VACUUM

where

501

is the affine connection. The covariant derivative of a two-component spinor

is represented by means of a set of

fundamental 2nd rank spinor fields called the spin-affine connection

plays the same role as

the tetrad field in the more well known tetrad tensor field formalism.

A mapping is needed between the space of spinors and tensors that allows us to represent tensors with full compatibility between actions carried out with tensor objects and the results one would obtain on first mapping into spinor space and then carrying out these same actions. ~ shall find that such a mapping can be found through a tetrad of fundamental fields objects that take a covariant tensor

into a 2nd rank spinor

which are mixed tensor/spinor by means of the simple mapping

Our compatibility requirement when applied to the action of the covariant derivative then requires that we obtain the same value on differentiation followed by mapping as mapping followed by differentiation. Hence we need to find tetrad fields that obey the equality

Applying Liebnitz's rule we obtain the requirement

which can only be fulfilled if

Thus~ a necessary condition on the existence of a compatible

spinor representation of a tensor is the existence of a tetrad field with this property. Given such a tetrad field, we need now also obtain the spinor affine connection that realizes this compatibility. Before proceeding we will need to also introduce the covariant second rank fundamental spinor

(the contravariant form being given by which plays a role for spinors similar to the fundamental metric tensor for the definition of an inner product and in the raising and lowering of spinor indices The covariant derivative of the fundamental spinor is obtained again by a correspondence principle. Given a spinor invariant formed via the metric property of this spinor, we require that the

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covariant derivative of this new spinor valued object to behave appropriately in analogy with the covariant derivative of a scalar field:

Carrying out the covariant derivative we obtain

Thank to the antisymmetric nature of

it can be easily shown that all components of the

first two terms cancel, leaving the requirement that

since

is arbitrary. It was shown by

Bergmann that the vanishing of the covariant derivatives of the tetrad quaternion field and of the metric spinor ere sufficient conditions to restrict the allowable solutions for the spin-affine connection to a unique solution in the case of a spinor space based upon the restriction of unimodular spinor transformations. By definition the fundamental quaternion fields and as a tensor with respect to the tensor index

transforms as a2nd rank covariant spinor

That is, we have that

Applying the Leibnitz rule, the covariant derivative of these fields can be written as follows if we assume that the connection for tensor objects takes the form appropriate for a

space which

is given by the Christoffel symbols.

or

where

are the Christoffel symbols.

denotes the time reversed quaternion field (so

named as the action that results is reversal of the sign of the

component in x)

where * denotes the complex conjugate. The fundamental spinor plays the role of the fundamental metric tensor in the raising /lowering of spinor indices and in the construction of the inner product and magnitude of spinors. Bergmann requires that the covariant derivative of the fundamental quaternion fields vanish, i.e. that transport ofthese ~ds and the fundamental spinor from one point in space to an infinitesimally near point are both globally parallel.

As shown by Bergmann, on obtaining a solution of these equations the resulting spinor-affine connection is uniquely obtained as

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503

Sachs introduces the new notion of an algebraic structure for Bergmann's tetrad fields. He shows that the

can be interpreted as a quaternion valued four vector and as such admits the

manipulations of quaternion algebra.

2.1 CLIFFORD ALGEBRA The real-valued quaternion algebra is the even sub-algebra of the Clifford algebra of 3-dimensional space Cl(3). Cl(3) is isomorphic to the algebra of 2 x 2 complex matrices and has also been called complex-valued quaternion algebra. The matrix representation of the basis of Cl(3) consists of 8 matrices: the identity matrix (rank 0),

Pauli matrices (rank 1)

the (rank 2) products

and the pseudo-scalar (rank 3)

The even rank elements of Cl(3) constitute the basis for the quaternion

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Since the second rank spinor fields have the form of 2 x 2 Hermitean matrices they may be represented as quaternions. As will be seen, Sachs exploits the associated algebraic structure to obtain his factorization of the metric.

3. Sachs' Factorization Sachs observes that the structure of the 10 parameter Poincare group, which includes translations, rotations and reflections, is represented in the Riemann geometry of conventional general relativity by the real-valued symmetric metric tensor

But reflection symmetry is not required by any of the

postulates of general relativity. If operations of reflection are removed from the Poincare group, the result is a 16 parameter group that Sachs calls the Einstein group.

does not provide a complete

representation of this group. But a faithful irreducible representation can be found in terms of the fundamental quaternion field. Hence the metric tensor can be written in the symmetric factored form

where the products of the field tensor components are understood as quaternion products. Now we can write the linear invariant infinitesimal line element as the quaternion differential

Thus ds is a quaternion-valued scalar invariant. In contrast to the conventional formulation this invariant no longer has any ambiguity of sign. It is invariant with respect to translations in space but has internal spin degrees of freedom. Sachs also defines the quaternion conjugate or time reversed quaternion field

from

Their product is the ordinary quadratic real-valued line element of Riemann space which is invariant with respect to changes in both spinor coordinates and translations. This factorization makes apparent "spin" degrees of freedom that are usually hidden. Sachs does not address the important questions of the number of degrees of freedom in the quaternion field that are preserved in the invariant interval differentials, By finding the rank of the Jacobian of eight differential components in ds and with respect to the sixteen coefficients in the quaternion field we find that there are exactly four degrees of freedom. This is suggestive of the form of the intrinsic spin four vector.

4. Shipov’s Tetrads Shipov concerns himself from the beginning of his development with associating angular reference frames to point-size entities. To accomplish t'lis he applies the concept of tetrads. We will briefly introduce this approach in this section. The method of tetrads or vierbien in the tensor analysis used in the early work on general relativity and unified field theories does not lead naturally to the fuli irreducible representation of the properties of higher order geometry. For example, in tensor notation, the Riemann metric is written as follows:

QUATERNIONS, TORSION AND THE VACUUM

where the tetrad

505

consists of four linearly independent covariant vector fields which provide a local

pseudo-Euclidean coordinate system at each point. We also have contravariant vector fields such that

and

We use Greek letters such as

etc. to denote tensor indices and Latin letters such as a., b

etc. to denote "tetrad" indices. Note that raising and lowering tetrad indices is done via the Minkowski metric metric

of the local coordinate system, while tensor indices involve the symmetric Riemann The invariant differential interval ds is written

The tetrad fields map a tensor into set of tetrad scalars

a is a “dead” index (Schouten) We may now corsider the covariant derivative of these tetrad scalars

are Ricci rotation coefficients As in the discussion of the covariant derivative of spinors above, we may ask when is the covariant derivative of the tetrad scalars compatible with tensor differentiation.

The analogous necessary condition is

which leads to the definition of the Ricci rotation coefficients

where the inverse tetrad is defined by

Hence identical prescriptions are used by Shipov and Bachs in the derivation of the

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connection for their respective geometries.

4.1 METRIC FACTORIZATION Using the tetrad bases we may form the four linear scalar invariants

and from these usual quadratic Riemann inetds

Thus the tetrad fields aliow a factorization of the invariant interval without sign ambiguity but within the context of Shipoy's formalism and without introduction of the spinor/quaternion calcuius of Sachs. Now since

and

are linearly related through a raising operation by the Minkowski

metric) by virtue of the construction given above, these represent only four degrees of freedom. Hence the Shipov differential ilivariants comprise the same number of spin degrees of freedom as the Sachs invariants In the Shipov construction, the raising/lowering operation in the internal Minkowski tangent space at each point in his geometry is the parallel of the process o~ quaternion conjugation in Sachs’

case. 5.

Conclusion

The resurrection of unified field theory, as originally envisioned by Einstein, Cartan arid many others following the development of general relativity, represents a clear alternative to the collection of phenomenological and mathematical procedures loosely referred to as the Standard Model. The approach to higher order geometry required to express absolute parallelism and exemplified by the spinor formulation obviate the need to appeal to physically unintuitive notions such as strings in 10 dimensional space. At this stage of our research program we conclude that there is a deep similarity between Sachs’ spinor and quaternion development and Shipov's tetrad based formalism, not only in general perspectives but at the level of metric factorization. On the other hand, there are such pronounced differences in notation and geometric formalism that further study will be required to determine whether or not the similarity extends to an isomorphism. It our intent to pursue this investigation to

such ends. References [1] General Relativity and Matter; A Spinor Field Theory from Fermis to Light-Years, Mendel Sachs, D. Reidel Publishing Co., 1982. [2] Quantum Mechanics from General Relativity; An Approximation for a Theory of Inertia, Mendel Saclis, Reidel Publishing Co., 1986. [3] A Theory of Physical Vacuum, G. I. Shipov, English edition, Russian Academy of Natural Sciences, 1998. [4] Mathematical Papers, by William Clifford, London, 1882. Lectures and Essays, Vol.1, London, 1879. [5] E. Cartan, Bull. Soc. France Math. 41, 53, 1913. [6J I. Infield, B.L. Van Der Waerden, Sitzber. preuss. Akad. Wiss., Physik-math. Ki, 380, (1933). [7] Two-Component Spinors in General Relativity, Peter G. Bergmann, Physical Review, Vol. 107, No.2, p.624. [8] Tensor Analysis for Physicists, J. A. Schouten, 2nd edition, Dover Publications Inc., 1989.

HOMOLOIDAL WEBS, SPACE CREMONA TRANSFORMATIONS AND THE DIMENSIONALITY AND SIGNATURE OF MACRO-SPACETIME An Outline of the Theory

M. SANIGA Astronomical Institute of the Slovak Academy of Sciences SK–059 60 Tatranská Lomnica, The Slovak Republic

1. Introduction

No phenomenon of natural sciences seems to be better grounded in our everyday experience than the fact that the world of macroscopic physical reality has three dimensions we call spatial and one dimension of a different character we call time. Although a tremendous amount of effort has been put so far towards achieving a plausible quantitative elucidation of and deep qualitative insight into the origin of these two puzzling numbers, the subject still remains one of the toughest and most challenging problems faced by contemporary physics (and by other related fields of human inquiry as well). Perhaps the most thought-provoking approach in this respect is the one based on the concept of a transfinite, hierarchical fractal set usually referred to as the Cantorian space, In its essence, is an infinite dimensional quasi-random geometrical object consisting of an infinite number of elementary (kernel) fractal sets; yet, the expectation values of its both topological and Hausdorff dimensions are finite. The latter fact motivated El Naschie [1,2] to speculate not only about the total dimensionality of spacetime, but also about its enigmatic signature. His reasoning goes, loosely speaking, as follows. It is assumed that the effective topological dimension of grasps only spatial degrees of freedom, whereas its averaged Hausdorff dimension, incorporates also the temporal part of the structure. These two dimensions are interconnected, as both depend on the Hausdorff dimension of the kernel set, And there exists a unique value of the latter, viz. for which (space) and (spacetime)! 507 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 507-510. © 2002 Kluwer Academic Publishers. Printed in the Netherlands

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2. Cremonian Pencil-Spacetimes

In our recent papers [3,4], we approached this issue from a qualitatively different, but conceptually similar to the latter, algebraic geometrical point of view. This approach is based on our theory of pencil-spacetimes [5–13]. The theory identifies spatial coordinates with pencils of lines and the time dimension with a specific pencil of conics. Already its primitive form, where all the pencils lie in one and the same projective plane, suggests a profound connection between the observed number of spatial coordinates and the internal structure of time dimension [5–7,9,11–13]. A qualitatively new insight into the matter was acquired by relaxing the constraint of coplanarity and identifying the pencils in question with those of fundamental elements of a Cremona transformation in a three-dimensional projective space [3,4]. The correct dimensionality of space (3) and time (1) was found to be uniquely tied to the so-called quadro-cubic Cremona transformations – the simplest non-trivial, non-symmetrical Cremona transformations in a projective space of three dimensions. Moreover, these transformations were also found to fix the type of a pencil of fundamental conics, i.e. the global structure of the time dimension. A space Cremona transformation is a rational, one-to-one correspondence between two projective spaces [14]. It is determined in all essentials by a homaloidal web of rational surfaces, i.e. by a linear, triply-infinite family of surfaces of which any three members have only one free (variable) intersection. The character of a homaloidal web is completely specified by the structure of its base manifold, that is, by the configuration of elements which are common to every member of the web. A quadro-cubic Cremona transformation is the one associated with a homaloidal web of quadrics whose base manifold consists of a (real) line and three isolated points. In a generic case, discussed in detail in [3], these three base points are all real, distinct and none of them is incident with the base line In the subsequent paper [4], we considered a special ‘degenerate’ case when one of lies on It was demonstrated that the corresponding fundamental manifold still comprises, like that of a generic case, three distinct pencils of lines and a single pencil of conics; in the present case, however, one of the pencils of lines incorporates and is thus of a different nature than the remaining two that do not. As a consequence, the associated pencil-space features a kind of intriguing anisotropy, with one of its three macro-dimensions standing on a slightly different footing that the other two. Being examined and handled in terms of the transfinite Cantorian space approach, this macrospatial anisotropy was offered a fascinating possibility of being related with the properties of spacetime at the microscopic Planck scale [4].

CREMONA TRANSFORMATIONS AND MACRO-SPACETIME

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If this spatial anisotropy is a real characteristic of the Universe, then its possible manifestations, whatever bizarre and tantalizing they might eventually turn out to be, must obviously be of a very subtle nature as they have so far successfully evaded any experimental/observational evidence. Yet, conceptually, they deserve serious attention, especially in the light of recent progress in (super)string and related theories [15]. For alongside invoking (compactified) extra spatial dimensions to provide a sufficiently-extended setting for a possible unification of all the known interactions, we should also have a fresh look at and revise our understanding of the three classical macro-dimensions we have been familiar with since the time of Ptolemy. 3. Conclusion

The concept of Cremonian spacetimes represents a very interesting and fruitful generalization of the pencil concept of spacetime by simply raising the dimensionality of its projective setting from two to three. When compared with its two-dimensional sibling, this extended, three-dimensional framework brings much fresh air into old pressing issues concerning the structure of spacetime, and allows us to look at the latter in novel, in some cases completely unexpected ways. Firstly, and of greatest importance, this framework offers a natural qualitative elucidation of the observed dimensionality and signature of macro-spacetime, based on the sound algebro-geometrical principles. Secondly, it sheds substantial light at and provides us with a promising conceptual basis for the eventual reconciliation between the two extreme views of spacetime, namely physical and perceptual. Thirdly, it gives a significant boost to the idea already indicated by the planar model that the multiplicity of spatial dimensions and the generic structure of time are intimately linked to each other. Finally, being found to be formally on a similar philosophical track as the fractal Cantorian approach, it grants the latter further credibility.

Acknowledgement–This work was partially supported by the NATO Collaborative Linkage Grant PST.CLG.976850. References 1. El Naschie, M.S.: Time symmetry breaking, duality and Cantorian space-time, Chaos, Solitons & Fractals 7 (1996), 499–518. 2. El Naschie, M.S.: Fractal gravity and symmetry breaking in a hierarchical Cantorian space, Chaos, Solitons & Fractals 8 (1997), 1865–1872. 3. Saniga, M.: Cremona transformations and the conundrum of dimensionality and signature of macro-spacetime, Chaos, Solitons & Fractals 12 (2001), [in press].

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4. Saniga, M.: On ‘spatially-anisotropic’ pencil-spacetimes associated with a quadro-cubic Cremona transformation, Chaos, Solitons & Fractals 12 (2001), [in press]. 5. Saniga, M.: Arrow of time & spatial dimensions, in K. Sato, T. Suginohara and N. Sugiyama (eds.), The Cosmological Constant and the Evolution of the Universe, Universal Academy Press, Tokyo, 1996, pp. 283–284. 6. Saniga, M.: On the transmutation and annihilation of pencil-generated spacetime dimensions, in W.G. Tifft and W.J. Cocke (eds.), Modern Mathematical Models of Time and their Applications to Physics and Cosmology, Kluwer Academic Publishers, Dordrecht, 1996, pp. 283–290. 7. Saniga, M.: Pencils of conics: a means towards a deeper understanding of the arrow of time?, Chaos, Solitons & Fractals 9 (1998), 1071–1086. 8. Saniga, M.: Time arrows over ground fields of an uneven characteristic, Chaos, Solitons & Fractals 9 (1998), 1087–1093. 9. Saniga, M.: Temporal dimension over Galois fields of characteristic two, Chaos, Solitons & Fractals 9 (1998), 1095–1104. 10. Saniga, M.: On a remarkable relation between future and past over quadratic Galois fields, Chaos, Solitons & Fractals 9 (1998), 1769–1771. 11. Saniga, M.: Unveiling the nature of time: altered states of consciousness and pencilgenerated space-times, Int. J. Transdisciplinary Studies 2 (1998), 8–17. 12. Saniga, M.: Geometry of psycho(patho)logical space-times: a clue to resolving the enigma of time?, Noetic J. 2 (1999), 265–274. 13. Saniga, M.: Algebraic geometry: a tool for resolving the enigma of time?, in R. Buccheri, V. Di Gesù and M. Saniga (eds.), Studies on the Structure of Time: From Physics to Psycho(patho)logy, Kluwer Academic/Plenum Publishers, New York, 2000, pp. 137–66 and pp. 301–6. 14. Hudson, H.P.: Cremona Transformations in Plane and Space, Cambridge University Press, Cambridge, 1927. 15. Kaku, M.: Introduction to Superstrings and M-Theory, Springer Verlag, New York, 1999.

PULSE INTERACTION IN NONLINEAR VACUUM ELECTRODYNAMICS A. M. IGNATOV General Physics Institute Moscow, Russia Email: [email protected] V.P. POPONIN International Space Sciences Organization San Francisco, CA, USA Email: [email protected]

Abstract. The energy-momentum conservation law is used to investigate the interaction of pulses in the framework of nonlinear electrodynamics with Lorentz-invariant constitutive relations. It is shown that for the pulses of the arbitrary shape the interaction results in phase shift only.

1. Introduction Although classical electromagnetic theory deals with linear Maxwell equations, there have been numerous attempts to bring the nonlinear phenomena into the stage. All relativistic and gauge invariant versions of electromagnetism are based on the Lagrangian density, L, which depends on the invariants of the field tensor. Generally, in terms of the electric (E) and magnetic (B) fields the Maxwell equations in absence of external charges may be written in a standard form:

where we put c = 1 and

The Lagrangian

depends on

Poincare invariants and J= EB only. The distinctive feature of Eqs. (1.1) is that since the Poincare invariants are identically zero for the plane electromagnetic wave, the latter is insensitive to vacuum nonlinearity and propagates without distortion. Of particular interest are the nonlinear corrections to the linear electrodynamics arising due to vacuum polarization in the strong electromagnetic field. In the ultimate case of slowly varying fields this results in Heisenberg- Euler electrodynamics [1]. 511 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 511-514. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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The main point of this paper is to describe the simplest, nonlinear vacuum process: the interaction of two electromagnetic waveforms propagating in opposite directions. 2. Maxwell Equations

We consider a linearly polarized wave propagating in the z direction of the with all other components being zero. In this situation, the second Poincare invariant vanishes,

so the Maxwell equations are written as

where the subscript denotes the derivative with respect to the corresponding variable and The Lagrangian in Eq.(1.2) is expanded in powers of I. Keeping the lowest-order nonlinear corrections we have

With the help of the appropriate scale transform, the coefficient may be reduced to ± 1. For the particular case of the Heisenberg-Euler electrodynamics, Of interest also is to keep in mind the Born-Infeld electrodynamics [2]with the Lagrangian

3. Energy-Momentum Tensor

The conservation laws for Eqs. (1.2) are given by

where the components of the energy-momentum tensor, namely, the energy density, W, the momentum density, N, and the stress, P, may be obtained using standard variation procedure e.g. [3]. Explicitly,

Usually Eqs. (1.5, 1.6) are thought of as a consequence of the Maxwell equations (1.2). However, we may consider the relations (1.6) as a constraint implied upon the components of the momentum-energy tensor, so there are two independent variables in Eqs. (1.5), for example, W and N. One can easily check that for the nontrivial solutions of Eqs. (1.2), i.e.

NONLINEAR VACUUM ELECTRODYNAMICS

513

for the Jacobian of the transform E, BW, N is non-zero. Thus, instead of looking for the solutions of Eqs. (1.2) we can solve Eqs. (1.5, 1.6) excluding the Poincare invariant I from Eqs. (1.6). 4. Solution

To exclude I it is convenient to introduce the invariants of the energy-momentum tensor, that is, its trace, S = P -W, and the determinant As it follows from Eqs. (1.6)

The latter relations implicitly define the dependence T = T(S). Substituting the Lagrangian (1.3) into Eqs. (1.7) we find that the first nonvanishing term of the expansion of T in powers of S is linear and it is provided by the quadratic term of the expansion (1.3): It is noteworthy that the Born-Infeld Lagrangian (1.4) yields exactly the linear dependence T (S) = - S. The relations (1.5) are resolved introducing the potential N= Restricting ourself with the linear relation between T and S, we obtain the Ampere-Monge type equation for

There are trivial solutions to this equation

with an arbitrary function F,

which correspond to the plane electromagnetic waveforms described by Eqs. (1.2) with I = 0. Besides these, implementing the Legendre transform[4] one can easily obtain the general integral of Eq. (1.8) valid for and, consequently, for As a result, we get the components of the energy-momentum tensor in a parametric form:

where

are arbitrary functions and

Consider, for example, two localized pulses of the arbitrary shape propagating in opposite directions. This corresponds to the following initial conditions:

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A. M. IGNATOV AND V.P. POPONIN

This initial condition is provided by the following choice

where

of

in Eqs. (1.9):

asymptote of the solution (1.9) at

The is then given by

where

is the net energy carried by the corresponding pulse.

5. Discussion Of interest is the geometrical sense of the obtained solution (1.9). The parameters and are, in fact, the light-cone coordinates disturbed by the electromagnetic field. One may say that the electromagnetic field alters the space-time metric due to the dependence of the speed of light on the field strength. In contrast with general relativity, the space-time remains flat. Another interesting point is that for the increase in the pulse amplitude results in delay in energy (and information) exchange between distant points, that is, the solution described by (1.11) is subluminal. This takes place for both the Heisenberg-Euler electrodynamics, which is currently the only one of physical sence, and for the elegant Born-Infeld theory, for which our results are exact. However, for the pulse propagation would be superluminal. From the viewpoint of nonlinear physics, the electromagnetic pulses in vacuum exhibit the soliton-like behavior: the collision results in a phase shift but the form of a pulse remains unchanged. The main interesting point with this respect is that unlike usual nonlinear equations, the shape of the soliton is arbitrary. Bibliography [1] [2] [3] [4]

Heisenberg, W., Euler, H. (1936), Z. Phys, 38, 714 Born, M., Infeld, L. (1934), Proc. Roy. Soc. (London), A144}, 425 Landau, L.D., Lifshitz, E. M. (1971), The Classical Theory of Fields, Oxford, New York, Pergamon Press, Courant, R. (1962), Partial Differntial Equations, N.-Y., London,

PROPOSAL FOR TELEPORTATION BY HELP OF VACUUM HOLES

CONSTANTIN LESHAN S. OCTEABRISCOE & R-L SINGEREI Kiev University MD-6233 Moldova Email: [email protected]

Abstract. We can teleport a body using geometrical properties of space. Teleportation would consist of sending a body outside the universe into zero-space in order to reappear at another point in the universe. Some teleportation properties we can see by a simple mechanical motion. Let body A have a linear and uniform motion in space. Body A passes distance ds during time dt without energy expenses. Consequently body A can be teleported without energy expenses. Energy expenses appear if body A moves between two points with different values of force field. Therefore teleportation is not possible if the startpoint and endpoint has a different value force field. How can time dt decrease? Unfortunately a body cannot move faster than light. There is a single solution. Superluminosity can occur in space where time properties do not exist. The same 0-space exists.

1. Zero-Space and Vacuum Holes As is well known the universe is curved and has a limited volume. After the Big Bang, the initial explosion giving birth to the universe, limited time would occupy a limited volume. This limited volume suggests boundary conditions, even if only in some points. What could be behind this border? Most authors assert "there is nothing outside the Hubble universe, neither galaxies, nor substance; absolutely nothing - neither space, nor time" generally [1]. In other words outside the universe could be a "point form space" where distance between two points is always equal to zero which we could name "zero-space" or "hole". Moreover this border can't exist in a single place because the cosmological principle, which states there are no privileged frames relative to another place or point in the universe, is violated. The border of the universe must pass through every point of space. Virtual holes in spacetime must exist in every point of the universe. This physical object is so-called "vacuum with holes" or "hole vacuum" [2]. Vacuum holes exist as virtual particles. What would happen if we sent body A outside the Universe? Since zero-space is a point and where time as a property does not exist, therefore it cannot contain body A and consequently body A will appear in the real universe at that same moment in time. With the distances between zero-space and any other point of universe being equal to zero, these holes can potentially exist in every point of universe. Therefore body A could appear at random in at point of in the Universe. Let us do the following thought experiment. In order to send body A outside of the universe we must first create a closed surface which consist vacuum holes around body A, for but an instant in time. Then we ask, where was the body A during that time dt that it was surrounded by vacuum holes? Inside of hole sphere body cannot exist because outside of universe cannot exist any body, consequently body A was existing all the time dt in other point of universe already. If we destroy the hole surface around body A we destroy in this way the channel that connect this two points and 515 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 515-516. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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body A will remain in new place. The body was teleported from point 1 to point 2 through zerospace. Inside of hole surface exist the non-Euclidean geometry of A. Poincare. This place is completely isolated system toward to external universe. Not exist any ray or other radiation able to penetrate through the hole in space and time. For internal observer the distance between centre and hole surface is infinite, as the distances between every two points decrease from centre to border and all distances is equal to zero near hole surface. The same properties have the A Poincare model of non-Euclidean universe from 1882 for case of 3-dimencional space. He proposed a model of non-Euclidean universe that is a precise copy of our hole sphere. For imaginary people that live inside of Poincare circumference the distance to border is an infinite. All universe for they are inside of this circumference. The transmitter of matter can be with internal or external hole production. First present a station that have a spherical room where is introduced the material body for teleportation (the sphere has a minimal area). On the external surface of sphere is equipment that produces vacuum holes around the body A. Second method present a station that produce a hole surface around oneself, therefore this station can repeatedly teleport oneself to at random points of universe. In this case the receiver of matter is not necessary. There is probability after a number of teleportations the station will appear on its planet. The equipment for hole production in this case is inside of sphere. The energy expenses are necessary for curvature of the space-time only but not for motion of body from point 1 to endpoint 2. First we must create the station with internal hole production. The station will be very durable as not have friction. During the exploitation term one will launch about thousand spacecrafts into very deep space, on the distance of hundreds or millions of light years. For example spacecraft can appear in Crab nebula or near Sirius. Unfortunately we don't know the endpoint after teleportation, but universe is uniformly and isotropic, therefore is all the same where look for a new planets for colonisation or extraterrestrial life. For hole teleportation we must create the equipment that able to produce the holes in space-time. For it is necessary to research interaction between vacuum holes and matter. Hole teleportation has a lot of advantages toward to quantum one. For QT we must collect all information about object therefore we must decompose one. Who agrees to be destroyed in transmitter in order to be rebuilt in the receiver? Would the soul be copied? So, quantum teleportation is not suitable for humans. The volume of information about a single human is enormous. Just how much information are we talking about anyway? (3) Well the visible human project by the American National Institute of Health requires about 10 Gigabytes (this is about ten CD ROMs) to give the full three dimensional details of a human down to one millimeter resolution in each direction. If we forget about recognizing atoms and measuring their velocities and just scale that to a resolution of one-atomic length in each direction that's about bits (a one followed by thirty two zeros). This is so much information that even with the best optical fibers conceivable it would take over one hundred million centuries to transmit all that information! It would be easier to walk! If we packed all that information into CD ROMs it would fit into a cube almost 1000 kilometres on a side! Enough said? The technology that permitted to build a human in receiver by help of information received from transmitter will appear only in a very far future. You see, QT is practically impossible. All this defects don't exist in hole teleportation. There is not necessary to decompose and rebuild a teleported bodies, therefore the HT is suitable for human teleportation. Objects are teleported faster that light at any distances, even to far stars. We must solve a single problem, how to create a closed hole surface.

References 1 I.D. Novicov, Evolution of universe, Science, Moscow, 1990 2. Conference proceedings, ICPS 94, S. Peterburg, 1994 3. http://www.sees.bangor.ac.uk/~schmuel/tport.html

COSMOLOGY, THE QUANTUM UNIVERSE, AND ELECTRON SPIN MILO WOLFF Technotran Press 1124 Third Street, Manhattan Beach, CA 90266 [email protected]

Abstract. Clifford, Mach, Einstein, Wyle, Schrödinger all pointed out that only a wave structure of particles (matter) can conform with experimental data and fulfill the logic of reality and cosmology. I describe a quantum Wave Structure of Matter (WSM) that satisfies their requirements and predicts the origins of natural laws. This structure is a simple pair of spherical outward and inward quantum waves convergent to a center. The wave pair is the physical origin of the electron quantum spin, which results when the inward quantum wave undergoes spherical rotation to become the outward wave. These two waves are a Dirac spinor, thus this physical quantum wave structure satisfies the theoretical Dirac Equation. But it also forms the structure of the universe! 1. Introduction - A True Science Odyssey

Discovering quantum wave structure is a beautiful adventure in which you find the origin of the natural laws, a powerful tool of science, and an exciting window on the quantum wave universe. Every electron, proton or neutron are quantum wave structures. Understanding their reality demands learning the quantum wave rules. It is easy but one has to think anew and discard the false notion of a material point particle, and recognize that location, charge and mass are properties of the wave structure. Since the quantum wave universe is not directly useful to our personal survival, nature has not equipped us to observe it as easily as apples and tigers. Lacking personal experience people imagine that the electron is a “particle” like a baseball. Laboratory evidence does not support this idea. Clearly, scientists must change their belief from particle to wave. The difficulty of that change is shown in the story below from Omni Magazine: Imagine that you are the commander of the fifth inter-galaxy survey unit and the survey captain reports to you, “They’re made of meat.” “You mean, meat?” “There’s no doubt about it. We picked several from different parts of the planet, took them aboard our vessels, probed them all the way through. They’re completely meat.” “That’s impossible. What about the radio signals? The messages to the stars?” “They use radio waves to talk, but the signals come from machines.” “So who made the machines? That’s who we want to contact.” “They made the machines. That’s what I’m explaining. Meat made the machines.” “Ridiculous! You’re asking me to believe in sentient meat.”

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“I’m not asking you, I’m telling you the results of our research.” “Okay, maybe they’re part meat, like the Weddilei, a meat head with an electron plasma brain inside.” “Nope. They do have meat heads but we probed them all the way through.” “No brain?” “Oh, there is a brain all right. It’s just that the brain is made out of meat!” “Oh? What does the thinking?” “You’re not understanding, are you? The meat brain does the thinking.” “Thinking meat? You’re asking me to believe in thinking meat?” “Yes, thinking meat! Conscious meat! Loving meat! Dreaming meat.” “Really? You’re serious then. They’re made out of meat.” “Finally! Yes. And they’ve been trying to contact us for a hundred of their years.” “So what does the meat have in mind?” “It wants to talk to us. Then I imagine it wants to explore the universe—contact other sentients, swap ideas and information. The usual.” “They actually do talk, then. They use words, ideas, concepts?” “Oh, yes. Except they do it with meat.” “I thought you just told me they used radio.” “They do, but what do you think is on the radio? Meat sounds. Singing meat.” “Omigosh! Singing meat! This is too much. Any true sentients in the galaxy?” “Yes, a rather shy hydrogen core cluster intelligence in a class nine star in G445 zone. Was in contact two galactic rotations ago, wants to be friendly again.” “And why not? How unbearably cold the universe would be if one were all alone!”

New truths of science are often unwelcome. Emotional rejection occurs if the new truth conflicts with established belief. Max Planck once said, “New scientific truth does not triumph by convincing its opponents, but because the opponents die and a new generation grows up unopposed to the new idea.”

2. The New Structure of Matter - Spherical Space Resonances Below is the new truth of the structure of matter that agrees with experimental facts. It is overwhelmingly simple because it uses only three principles to establish the Wave Structure of Matter as the basis of all scientific laws. The proposal that mass and charge were properties of a wave structure in space was consistent with quantum theory since quantum mathematics does not depend on a belief in particle substance or charge substance. The reality is that space waves are real while mass and charge points are mere appearances, “Schaumkommen” in the words of Schroedinger. The famous English geometer, William Clifford (1876), wrote “All matter is simply undulations in the fabric of space.” Einstein and Ernst Mach reasoned that particles must be “spherically spatially extended in space.” Einstein wrote, “ ..hence the material particle has no place as a fundamental concept in a field theory.” Paul Dirac was never satisfied with the point particle because the Coulomb force law had to be corrected by “renormalization.” He wrote (1929) “This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity because it turns out to be small, not neglecting it because it is infinitely large and you do not want it!” Wheeler and Feynman (1945) modeled the electron as spherical inward and outward electromagnetic waves, seeking to explain radiation forces, but encountered difficulties because there are no spherical solutions of vector e-m wave equations. Nevertheless their

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particle sends quantum waves outward and

Milo Wolff (1990-93-97), using a scalar wave equation with spherical wave solutions, found the Wave Structure of Matter (WSM) described here. It successfully predicted the natural laws and all properties of the electron, except its spin. This paper provides a physical origin of spin which accords with the Dirac Equation. 2.1 THE NATURAL LAWS ARE FOUND IN THE WAVE STRUCTURE Our knowledge of science and the universe is based on natural laws, the rules for calculating electricity, gravity, relativity, quantum mechanics, and conservation of energy and momentum. The origin of these laws was unknown. Now their the origin is found to be a quantitative result of the WSM. The wave-structured particle, Figure 1, is termed a space resonance (SR). The medium of the waves and the leading player in the new scenario is space. Space resonances and the laws they produce are derived from properties of the wave medium. Thus, this single entity, space, described by the three principles, underlies everything.

3. Principle I - A Wave Equation This equation describes how quantum waves are formed and travel in the space medium. If the medium is uniform, typical nearly everywhere, only spherical waves occur. If the medium is locally dense, as in the central region of a proton, waves can circulate like sound waves in a drum or sphere. If observed in relative motion, Doppler modulation and elliptical waves appear. Principle I is: Quantum matter waves exist in space and are solutions of a scalar wave equation. The wave equation is: Where AMP is a scalar amplitude, c is the velocity of light, and t is the time. A pair of spherical in/out waves forms the simple structure of the electron or positron. The mathematical properties of this combination display the laws of: mechanics, quantum mechanics, special relativity, and e-m. The waves decrease in intensity with increasing radius, like the force of charge. The inward and outward waves move oppositely, thus

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forming a resonant standing wave. Arriving at the center, the IN-wave rotates, producing ‘spin’ and becoming the OUT-wave. All properties of the waves depend on their medium, space, as is true for all oscillators. There are two combinations of the resonances, electrons and positrons. Thus matter is constituted of two binary elements inward and outward waves. It is curious that the universe, like computer hardware, is binary. Does this have a profound meaning? The rules of quantum mechanics (QM) and special relativity (SRT) are the result of the motion, with a velocity b = v/c, of one SR relative to another, which produces a Doppler shift in both the IN- and OUT- waves. All parameters of QM and SRT for a moving particle appear as algebraic factors in the Doppler-shifted waves; that is, the deBroglie wavelength of QM, and the relativistic mass and momentum increases, exactly as experimentally measured. This can be shown by writing the amplitude received at either SR - both are alike: Received amplitude = 1/r {2 AMP-max} exp[ikg (ct + br)] sin [kg (bct + r)]. This is an exponential carrier oscillator modulated by a sinusoid. In the carrier: Wavelength = h/gmv = deBroglie wavelength with relativistic momentum. = mass frequency with relativistic energy. And in the sinusoid: Wavelength = h/gmv = Compton wavelength with relativistic momentum. Frequency = = b x (mass frequency) = relativistic momentum frequency. There are two solutions of Principle I and two combinations which correspond to electrons and positrons. Charge properties depend on whether there is a + or - amplitude of the IN wave at the center. If a resonance is superimposed upon an anti-resonance they annihilate. The amplitude at the center is finite as observed, not infinite as in the Coulomb rule. They obey Feynman’s Rule: “A positron is an electron going backward in time.” See this by replacing the variable t with a -t in an electron resonance; a positron resonance is obtained. The change of t has exchanged the roles of the IN and OUT waves. 4. Energy Transfer And The Action-At-A-Distance Paradox It is essential to recognize that communication or acquisition of knowledge of any kind occurs only with an energy transfer. Storage of information, whether in a computer disk or in our brain, always requires an energy transfer. Energy moves a needle, magnetizes a tape, and stimulates a neuron. This rule of nature is embedded in biology and our instruments. The energy transfer mechanism is found in Principle II below. One major failure of the classical force laws is they have no physical mechanism for energy transfer. This is the fault of the static point particle model, which contrasts with the dynamic, spatially extended Space Resonance. Ernst Mach observed positive evidence of cosmological energy transfer in 1883, noticing that the inertia of a body depended on the presence of the visible stars. He asserted: “Every local inertial frame is determined by the composite matter of the universe." His

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concept arises from two different methods of measuring rotation. First, without looking at the sky, one can measure the centrifugal force on a rotating mass m and use the inertia law f=ma, to find circumferential speed v. The second method is to compare the object’s angular position with the fixed (distant) stars. Both methods give exactly the same result! Mach’s Principle was criticized because it appears to predict instantaneous action-at-adistance. How can information travel from here to the stars and back again in an instant? The solution lies in the space resonance. Space is not empty because it is filled with the waves of every particle in the universe (Principle II below). Inertia is an interaction with the space medium. There is no need to travel across the universe.

5. Principle II - Space Density Principle (SDP) This principle defines the medium of quantum waves in space. It is very important because the natural laws depend on the waves of the electron-positron which in turn depend on the medium. Thus the medium is the actual origin of the natural laws. Principle II is: Waves from all particles in the universe combine their intensities to form the wave medium of space.

Specifically, the frequency f, or mass m, of a particle depends on the sum of squares of all wave amplitudes, from the N particles in the Hubble universe, which decrease inversely with range squared. This universe exists inside a radius R = c/H, where H is the Hubble constant. This principle contains a quantitative version of Mach’s Principle because the space medium is the inertial frame of the law F=ma. Energy exchange takes place between the mass m and the surrounding space medium. Because particles in the Hubble universe, the medium is nearly constant everywhere and we observe a nearly constant speed of light. But near a large body, a larger space density produces a tiny curvature of the paths of quantum waves (and thus of light). But note that the self-waves of a resonance are counted too. Thus space becomes dense near the resonance center due to its own amplitude. Space is non-linear at the central region, which provides the coupling between two resonances needed for energy transfer. We observe this and call it “charge.” Can this principle be tested? Yes. If a resonance’s self waves affect space density, then the intensity of self-waves at some radius, ro, must equal the total intensity of waves from the other N particles in the Universe. Evaluating this equality yields This is called the Equation of the Cosmos, a relation between the size ro of the electron and the size R of the Universe. Astonishingly, it describes how all the N particles of the Hubble Universe create the space medium and the “charge” of each electron as a property of space.

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Continue the test. Insert values into the equation above. meters. particles. Then meters. This should be near the classical radius, of an electron, which is meters. The test is satisfied. Let's discuss Energy Transfer and the Conservation of Energy. Typically energy transfer occurs between two atomic or molecular quantum states: a source and a receiver. In the source, an electron's energy shifts downward; in the receiver, there is an equal shift upward. Only oscillators with similar frequencies ‘tuned’ to each other can couple and shift frequency. Accordingly, the frequency (energy) changes must be equal and opposite. This is exactly the content of the Conservation of Energy law. 6. Principle III - Minimum Amplitude Principle (MAP)

The third principle is a powerful law of the universe that determines how interactions take place and how wave structures will move: The total amplitude of all particle waves in space always seeks a minimum.

Thus energy transfers take place and wave-centers move in order to minimize their total wave amplitude. This principle is the disciplinarian of the universe. Amplitudes are additive, so moving two opposite resonances closer together will minimize amplitude. Thus, this principle dictates “Like charges repel and unlike charges attract.” because those rules minimize total amplitude. The MAP produces the Heisenberg Exclusion Principle, which prevents two identical resonances (fermions) from occupying the same state. This is disallowed because total amplitude would be maximum, not a minimum.

7. Spin And The Dirac Equation

The physical nature and cause of electron spin was unknown before the WSM. However, a successful mathematical theory of spin had been developed by Nobel laureate Paul Dirac (1926, see Eisele, 1960). It predicted the positron (Anderson, 1931) and a spin of h/4pi angular momentum units. Dirac was seeking a connection between Schroedinger’s quantum equation and the conservation of energy given by

Unfortunately, Eqn (1) uses squared terms and Schrödinger equation cannot. Dirac had a crazy idea: “Try replacing Eqn (1) with a 4-matrix equation.”

Where [Identity], [alpha], and [beta] are new 4-matrix operators. This worked and his Equation (2) became famous. As a result Dirac realized that only two wave functions were needed for the electron. So Dirac simplified the matrix algebra by introducing number pairs, termed spinors, creating a two-number algebra instead of our common single number algebra. His spinor algebra, gave no hint of the physical structure of the

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electron. But now, we see that the in/out wave pair are real spinor waves, the physical counter-part of Dirac's theoretical spinor functions. Spin occurs when the IN wave rotates to become an OUT wave. It uses a 3D property known as ‘spherical rotation’ (Gravitation, Misner et al, 1973) in which space returns to its initial state after two turns. It is necessary that space return, otherwise it would twist up without limit. Two turns produces an angular momentum of ± h/4pi, exactly what is observed. There are only two ways to rotate, CW and CCW. One is the electron; the other, the positron. This is why each charged particle has an anti-particle. Analysis of spherical rotation by Batty-Pratt & Racey (1980) showed that exponential oscillators were spinors. Wolff (1990) realized that the in-out waves of the WSM were real spinors, satisfying the Dirac Equation. The SR also displays other physical properties of an electron including CPT and conversion to a positron. To see this start with the two solutions of the wave equation in spherical coordinates which are:

You can experiment with the CPT inversions. To perform a Time inversion, change t to t, which converts the positron into an electron. To perform a mirror inversion (Parity), imagine that the waves are viewed in a mirror; a positron is a mirror image of the electron. To change a particle to an anti-particle (Charge inversion), switch the in-waves and the out-waves, and the spin . The successive C, P, and T inversions return an electron to its initial state proving the CPT rule, as a property of the wave structure. Another physical property of the quantum wave electron is that inverting its spin axis is not equivalent to reversing its spin, in contrast with cylindrical rotation. Test this. The electron spinor is To reverse the spin, change t to -t. The spin reversed spinor becomes which exchanges the outgoing wave with the incoming wave. Compare with inverting the spin axis of the electron spinor using the inversion matrix:

Contrary to our human intuition, inversion and spin reversal are not the same, verifying the quantum wave structure of charged particles.

8. The Origin Of The IN Waves And The Response Of The Universe At first thought it is a puzzle where the IN waves come from. But we have ignored the waves of all other particles in space. To find reality and a rational origin of the inward waves, we must deal with the wave-filled universe. Christian Huygens, a Dutch

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mathematician, found that if a surface containing many separate wave sources was examined at a distance, the combined wavelets of the sources appeared as a single wave front with the shape of the surface. This is termed a ‘Huygens Combination’ of the separate wavelets, Figure 2.

When the outgoing wave encounters other particles in the universe, their out-waves are mixed (“reflected’) with a component of the initial out-wave. These components return, in phase, to the initial center. Thus waves from all the other particles combine to form a Huygens Combination wave front that is the in-wave of the initial particle. We should not imagine each particle as one pair of IN and OUT waves, because one pair cannot exist alone. Each particle depends on all others particles in the universe to create its IN wave. We have to think of ourselves as inextricably joined with other matter of the universe. In conclusion, the value of studying the WSM is the insight it provides to deeply analyze quantum wave structure, the cosmos, the natural laws and their application, especially to ICs, computer memory devices, and energy. References E. Battey-Pratt and T. Racey (1980), Intl. J. Theor. Phys. 19, pp. 437-475. Louis Duc de Broglie (1924), PhD Thesis “Recherché sur la Theorie des Quanta,” U. of Paris. William Clifford (1876), in The World of Mathematics, Simon & Schuster, NY, 1956. Paul Dirac (1929), Proc. Roy. Soc. A 117, p. 610. Paul. Dirac (1937), Nature, London. 174, p. 321. Albert Einstein (1950), Generalized Theory of Gravitation. John A. Eisele (1960), Modern Quantum Mechanics with - Elementary Particle Physics, John Wiley, NY. C.W. Misner, K. Thorne, and J.A. Wheeler (1973), Gravitation, W.H. Freeman Co., p. 1149. Walter Moore, (1989), Schroedinger - Life and Thought, Cambridge U. Press, p. 327. J. Wheeler and R. Feynman (1945), “Interaction with the Absorber..” Rev. Mod. Phys. 17, p. 157. Milo Wolff (1990), Exploring Physics of Unknown Universe, ISBN 0-9627787-0-2,, Technotran Press, CA. Milo Wolff (1991), Invited paper at 1st Sakharov Conf. Phys, Moscow, May 21-31 p. 1131 Nova Publ., NY. Milo Wolff (1993), “Fundamental Laws, Microphysics and Cosmology,” Physics Essays 6, pp. 181-203. Milo Wolff (1995), “A Wave Structure for the Electron,” Galilean Electrodynamics 6, No. 5, pp. 83-91. Milo Wolff (1997) “Exploring the Universe..” Temple University Frontier Perspectives 6, No 2, pp. 44-56. Milo Wolff (1997) “The Eight Fold Way of the Universe,” Apeiron 4, No. 4. Milo Wolff (1997) “Mass Change and Doppler Shift ..” Galilean Electrodynamics 8, No. 4. Milo Wolff, “ Quantum Science Corner” The website: http://members.tripod.com/mwolff

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THOMAS D. ANGELIDIS Centre for Mathematical Physics 19 Cheval Place, Suite 5014 London SW7 1EW, United Kingdom. E-mail: [email protected]

À propos Writing this paper in honour of Jean-Pierre Vigier’s 80th birthday, brings to mind the time when I was first introduced to him by my late friend Karl Popper in 1979 at a seminar. Since then I found in Vigier not only a most captivating person, but also a thinker utterly devoted to his problems. We both, I trust, belong to the fraternity of critical rationalists, the fraternity of those who are eager to argue, to learn from one another, and who have the intellectual courage in changing their mind, under the influence of criticism, even on points of fundamental importance to their cherished conjectures and beliefs.

1. Locality Versus Nonlocality: A Verdict On “Nonlocal Interactions”

The central theme in Vigier’s diverse work [1] has been his battle against the Copenhagen interpretation of the theory of the quantum formalism and, partly following the steps of de Broglie, Einstein, Schrödinger et al., his valiant quest to construct a stochastic interpretation of This is a realistic interpretation of where particles are presumed to have definite timelike trajectories in space-time (in contrast to the Copenhagen interpretation which denies this), based on Dirac’s notion of a (covariant) aether, presumed to be a real physical (stochastically fluctuating) field, mediating “real interactions among particles” induced by the so-called “quantum potential” Q presumed to act at any distance. Vigier’s interpretation (like Bohm’s [2]) fails to provide a local explanation of the quantum statistical correlations exhibited in the Einstein-Podolsky-Rosen (EPR) [3]Bohm (B) [4] ideal experiment. In place of this hiatus, and deviating from the steps of de Broglie, Einstein, Schrödinger et al., Vigier (like Bohm) advances the notion of interactions among particles at any distance, now referred to as “nonlocal interactions” induced by the “quantum potential” Q, as a necessary assumption for a “nonlocal yet causal” explanation of the EPRB ideal experiment. 525 R.L. Amoroso et al (eds.), Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 525-536. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Four questions arise here: (1) Do particles have timelike trajectories in space-time?; (2) Are “nonlocal interactions” a necessary assumption?; (3) Are “nonlocal interactions”, if they exist, compatible with special relativity?; (4) Do such “nonlocal interactions” exist ?. Our arguments in support of our replies to the four questions are as follows. Our reply to question (1) is: Yes. We have shown [5] that the demonstration of conservation of momentum as a theorem of requires that particles do have timelike trajectories in space-time. Furthermore, the recent results of Popper’s proposed experiment [6-8] speak for particles having trajectories in space-time as itself requires and against the Copenhagen interpretation of which denies this (we will not elaborate further here). Our reply to question (2) is: No. The consistent theory gives a local and causal (“common cause”) interpretation of the EPRB ideal experiment [9,10]. The theory shows that local action suffices to explain all that predicts for the EPRB ideal experiment and its experimental realization by Aspect et. al. [11]. And so there is no need to introduce “nonlocal interactions” in order to explain the quantum statistical correlations exhibited in the EPRB ideal experiment. Whence, the assumption of “nonlocal interactions” is not necessary. Our reply to question (3) is: No. If assumed to exist, “nonlocal interactions” are not compatible with special relativity. In the Minkowski space-time of special relativity, the “quantum potential” (the subscript m in stands for “Minkowski”) is interpreted by Vigier [1] and Bohm et al. [12] as inducing influences (“nonlocal interactions”) connecting spacelike-separated events, events which lie outside each other’s light cones. As we have alluded in [10, p. 1636], it is not difficult to show that, if it exists, any influence connecting spacelike-separated events clashes with the causal structure (order) of events in The formal demonstration is as follows: According to Vigier et al. (Ref.1, pp.95-100), and Vigier has recently confirmed this to us, is a symmetric function on Since a function is a one-to-one relation, is a symmetric relation on Let ‘(x)’ and ‘(y)’ stand for and respectively. The definition of a symmetric relation is, in symbols, In words, for any event x and any event y, if holds between x and y, then holds between y and x. By Zeeman’s theorem [13], relativistic causality R is a partial ordering on and, therefore, all (elements) events in must satisfy, inter alia, the antisymmetric relation where is the negation sign. In words, for any pair of distinct events x and y, if x and y are causally connected (R holds between x and y), then y and x are not causally connected (R does not hold between y and x). Now let hold between a pair of distinct events x and Then, by modus ponens, from and the conditional we deduce Assume that the “causal connection” is consistent with the R-causal connection, that is, whenever holds between x and y, R holds between the same x and y, in symbols, or, equivalently, where this equivalence is deduced by contraposition and by an alphabetic change of the bound occurrences of x and y. By modus ponens, from we deduce Rxy. By modus ponens, from and Rxy and

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we deduce By modus ponens, from and we deduce Whence, from the assumption that holds between a pair of distinct events x and y and from the assumption that the “causal connection” is consistent with the Rand causal connection, we deduced both an obvious contradiction. As a consequence, if the “quantum potential” is interpreted in Vigier’s sense as inducing “nonlocal interactions” connecting spacelike-separated events and, if as Vigier [1] and Bohm et. al. [12] assume such “nonlocal interactions” exist, then such “nonlocal interactions’ are not be compatible with the causal structure (order) of events in We are aware of the claim by Vigier et. al. (Ref.1, pp.95-100) that such “nonlocal interactions” satisfy Einstein’s causality. However, their attempt to establish their claim nowhere addresses, nor considers, nor brings in explicitly the (underlying) causal structure of events in And so their attempt apparently fails to establish their claim. We are also aware of suggestions in the literature [12] that such “nonlocal interactions” do not manifest themselves at the statistical level in the shape of a signal being exchanged faster than light and thus no relativistic prohibitions are violated. However, such suggestions do not address the real issue. In reply, we note that our (coordinate-free) demonstration above is not only independent of any signalling assumption, but also shows that the conflict with relativistic causality lies deeper than the statistical level. It lies at the level of individual events and of their particular outcomes in where, if it exists, the “nonlocal interactions” are presumably at work by exerting changes at any distance in the physical properties attributed to individual particles located in spacelike-separated regions. There is no “peaceful coexistence” [14] between special relativity and such “nonlocal interactions”, if they exist. And, as we have alluded (Ref.10, p.1637), for any realistic interpretation of such “nonlocal interactions”, if they exist, lead to an impasse which can only be resolved by experiment. Fortunately, such an experiment is now at hand: Popper’s experiment shows that such “nonlocal interactions” do not exist. Our reply to question (4) is: No. Popper’s experiment, which is essentially Aspect’s experiment with slits rather than with polarizers, speaks against Vigier’s prediction that the scatters of individual particles would be correlated by “nonlocal interactions” induced by the “quantum potential” Q. That is, when the particle moving to the left scatters upwards, the particle moving to the right scatters downwards since, according to Vigier, the measurement by the left slit on the left particle has a “nonlocal” influence on the right particle, even with the right slit removed. In a discussion with Popper (in May 1983), Vigier argued that his prediction would correspond to the situation in the Bohm version (measurement of polarization or spin rather than position) of the EPR experiment. In Vigier’s own words [15]: “ Now the question is: Are these measurements correlated or not ?.... Therefore, also in your [Popper’s] experiment there will be nonlocal correlations between the two photons. If one slit twists the spin of one photon, there will be an immediate action on the other photon going in the opposite direction. This implies that we have nonlocality... we have shown that the quantum potential gives rise to an action at a distance between the two photons.”

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In sharp contrast to Vigier’s prediction, our prediction - based on and 0 -was that Popper’s proposed experiment would show a “null result” - no deflection of particle trajectories [16,17]. Nothing would happen. The right particle (photon) would go on undisturbed in the absence of the right slit B, precisely as locality and require. The recent results [18] of Popper’s experiment clearly show that nothing happens: the right particle (photon) goes on undisturbed in the absence of the right slit B. This “null result” shows that there are no “nonlocal actions” between the left particle (photon) and the right particle (photon) causing the right particle (photon) to scatter (to “twist”) in the absence of the right slit B. Popper’s experiment gives a striking demonstration of locality and of our prediction of a “null result” (we believe a long-standing bet of $5 with Vigier as to whose prediction would be falsified by Popper’s experiment is now due!). Sudbery [19] (with Redhead [20] following) claims that “there must be some flaw” in Popper’s deduction from the Copenhagen interpretation, namely, “narrowing slit A increases the spread of momentum at B”, and that “this would be a striking nonlocal effect...shared by any interpretation that incorporates the projection postulate...”. Sudbery’s own deduction - “in order to implement the projection postulate” - is that “narrowing slit A does not increase the spread of momentum at B because it is already infinite ”. So the “flaw” Sudbery attributes to Popper is that the “spread of momentum at B” is infinite rather than finite, and presumably the “spread of momentum at B” cannot possibly be further increased. But i f so, then the actual “null result” obtained, almost “no spread of momentum at B” (cf. Fig.5, Ref.18), decisively refutes Sudbery’s own deduction of an infinite “spread of momentum at B”. Popper’s “own conjecture” [7,8] was the same as our prediction of a “null result”. It was essentially based on preliminary work in the long quest to realize Einstein’s hope of strengthening into a realistic and local theory (like Popper was aware of our work in its most intimate details since its inception, and he steadfastly supported it to his last days. At the same time, he invited others to apply to it the kind of earnest critical scrutiny Popper himself had applied to it in all sincerity. Popper’s invitation has so far been met with an (uneasy, perhaps) official silence, broken only by a few idle rumours and fleeting whispers which, as far as we know, have not even begun to take the shape of an argument [21]. The need to carry out Popper’s proposed experiment and its crucial importance, inter alia, for the issue of locality vs. nonlocality was for almost 20 years summarily dismissed by most physicists nurtured in the Copenhagen doctrine as being part of “rubbish of a most stimulating kind” [22]. Setting aside such disrespectful remarks, it turned out that what had been relegated to “rubbish” was an unrecognised nugget of wisdom waiting to be retrieved. It was our unwavering promise to Popper to see that his proposed experiment was carried out (knowing how close it was to his heart). It required a lot of dedication and perseverance to overcome the stagnant, if not hostile, initial situation. Our arguments persuaded some experimental physicists to suspend their disbelief (albeit not for long), weaned them from some mistaken criticisms of the proposed experiment, steered them clear of what was mistakenly considered to be Popper’s proposed experiment, and tutored them to understand and focus on the real issues involved in Popper’s experiment [23].

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This eventually led to the realization of the proposed experiment almost as intended. Yet, even after the experiment was done, a lingering disbelief has remained as is evident from the statement: “Indeed, it is astonishing to see that the experimental results agree with Popper’s prediction.” [18]. Why the astonishment? In a real local world, as envisaged (say) by the theory this is as expected. Rather it is the Copenhagen doctrine that cannot come to terms with our prediction of a “null result” which gives rise to the exclaimed astonishment at, and the lingering disbelief in, the experimental results actually agreeing with Popper’s prediction. Notwithstanding the results, and without any explanation, later the authors suddenly reverted to the trail of mistaken attributions to Popper, and asserted that [18]: “Popper and EPR made the same error...”. What “error” ?. The “error” of allegedly confusing the “twoparticle entangled state” with the “state of two individual particles”. We find nowhere an argument in support of this allegation except some rather dubious assertions that “the conditional behaviour” of a particle “in an entangled two-particle system is different” and that “the uncertainty principle is not for ‘conditional’ behaviour”. We find it difficult to decipher such assertions, let alone argue either for or against them (incidentally, the authors denied our request for a note in their paper [18] disclaiming any implied agreement with their assertions). Instead of their attempt to explain the results away (à la Copenhagen with nothing more on the menu than “click-click” measurements), we invite the authors [18] to make the earnest effort (like Sudbery [19]) to enlighten us with a clearly formulated argument that could at least be sufficiently understandable to enable an assessment.

2. A Local Extension of

A Verdict on “Impossibility Proofs”

The EPR argument is that either we must give up locality or we must admit the incompleteness of the theory of the quantum formalism [3]. We strengthened by adding four postulates to it (Section 5). And by making manifest that the extended theory is local we thereby showed [9,10] that the (unextended) theory is incomplete as the EPR argument would have it. The strengthening of affects only the universe of discourse of by extending the range of its variables in a demonstrably consistent way. The consistency proof proceeds by giving a model of the extended theory in the model-theoretic sense [10]. The theory is a proper extension of in symbols, (proper inclusion). This strengthening of seems to be minimal. The suggested “impossibility proofs” of a consistent local extension of not only fall short of their stated goal, but are apparently refuted in the presence of the consistent local theory In the class of “impossibility proofs” belong the Bell “impossibility proof” [24], the Kochen-Specker (KS) “impossibility proof” [25], and the GreenbergerHorne-Zeilinger (GHZ) “impossibility proof” [26], and their variants [27]. We know that the KS argument adds to certain formal constraints, which not only cannot be deduced from (and, therefore, do not belong to but also apparently go

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well beyond the three formal postulates of locality (or locality constraints) enunciated by Bell et. al. (Section 3). Setting aside the issue of the unclear physical significance of the KS constraints, we know that the KS theory, obtained by adding the KS constraints to is inconsistent. Now an inconsistent set of statements, say, are just those which have every statement A as a consequence is inconsistent for every statement A). So from the inconsistent KS theory one could deduce locality and nonlocality too (and perhaps, if one likes, that the Moon is made of green cheese). It is almost trivial that the inconsistent KS theory is stronger than the weaker consistent theory (obtained by adding the locality constraints to In other words, the KS “contradiction” cannot be attributed to locality. And locality, as incorporated in the weaker consistent theory is safe and well. The theory and locality cannot possibly be the target of the KS “impossibility proof, let alone be refuted by it. In the same vein, and locality cannot possibly be the target of the GHZ “impossibility proof, let alone be refuted by it. We therefore question the validity of the claims [27] that these “impossibility proofs” refute locality. It is not difficult to show (we shall not do so here) that there is a common flaw in the arguments leading to the KS and the GHZ “impossibility proofs”. The flaw lies in a certain misconception of the link between probability functions and semantic notions. Bell’s argument is based on the so-called “violated Bell inequality”, call it and purports to show that itself is nonlocal. In Sections 7 and 8, we show that and, in the presence of that With this, Bell’s argument fails and so does his “impossibility proof”. And with it fails Bell’s conjecture of nonlocality. As a consequence, all experiments based on do not establish that itself is nonlocal since does not belong to Apparently, all such experiments have altogether missed their intended target, namely, locality. On the other hand, there is a proper Bell inequality deducible in In other words, satisfies the inequality We have called [28,29], and here we call again, for an experiment to test the validity of the proper Bell inequality and submit it as a crucial test against the apparently false yet spellbinding conjecture of nonlocality. Before Bell the most one could assert was that the theory itself was silent on the issue of locality. In fact, Einstein was right never to have claimed that was nonlocal. But Einstein did claim that the Copenhagen interpretation was nonlocal. Einstein clearly did not identify with its Copenhagen interpretation. Einstein drew this distinction in order to use his Principle of Local Action against the Copenhagen interpretation and not against Einstein’s distinction is now affirmed by the fact that and What the Copenhagen interpretation adds, inter alia, to is von Neumann’s famous projection postulate asserting an ‘acausal’ state transition (‘acausal’ in the sense of not falling under the remit of a state-evolution equation, say, the Schrödinger equation), also referred to as “collapse of the wave packet” or as “state reduction”. It is the addition of this postulate that renders the Copenhagen interpretation nonlocal in the sense of action-at-adistance and (historically) gave rise to the spellbinding conjecture of nonlocality.

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In the presence of not only the consistent local theory but also itself is apparently incompatible with the addition of von Neumann’s projection postulate and its Copenhagen interpretation. And, as hinted in our reply to Sudbery (Section 1), the “null result” of Popper’s experiment not only gives a striking demonstration of Einstein’s principle of locality, but also refutes von Neumann’s projection postulate and with it the nonlocal Copenhagen interpretation. For a non “null result” (deflection of particle trajectories) would be the prediction of any interpretation that adds to the projection postulate (as Sudbery apparently admits).

3. The Formal Postulates of Locality

Bell et al. [24,30] enunciated three conditions for locality. We re-state them here in the shape of three formal postulates of locality which any theory T, with some of its postulates expressed by the quadruple of specified functions, must satisfy if T is to qualify as a local theory in the sense of Bell et al. (L1) Any joint probability function must be defined as a specified instance of the syntactical form where any specified function must not depend upon the variable and where any specified function must not depend upon the variable The product form is known as the “factorizability condition”. (L2) Any specified function variable (L3) Any specified range or upon the variable

must not depend either upon the variable

of the variable

or upon the

must not depend either upon the variable

Note well that (L1) stipulates that the values of the functions and must be bounded by 0 and 1, as probabilities should be. And note well that (L2) does not exclude the possibility that the function may be chosen to depend upon some other variable, say provided is a variable distinct from both variables and

4. Bell’s Conjecture of Nonlocality

Bell’s “impossibility proof”, which we shall here call Bell’s conjecture of nonlocality, purports to show “the incompatibility of any local hidden variables theory with certain quantum mechanical predictions” [31]. We shall slightly sharpen Bell’s own formulation here. Let T be a theory with some its postulates expressed by the quadruple where is the range of the

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variable is a specified function defined on and are specified functions and defined on and D is the range of the variables Let be the logical conjunction of the three formal postulates of locality where the symbol stands for the (truth functional) conjunction. Then, Bell’s conjecture of nonlocality asserts that:

holds. Or, in Bell’s [24] own words, the QF probability function

defined by

“cannot be represented, either accurately or arbitrarily closely, in the form of Eq(1)”.

5. The Postulates Added to the Theory

By a theory (in some formal language) we mean a set of sentences (well-formed formulae, abbreviated to “wwf”) which is closed under deducibility, that is, such that for each sentence (wff) if then A subset of a theory T is called a set of postulates for T if for every Let be the set of postulates of the quantum formalism (QF). Then the theory of the quantum formalism is the set defined by:

We have added to the four following postulates and gave a model thereby establishing the consistency of the extended (or strengthened) theory [10]. We have shown that the theory is a proper extension of since there are sentences, say, which belong to but not to in symbols, (proper inclusion).

The added postulates manifestly satisfy Their physical interpretation has been given elsewhere [9,10]. By the rules of substitution, the range of the variable must be The values of and are

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bounded by 0 and 1, as probabilities should be. This answers Feynman’s “fundamental problem” [32]. The theory is consistent, as the theory of the model is always a consistent theory (Ref. 10, p.1652). And from one deduces [9,10] a family of functions which converges uniformly to a unique limit function identical with the QF function for given by Eq(2), as the syntactical form of Eq(1) precisely requires. If we understand Bell’s own words correctly, the possible existence of a family of functions endowed with this property was denied by him. Whence, the consistent theory refutes Bell’s conjecture of nonlocality. In the presence of the sum of products of probability amplitudes in QF, corresponding to the two mutually exclusive alternatives involved in the entangled state can be transcribed into the sum of products of conditional probabilities in for the alternatives in question with the predictions of QF preserved. This is seen from:

deduced from (L1) using the postulated distribution given by Eq(3) shows that each belonging to can be written as the sum of two real-weighted products of conditional probabilities corresponding to the two mutually exclusive alternatives [9,10]. Furthermore, Eq(3) shows how Bell’s hypothesis of “local causality” or of “no action at a distance” [33], formulated as the “factorizability condition” (L1), is satisfied in Incidentally, from Popper’s [34] formal theory of (conditional or relative) probability, all the theorems of Boolean algebra can be deduced (Kolmogorov’s theory of probability fails to do this). And Boolean algebra can of course be interpreted in many ways including, if one likes, “classical” ways. As a consequence, we have questioned [35] the validity of a ‘general proof’ by Deutsch and Ekert purporting to have established that entangled quantum states “generally have no classical analogue”, that is, without exception. For, if one likes, Eq(3) can be interpreted as the “classical analogue” or counterpart in of the entangled quantum state

6. The Proper Bell Inequality

Proposition: Proof: The proper Bell inequality (first-order sentence) is a theorem of or, equivalently, is a model of The proper Bell inequality is deduced from the conjunction of together with an arithmetical lemma [30] and reads:

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where the symbols occurring in are individual constants which the structure maps to designated individuals objects (numbers) of the domain G of (i.e. is a real number etc.) [36]. This mapping under must be distinguished from a value assignment in which does not affect the denotation of individual constants [36]. The designated individual objects (numbers) are determined by and are here interpreted as the distinct directions of the settings of the polarisers. We have shown [28,29] that in all the values of the QF probability function satisfy the proper Bell inequality

7. The “Violated Bell Inequality”

Now a more pressing question may be the following: Does the so-called “violated Bell inequality”, call it belong to Reply: No, does not belong to Proposition: Proof: Let be the structure that maps the individual constants to the designated individual objects (numbers) of called [37] the “maximum violation values”. Then, in the false propositional sentence (“violated Bell inequality”):

would be deducible from the true Bell sentence (inequality) i f there existed a value assignment in such that the variable is assigned the value in the first two terms occurring in AND is assigned the value in the last two terms occurring in with (due to the distinct settings of each polariser), that is:

Going from right to left in Eq(4), we would then have under one and the same value assignment in which is impossible as no number is different from itself, an obvious contradiction. Thus, no such value assignment in exists. And since the consistent theory is closed under deducibility, it follows that What went wrong ? Bell’s “substitution” leading to the false sentence (“violated Bell inequality” distinct from the true Bell sentence (proper Bell inequality would amount to the existence of a value assignment in under which a number is different from itself. We would respectfully submit that such an absurdity as Bell’s “substitution” has nothing to do with locality or with the quantum formalism. For arbitrary substitutions are not always admissible in derivations (in sharp contrast to proofs, where the premisses are logically true): A derivation cannot admit a step which depends on an arbitrary substitution because an initial formula generally does not logically imply a substitution variant of itself [38] (we will not elaborate on this point further in this paper).

LOCAL THEORY 8. Bell’s Argument Based on

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Fails

What does the proposition tell us about Bell’s argument purporting to show that the theory of the quantum formalism itself is nonlocal ?

Proposition: Proof: Since and it follows that Since Bell’s argument fails to show that itself is nonlocal. And with it fails Bell’s ‘impossibility proof’ and his conjecture of nonlocality - this conjecture having been already refuted by the theory As a consequence, all experiments based on do not establish that itself is nonlocal since does not belong to Apparently, all such experiments have totally missed their intended target, namely, locality. Fortunately, not all is lost. Although the “null result” of Popper’s experiment gives a striking demonstration of Einstein’s locality, we call for an experiment to test the validity of the proper Bell inequality and submit it as a crucial test against the apparently false conjecture of nonlocality, should there still be a lingering disbelief in locality.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

17.

18. 19.

Vigier, J.P., in Jean-Pierre Vigier and the Stochastic Interpretation of Quantum Mechanics, eds., Jeffers, S., Lehnert, B., Abramson, N., and Chebotarev, L., Apeiron, Montreal, 2000. An anthology of Vigier’s papers. Bohm, D.J., Phys. Rev. 85, 166, 180 (1952). Einstein, A., Podolsky, B., and Rosen, N., Phys. Rev. 47, 777 (1935) and Einstein, A., in Albert Einstein: Philosopher-Scientist, ed. Schlipp, P.A., La Salle (3rd edn.), Open Court, 1970, pp. 81-87. Bohm, D.J., Quantum Theory, Prentice-Hall, Englewood Cliffs, NJ, 1951, pp 611-623. Angelidis, Th. D., Found. Phys. 7, 431 (1977). Popper, K.R., Quantum Theory and the Schism in Physics, Hutchinson, London, 1982, pp. 27-30. Popper, K.R., in Open Questions in Quantum Physics, D.Reidel Publishing Co., Dordrecht, 1985, pp. 5-11. Popper, K. R.,in Determinism in Physics, eds., Bitsakis, E. et.al., Gutenberg, Athens, 1985, pp. 13-17. Angelidis, Th. D., Proc. Athens Acad. 66, 292 (1991). Angelidis, Th. D., J. Math. Phys. 34, 1635 (1993). Aspect, A., Dalibard, J., and Roger, G., Phys. Rev. Lett. 49, 1804 (1982). Bohm, D.J., and Hiley, B.J., The Undivided Universe, Routledge, London. 1993. Zeeman, E.C., J. Math. Phys. 5, 490 (1964). Shimony, A., Search for a Naturalistic World View, Vol. 2., Camb. Univ. Press, 1993, pp. 151-154. Vigier, J.P., in Open Questions in Quantum Physics, D.Reidel Publishing Co., Dordrecht, 1985, pp. 26-27. Popper’s dedications (Ref.17) to me were, inter alia, in recognition of my contributions in our discussions towards a sharper formulation of his proposed experiment: (a) my proposal for the removal of one of the two slitted screens, which I considered to be a crucial test for a “null result” (no deflection of particle trajectories), and (b) that a “point source” was not necessary to carry out the experiment (only the individual emission events need be localized, as they actually are, within a somewhat extended real source). In his book, Quantum Theory and the Schism in Physics (Ref.6), Popper wrote by hand (as always): “With all good wishes for Thomas Angelidis from Karl Popper, 7-12-83, a date at which this book was superseded”. In his copy of the Proceedings of the Bari Workshop (May 1983), Open Questions in Quantum Physics (Ref.7), Popper wrote: “To Thomas from Karl in friendship. March 15, 1985”. And on one of his papers, Popper wrote: “For Thomas, with admiration and love, from Karl, 4-2-1987”. Kim, Y.H., and Shih, Y.H., quant-ph/9905039 v2, 19 October 1999, available from the Web site: http://xxx.lanl.gov, to be published in Foundations of Physics. Sudbery, A., in Microphysical Reality and Quantum Formalism, Vol. 1, eds., van der Merwe et. al., Kluwer Academic Publishers, Dordrecht, 1988, pp. 267-277. In the quoted text, we interchanged ‘A’ and ‘B’ for agreement with Refs. 6-8,18. Also see references therein to other similarly mistaken arguments.

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20. Redhead, M.L.G., in K. Popper: Philosophy and Problems, ed. O’Hear, A., Cambridge University Press, Cambridge, 1995, pp. 163-176. Here Redhead claims (p.168) “the flaw in Popper’s argument is that he misunderstands the nature of the EPR correlations” and refers to Sudbery’s (Ref.19) “critique..making broadly similar points”. Redhead also writes (p.163) that “my great regret is that he cannot respond to this paper with criticisms of my arguments!”. The results (Ref.18) of Popper’s experiment decisively refute Redhead’s own prediction, and we believe this would have been Popper’s reply to Redhead. 21. In Ref. 20, Redhead refers (pp.175-176) to Popper’s “support to the work of Thomas Angelidis”. The rest of Redhead’s story calls for some frank clarifications. (A) In a rather extensive correspondence (from February 94 to April 94), Popper did reply in detail to Redhead’s comments on my work (Ref.10). (B) Redhead does not mention Popper’s letter of 16 April 1994 where Popper wrote “both Thomas and I should very much like” Redhead to write up “a short critical paper” and submit it to J. Math.Phys. together with my reply. That was more than six years ago and Redhead has yet to do it. (C) As for Redhead’s comments: (i) he multiplied my postulate n3 with an ad hoc expression, which he had not realized its value was zero, and he called it a “generalization”!; and (ii) he asserted that a substitution of one variable for another amounts to an identity. Presumably, this may explain Redhead’s reluctance to publish his comments on my work (Ref.10). Also see Note Added in 2002 below (Ref. 39). 22. Mermin, N.D., Boojums All the Way Through, Cambridge University Press, Cambridge, 1990, p. 195. 23. Shih,Y.H., private communication (e-mail, 20 October 1997), “Dear Thomas,...I remember these discussions and thank you for teaching me about Professor Popper’s experiment. One of my student(s) started that experiment two weeks ago. It is not an easy experiment...I will keep asking you questions about Popper’s experiment..”. Yet my essential contributions (described as “important suggestions” [18]), which played a key role in bringing about the realization of Popper’s experiment, remain unacknowledged. 24. Bell, J.S., Physics (N.Y.) 1, 195(1964). 25. Kochen, S., and Specker, E.P., J. Math. Mech. 17, 59-87 (1967). 26. Greenberger, D.M., Home, M.A., Zeilinger, A., in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, ed., Kafatos, M., Kluwer Academic Publishers, Dordrecht, 1989, pp. 73-76. 27. Mermin, N.D., Phys. Rev. Lett. 65, 3373 (1990). 28. Angelidis, Th. D., in New Developments on Fundamental Problems in Quantum Physics, eds., Ferrero, M., and van der Merwe A., Kluwer Academic Publishers, Dordrecht, 1997, pp. 17-29. 29. Angelidis, Th. D., in Causality and Locality in Modern Physics, eds., Hunter, G., Jeffers, S., and Vigier, J.P., Kluwer Academic Publishers, Dordrecht, 1998, pp. 451-462. 30. Clauser, J.F., and Home, M.A., Phys. Rev. D 10, 526 (1974). 31. Shimony, A., Found. Phys. 19, 1426 (1989). 32. Feynman, R.P., Intl. J. Theor. Phys. 21, 467 (1982). 33. Bell, J.S., J. de Phys. (Paris) 42, C2, 41 (1981). 34. Popper, K.R., The Logic of Scientific Discovery, Hutchinson, London, 1972 (6th impression), p. 329 and Appendices *iv and *v. 35. Angelidis, Th. D., in Mysteries, Puzzles and Paradoxes in Quantum Mechanics, ed., Bonifacio, R., American Institute of Physics (Conference Proceedings 461), New York, 1999, pp. 255-259. 36. Bell, J.L., and Machover, M., A Course in Mathematical Logic, North-Holland, Amsterdam, 1977, pp. 10, 49-50, 162-163. 37. Clauser, J.F., and Shimony, A., Rep. Prog. Phys. 41, 1881 (1978). 38. Carnap, R., Introduction to Symbolic Logic, Dover, New York, p. 49. 39. Note Added in 2002: I take Redhead to have laid to rest his two stillborn comments (Ref. 21), since he nowhere upholds them in his later joint paper [J. Math.Phys. 40, 4290 (1999)].The thesis of the third comment is also stillborn since the alleged counterexample is not a counterexample. The proof of uniform convergence rests on the inequality valid for and (Ref. 10, p. 1651, line 3). Substitute the given “counterexample” values, namely, “for any let and let Then and Since their alleged “counterexample” values clearly satisfy the inequality and thereby satisfy the sentence (the formal definition of uniform convergence). Whence, the sentence is true. What has gone wrong? Redhead and Melia would have established their presumed failure of uniform convergence had they succeeded to show (but did not) that (the negation of is true, where is And the irony is that their alleged “counterexample” values show that is false! Upon instantiation, from one deduces The single value for suffices to render the first conjunct false and thereby show that is false. What else has gone wrong? Redhead and Melia failed to correctly negate the sentence since they write in their paper (p. 4293, line 8 from below) that “..there is some such that for every there is some with...”. But there is no universal quantifier occurring in the prefix of Only the existential quantifier occurs in the prefix and reads “there is a ”. Redhead should have known that, upon negation, a universal quantifier is replaced by an existential quantifier!. The remainder of their paper exhibits an even more confused, if not misleading, travesty of the rather straightforward textbook notion of uniform convergence and of the physical interpretation of the local theory Having fairly and squarely refuted their main thesis here (refuting other sophistries is left as an exercise), I would remind Redhead that we had a gentleman’s agreement (Ref. 21) to have his comments and my reply published together. Had Redhead honoured our agreement, I would have pointed out the pitfall in this third stillborn comment before the rush to publish it (in my ignorance) and thereby averted such a public display of nonsense.

INDEX

Absolute space, 58-60 Absorber Theory of Radiation, 81, 82 Action at a distance, 81, 520 Advanced waves, 60 Aether, 242-3 Affine connection, 1 Affine structure, 298, 369, 481 Affine Theory of Gravity, 295, 299 Aharonov-Bohm effect, 3, 242, 257 Amoroso, Richard L., 27, 59, 147, 241, Angelidis, Thomas, 525-536. Angular momentum, 245, 254, 350 Anomalous Red Shift, 22-3 Antiphoton, 235-8 Arp, H., 23, 104 Arrow of Time, 74, 324, 491, 494-6 Aspect experiment, 149 Astrophysical Processes, 111, 119 Atomic Hydrogen, 207 Axisymmetric model, 126, 133-4 Backaction, 424 Beil, Ralph G. , 233 Bell, Sarah B., 303 Big Bang, 14, 42, 33, 46, 51, 59, 65 Birkoff’s law, 28 Blackbody radiation, 27, 29, 59, 65, 113, 117 Blackett effect, 332 Black hole, 28, 103, 427 Blueshift, 103, 106 Bogomolny sections, 358 Bohm, D., 85, 152, 155, 405, 419 Bohr orbits, 28, 214, 217, 220, 361, 364-65 Bohr-Sommerfeld atom, 305-6 Borzeszkowski, Horst V., 295 Bosons, 341-3 Brandenburg, John, 267 Brownian morion, 96-7, 100 Burns, Jean, 491 Cartan, E., 1, 197 Casimir effect, 115, 459 Cauchy problem, 127 Causality, 377, 409-10 Cavity QED, 27, 35, 58, 113-15 Cavity-QED Resonance, 27, 35

Chaotic-stochastic Atom, 357-66 Charge-Neutral, 469 Chew, Geoffrey, 51 Chouinard, Edmond, 223 Ciubotariu, Ciprian, 85, 357 Ciubotariu, Corneliu, 85, 357 Classical Electromagnetism, 197 Clifford algebra, 502 CMBR, 27, 37, 60, 138 COBE, 68 Cohomology, 199 Cole, Daniel C., 111 Compactification, 35, 36, 60, 62 Complex Minkowski space, 28, 62-3, 184-6, Composite Particle, 469 Compton wavelength, 434 Consciousness, 60, 223, 425 Continuous State, 59, Continuous State Universe, 28, 59, 62 Copenhagen interpretation, 149, 401 Cosmological constant, 41, 68, 262, 315, 319 Cosmology, 39,59, 65, 313, 517 Coulomb force, 518 Coulomb potential, 241, 250 Coxeter graphs, 188 Cramer,J.G., 60 Creation of Matter, 11-26, 58 Cremona Transformations, 507 Croca, Josee R., 377, 385 Crowell, Lawrence B., 321 Cullerne, John P., 279, 303 Curved spacetime, 98, 303 cyclotron resonance, 361 Cyganski, David, 499 Dark energy, 60 Dark matter, 19, 103 Datta, S., 103 De Broglie, 153, 155, 242, 360, 377, 419 Dechoum, K., 393 Di Pietro, Vincent, 267 Dialectric constant, 440 Diaz, Bernard M., 303 Dimensional reduction, 27, 59 Dimensionality, 314, 406, 507

537

538 Dirac equation, 304, 417, 522 Dirac Vacuum, 31, 37, 57, 147, 252-3, 257 Dispersion, 147 Divergence of electromagnetic field, 128, 132, 251 Doppler effect, 24, 58, 106, 110 Double-slit Interferometry, 147, 150-51 Dragic, A., 349 Dualism, 57 Dvoeglazov, Valeri V., 167 Eddington, A., 33 Einstein, Albert, 148, 518 Einstein Action, 259, 368 Einstein-Yilmaz tensor, 443 Electromagnetic fields, 189 Electromagnetic Zero-point Field Electromagnetic Field Theory, 197 Electron spin, 517 Electron mass, 485 Engineering the Vacuum, 459 Engler, Fritz, 157 Entropy, 76, 491-6 EPR correlations, 421 Equivalence principle, 426 Euclidian metric, 41,59, 62 Evans-Vigier Field, 167 Expanding Universes, 39, 121 Extended Electromagnetic Theory, 125 Far Field, 147, 154 Fermi sea, 338 Feynman, R.P., xii, 51 Finsler geometry, 3, 89, 369 Flat spacetime, 43, 66, 95, 98, 248, 257, 310, 437 Floyd, Edward R. , 401 Fourier transforms, 224, 386 Fractal Universe, 85, 90-2, 97 Franca, Humberto, 393 Friedman universe, 11, 39, 44 General Relativity, 254, 303, 310, 429, 431, 443-4 Geometrodynamics, 457-58 Geon, 39, 62, 63 Grand Unification, 279, 282 Gravitation Theory, 56, 259, 271, 279, 287 Gravitational Energy-Momentum, 291 Gravitational Magnetism, 331, 335 Gravitational potential, 251 Gravitational redshift, 24 Gravity and Electromagnetic Field, 241, 267 Grebeniuk, M.A., 313 Haisch, Bernard, 268, 447 Harmonic oscillator, 365 Hawking radiation, 31, 427 Hawking, S.W., 480

INDEX Heisenberg’s Uncertainty Limits, 385 Higgs boson, 447, 456 Hiley, B., 422 Holographic connection, 227 Holographic Mind, 223, 424 Homaloidal Webs, 507 Homogeneity, 43 Hopf map, 200, 203 Hoyle, F., 39, 83 Hubble constant, 42, 262 Hubble radius, 269, 487 Hunter, Geoffrey, 147, Ibison, Michael, 483 Ignatov, A. M. ,511 Imaginary dimensions, 61, 183 Inertia, 255 Inertia Reaction Force, 449, 452, 455 Inertial mass, 456, 485 Inflation, 39, 44, 66 Interferometry, 147, 377 K-mesons, 495 Kafatos, Menas, 65, 147 Kaluza-Klein theory, 3, 268, 271, 334, 370, 419 Kerr-Newman solution, 427 Kiehn, Robert M., 197 Klein-Gordon equation, 93, 153, 329, 416 Kline, J. F. 267 Kostro, Ludwik, 413 Kowalski, Marian, 157 Kugelstrahlung, 148, 151, 234 Lagrangian density, 438-40 Landau-Ginzburg theory, 456 Lange, Bogdan, 409 Larmor equation, 212 Larmor radius, 364-65 Lehnert, Bo, 125, 426 Length scale, 248, 488 Leshan, Constantin, 515 Light nuclei, 18-9 Line element, 16, 61 Lithium seven, 149 Long-range interactions, 143 Lorentz force, 440 Lorentz transform, 167, 170-1, 245, 291, 453 M-theory, 314, 328, 334, 419 Macgreggor, Malcolm H., 337 Mach’s principle, 426, 457-58, 521 Mgnetic resonance, 349, 351 Magnetic monopoles, 125, 130 Malta, C. P., 393 Mani, R., 147 Maric, Z., 349

GRAVITATION AND COSMOLOGY

539

Poponin, Vladimir, 511 Marshall, Trevor, 459 Post-Quantum Physics, 419 Marto, Joao, 377 Preferred frame, 370 Mass-Neutral, 469 Pregeometry 477-82 Matter creation, 60, 88 Maxwell’s Equations, 147, 184-5, 250, 370, 431, 475, Preons, 470-1, 475 Pribram, K., 424 512 Proca equation, 93, 128, 173, 175, 252 Meat brain, 517-8 Puthoff, Harold E., 249, 268, 431 Melnikov, Vitaly N., 313 Pythagorean theorem, 427 Metric tensor, 435 QED, 265, 460 Microwave background, 17 QSO Redshift, 103 Milne’s cosmology, 42, 314 Quantization, 307 Minkowski space, 40, 62, 183, 289 Quantum Gravity, 259, 303, 311, 321 Minkowski time, 229 Quantum Hall effect, 337-48 Modanese, Giovanni, 259 Quantum Mechanics, 401, 409, 367, 413 Mossbauer effect, 437 Quantum Theory, 211 Multidimensional Gravity, 313 Quantum Hall effect, 281, 337 Multiple Scattering Theory, 103-110 Quantum Measurements, 389 Munera, Hector A., 469 Quantum potential, 421, 423 Nadelstrahlung, 148, 151, 154 Quasars, 23, 104, 110 Narlikar, Jayant V., 11, 81, 103 Quasi Steady-state cosmology, 15, 18, 20-2 Naturalism, 57 Quasiparticles, 341 Near-field, 147, 154 Quaternions, 499 Nester, James M., 288 Radiation resistance, 216, 395 Non-Abelian Gauge Groups, 183 Radiation theory, 210 Nonlocality, 72, 142 Random walk, 95, 98 Nucleosynthesis, 18 Observation in Cosmology, 65 Rauscher, Elizabeth, 60, 183 Occam’s razor, 458 Raws, W., 477 Olber’s paradox, 60 Redshift, 11, 24-25, 60, 103, 106, 138, 256, 437 Page, William S., 499 Retarded waves, 60, 421 Pallikari, Fotini, 95 Reissner-Nordstrom metric, 438, 442 Particle horizons, 43-4 Relational space, 58, 59 Path integrals, 83, 87-8, 421 Riemannian curvature, 2 Peano-Moore curve, 89 Richardson, Tim, 157 Peebles, P.J.E., 57, 62 Rindler, Wolfgang, 39 Permittivity, 248 Rowlands, Peter, 279 Penrose, R., 63, 205 Roy, Sisir, 103 Phase, 46, 190, 464 Rueda, Alfonso, 119, 268, 447 Phase space, 365 Runge-Kutta algorithm, 364 Photon, 133, 135, 147, 163, 205, 212, 233, Rydberg states, 34, 210 373-4, 469, 496 Sachs, M., 503-4 Photon Diameter, 157 Sagnac effect, 127 Photon Emission , 149, 207, 214, 222 Sakharov-Puthoff theory, 8, 256-7, 268 Photon gas, 127, 137 Salam, A. 334 Photon mass, 60, 128, 137, 147, 250 Saniga, Metod, 507 Photon Spin, 197 Sarfatti, Jack, 419 Photon Wave-packet, 151 Schrodinger equation , 351, 393, 415 Physical Vacuum, 27, 59, 241 Schwarchild radius103, 261, Pilot wave, 422 Schwarchild solution, 444 Planck Constant, 1-10, 323, 413 Shipov, G., 504 Plane waves, 132 Simultaneity relations, 367-68, 371 Polarizable-vacuum, 247, 431-34, 438, 443-45

540 Singularity, 64 Sirag, Saul-Paul, 331 Solitons, 158, 254, 373 Sommerfeld theory, 207 Spacetime neighborhoods Space resonance, 518, 519 Spacetime, 33, 95 Spectral lines, 105 Spin, 32, 198, 199, 355, 523 Spinors, 289, 291, 295, 500 SQUID, 332 Stancu, Viorel, 85, 357 Standing wave, 60, 63, 64, 360 Stapp, H., 55, 420 Static universe, 11, 58 Statistical Mechanics, 111 Steady-state cosmology, 12, 16, 39, 58 Stefan-Boltzman law, 118 Stochastic Electrodynamics, 393, 491, 493 Stream of consciousness, 423 String theory, 37, 328 Structuralism, 59 Stuckey, Mark, 477 Substantivalism, 59 Superluminal, 140, 189, 421 Syncronization, 367 Tattva dimensions, 226-7 Teleportation, 372, 375, 515 Telleparallelism, 3-6, 8 Thalesium, 308-10 Theoretical Cosmology, 11, 27, 39, 59, 81, 85, 95 Thermodynamics, 111, 114 Tight Bound States, 349 Tired light, 58 Topology, 88, 197, 199

INDEX Torr, Douglas G., 1, 367 Torsion, 5, 199, 334, 499 Trajectory representation of QM, 401-2 Transverse field, 190,192 Treder, Hans-J., 295 Tully-Fisher relations, 108 Tung, Roh S.,288 Twistor, 63, 205 Uncertainty principle, 385, 387-9, 460, 491, 494 Unified Field Theory, 188, 267, 419, 426 Vacuum conductivity, 127, 130 Vacuum Dynamics, 452, 511 Vacuum Holes, 241, 515 Vacuum radiation, 491, 494 Vargas, Jose G., 1, 367 Vigier, Jean-Pierre, xi, 27, 85, 147, 148, 167, 241, 349, 419 Von Neumann, J., 410 Wadlinger, L.P., 147 Walker, William D. ,189 Wavepacket, 147, 149 Wave structure, 519 Weak field limit, 261 Weinburg, S., 167, 173, 179, 478 Wheeler-Dewitt equation, 324 Wheeler-Feynman theory, 28, 64, 82, 518 Wheeler, J.A., 62, 429 Whitehead, 51 Wolf’s Mechanism, 103-4 Wolff, Milo, 517 Zero point field, 112, 122, 214, 258, 268, 270, 397, 448, 450-455, 459, 483-90 Zitterbewegung, 56

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