Mathematical Aesthetic Principles Nonintegrable Systems

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MATHEMATICAL AESTHETIC PAINCIPLES/NONINTEGAHBLE SYSTEMS

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MRTHEMRTICHL AESTHETIC PRINCIPLES/NOHINTEGRRBLE SYSTEMS

Murray Muraskin Physics Department University of North Dakota Grand Forks, North Dokota 58202

Wor!d Scientific Singapore »New Jersey London* Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

MATHEMATICAL AESTHETIC PRINCIPLES/NONTNTEGRABLE SYSTEMS Copyright © 1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE In our research conducted over many years, we have developed and often initiated studies in these basic areas: (1) Mathematical Aesthetics (2) Nonintegrable Systems (3) Construction of Model Universes Using the Computer Articles on these subjects are spread out throughout the literature. In the references we list articles published since 1975. In the fall of 1991 and the fall of 1992 this author gave talks at Iowa State University entitled "Nonintegrable Systems" and "Mathematical Aesthetics and the Simple Sine Curve". Another talk was given at the University of Manitoba in the fall of 1993 on "Mathematical Aesthetics, Sine Curves and Wave Packets" The core of this text is modelled after these mathematical physics and applied mathematics seminars as well as talks given at the 7 th ,8 th , and 9th International Conferences on Mathematical and Computer Modelling. I wish to thank the Iowa State people, in particular C. Hammer, D. Pursey, R. Leacock, D. Ross, B. Young for there hospitality and interest in this project. At Manitoba the author wishes to thank Witold Kinsner for his hospitality and interest. Prof. Avula is thanked for organizing the conferences on mathematical and computer modelling. In the course of our research it was recognized that the basic equations should be nonintegrable. By this we mean that integration from our initial point, called the origin point, to any other point depends on the integration path between the points in general. Thomas E. Phipps Jr. also studied nonintegrable systems in the sense above in his book "Heretical Verities: Mathematical Themes and Physical Description" (see page 326 and following of his book 1 ). From his book and from a private communication it is clear to the author that Dr. Phipps appreciated the importance of nonintegrability in fundamental physics for some time. I would like to personally thank Thomas Phipps Jr. for his insights and encouragement. The 3 basic areas listed in the begining have been blended together in our research. However nonintegrable systems stands on its own. That is, it is not necessary to discuss mathematical aesthetics in order to discuss nonintegrable systems. We use a simple set of equations, called the ABJL equations to introduce the subject of nonintegrable systems in Chapter 2. Even though the ABJL equations are a special case of the mathematical aesthetics program, this point is not essential in the theory of nonintegrable systems. The reader can thus skip to the section on the ABJL equations if he is only interested in nonintegrable systems. How significant is the effect of nonintegrability on the systems we study? We shall find that no integrability opens up whole new "worlds" to study. In fact all the figures in this book (with one exception, Figure 1.3, which involves 6 dimensional space) are associated with nonintegrable systems. We have still not been able to obtain multi maxima and minima solutions to the 4 (or less) dimensional mathematical aesthetic equations when the integrability equations are satisfied. The ABJL system becomes

Math Aesthetics/Nonintegrability

vi

uninteresting when the integrability equations are satisfied (some of the variables do not change from point to point while other variables are unbounded). Thus, we lose the sinusoidal dependence along any path segment when integrability is satisfied. In addition the 3 component lattice system (Equation 3.18) becomes trivial when we require integrability. Also, the sine within sine system Eq. (5.19) and Eq. (5.25) again becomes trivial when the parameters involved are taken to be Eq. (5.27), the choice of which forces the system to now be integrable. Thus, the study of nonintegrable systems enables us to obtain a variety of interesting situations which would be lost if the equations were required to be integrable. In addition, nonintegrability can be considered more natural than integrability as different paths traverse different environments, so there is no reason to expect results of integration to be independent of path. Thus, we can expect nonintegrable systems to receive greater attention as the years go by. We recall, basically on grounds of expediency, that linear systems eclipsed nonlinear systems, as well, until more recent times. In this book we present new techniques to deal with nonintegrable systems developed by the author over many years. I wish to thank my colleagues here at the University of North Dakota, in particular W. Schwalm, G. Dewar, and L. Jensen for going over my talks with me before presentation. The administration here at North Dakota has also been very helpful in supplying the computer time necessary for the project as well as providing an excellent climate to perform research. In this regard I would like to thank T. Clifford, A. Clark, B. O'Kelly, B. S. Rao, W. Weisser, H. Bale, D. Rice, A. Koch, E. Strinden, K. Dawes, G. Kemper, D. Vetter, and A. W. Johnson. At the computer center I would like to thank D. Bornhoeft, A.Lindem, D. Dusterhoft, and R. Johnson. S. Nemmers and D. Home helped in getting us started using the scientific computer language EXP. I also wish to thank G. Adomian and L. Smalley for their support. B. Ring helped with the computer work prior to 1976. My former students M. Ramanathan and C. Weyenberg also contributed to the research. R. Molmen, M. Brown, and D. Rand also helped with regards to the supercomputer project. Also, I give thanks to my wife Margaret whose emotional support made this work possible. I am also indepted to her for her tireless work on the word processor. S. Krom was instrumental in the preparation of the book. C. Cicha also has helped in the typing aspect. P. Erickson and R. Snortland are responsible for the preparation of the illustrations. I would like to also thank P. Erickson for spotting typographical errors in the manuscript. The text begins with a discussion of mathematically aesthetic principles (Chapter 1), including the reasoning behind such a study. Needless to say there exists mathematical principles that can be classified as being "aesthetic". It never ceases to impress how much is implied by a few simple mathematically aesthetic ideas. That is, with such a small imput we obtain, in a mathematical sense, lattice systems, soliton systems, closed string particles, instantons, chaotic looking solutions, as well as wave packet systems among other things as output. Arguments making use of mathematical aesthetics have long been used in developing physical principles. For example, in obtaining the Dirac Equation of relativistic quantum mechanics, Dirac required that all first partial derivatives be treated in a uniform manner.

vii

Preface

Of course, mathematical aesthetics was not the sole ingredient in obtaining the Dirac Equation. On the other hand, in this book we elevate the study of mathematical aesthetics to a discipline all of its own. There are no "physical" type arguments to be used in conjunction with the mathematical aesthetics. In other words, we study, here, mathematical aesthetics for its own sake. We find that the mathematical aesthetic ideas can be cast into a set of nonlinear equations whose solutions have considerable content as mentioned above. In our early work, as described in reference 2, our results were very limited. It was only after we recognized that the nonlinear system of equations implied by the mathematical aesthetic principles should be nonintegrable as well did we obtain interesting mathematical model universes. Although the studies of mathematical aesthetics, nonintegrable systems and construction of model universes using the computer is mathematical in scope, the expectation is that these studies will contribute to the understanding of some of the unsolved problems in basic physics. A considerably greater commitment of computer resources would be needed in this regard. Even without such resources we shall still be able to obtain certain insights useful to physics. In this book we shall study how these mathematical principles relate to such problems as the arrow of time and the concept of nonlocality. In order for a deterministic theory to account for quantum effects it is necessary that the theory be nonlocal (Bell's theorem). We shall see that the wave packet solution arising from the aesthetic principles leads to a situation that can be described as nonlocal. The traditional way of obtaining basic equations, at present, is to generalize equations that have been shown to be valid in some domain. We recall, even equations such as Maxwell's Equations need to be generalized. This is because Maxwell's Equations are quantized, which means that they are only valid in an average sense. However, there are an infinite number of ways to generalize existing equations. How are we to decide between all the possible generalizations as there are inherent limitations to empiricism? As it is inconceivable to us that the foundation for physics is of a capricious nature, we make the hypothesis that the basis of physics lies in mathematical aesthetics. Only modest steps in such a program have been studied and are recounted here, much more awaits future research. A discussion of outstanding problems yet to be done is found in Chapter 7, section 5. This book is written so that a graduate student or advanced undergraduate student in physics or applied mathematice should be able to master the material. The material is taught as part of my graduate course in Mathematical Methods of Physics. In particular Chapters 1,2, part of 3, 4 and 5 and the appendices are made use of in this course. The appendices are included so as to make the book self contained, and to show how the mathematical aesthetics program relates to standard mathematical subjects. The book, especially Chapters 6 and the first three sections of Chapter 7, also serves as a cohesive record of a research program that obtains sine solutions, lattices of different varieties, soliton behavior, instanton behavior, closed strings, sine within sine curves, wave

Math Aesthetics/Nonintegrability

viii

packets, an agitated vacuum, irregular oscillations, i.e. basic building blocks, with virtually no imput. All that is needed are a few mathematically aesthetic principles. A note in using this book, numbers taken off the computer are truncated rather than rounded off. Our notation used is such as to conform to the notatation in the research articles. We use r j k to refer to the change function. We work in terms of a Euclidean space and use Cartesian coordinates in all our computations. Thus, we emphasize, as we do in the text, that r j k has nothing to do with Christoffel symbols, gij in the text refers to a dynamical field, and is not related to the metric tensor. On the other hand, in Appendix E, we discuss curvilinear coordinates. In this appendix Gy refers to the metric tensor, and Ajk refers to Christoffel symbols, also referred to as the connection. Vectors (tensors), as contrasted with components of these quantities, are written in bold face or with an arrow over the vector. The fourth component is written interchangably as a zero component, as the meaning is to be the same. A summation convention is assumed when upper and lower indices are the same. References 1. T. E. Phipps Jr., Heritical Verities: Mathematical Themes and Physical Description, Classical Nonfiction Library, Urbana Illinois, 1987 2. M. Muraskin, Particle Behavior in Aesthetic Field Theory, Intl. J. of Theor. Physics, 13 303,1975.

ix

Preface

Math Aesthetics/Nonintegrability

x

Figure Caption Wave packet solution to the set of equations based on mathematical aesthetics (see Chapter 1). The origin point data is described in Chapter 5, section 10. Plot is for a representative component, r'n, along the x axis. This is a "big" picture type plot where 24000 points along x are compressed onto a single computer page. The plot is at z=-5, y=15 (units of y and z are 0.005). The grid along x is 0.0005859375 and the spacing between x points is 0.075. The grid along y and z is 0.00005. The system of equations in addition to being nonlinear, is also nonintegrable (see Chapter 2 and 3). In order to obtain this plot, we made use of the path specification approach, and integrated first along z, then y, then x to reach any point. This plot demonstrates the great wealth of information contained in a few mathematically aesthetic ideas. The "vacuum" between wave packet structures also is anything but empty and may have lessons to teach us. A study of the vacuum in such a wave packet solution is found in Chapter 5, section 8. Greater computer resources would be needed to understand the mechanice obeyed by such wave packet structures.

Table of Contents Preface Chapter 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

v

1. Mathematical Aesthetics Why Mathematical Aesthetics All Things are Numbers Flat Space versus Curved Space Some Nonlinear Equations Mathematical Aesthetic Principles The Aesthetic Field Equations (Gamma Equations) Other Relations Obtained from the Basic Principles The Integrability Equations Computer Project Sinusoidal Wave Solutions to the Gamma Equations Multiparticle Solutions

1 1 1 1 2 2 4 6 7 10 11 12

Chapter 2. Nonintegrable Systems 1. The ABJL Equations 2. A Formulation of No Integrability Theory 3. Generalized Derivatives 4. Summation Over Path Method 5. Second Approach to Nonintegrable Systems 6. Computer Results for a Soliton Lattice System

17 17 20 21 23 26 28

Chapter 3. Commutator Method 1. The Commutator Mathod and the ABJL Equations 2. Commutator Method for a Three Component Lattice 3. The Product Method 4. Random Path Approximation

40 40 45 50 51

Chapter 4. Nonintegrability and the Arrow of Time 1. The Arrow of Time

54 54

Chapter 5. The Gamma Equations as a Source of Fundamental Building Blocks 59 1. Introduction 59 2. Sinusoidal Behavior Along any Path Segment Arising from 60 Mathematical Aesthetics Program 3. Linear Sine Within Sine System 63 4. Another Linear Sine Within Sine System 65 5. Nonlinear Sine Within Sine System 66

Math Aesthetics/Nonintegrability

6. 7. 8. 9. 10. 11.

xii

Nonlinear Sine Within Sine Behavior and the Aesthetic Fields Program Wave Packet Solution A Study of Properties of the Vacuum for the Wave Packet Solution Assignment of Origin Point Data A More Viable Wave Packet Solution Features of the More Viable Wave Packet Solution Solution

69 75

79 82 85 87

Chapter 1. 2. 3. 4. 5. 6. 7. 8.

6. A Study of Some Additional Solutions to the Gamma Equations A Lattice of Closed String Solitons Ladder Symmetry Loop Lattice with a Doublet Basis Concept of Imperfect Lattice "Chaotic" Looking System Instanton Solution A Dynamical Lattice Irregular Oscillations along any Path Segment

112 112 116 119 120 125 126 133 141

Chapter 1. 2. 3. 4. 5. 6.

7. Mathematical Aesthetics: Additional Topics Component to Represent the Particle System Lorentz In variance of the Gamma Equations Higher Dimensions A Brief Summary Outstanding Problems Mathematical Aesthetics and Epistemology

147 147 150 150 157 159 160

Appendix A. Elements of the Calculus

164

Appendix B. Theorem of the Calculus Requiring Path Independence for Integrable Systems

167

Appendix C. Elements of Tensors 1. Tensors and Coordinate Transformations 2. The Importance of Tensors 3. Rotations of a Cartesian System 4. Raising and Lowering of Indices and Mathematical Aesthetics

171 171 173 176 178

Appendix D. Elements of Determinant Theory 1. Transformation Law for a Determinant of a Second Rank Tensor 2. Expansion of a Determinant in Terms of Cofactors 3. The Equations for the Change of g 4. Relationship with the Aesthetic Fields Program

181 181 184 186 187

xiii

Table of Contents

Appendix E. Curvilinear Coordinates 1. Parallel Transport and the Connection 2.Mathematical Aesthetics and General Coordinates 3. A Non Dynamical Tensor 4. A Restatement of the Mathematical Aesthetic Principles

189 189 194 196 206

References to More Recent Research Articles by the author

208

CHAPTER NO. 1 MATHEMATICAL AESTHETICS 1. Why Mathematical Aesthetics The underlying hypothesis we make is that the foundation of physics lies in mathematical aesthetics. Otherwise, how can we justify one equation rather than another in view of the limits of empiricism. Although our hypothesis may appear eminently reasonable, the implication of such a program is too broad to handle all at once, so we will focus, at least at the outset, on a few simple questions which can be readily adressed: (1) Do there exist principles which can be classified as mathematically "aesthetic" that can be written down? (2) If so, can these principles be cast into a set of nonlinear equations describing change? (3) If so, what sort of solutions can be obtained from these equations? Can we get multiparticle solutions? 2. All Things are Numbers To Einstein is attributed the far reaching idea that all physics is geometry. A more conservative principle would be embodied in the statement that all physics is mathematics. Or perhaps, getting down to basics "all things are numbers" This latter belief has been attributed to the Pythagoreans. Within the framework of field theory such an idea can be expressed by the absence of a need for units associated with the set of numbers assigned to each point defining the field. Units arise once one has particles with certain attributes. Rods to measure length are to be constructed from aggregates of these particles. Clocks to measure time exploit the periodic motion of particles. Mass and charge are properties assigned to the particles. Thus, we can say that units reflect an underlying particle substructure. In a basic theory the expectation would be that particles are constructs from the basic fields. If so, the need for units at the fundamental level would be done away with. Units represent entities that are not reducible in terms of more basic quantities. Thus, to admit a need for units can be looked on as an additional complication for the theory. Why ask for a more involved theory than would absolutely be necessary? 3. Flat Space Versus Curved Space The notion "all things are numbers" is more general than the notion "all physics is geometry". In addition we shall observe later that nonintegrability is more natural than a theory based on integrability. Nonintegrability is not intrinsically a geometric concept. This offers an example of the restrictive character of the geometric point of view. Furthermore geometric theories have been pursued by Einstein, Schrodinger, Weyl, Eddington and many others without compelling results. Additional arguments can be given favoring a flat space: (1) Flat space is simpler than curved space. One does not usually consider a more complicated situation without fully investigating the simpler alternative. (2) Microscopic physics does not call for curved space, at least not yet. (3) Gupta1 has formulated gravitational-fheoiy rn flat space.

Math Aesthetics/Nonintegrability

2

(4) But perhaps most importantly, one can always imbed a curved space in a flat space of sufficiently high dimension. Thus, there is no loss of generality by working in flat space. We shall work, at least at the outset, in a Euclidean space using a Cartesian system of coordinates. Such a coordinate system has the attraction of simplicity and we do not want the coordinate system to simulate any dynamics as would be the case if we adopted a curvilinear system of coordinates. It is implied that the equations should have the same structure regardless of the Cartesian system under discussion (invariance of the equations under 4 dimensional rotations). We will work in four dimensions unless circumstances suggest a greater number of dimensions would be useful. Sometimes we will work with a smaller number of dimensions for calculational ease. We call the x° direction the time axis. 4. Some Nonlinear Equations Consider the equations dx _ • ax -bxy dt ~ dy_ -cy + dxy. dt

(1.1)

These are the Volterra Predator-Pre)' relations. For a,b,c:,d > 0 we get oscillatory behavior for x and y. Consider the equations dx = -X -2xy dt " dy_ = -y. 2 , 2 ■x + y dt

(1.2)

These are the Henon-Heiles equations2 For particular values of the origin point data we get random looking behavior in phase space. What do these equations have in common? They are first order coupled nonlinear equations with quadratic terms on the right hand sides. The values of x and y are given in terms of xo and yo, where xo and yo are the values of x and y at t = 0. Already we see that equations of this type have solutions with an interesting and varied behavior. We mention these equations since they bear some resemblance to the type of equations we will be studying. Equations such as Eqs. (1.1) and (1.2) are not suitable as field equations since they have no terms involving space derivatives of the variables. 5. Mathematically Aesthetic Principles What we seek is particle solutions of nonlinear equations. The first question we may ask is which equations should we choose to study. In formulating field equations one

3

Mathematical Aesthetics

normally starts with a generalization of an existing set of equations which are valid in some domain. Empirically an electron is considered a point. How then does one generalize a point? Thus, there are problems in potential generalizations: (1) There are an infinite number of possible generalizations. Without empirical justification how do we decide between these generalizations? One could think of all these possible equations each written on a separate piece of paper and then placed in a hat. Then one could say, "God does not play the lottery". (2) The generalizations are nonlinear. Thus, in many instances it is not clear what the generalizations really imply. However, the difficulties go deeper than this. Consider the Einstein-Schrodinger generalizations of the Einstein gravitation equations obtained from an action principle: « 5 / £ v / ^ d 4 x = 0.

(1.3)

What is done is to arbitrarily exclude higher derivatives from the Lagrangian density so as to make the equations for gy no higher than second derivative. But there is nothing "wrong" with higher derivatives. One could say that an almost too cavalier approach has been taken with respect to higher derivatives. There may be some local justification for omitting higher derivatives, but it is hard to look at this omission as anything but ad hoc, at least at this stage. (3) Thus the difficulty is one of mathematical inelegance. Consider the equations dip , i?i -rr =[ca ■ P + 0mc2]ip. (1.4) at This is the Dirac equation. Here all first derivatives are treated in a uniform way. There are two natural ways if we wish to treat all higher derivatives in a uniform way: (1) We can have all derivatives appearing in a single equation (or in a few equations). (2) We can have an infinite number of equations. We will consider the latter of these possibilities in what follows. Next we recognize that the basic equations that are currently in use always involve low rank tensors. When was the last time you had to deal with, say, a 6,142,371 rank tensor? Granted, low rank tensors are simpler to deal with compared to high rank tensors. However, this is not sufficient reason to promote a low rank tensor over a high rank tensor on conceptional grounds. The question then is whether it is possible to formulate a set of field equations according to mathematically aesthetic principles. What might these principles be? We list some of the principles we have been working with in our original formulation4. The list of aesthetic principles is meant to be kept flexible at this stage. (1) All derivatives of tensors are treated in a uniform way (with respect to change).

Math Aesthetics/Nonintegrability

4

(2) All dynamical tensors are treated in a uniform way (with respect to change). A dynamical tensor in a Cartesian space is one that changes from point to point. Examples of a nondynamic vector are the Cartesian unit vectors i,j,k. (3) The field is continuous and singularity free and possesses a Taylor series. (4) We require self-consistency. The equations are obtained using Aristotelian logic. (5) We seek a theory without arbitrary functions. In this way we exclude hyperbolic theory. In hyperbolic theory the field and its time derivative are arbitrary on, say, a t = 0 hypersurface. But this would mean that particle structure is arbitrary on this hypersurface. If the aim of the theory is to obtain particle structure then arbitrariness on a hypersurface can be looked at as an incompleteness of the theory. Also, in view of possible conservation laws, how are we to determine the number of particles on a hypersurface if we have arbitrariness there. The rejection of hyperbolic equations with arbitrariness on a hypersurface does not imply that wave solutions do not appear in the theory as we shall see. From a different point of view, we wish to minimize arb itrariness in the theory. For these reasons, we shall assume arbitrariness at but a single point. This is similar to the Volterra and Henon-Heiles equations introduced before. (6) As all our computations will be done in a Cartesian system, we require here that the equations have the same structure when we change from one Cartesian system to another. The equations should then be tensor equations. In Appendix C we discuss elements of tensor calculus. We extend the coordinate system to curvilinear coordinates in Appendix E. 6. The Aesthetic Field Equations (Gamma Equations) As most theories admit a vector field we shall start off by assuming the existence of a vector Aj (tacitly we have supposed that tensor fields exist from the second mathematically aesthetic principle). We write for the change of Aj between neighboring points dAi = T\k Aj dxk

(1.5)

Tjk are a set of coefficients called the change function as they determine the change of Aj. We have dropped terms of higher order in dxk. For a second vector field we write dBj = rj k Bj dxk.

(1.6)

Thus, we are assuming that r j k is a universal change function that determines the change of all vectors in a uniform way. That is, one set of numbers corresponding to a vector should not be treated any differently from any other vector set of numbers. Said in another way, one vector does not have "something painted on it" that says it is any more significant than any other vector. We required that Aj appear on the right in Eq. (1.5) so that the sum Aj+B; would behave like any other vector with respect to change. From the product AjBj we have from Eq. (1.5) and Eq. (1.6) d(A( Bj) = (T'k A, Bj + Tjk Aj Bt) dx k .

(1.7)

5

Mathematical Aesthetics

Now AjBj is an example of a second rank tensor. According to our principles, we then require that r ] k determine the change of all second rank tensors in a uniform way. For a second rank tensor gy we then get (replacing A,Bj by gtj and AjBt by gjt) d

gij = ( r !kgtj + r] k gi,)dx k

(1.8)

Going one step further, an nth rank tensor is taken to behave like a product of n vectors. From Aj, gy, we define A' by means of the equation (tensor calculus allows the introduction of upper indices even for Cartesian systems as we emphasize in Appendices C and E) Ai=gljAJ.

(1.9)

Then from Eq. (1.5) and Eq. (1.8), provided g;j has an inverse, we get cLV = -rj k Aj dx k .

(1.10)

In a Cartesian system the difference between two vectors, dAj, even at different points is still a vector. Thus, from Eq. (1.5) I \ is a third rank tensor, and thus behaves like A'BjCic- Therefore the change of T'k is given by (using the same rules as before A'BmCk is replaced by TJ^, etc.)

drjkKrur^ + ^rs-r^r^dx 1 . Then, since T':k are assumed well behaved (aesthetic principle (3)) we have dT' arj,i d x dI

* = id

'

(i.ii)

FromEq. (1.11) and Eq. (1.12) we then get dT'

^

= rjnkr]]1 + rj n ,rs-r]rri n ,

(1.13)

These are 256 equations for the change function alone. We note from Eq. (1.13) that derivatives of gamma are given by products of gamma. We may call these equations the Aesthetic Field Equations, or more simply the gamma equations. We note that we can obtain Eq. (1.10) by requiring that r[ k behaves like D k . Thus, it is not necessary that gy has an inverse to obtain Eq. (1.10). We summarize what we have done. We have introduced a change function that determines the change of all functions. But the change function is itself a function. Thus, it must determine its own change by Aristotelian logic (aesthetic principle 5). The functions we are dealing with are Cartesian tensors and the manner in which T]k

Math Aesthetics/Nonintegrability

6

determines change is such that all Cartesian tensors are treated in a uniform way with respect to change. We have thus obtained a set of partial differential equations from a small amount of input, figuratively we can say we have obtained "something out of nothing" or at least almost nothing. Equation (1.13) arises from what Schwalm5 calls logical self-implication in his article "The Lure of Mathematical Science". We note that Eq. (1.13) is a set of first order coupled nonlinear equations with quadratic type terms appearing on the right hand side. Fjk is given in terms of rj k at a single point. Thus Eq. (1.13) has similarities with the equations such as the Volterra Predator-Prey equations and the Henon-Heiles equations mentioned earlier, which we know have considerable content. Other Relations Obtained from the Basic Principles From Eq. (1.8) we get dgjj

<9x k

r ; k g t J - r j k g i l = 0.

(1.14)

Introducing a set of four independent vectors (the a index refers to which vector we are dealing with) we get by the previous rules de^r^dx*

(1.15)

^ - ki = T Jlk eQ dx ' '

(1.16)

Then we get

after use is made of (1.17)

dxk We can then obtain using Eq.(l .16) r 1 = P1

L

(1.18)

provided there exists an inverse field e e

a j -
(1.19)

Thus, if the inverse exists the gammas can be expressed in terms of vector quantities.

7

Mathematical Aesthetics

We see we have a means to treat all dynamical tensors in a uniform way. (dx' is not a dynamical vector. By this we mean d(dx')=0. Also the Cartesian unit vectors i,j,k do not change from point to point. The Cartesian unit vectors are not dynamical vectors either, as mentioned before.) Does the fact that we are dealing with an infinite number of tensors lead to additional restrictions on the theory? For example, r]kgti behaves like the quantity A'BjC k D t E|. On the other hand d(Tjkgt|) is determined by Eq. (1.14) and Eq. (1.13). Thus, we have two ways to calculate d(Tjkgt|). However, we readily see that the two methods give the same result. This is true as well when we consider expressions involving contractions. The results are general. Thus, we have no additional conditions on the theory by treating all tensors in a uniform way with respect to change (see also Appendix D). What about derivatives? Consider, for example, d(<9mrjk). But since derivatives are given by products (note Eqs. (1.13), (1.14), (1.16)), the above considerations imply that there are no additional restrictions by requiring that all derivatives of tensors be treated in a uniform way with respect to change. We realize this anyway since in Cartesian space derivatives of tensors are themselves tensors. Thus, we could have included aesthetic principle 1 within aesthetic principle 2. We have only separated these two principles for the purpose of emphasis. In a basic field theory we have entities—which we take to be Cartesian tensors. The next question is how do these tensors change? This is described by derivatives of tensors. In our approach all tensors are treated in a uniform way regardless of rank and all derivatives of tensors are treated in a uniform way regardless of order, when we consider changes. Another feature of the theory is that all (dynamical) scalars are constant. For example, we may calculate d(A'Bj) using Eq. (1.6) and Eq. (1.10) to verify that we get zero. If such scalars were not constant, then a tensor and another tensor just like it but multiplied by such a scalar, would behave differently with respect to change, contrary to the aesthetic program outlined previously. A greater discussion of this feature appears in Appendix C. Note, we cannot say that there are no other equations besides Eq. (1.13) that can be obtained from mathematically aesthetic ideas. We have made an effort in this regard, but we have not been able to come up with another reasonable set of equations. Basically a fundamental theory involves entities and how these entities change. This would at any rate, indicate to us, that the number of mathematically aesthetic principles leading to a set of partial differential equations is at most not large. 8. The Integrability Equations For regular gy we have

d% dx'dx 1

_ d% dx'dx'

(120)

8

Math Aesthetics/Nonintegrabiliry

Then from Eq. (1.14) we get gilRJlmk+ghlR,tmk = 0,

(1-2D

where pt

From

_

5r

ik

drim

m

dxk

dx

j

,

j

a2rjk1 _ a2rjk

dx'dx

[l.W\

t

dx'dx' '

(1.23)

we get

A]kp, = r u i g + r]m RJJ, - r - < „ = o.

(i .24)

From d2e? dx'dx1

,2-a dle[ dx'dx1 '

(1.25)

we get R U = 0.

(1-26)

We note that Eq. (1.21) and Eq. (1.24) are satisfied if Eq. (1.26) is satisfied. If we consider products offields,for example,

a2(rjsk i ji ) _ a2(rjk gs j ax ax ax'9x '

l

'

;

we see that Eq. (1.27) is also satisfied if Rjk] = 0. In this manner we see that all mixed derivatives of fields are symmetric provided Eq. (1.26) is satisfied. Thus, we do not get a never ending system of integrability conditions by requiring that all fields have symmetric mixed derivatives. If all fields are constructed from TL, and not from e^1 (e" may not be assumed to have an inverse at all points so Eq. (1.18) no longer holds), which is an intriguing hypothesis, we see that all products involving FL including contractions have symmetric mixed partial derivatives provided A]kpl = 0.

(1.28)

9

Mathematical Aesthetics

The equations for the various fields can be put in compact form by introducing a colon derivative

gij:k = 0

rj k: , = 0.

(1.29)

The colon derivative has the same formal structure as a covariant derivative, however, its meaning is different, as here we are dealing with the change of tensors in a Cartesian space and not with curvilinear coordinates. In a similar vein, r j k , we emphasize, has nothing to do with Christoffel symbols. For a general tensor we have T ^ " mnp..,, = 0 ,

(1.30)

where Tljk mnp... is any function of the simple fields gy, r j k ej\<9p. Eq. (1.30) represents the infinite system of equations that we previously alluded to. Furthermore, these infinite equations involve all infinite orders of derivatives. In practice we do not have to concern ourselves with an infinite number of equations since the system can be said to close. By this we mean all changes of tensors are determined from r j k . However, the equations for rj k only involve r j k . Thus, we only need to focus on the equations for the change of TL, Eq. (1.13). Once we have determined Fj k we can get the change of any other tensor field. The field equations Eq. (1.30) are augmented with the integrability equations Rjkl = 0

(1.31a)

Ajkpl = 0.

(1.31b)

or

The integrability equations are restrictions on the origin point data. inserting Eq. (1.13) into Eq. (1.22). This gives

Rln* = 4 i t - rj, r ^ + 4 rjk - r;k r]m.

We see this by

(1.32)

We establish the following theorem: If R ^ = 0 at one point it is zero at all points (provided T'jk and all its derivatives exist at the origin). R'imk changes according to A'BjCmDk. Thus, we have

- ^ = -rmp Rj& + r™ R U , + IT P Rjml + r - RJkm.

(1.33)

Math Aesthetics/Nonintegrability

10

Since Rjk|= 0 we get from Eq. (1.33) that dRj kl /dx p =0, then from these results and taking a/ax' of Eq. (1.33) we see d 2 RJ kl /ax'ax m = 0, etc. Thus, all the derivatives of Rj kl are zero, so R'jk! does not change. All tensor equations in the manner above can be shown to be maintained by the field equations at all points. For example, consider the tensor equations T[k= Tkl. We wnte this as V'Jk — r[ k -r k t =0, then with the same argument as above we see that all derivatives of i/j\k are zero, so ^ 4 does not change. 9. Computer Project The computer project makes use of the total differential equations (Eqs. (1.11)), which are written below,

rjk(Q) = r]k(P) + (Vmk i™ + rj m r£ -17 rJjpAx 1

(1.34)

Thus, from the field at P we can calculate the field at Q. From the field at Q we can calculate the field at the next point as Eq. (1.34) holds at all points. In this manner we can calculate Tjk at all points in space. As Ax1 is finite we need an approximation scheme. We generally use the 4th. order Runge-Kutta method, although we have programmed the equations using Adams' method and Ramanathan and Weyenberg, in their graduate theses, have programmed the equations using the IMSL routine DGEAR. To have confidence in the myriad of numbers coming off the computer we have done the following: (1) We reduce the grid to see if we get the same results to some tolerance. (2) When integrability is satisfied we have a set of algebraic equations that are maintained at all points by the field equations (note previous section). A| tol = 0 is 384 nonlinear restrictions, and RL = 0 is 96 conditions. We periodically check if the integrability equations are satisfied. If we mix up two lines of data, as an example, this shows up rather dramatically in the integrability test. In a representative situation we get A jkpi ° f t n e order of 10"14 when integrability is satisfied (values of r j k are taken from the range .1 to 1.0). When two lines are mixed up we get A'k , of the order of 10"' and 10"2 (3) In symmetric situations, such as lattice solutions, we see that the symmetry is maintained for the grid size used. We would not expect to preserve the symmetry if errors were a factor. Symmetric lattice solutions first appeared in Muraskin6 (4) For soliton solutions the magnitude of the maxima (minima) is preserved in time. Deviations from this are a sign that errors are a factor. Soliton solutions will be obtained later. By soliton solution we mean the magnitude of maxima (minima) do not change in time. (5) Exact solutions, as in the next section can be used to test numerical results.

11

Mathematical Aesthetics

10. Sinusoidal Wave Solutions to the Gamma Equations Consider the following origin point data:

r 23 = -r, 3 = A r2 = .r1 = A 1 1 10 20 p 3 ___1 pO _ 31 3I

A

T-IO

x

l

0\

_ i -p3 __L *-* " 01 -

_

r ^ = r° 2 = -r°2 = -r^2 = o,

(1.35)

with the other r j k zero. This structure is maintained by the field equations although the zero on the right hand side of Eq. (1.35) is not maintained at all points. We can see this by integrating the gamma equations from the origin to any other point. The A-., = 0 integrability equations are satisfied by the above set of data. The Equations (1.13) collapse into the following for Tj, and T ^ : d r 3 i _ _2 3 1,3l3Z dx 3

dTn M

^

^ f = r20r3,

= H0H2

_ ~

, 23

3 31

-

= o,

(1.36)

with a similar type of equations for the pairs vM)i> ^ 0 2 ) ' ( M I , r 3 2 ) , ( 1 0 | , r 0 2 ) .

The equations uncouple to read

<9(x3)2

d(x 0 ) 2

d(x 3 ) 2

d(x0)2'

K

'

'

with similar results for other pairs of components listed above. The solutions of Eq. (1.37) are l?3,, = C cos [A (x3 - x 0 )] T^2 = C sin [A (x3 - x 0 )].

(1.38)

The conclusion is that sinusoidal waves are exact solutions to the field equations for a particular choice of the origin point data. Note, the field equations (1.13) do not resemble the wave equation, nor were they motivated to give wave solutions. Note, the type of

Math Aesthetics/Nonintegrability

12

solution we get is critically related to the choice of origin point data. This is true for the Henon-Heiles equation as well, where we get random type behavior for certain choices of origin point data. We will be identifying maxima and minima in the field with particles (at least for now). Thus, the results are suggestive that multiparticle solutions of Eq. (1.13) may be possible based on a few parameters. 11. Multiparticle Solutions Our expectations of multiparticle solutions have been confirmed. Consider the set of origin point data

ri, ri, = i.o . -=i i.o . u i 3 2

3 3

cm = -i.o -1.0 rr<j0 3 = -i.o, rg,

(1.39)

with the other components zero. We get a more complicated set of data using Eq. (1.39) for PL and from

rj^aUfa^.

(1.40)

We have chosen a? to be

(1.41) Figure 1.1 shows an x,y map for the representative component r j , . We call such a solution a lattice solution for obvious reason. All 64 I\'k components show a lattice structure so we do not concern ourselves with which field component should be taken to represent the particle. In our early work we introduced a gy and the maxima and minima of the determinant of gy, denoted by g, were taken to represent particles. The quantity g is unchanged by four dimensional rotations. Another possibility is to represent the particle system by T^, as this quantity is unaltered by three dimensional rotations. We cannot say at this time which of these quantities should be taken to represent the particle system. A lattice can be obtained with different sets of origin point data. For example, using Eqs. (1.40), (1.41) and

r 32 = i.o Io, =-1.0,

(1.42)

we can also obtain a lattice. This is illustrated in Figure 1.2 for r ] , . Maps for different z show a symmetric pattern of three dimensional maxima and minima, likely an infinite number due to the symmetry. The sets of data Eq. (1.39) and Eq. (1.42) do not satisfy the

'3

Mathematical Aesthetics

Figure 1.1. Lattice solution obtained by using Eqs. (1.39),(1.40),(1.41) for origin point data and specifying an integration path by first integrating along y and then x. Numbers in the figures in this chapter are 100 times actual numbers.

integrability equations. The maps given in the figures arise by prescribing an integration path. We first integrate along x°, then x 3 , then x2 and finally x1 to get to any desired point. We specify an integration path since when the integrability equations are not satisfied the results depend on integration path (Appendix B). We can look at the specification of integration path as detracting from the "aesthetics" of the theory. What to do? We have made an extensive search for systems in four dimensions that satisfy the integrability equations and give rise to the lattice. We have not found any such system. This does not mean that such a system does not exist, it just means that if such a system exists it is not so easy to find. An example of a two dimensional lattice satisfying integrability but in 6 dimensional space is shown in Figure 1.3. We have also found a 3 dimensional lattice satisfying integrability but in a 9 dimensional space. Even if we found a lattice system satisfying integrability in four dimensional space we could not expect to use it as a base for a perturbation calculation. This is because any pertubation is likely to destroy integrability. As early as 1977 we recognized that if we drop integrability we get solutions of considerably greater structure7. This serves as an introduction to what we call No Integrability Aesthetic Field Theory. A discussion in detail of nonintegrable systems will be deferred until later. Note,

Math Aesthetics/Nonintegrability

14

Figure 1.2. Lattice solution obtained from Eqs. (1.40),(1.41),(1.42) as origin point data and specifying an integration path as in Figure 1.1.

Figure 1.3. Lattice solution obtained in Muraskin, Mathematical and Computer Modelling 9 (1987) 883. Here the integrability equations are satisfied, but the system is 6 dimensional.

15

Mathematical Aesthetics

if the aim of the theory is a set of numbers at each point we have already done so by construction. We do not require that the integrability restrictions be satisfied and we specify an integration path. A basic tenet of our approach (aesthetic principle 5) is that data is arbitrary at a single point rather than a hypersurface. A certain amount of arbitrariness is needed since the location of the origin point is arbitrary and an arbitrary rotation of the Cartesian axes leaves the system still Cartesian. A general linear transformation could also be allowed as this will not simulate any dynamics. With data arbitrary at a single point there is no reason for results to be independent of integration path. After all, the different paths travel in different terrains. One can expect to pay an awfully high price for independence of path. The A\ , = 0 integrability equations are 384 conditions on the 64 pieces of origin data. Even in the simpler version of the theory, R'jkl = 0 represents 96 conditions on the origin point data. These complicated nonlinear conditions can be looked at as unduly restrictive of the theory. We note in the Henon-Heiles system for some choices of origin data we get random type behavior in phase space. For other choices of origin point data we do not see this effect. So to restrict the origin point data is to restrict the theory. However, integrability arises from the symmetry of mixed derivatives Eqs. (1.20), (1.23), (1.25), and (1.27) and the symmetry of mixed derivatives is a basic feature of the calculus. After all, an elementary theorem says that regular functions automatically have their mixed derivatives symmetric. Thus, if we are to investigate a theory based on no integrability we have to face up to some pointed questions: (1) Can we formulate a theory based on no integrability in an aesthetic manner? We have already raised the point that specification of integration path can be looked at as detracting from the aesthetics. (2) How do we handle a no integrability theory? (3) How do we avoid the restrictions implied by the theorem of the calculus that says all regular functions automatically have their mixed derivatives symmetric? (4) Provided we can answer the above questions, we can ask what sort of things is the theory capable of describing? Do we get multiparticles? Can we get computer pictures of the solutions? We end this chapter with the following conclusions: There are such things as mathematically aesthetic principles; these principles can be cast into the form of a set of nonlinear equations; these equations have considerable content and can be treated using the computer; and finally, nonintegrability is an important ingredient in such a program. References 1. S.N.Gupta, Einstein and Other Theories of Gravitation, Reviews of Modern Physics 29 (1957) 337 2. Henon M. and Heiles C , The Applicability of the 3rd Integral of Motion: Some Numerical Experiments. Astr. J. 69 (1964) 73. 3. A-Einstein, The Meaning of Relativity, Third edition, Princeton Univ. Press, (1950).

Math Aesthetics/Nonintegrability

16

4. M. Muraskin, Particle Behavior in Aesthetic Field Theory, International Journal of Theoretical Physics 13 (1975) 303. 5. W.A.Schwalm, The Lure of Mathematical Science, N.D.Quarterly 54 (1986) 212 . 6. M. Muraskin, Aesthetic Field Theory: A Lattice of Particles, Hadronic Journal 1 (1984) 296. 7. M. Muraskin and B. Ring, Increased Complexity in Aesthetic Field Theory. Found. ofPhysics 7 (1977) 451 .

CHAPTER NO. 2 NONINTEGRABLE SYSTEMS 1. The ABJL Equations In this chapter we discuss nonintegrable systems in it's own right. We shall use a simple system below for illustrative purposes. One feature required of the equations is that derivatives are given by products (this is a property of the equations appearing in the previous chapter). The equations we study here are:

aA

ax"

= k,J

dA = k,L dy'

dB

^-=k,J

ax

di <9L — =k2A — =k2A ax

ax

dB dJ — =k)L ^ - = k 2 A ay ay

dL ^-=k2A, ay

(2.1)

with k, = A-B k2 = J-L.

(2.2)

Since the change of A and B satisfy the same equations ki is constant. Since the change of J and L satisfy the same equations k2 is constant. We see this by taking derivatives of Eq. (2.2). We shall call the above equations the ABJL equations. Arbitrary data is specified at a single point called the origin point. We shall find that the ABJL equations can be obtained from the gamma equations,

^r^rji' +r^r^ii,,

(2.3)

when a certain choice of origin point data is taken. More about this in Chapter 5. By taking derivatives of Eq. (2.1) we obtain expressions for the mixed derivatives such as (a x = — , Oy = — ) ax ay a x y J=k,k 2 J Oy X J=k,k 2 L.

(2.4a) (2.4b)

And, thus, we have ( ^ - ^ ) J = k,k2.

(2.5)

The rules of the calculus are very explicit here. Since mixed derivatives of regular functions are symmetric it follows that k, = 0 or k2 = 0. But if k, =0, then A,B are constant and J and L are unbounded. If k2 = 0, then J,L are constant and A,B are unbounded. Thus, the system Eqs. (2.1), (2.2), is without interest.

Math Aesthetics/Nonintegrability

18

When k| ^ 0 and k2 ^ 0 the equations are said to be nonintegrable. This means that in integrating the field equations the results depend on integration path. This simple result follows from an elementary theorem of the calculus and can be easily demonstrated numerically on the computer to some given tolerance. The simple theorem of the calculus is presented in Appendix B. Note our usage of the term ''nonintegrable" is not the same as other authors. Our usage is the same as Phipps1 as appears in his book. One can argue that it is unduly restrictive to require integrability on the following grounds: (1) Different paths traverse different terrains. There is no reason for integration results to be the same for different paths. (2) The integrability equations, k| =0 or k2=0 (note all mixed partial derivatives are zero if these integrability equations are satisfied), unnecessarily restricts the origin point data. (3) Nonintegrable systems can be treated as we shall see and have considerable content. Even though the Eqs. (2.1) and (2.2) do not appear interesting as the calculus would have us believe, we shall study them anyway, but in the case that the integrability equations are not satisfied. When integrability is not satisfied the results of integration depend on the integration path. The simplest thing we can do is to specify an integration path. We can first integrate along y and then x to get to any desired point for the two dimensional system Eqs. (2.1) and (2.2). This procedure, although too simplistic, is as in the last chapter called the specification of path method. We can integrate A,B, J, L to any point, given data at the origin using the 4th order Runge-Kutta Method. The resulting map for the quantity A is given in Figure 2.1. We used A= -1, B=0, J=l, L= -1 for origin point data. We get similar maps for B,J,L. A property of this data is k)k 2 is less than zero. For obvious reason again we call such a solution a lattice solution. Note lattice solutions also appeared starting with the gamma equations in Chapter 1. The resulting lattice (Figure 2.1) is not such a surprise. From Eq. (2.1) we get

dy22

=k,k2A.

(2.6)

The lattice solution we have obtained is characterized by sinusoidal dependence along any integration path segment provided ki k2 < 0. B,J,L also are characterized by a sine curve or a displaced sine curve along every path segment. A displaced sine curve occurs in the case of J as a function of y. Note Equation (2.7), d2j

^

eh dy In this last instance J-k2 is a sine curve

= k

'

k

'

J

2=k,k2(J-k2).

(2.7)

19

Nonintegrable Systems

Figure 2.1. The quantity A in the ABJL system when we specify an integration path, showing the lattice structure. In all figures of this chapter numbers are 100 times actual numbers.

Thus Figure 2.1 shows that the ABJL equations are interesting provided the integrability equations are not satisfied. The following questions come to mind at this stage: (1) As we cannot justify using one integration path rather than another, can we handle nonintegrable systems in a way that does not favor one integration path rather than another? We remember the specification of path procedure favors an integration path. (2) Mixed derivatives of regular functions are defined to be symmetric. How then are we to deal with equations like Eq. (2.5)? To answer these questions we will generalize the notion of derivative. Our methods will agree with that obtained using usual derivatives in the special case that the

Math Aesthetics/Nonintegrability

20

integrability restrictions (such as k] = 0 or k2=0 in the case of the ABJL equations) are satisfied. We note nonintegrable systems, where results of integration depend on path, appear in other branches of physics such as thermodynamics and relativity. We mention, as an example of the latter situation, the article by Newburgh and Phipps2. A simple example in relativity is the experiment where clocks are flown around the world and then compared with stationary clocks. The procedure used here is "you tell me what path and I'll give you an answer" This is what we call the specification of path method. This method has its limitations in a field theory where numbers obtained in integration from a point are needed in calculating the field at another point (as in Eq. (2.1)), and it is a reasonable question to ask why one chooses one integration path rather than another. 2. A Formulation of No Integrability Theory We make the following hypotheses: (1) The manner in which we calculate the field should be independent of how we assign the arbitrary data associated with the equation. This is true for the Henon-Heiles equation, the Volterra Predator Prey equations, mentioned in the previous chapter, as well as other equations we routinely deal with such as hyperbolic equations. As integrability is a restriction on the origin point data (remember for the ABJL equations we have ki = 0 or k2 = 0), we conclude that the way we calculate the field should be independent of whether integrability is satisfied or not. (2) No order of mixed derivatives should be favored over any other ordering. Said in another way, no integration path should be favored over any other integration path. (3) We need a set of basic equations, such as the ABJL, or gamma equations where derivatives are given by products. We shall show that we can calculate, for example, A, B, J, L, and T]k at all points using these assumptions. We first consider the case when the integrability equations are satisfied. When integrability is satisfied we may use the formulas encountered in the traditional calculus. We can from the notion of derivative express, for example, J at any point in terms of a series involving derivatives of J evaluated at the origin (see Appendix A):

J(x + Q! dx, y + a2 dy) = J(x,y) + Q] dx J dx + a2 dy J dy + a,a 2 d xy J dx dy + Q | ( ^ r l ) dxxj dx dx + g ^ g g l J ^ j dy dy +

QI(QI-1)(QI-2)

~

ce$ot\-$

+— +

a2(a2- l)(a 2 -2)

02(0:2-1)

a2<9xxyJ dx dx dy + a , —

^ J dy dy dy

<9xyyJ dy dy dx (2.8)

21

Nonintegrable Systems

This formula can easily be extended to any dimension. ot\ is the number of segments along x, and aj is the number of segments along y. All mixed derivatives are symmetric once the integrability equations are satisfied so the right hand side is not altered by replacing all mixed derivatives by symmetrized derivatives. Thus, we can make replacements in Eq. (2.8) of the type

x

>7

* r(6'xyy + 6'yxyT"6'yyxJ

etc.

(2.9)

Now, from the basic equations we can replace all derivatives by products. Thus J at any point is determined by products of A, B, J, L evaluated by the origin point. This is true whether integrability is satisfied or not by hypothesis 1. Hypothesis 2 is satisfied by working with symmetrized derivatives and hypothesis 3 is satisfied by working with the ABJL equations. This method is called the product method. It is not a practical approach unless the basic equations are simple so that we can write down formulas for derivatives for any order we please. In the case of the ABJL equations the product method as well as the commutator method are feasible. We shall discuss these methods in the subsequent chapter. For the gamma equations the product method yields rj k (x+a,dx, y+a 2 dy...)= rj k (x I y...)+5>(T; n k V$ i

+rj m r g - rjj TJJpdx'+tproducts involving 3 or more gammas)p

(2.10)

For nonsimple looking equations, as the case of Eq. (2.10) in general, the products become unwieldy as we move farther away from the origin, so we seek an alternative but equivalent method. We shall here demonstrate that one can define generalized derivatives, which are not in general symmetric, so that Eq. (2.5) is consistent. These derivatives agree with conventional derivatives in the case that the integrability equations are satisfied. 3. Generalized Derivatives From equations of the type Eq. (1.11) we have for changes along x and y respectively dJ=Ak 2 dx dA=Lk,dy.

(2.11a) (2.11b)

Math Aesthetics/Nonintegrability

22

We define first derivatives in the usual way as the nonsymmetry of mixed derivatives does not enter here. Derivatives are defined as finite differences with an eye to our numerical work. Derivatives are defined with respect to the origin point P where the data is given. For a neighboring point along x we then have <9XJ dx = J(Q) - J(P). Consider next the following four points: R. P.

.S .Q

We can naturally define a mixed partial derivative as: d y j d x d y = J,(S)-J(R)-J(Q) + J(P)

(2.12)

or d y j d x d y = J2(S) - J(R) - J(Q) + J(P).

(2.13)

Here J](S) is the value of J obtained by integrating from P - » R - > S , while J 2 (S) is the value of J obtained by integrating from P —> Q —► S. If integrability is satisfied then there is no distinction between Eq. (2.12) and Eq. (2.13), since J(S) = Ji(S) = J 2 (S). Which of the two possibilities, given by Eq. (2.12) and Eq. (2.13) should be chosen for dyj dx dy? The criterion we use is consistency with the system Eqs. (2.1) and (2.2), where the role of Eq. (2.1) is not to determine A(x,y),B(x,y),J(x,y), L(x,y), but instead to determine change along integration paths. From Eq. (2.1 la) we evaluate the following combinations: J,(S)-J(R) = A(R)k 2 dx -J(Q) + J(P) =-A(P)k2 dx .

(2.14)

Then we use Eq. (2.1 lb) to get Ji(S) - J(R) - J(Q) + J(P) = k,k 2 L(P) dx dy

(2.15)

Then, if we use the definition Eq. (2.12) we get < V = k|k2L,

(2.16)

which is the same as Eq. (2.4b). That is, the choice Eq. (2.12) gives consistency with the basic Equations (2.1) and (2.2). In a similar way we define d x y Jdxdy = J2(S) - J(Q) - J(R) + J(P).

(2.17)

23

Nonintegrable Systems

Using a similar procedure as before we get: < V = k,k2J,

(2.18)

which is the same as Eq. (2.4a). We can define mixed derivatives of any order, 9yx , to be consistent with the basic Equations (2.1) and (2.2) by the following rule. We first take finite differences with respect to the index on the far left, then we take finite differences with respect to the adjacent index, etc. The reason we take finite differences with respect to the index on the far left first, is that this is the opposite of what we did before when we took differences as we did in Eq. (2.1 la) and Eq. (2.1 lb). From Eq. (2.12) and Eq. (2.17) we see that second mixed partial derivatives are not in general symmetric. Finally, from Eq. (2.16) and Eq. (2.18), and with the use of Eq. (2.2), we find that Eq. (2.5) is satisfied. Not only is Eq. (2.5) now satisfied, we also have that the origin point data is unrestricted (we no longer have k]=0 or k2 =0) and the results of integration from the origin point depend on path. On the computer it is a simple matter to check numerically that the results of integration now depend on path. Thus, we have seen, in the manner above, that all orders of derivatives can be defined to be consistent with the basic Equations (2.1) and (2.2). Furthermore, in the special case that the integrability equations are satisfied, the generalized derivatives are the same as the conventional derivatives of Appendix A. The conclusion we reach is that the manner of defining mixed derivatives above together with equations of the type Eq. (2.11) gives the same results as working with the basic equations, such as Eq. (2.1) or Eq. (2.3), where derivatives can be replaced by products. 4. Summation Over Path Method For illustrative purposes we consider the situation in Eq. (2.8) when a\ = 1 and ai = 1. Then Eq. (2.8) with symmetrized derivatives on the right hand side becomes (we make use of the same set of 4 points P, Q, R, S drawn previously) J(S)=J(P)+dxJ dx + dyJ dy + |(dxy+dyx)J dx dy.

(2.19)

Now instead of replacing derivatives by products using Eq. (2.1) as we did previously, we replace derivatives by their definition in terms of finite differences using the generalized derivatives of the previous section. Then Eq. (2.19) becomes J(S)4((J,(S)-J(R))+(J(R)-J(P))+J(P)) +|((J2(S)-J(Q))+(J(Q)-J(P))+J(P)).

(2.20)

Math Aesthetics/Nonintegrability

24

where the inner bracket terms are evaluated using equations like Eq. (2.11). There are two terms in Eq. (2.20) multiplied by \. The top term gives the contribution to J from the path

r

and the bottom term gives the contribution from the path

P

T

Q

By considering points farther from the origin then the above case where a>\ = 1, a2 = 1, we get in a similar fashion J(U)=1/N^ (contributions from each path). P ia s

(2.21)

N refers to the number of paths. We note from Eq. (2.20) we get no contributions from paths like

In general in constructing paths we do not allow for "backtracking". Segments of paths are parallel to the x,y coordinate axes. This reflects the fact that Eq. (2.1) are given in terms of Cartesian coordinates. Eq. (2.21) is called the summation over path method. Eq. (2.21) is such a simple formula that one may wonder if we can obtain it from other considerations. The answer is that we can. In integrating from the origin to a fixed point the results depend on path in general. But no path should be favored over any other path. Thus all the path contributions to the field should be weighted in the same way. Thus, we can write down: r(U)= k£](contributions from each path) where k is constant. We set k = 1/N since when integrability is satisfied, which means that all contributions are the same, we wish to obtain the same answer as when we integrate along a single path. That is the above equation should hold regardless of whether or not the integrability equations are satisfied. If we allow backtracking when constructing a path, using a finite grid, we could construct paths having an infinite number of segments. Also we could construct an infinite number of paths from the origin point to any finite point. By disallowing these kinds of infinities we limit ourselves to paths without backtracking.

25

Nonintegrable Systems

We emphasize that the the role of Eq. (2.1) is not to determine A(x,y), B(x,y), J(x,y), L(x,y). Instead the role of the equations here is to determine changes of A, B, J, L along an integration path. That is, we determine quantities like J](S), J2(S). We emphasize that the theorem of calculus that requires symmetric mixed derivatives for regular functions does not apply to path dependent quantities like J|(S), J2(S). Once we perform a summation at each point we obtain regular functions (left hand side of Eq. (2.21)), and mixed derivatives, defined in the standard way, for these quantites are symmetric. However, we emphasize, the basic equations like Eq. (2.1) are given in terms of the generalized path dependent derivatives. We see that by introducing Ji(S), J2(S), etc., we avoid restricting the origin point data, which in turn limits the content of the theory. The quantities Ji(S), J2(S), which are path dependent in general also enables us to avoid the situation, which can be considered unnatural, where results of integration are independent of path. The price to be paid is a superposition principle at each point. However this feature can be looked at as an additional degree of freedom which may be useful in seeking an understanding of some of the unsolved fundamental problems of physics. An interesting result in this regard is discussed in Chapter 4, in the case where the precise form of the basic equations is immaterial. We can say that use of conventional derivatives leads to a restrictive situation. Consider again ( 3 ^ ) 1 = kikj.

(2.22)

The calculus says we must take ki =0 or k 2 =0 for regular functions. This means that we can not have any choice for arbitrary data. But we may not wish to limit our choices for the origin point data. As an example, consider the Henon-Heiles equations. For some choices of origin point data, as mentioned before, we see random type behavior in phase space. For other choices of origin point data we do not see this effect. Thus we recognize that to limit the origin point data is restrictive for the theory. Secondly the calculus says that results of integration from the origin are independent of path for the system Eq. (2.1) and Eq. (2.2). There is no flexibility on this point. But we may not wish for integration to be independent of path. Consider an interesting analogy, which can be called "the muddy shoes theorem". In North Dakota in the spring my wife can tell which path the author has taken by simply looking at the carpet. This simple example enforces the intuitive idea that without evidence to the contrary, path should be considered important in general. We may argue that any procedure that requires the integrability equations be satisfied is too restrictive an approach. With the generalized derivative, equations such as Eq. (2.22) are satisfied, and we need not restrict the origin point data in any way, as well as having results of integration depend on path. Prior to the development of the study of nonintegrable systems the "knee-jerk" reaction on being given equation Eq. (2.22) is to set k, or k2 equal to zero. But then we see from the study of the ABJL equations that we lose the lattice and the ABJL equations are rendered totally uninteresting. We shall see this effect again in the next chapter as

Math Aesthetics/Nonintegrability

26

well as in Chapter 5. In Chapter 5 we study the case of nonlinear nonintegrable sine within sine behavior. If we force the origin point data to satisfy the integrability equations, we end up with a trivial solution. Thus, the effect of sine within sine is lost! Thus we see by considering nonintegrable systems and not doing the "knee-jerk" operation we open ourselves to a world of interesting new possibilities. The ABJL equation may be very simple equations, but they nevertheless offer a good example of how nonintegrable systems can lead to new horizons. It would be in our view short sighted to ignore such systems by limiting ourselves to systems obeying the integrability conditions, especially in view of the techniques developed to deal with nonintegrable systems. Furthermore the use of high speed computers enables such systems to be seriously studied. Derivatives in this section are defined at the origin point where the arbitrary data is given. Derivatives can be defined at other points below. These derivatives can be said to define a generalized calculus. Consider the points drawn below (we call the axes x and t, rather than x and y, as this will be useful in future discussion), t "

T U R S P Q »

X W V x

Then, we can define the following derivative at the point S dt (fltJ) s . P ^ R ^ s = J,(U)-J,(S).

(2.23)

Here Ji(U) is the value of J obtained from integrating from P —» R —> S —» U, and Ji(S) is the value of J obtained by integrating from P —> R —> S. Ji(S) appears on the right hand side of Eq. (2.23). This is reflected on the left hand side by the subscript P —> R —> S. This definition of mixed derivatives away from the origin agrees with the usual derivatives when results of integration are independent of path. This definition is somewhat cumbersome so the use of these derivatives is expected to be limited. We note that from derivatives defined at P we can obtain for example, J, at all points of space via the basic Eq. (2.1) together with Eq. (2.8) and Eq. (2.9). In this chapter we have discussed two equivalent methods called the product method and the sum over path method. In Chapter 3 we will discuss a third equivalent method called the commutator method. Here we integrate along a single path and supplement this result with an algebraic term obtained from the commutator of mixed derivatives. The different methods allow us to check the numerical calculation. 5. Second Approach to Nonintegrable Systems When results of integration depend on path there are many ways that we can proceed. After all we can weight different paths in any way we please, and thus there are an infinite number of ways to deal with nonintegrable systems. However, it is necessary that results

27

Nonintegrable Systems

agree with traditional methods once the integrability equations are satisfied. Furthermore, we do not wish to introduce any scheme that can be considered artificial. This will limit us to what appears in the previous section, as well as what will appear in this section. In the previous section we integrated and then summed results for different paths at the last step. In the present section we sum as soon as possible. This method can be illustrated by showing how to calculate the field at a few points close to the origin in two dimensions (we take x —> t, and y —> x in equation Eq. (2.1) for demonstrative purposes). We use the same points P,Q,V,R,S,W,T,U,X drawn in the last section. We use the basic Eq. (2.1) to calculate the field at Q and V. The information at P, Q, V, can be stored on tapes. Then from information at P, Q, V we can calculate the field at R, S, W in the following manner. The calculation of the field at R, involves no superposition since it lies along a coordinate axis, and no backtracking is allowed for the same reason as before. In integrating to S we get different results depending on which path, as the integrability equations are not required to be satisfied. There is no reason to favor results from one integration direction rather than another, so we write J(S)=j((contribution from R)+(contribution from Q)).

(2.24)

We next calculate the field at W in a similar manner J(W)=j((contribution from S)+(contribution from V)).

(2.25)

In this way we have computed the field at R, S, W, which can again be stored on tapes if we wish. From this information we can next calculate the field at T, U, X, etc. Thus the integration scheme can be written J(U)=1/N2 contribution from a neighboring point where the field has already been determined

(2.26)

Here N is the number of integration directions. The summation is over integration directions. In the case above N is at most 2. In four dimensions N is at most 4. The rule Eq. (2.26) can easily be extended to as many dimensions as we wish. Thus, we have a well defined procedure to calculate the field at any point in space given origin point data. In numerical work we check that solutions agree to a prescribed tolerance when we reduce the grid, as we do in section 4 as well. In four dimensions P, Q, V is extended to a hypersurface (actually a portion of a hypersurface). Then from information on a hypersurface we can calculate the field on succeeding hypersurfaces, similar to the case of hyperbolic equations. (Although we stress that data is arbitrary at a single point here, rather than being arbitrary on a hypersurface.) This approach is in contrast to the sum over path approach, where past history is needed to calculate the field at any point. In the sum over path method one needs to return to the origin point in calculating a field contribution for a representative path. However, in the present approach it is not needed to return back to the origin point

Math Aesthetics/Nonintegrability

28

in calculating the field at any particular point. From information stored on tapes one can integrate to succeeding hypersurfaces. The information on tapes can be displayed in the form of maps if desired. We may call the coordinate in the direction of flow of information from one hypersurface to another as the "time" (shown in preceding diagram of location of points nearby to P in section 4). Directions perpendicular to the time direction are called space coordinates. The approach of the previous section, has the property that derivatives of all orders can be introduced consistent with the basic Eq. (2.1). With the present approach only second mixed partial derivatives can be introduced consistent with the basic equations. The reason this is so is that once one performs a summation (superposition principle) we are introducing information above and beyond the basic equations, and here one is performing the superposition as early as possible. Second mixed partials are defined (at the origin) in the same way as previously Eq. (2.12) and Eq. (2.17), so as to get consistency with Eq. (2.5). We note that this integration scheme as well as the integration scheme of section 4 agrees with conventional treatment, in the case that the integrability equations are satisfied (results of integration are independent of path). Note Newtonian mechanics favors first and second derivatives for reasons which are not apparent. In the present approach to nonintegrable systems we see that there is a reason to favor such low order derivatives. In Chapter 1, one of our basic aesthetic principles was to treat all derivatives in a uniform manner. Now that we recognize the significance of nonintegrable systems we would change the basic principle to read: The way all dynamic Cartesian tensors change should be treated by a uniform procedure regardless of rank. In dealing with the second approach to nonintegrable systems we do not introduce derivatives higher than the second (this is in contrast to the first approach to nonintegrable systems where all orders of derivatives can be introduced). The integration scheme of section 4 implies that we have a means to calculate change along extended paths. If we have no such means then the approach of the present section should be used. Here the role of the basic equations is to determine change between neighboring points (neighboring points are connected by path segments that are parallel to the coordinate axes), but no further. We can say that the second approach to nonintegrable systems does not elevate the concept of extended path to a prominent position in the theory. 6. Computer Results for a Soliton Lattice System We will further look at the ABJL system in future chapters. The ABJL system represents the simplest lattice system when we speoify an integration path. We call such a system a two component lattice, as B and L are given in terms of A and J by Eq. (2.2). When use is made of the sum over path method we have not been able to observe any two dimensional maxima or minima in the region mapped. A discussion of the second approach to nonintegrable systems and the ABJL system will appear in Chapter 4. In this section we will discuss the gamma equations Eq. (2.3) in conjunction with the following

29

Nonintegrable Systems

intrinsically three dimensional origin point given in Eq. (2.27). We shall call the z axis the "time" axis. We work with this system as it is a more complicated lattice system (it is a 5 component lattice) when an integration path is specified. We wish to observe how the integration schemes Eq. (2.21) and Eq. (2.26) affect the lattice. The origin point data we choose to work with is given by

r] 2 =i.o

11,=-i.o

r]2=-i.o

r|,= i.o,

(2.27)

with the other gammas zero. We specify a path by first integrating along y, then x, and then z to get to any point, as the integrability equation are not satisfied. A map in the x,y plane is given in Figure 2.2. Again we see a lattice, although the system is more

Figure 2.2. Map of r j 3 using the gamma equations with the data Eq. (2.27) using a specification of path.

complicated than the lattice associated with the ABJL system. Not only does it involve more components, it is a nonlinear system as well. The ABJL system is a linear system as can be seen from Eqs. (2.6) and (2.7). We shall find what equations the present system collapses into in Chapter 5. The gammas can be expressed in terms of 6 quantities in this case with one linear restriction between the 6 quantities. Hence, we call such a system a 5

Math Aesthetics/Nonintegrability

30

component lattice. Figure 2.2 is a map for the representative component r 3 3 . All the components of gamma that vary show the lattice structure, so we will not concern ourselves here with which component to study. The maxima (minima) in the figure all have the common magnitude 0.49 ± 0.01 (note numbers in the figure are 100 times the actual numbers). In Figure 2.3 we give an example of the raw data coming off the computer. We see the maxima (minima) values of 0.49 ± 0.01 in a regular array. There is C4M4PG4M

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Figure 2.3 Raw numbers off the computer for the system of Figure 2.3 at z= -5 in units of .3. Path specification method is used.

31

Nonintegrable Systems

an uncertainty of at least one unit (a unit is .3) in the location of the maxima (minima) as the largest magnitude can occur somewhere within the spacing of the points which is 0.3. Figure 2.3 is not at z=0, as is the case of Figure 2.2, but at z= -5 units. We see that the magnitude of the maxima (minima) is the same at this "time" as it was at time zero. Furthermore, as we alter z we see this property is maintained. A maximum or minimum can be called a particle. In Chapter 5 we shall consider more sophisticated kinds of particle but for now we shall be satistified with such a simple notion of particle. Furthermore since the magnitude of the maxima (minima) does not change in time, so far as our numerical work tells us, we shall call these particles solitons as there is no attenuation of magnitude as time evolves. We next examine the motion of these point soliton lattice particles as a function of time. The results are given for the trajectory of a particular lattice particle in Figure 2.4. An uncertainty of one unit in the positive as well as negative direction for the location of the particle is shown in the figure. We see that the lattice particle undergoes what appears numerically to be simple harmonic motion

-15

iH { i ' n V

T

-9 -6

i

Y

—« i

-

■

-z

-3

1

Z

*-

1

1

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-15

jlHHl,

-12

i!

lHHIfI

'"IK: -6

, X

—« i

i

-z

-3

1

z ^1

1

Figure 2.4. x and y Location of a soliton lattice particle as a function of z (time coordinate). The results are consistent with simple harmonic motion for x and y. Path specification method is used. The system is that of Figure 2.2.

Math Aesthetics/Noninteerabilitv

32

in both the x and y directions giving rise to an elliptic type motion. The other lattice particles all undergo similar motions. We can follow the motion of a soliton lattice particle for as long as we wish, at least so far as the numerical work is concerned. In our aesthetic fields program, at the outset, we do not assume multiparticles, quantum mechanics, special relativity, curved space, Lagrangians, etc., all we have are mathematically aesthetic principles. These above phenomena would be expected to come out eventually as secondary concepts. This project remains a long range goal. As we mentioned in the beginning of Chapter 1, we only focus, to start, on a few questions that can readily be addressed. We see now on having obtained simple harmonic motion that it is possible to, at least, introduce the concept of an effective Lagrangian that has the effect of leading to simple harmonic motion. We recognize that to introduce the effective Lagrangian as a first step, in order to obtain the location of the particles, would only mask the underlying aesthetics of the theory. To get away from the simple numbers in Eq. (2.27), that is 1.0 and -1.0, we can integrate the equations to some point, and then use the resulting gammas as origin point data. If we do this we see that we still get soliton lattice structure, with the same soliton magnitude of 0.49. An example of this procedure is adopted when we integrate the system along y, then x, then y, then x, and then y in segments of .3 using a grid of .003. The resulting gammas are then used as origin point data. We then look at the system using the summation over path method Eq. (2.21) to see how the superposition principle alters the lattice. The resulting map is given in Figure 2.5 for the quantity Tj 3 . We see maxima of values of 0.48 and 0.45 and a minimum with the value of -0.49. These values could be consistent with the soliton value of 0.49. This is because we used a very coarse grid of 0.24 in this map. The reason we used such a coarse grid, at the expense of numerical accuracy, is that we wanted the map to include several maxima and minima, and the number of integration paths becomes astronomic as we proceed away from the origin. In fact, the reason for the shape of the map (Figure 2.5) is that the number of paths is largest in the diagonal directions. The number of integration paths in the x,y plane is given by

where n= nx + ny and nx is the number of segments along x and ny is the number of seqments along y. Thus, it is clear that in order to proceed it is necessary to have an approximation scheme to get around this path proliferation problem. We call the resulting method the random path approximation. Here, we take a subset of paths arrived at in a random fashion. The results of this method were compared satisfactorily with results from the commutator method and will be discussed further in Chapter 3. To generate random numbers we made use of the IMSL routine GGUD. The use of random paths also appears in Doll, Coalson, and Freeman3. In Figure 2.6 we study the same system as appears in Figure 2.5 using the random path approximation. We see the same qualitative picture as Figure 2.5 (two maxima and one minima close to the origin). In addition, as we are studying a larger region, we see additional maxima and minima. We

33

Nonintegrable Systems

observe that the soliton magnitude 0.49 ± 0.01 is preserved by the superposition principle but the solitons are rearranged by the integration scheme, into a more complicated pattern than the lattice of Figure 2.2, although symmetries are still present. We note all the maxima (minima) appear to lie on two lines. The resulting map is given in Figure 2.6. We note, the origin is in the center of the figure. To summarize, we see that the superposition principle Eq. (2.21) alters the lattice system in a nontrivial way but still manages to keep the soliton property, although a beneficial use for the additional degree of freedom Eq. (2.21) is as yet unclear. We emphasize that the quantities on the left hand side of Eq. (2.21) are functions. Furthermore from Figure 2.6 we see we are talking of well behaved functions. Thus at the stage of observing maps, functions appear and mixed derivatives of these functions are

Figure 2.5. Sum over path method for the system of Figure 2.2 when we integrate Eq. (2.27) from the origin as discussed in the text and then use the resulting gammas as origin point data.

Math Aesthetics/Nonintegrability

34

symmetric. By solely looking at the map (that is, not knowing how the map was arrived at), one could not draw the conclusion that a superposition principle at each point was employed. From the material found in the map one could then introduce a phenomenological theory that seeks to describe what is in the map, without ever concerning oneself with the manner in which the map was made. Thus, the quantities

Figure 2.6. Random path approximation to the sum over path method. The system is the same as in Figure 2.5.

such as J)(S), J2(S) of section 3, can be thought of as "hidden" at the stage of studying the content of maps. Although these quantities can be thought of as hidden once we have performed the summation, quantities such as J| (S), J2(S) are very important as they allow us to capture the information contained in the system, without restricting the origin data (which restricts the theory) in any way. The situation is similar in Feynmans 4 sum over path approach to quantum mechanics. By only studying maps of ip, we can not tell that use of a superposition principle has been made. We next considered the effect of the second approach to nonintegrable systems Eq. (2.26) on the lattice system Eq. (2.27). To make use of historical developments we integrate the system Eq. (2.27) along x, then y, then z, then y, then x, and then z, going

35

Nonintegrable Systems

Figure 2.7. Second approach to nonintegrable systems for the system of Figure 2.2 when we integrate Eq. (2.27) away from the origin as discussed in the text. Map is at z=0.

700 points for each segment with a grid size 0.003. The resulting gammas were then used as origin point data. We have argued that such an operation preserves the soliton lattice with the same soliton magnitude. The resulting map at z=0 is given for T\3 in Figure 2.7. Numbers in the figure are 100 times actual numbers. This figure was obtained with a grid of 0.009375. We note magnitudes of .49. We also see magnitudes of .45, .46, .47, and .48. Smaller grids (as small as .00234375) show that these magnitudes are, in fact, .49. Thus, the soliton magnitude is preserved by both the superposition principles Eq. (2.21) and Eq. (2.26). The system is more complicated than the lattice although there are symmetries. When x —> -x, and positive numbers go to negative numbers, and z ^ - z w e see a similar set of maxima and minima. In addition in Figure 2.7 we note maxima (minima) with the magnitude of .23 ± 02. Smaller grids tell us the magnitude is .23. We have studied what happens to one such minimum close to the origin in the - - quadrant. In Figures 2.8-2.12 we see the system at z= -2, z = -6,z= -8,z=4,z=10 respectively (units of z are 0.084375). At z= -8 the minimum under consideration has totally disappeared. At z=10 a soliton of magnitude .49 appears where none was present previously and this soliton persists for greater "time",at least for

Math Aesthetics/Nonintegrability

36

those times studied. Figures 2.8-2.12 made use of a grid of 0.028125 which is a coarse grid. Nevertheless, the presence of magnitudes greater than .40 farther from the origin gives us some confidence in the results (deviation from the soliton value of .49 gives us a handle on numerical errors). Also the appearance of the soliton magnitude .49 in Figure 2.12, which persists in time gives us additional confidence. Additional computer time would be useful to study the system with greater accuracy. Our observations are consistent with a soliton appearing in a region where no soliton was found previously. This effect was found in other instances as well for this system. Also solitons were found to disappear. Thus, the trajectories of solitons cannot in general be followed for as long as one wishes due to the appearance and disappearance of solitons. This is contrasted to the specification of path results where we could follow the solitons for as long as we wish. Thus we see that the way we integrate a nonintegrable system can alter a situation where solitons have well defined trajectories to a situation where they do not have well defined trajectories. This feature may be useful in dealing with quantum mechanics where we know particles do not have well defined trajectories, although no definitive conclusions can be drawn at this time in this regard. Although in this chapter we have developed techniques to handle nonintegrable systems, which will be continued in the next chapter, and have applied these techniques to a soliton lattice system, it remains an outstanding and important question what kinds of effects can be attributed to the superposition principles. Can these superposition principles be used to help understand some of the unsolved problems of physics? Clearly in this section we have only scratched the surface of this problem. This problem of the practical usage of a superposition principle at each point will be brought up again in Chapter 4. In general the effect of the superposition principles will depend on the system under study. There are physical concepts that are present regardless of the equations used.

Figure 2.8. The system of Figure 2.7 using the second approach to nonintegrable systems for minimum close to the origin in the - - quadrant at z= -2 (in units of 0.084375). The -0.23 minimum at z=0 appears here as -.15.

37

Nonintegrable Systems

An example of this is the notion of the "Arrow of Time". We shall see in Chapter 4 that the second approach to nonintegrability can give an account for this effect, such that the effect disappears once the integrability equations are satisfied.

Figure 2.9. The system of Figure 2.8 at z= -6.

Figure 2.10. The system of Figure 2.8 at z= -8. The minimum under consideration has totally disappeared.

Math Aesthetics/Nonintegrability

38

Figure 2.11. The system of Figure 2.8 at z=4.

Figure 2.12. The system of Figure 2.8 at z=10. A soliton of magnitude .49 appears here where none was present previously.

39

Nonrntegrable Systems

References 1. T.E. Phipps Jr. Heretical Verities: Mathematical Themes and Physical Descriptions, Classical Nonfiction Library, Urbana 111. (1987). 2. R.G. Newburgh and T.E. Phipps Jr., Relativistic Time and the Principle of Caratheodory, // Nuovo Cimento 67 (1970)84. 3. J.D. Doll, R.D. Coalson, D.L. Freeman, Fourier Path-Integral Monte Carlo Methods: Path Averaging, PRL 55 (1985)1. 4. R.P. Feynman, The Space-Time Approach to Non-Relativistic Quantum Mechanics, Rev. of Mod. Physics 20 (1948) 367.

CHAPTER NO. 3 THE COMMUTATOR METHOD 1. The Commutator Method and the ABJL Equations The commutator method is equivalent to the sum over path method and the product method. The commutator method enables one to integrate along a single path. One then augments the answer with an algebraic term, dependent on the commutator for mixed derivatives. The algebraic term is different for each point. The commutator method is only useful when we can calculate the commutators for the mixed derivatives and they take on a simple form. Thus, the utility of the commutator method, like the product method, depends on simple structure for the basic equations. We will illustrate the commutator method using the ABJL Equations (2.1) and (2.2). For the ABJL system we can obtain formulas for an arbitrary order derivative and from this we can arrive at the commutator for all mixed derivatives. By taking derivatives of Eq. (2.1) we find d[2„]A=(k,k2)nA,

(3.1)

where [2n] means that there are 2n indices with n equal to an integer starting with one. It does not matter what the indices are (for the two dimensional ABJL system the indices are either x or y). Thus all second derivatives of A are the same, all fourth derivatives of A are the same, etc.. We also get 9 [2 n + i]A=k I (k,k 2 ) n J (3.2a) a [ 2 n + 1 ] A=k,(k,k 2 ) n L,

(3.2b)

where [2n+l] represents an odd number of indices. We get Eq. (3.2a) if the first index on the left in d is x, and we get Eq. (3.2b) if the first index on the left in d is y. Thus we can rewrite Eqs. (3.2a),(3.2b) as d x [ 2 n ] A=k,(k,k 2 ) n J 5 y[ 2n]A=k,(k 1 k 2 ) n L

(3.3)

All derivatives of A of a certain order starting with x are equal, and all derivatives of a certain order starting with y are equal. As for the derivatives of J we get in a similar way 9 [ 2 n + l ] J=k 2 (k,k 2 ) n A.

(3.4)

4]

The Commutator Method

Thus all odd order derivatives of J are the same for a given n. We also have dmJ=Qnk2)nJ

(3.5a)

d [2 n]J= (kik 2 ) n L.

(3-5b)

or

We get Eq. (3.5a) when the first index on the left in d is x, and we get Eq. (3.5b) when the first index in d is y. Thus Eq. (3.5) can be written dx[2n-l]J=(k,k 2 ) n J

ay[2„.,]j=(k,k2)nL.

(3-6)

We next obtain the commutation relations from Eq. (3.3) and Eq. (3.6) as follows (<9xy[2n-l]-dyx[2n-l])A= (k,k 2 ) n+l (<9xy[2n-2]-<9yx[2n-2])J= (klk 2 ) n k2.

The indices in dxy

need not be the same as in <9yx

(3.7)

Some: simple commutators are

[<9x,dy]J = (dxy-dyx)-^ (kik 2 )k 2 , (3xyx-5yxx)A=(k| k 2 )(k,k 2 ), (9xyy-3yxy)A= (k,k 2 )(k!k 2 ), (<9xyxx-dyxxx)J= (kik 2 )(k,k 2 )k 2 , (dxyyx-Syxyx)^ (k, k2)(ki k 2 )k 2 .

(3.8a) (3.8b) (3.8c) (3.8d) (3.8e)

are zero. For For example, All commutators not appearing ini Eq. (3.7) are

(a xy -a yx )A=o, (dxj-x-dyxxJ^ 0,

(5xyy-ayxy)J= o.

(3.9a) (3.9b) (3.9c)

We note that all nonvanishing commutators depend on the constants k| and k2. To illustrate the commutator method we consider the following points R. .S P. .Q Here the number of segments along x (the quantity « i ) is one, and the number of segments along y (the quantity a 2 ) is also one. Then Eq. (2.8), using symmetrized derivatives becomes for the quantity J J(S)= J(P)+ | ^ dx + | ^ dy + i (dXy+dy*)} dx dy. dx. dy 2

(3.10)

Math Aesthetics/Nonintegrability

42

From the commutator of second mixed derivatives we have [a x ,a y ]J=k,k2.

(3.11)

a yx J = 9 x y J - k , k i

(3.12)

Then solving for dy* we get

This is then inserted into Eq. (3.10) giving dJ di , 1 J(S)= J(P)+— dx + — dy +d„J dx dy -k,k\ dx dy - . ox dy 2

(3.13)

Eq. (3.13) then gives the contribution from the path P —> Q —► S, which we denote plus a correction term.The correction term here is k,k 2 ._L2dxdy.

byJ,

(3.14)

To establish this statement we express Eq. (3.13) without the correction term in terms of finite differences as we did in Chapter 2, section 4. This gives J(S)= [J2(S)-J(Q)] + [J(Q)-J(P)]+ J(P).

(3.15)

The right hand side of Eq. (3.15) is just the contribution to J from the path going from P -> Q -> S. In a similar fashion we can obtain the correction to the path _t in the general situation where at\ and a 2 can be any positive integers. We use the commutation relations for the mixed partials in order to express all derivatives so that the x indices always appear on the far left. We then use Eq. (2.8) with Eq. (2.9) to obtain J at any point. In this way we see that the value of J at any point is given by the contribution from the single path -5, augmented by the infinite sum (we write it down to order 6 in the grid size): 1 1 33 -{a iQ2k2(k,k2)dxdy - + — a2(a2-l)(a2-2)a ,k2(k,k 2 ) 2 dx (dy)3 2 3 4 -{a1l a 2 k 2 (kik 2 )dxdy - + -a 2 (a 2 -l)(a 2 -2)a,k 22(k,k 2 ) dx(dy) 2 2 2 f— Q l ( Q l -I)^a2(a2-l)k2(k,k 2 ) (dx) (dy) 4 2 2 2! + ^1a , ( a , - l ) ^ a 2 ( a 2 - l ) k 2 ( k , k 2 ) (dx) (dy)2 ? 2 3 f— Q I ( » I -l)(a,-2) Q 2 k 2 (k,k 2 ) (dx) dy^ + ^3!1a 1 ( a 1 - l ) ( a , - 2 ) a 2 k 2 (k,k 2 ) 2 (dx)3 dy i 5 3 t — a 2 (a2-l)(a2-2)(Q2-3)(Q 2 -4) a\ k 2 (k,k 2 ) d x ( d y ) ^ 3 5 0 5!1 2 (a 2 -l)(a 2 -2)(a 2 -3)(Q 2 -4) Q l k 2 (k,k 2 ) dx (dy) +-Q 4 (-— a\(a\ -1) -a2(Q2-D(a 2 -2)(Q 2 -3) k 2 (k,k 2 ) 3 (dx) 2 (dy) 4 J ! 0 2! 6 1 3 3 +t — - aQ, (| (aQ,,- -l l) K-a (dx) 2 (dy) 4 Q2.(a ^ 2- -l)(a Q z t 2e-2)(a - l K ^2 -3)k ^ k j C2 (k,k k . k2;) 3>)W(dy) 11

1 1 3 + - a , ( a , - l ) ( Q i - 2 ) -a 2 (a 2 -l)(a 2 -2)k 2 (k,k 2 ) 3 (dx) 3 (dy) 3 j\

5\

h

5

-

43

The Commutator Method

1 1 , 2 + - a 1 ( a , - l ) ( a 1 - 2 ) ( a 1 - 3 ) - a 2 ( a 2 - l ) k 2 (k,k 2 ) 3 (dx) 4 (dy) 2 + - a , ( a , - l ) ( a , - 2 ) ( a 1 - 3 ) ( a | - 4 ) a 2 k 2 (k,k 2 ) 3 (dx) 5 dy \ } +

(3-16)

The pattern is clear, so the correction can be written for any power of the grid size. The term with (dx) 2 (dy) 4 in Eq. (3.16), for example, comes from the derivative term containing 9Xxyyyy- The coefficient \ in this term is just the number of y indices in dxxyyyy divided by the total number of indices in <9Xxyyyy- This rule can be seen to be general. In a similar way we can write down the correction at each point in the calculation of A. The correction is again to the path _ t . The infinite series for the correction to A is {^a](a]-l)

a2(hk2)2(dx)2

l

dy

-

+ -a2(a2-l)a,(kik2)2dx(dy)2^ + - a , ( a 1 - l ) ( Q 1 - 2 ) ( a , - 3 ) a 2 (k,k 2 ) 3 (dx) 4 dy i + i a , ( a i - l ) ( a r 2 ) i a 2 ( a 2 - l ) (k,k 2 ) 3 (dx) 3 (dy) 2 1 + ^ a , ( a i - l ) ^a 2 (Q 2 -l)(a 2 -2Xk,k 2 ) 3 (dx) 2 (dy) 3 I +a, ^ a 2 ( a 2 - l ) ( a 2 - 2 ) ( a 2 - 3 ) (k,k 2 ) 3 dx (dy)4 ~ +]-a,(a 1 -l)(Q 1 -2)(a,-3)(a 1 -4)(a,-5)a 2 (k,k 2 ) 4 (dx) 6 dy^ +^a 1 (a,-l)(a 1 -2)(a 1 -3)(a,-4) ^ a 2 ( a 2 - l ) (k,k 2 ) 4 (dx) 5 (dy) 2 1 + ^ a , ( a , - l X a i - 2 ) ( a 1 - 3 ) ^a2(a2-l)(a2-2)

(k,k 2 ) 4 (dx) 4 (dy)3 -

+ i a , ( a , - l ) ( a , - 2 ) ia 2 ( a 2 -l)( Q 2 -2)(a 2 -3)(k,k 2 ) 4 (dx) 3 (dy) 4 * + 2 1 , a i ( a | - l ) ^a 2 (a 2 -l)(a 2 -2)( Q 2 -3)(a 2 -4)(k,k 2 ) 4 (dx) 2 (dy) 5 +o, ia 2 (a 2 -l)(a 2 -2)(Q 2 -3)(a 2 -4)(a 2 -5) (k,k 2 ) 4 dx (dy)6 - } 6!

/

5

-

(3.17)

+ In integrating along a single path we continually reduce the grid until we have a convergence to a desired tolerance. Then in dealing with the correction term we may also, for example, continually double a\ and a2 and at the same time half the grid till we get convergence to a desired tolerance. As we went farther away from the origin and the grid size was reduced, so a\ and a2 become very large, we found a need to keep more terms

Math Aesthetics/Nonintegrability

44

in the series for the correction Eqs. (3.16),(3.17) to maintain our accuracy. We studied the quantity A in a portion of the x,-y quadrant. We were unable to find any maxima (minima) in a region of 15x14 points with separation between points of .3. In looking for a two dimensional maxima or minima in the region mapped, we observed that the largest magnitude for A, for each value of y, appeared to lie on a straight line as indicated in Table 3.1 (the units for x and y are .3). Table 3.1 Location of Minima for A for each y.

-y 0

I 2 3 4 5 6 7 8 9 10 11 12 13 14

X

3 3 4 5 6 6,7 7 8 9 10 11 12 13 14 15

A -0.70 -0.98 -1.25 -1.53 -1.82 -2.10 -2.39 -2.68 -2.98 -3.29 -3.57 -3.87 -4.18 -4.47 -4.77

The value of A in Table 3.1 continued to grow at a steady rate. Numerical studies cannot say that this pattern will continue indefinately. What we see is that no two dimensional minima or maxima were found in the region mapped. Furthermore the steady increase of A offered no sign of any impending maxima or minima. Thus, the degree of freedom associated with the sum over path superposition principle did not lead to any beneficial effects for the ABJL system in the region mapped. We see that the type of results arising from this superposition principle depend on the type of lattice under consideration. The two component lattice associated with the ABJL system is the simplest lattice we have studied. We will find that when we study a three component lattice solution of the gamma equations, in the next section, that two dimensional maxima (minima) do exist, and the five component lattice of Chapter 2, section 6, also has maxima and minima when we make use of the sum over path degree of freedom.

45

The Commutator Method

2. Commutatator Method for a Three Component Lattice We seek a nonintegrable lattice system obeying simple equations for which the commutator method leads to two dimensional maxima (minima). Consider the following equations dB —=-CL dx

dC

ax"

aT 0

dB —=-DL dy

ac

dD —=BL

dD

= BL

dy"

=0

ay

ac_

dB

DL dz with L constant.These equations uncouple to give

aT0

aD dz

(3.18)

d2B _ 2 -L B ax 2 " -

a2B = -L2B 'dy1

d2C _ 2 -L C ax 2 " -

a2c

d2D _

dz2

= -L2C

a2D = -L2D. (3.19) ~df~ az2" which have sinusoidal solutions. The system is nonintegrable since we have nonvanishing commutators such as (ayx-axy)C=-L2D (dyX-dxy)D=L2C. (3.20) -L2D

All the nonvanishing commutators are zero if L=0, but L=0 makes the system Eq. (3.19) trivial. This is our second example where we see that requiring that the integrability equations be satisfied renders the system uninteresting. When we specify an integration path by first integrating along y and then x, a map of B in the x,y plane gives rise to a lattice, although it is a two dimensional lattice, since B does not change in z. Despite this limitation we will be able to obtain a numerical improvement over the sum over path method when we use the commutator method. Furthermore this situation does lead to two dimensional maxima (minima) unlike the case in the previous section. The Equations (3.18) can be obtained as a special case of the gamma equations by choosing the origin point data as follows r1 = -r2 == -E x

20

1

13

x

10

r = r1 = A 2

1

1

23

3

r = r = r° 1

30

1

13

1

13

x

33

l

10

L

10

*-

Math Aesthetics/Nonintegrability

46

p1 2 _ l-p3 _ L-pO _ - i-p2 _ - 1-p3 _ - 1 T-IO _ -pj 30~ 23~ 23~ 33~ 20~ 20~ u

r] 2 =rL=r5,= -r^=-r?2=-r?2=L,

(3.21)

with the other gammas zero. This structure for gamma is preserved by the equations for gamma (Eq. 1.13). We can see this by integrating from the origin to any point. In our numerical work we chose L=1,B=.5,C=.6,D=.4. We also found that A at all points obeys A=L-B,

(3.22)

with L constant. We may integrate the gamma equations using the origin point data Eq. (3.21) to, for example, x=.3, then y=.6, and then z=.3, using a grid of 0.003. We may then use the resulting gammas as origin point data. When we specify an integration path by first integrating along z, then y then x we find a two dimensional lattice in the x,y plane for B. There is no change for any of the variables in x° so dynamically we have an intrinsically three dimensional system. As the dynamics of our system is three dimensional, we shall treat z as the "time" We see that the magnitude of the two dimensional maxima (minima) for the quantity B preserve the same value as a function of z (soliton behavior). The numerical work is consistent with x(z) and y(z), where x(z) and y(z) represent the location of a maximum or minimum, being sinusoidal and giving rise to elliptical trajectories. All the maxima and minima for B that were observed undergo this type of motion. We saw solitons with similar motions for the more complicated 5 component lattice in Chapter 2, section 6. However this latter soliton system is of greater complexity so the use of the commutator method is not realistic. We note an interesting feature of the current system is that the gammas in Eq. (3.21) are four dimensional. However the dynamical system described by Eq. (3.18) is three dimensional. This situation is still valid if we integrate to some point in x,y and z and then use the subsequent gammas as origin point data. Thus we may argue that higher dimensions for gammas do not mean that the dynamics cannot be of lower dimension, leading to an interesting possible role for higher dimensions. A limitation of the above system is, for example, that B does not change along the z axis even when we integrate to some point and then use the resulting gammas as origin point data. This feature can be traced to the condition that B does not change in z in Eq. (3.18). Even in the situation of integrating along x, y and z and then using the resulting gammas as origin point data we see that the quantity D does not have maxima and minima in the x,y plane. Thus the system Eq. (3.18) is not without shortcomings. At any rate we shall now study the commutator method for the system given by Eq. (3.18) with the origin point values given by B=.5,C=.6,D=.4,L=1.0. We shall only concern ourselves with an x,y plot for B. We define n=nx+ny, where nx is the number of x derivatives and ny is the number of y derivatives. When n is even the nonvanishing derivatives of B, from Eq. (3.18), have the form d(xx)(yy)(xx)(xx)... B= (-l)2L n B.

(3.23)

47

The Commutator Method

The nonvanishing derivatives involve a succession of xx and yy pairs. The order of pairs makes no difference. The parentheses in the subscript for d are used to emphasize the pair structure. We note, then, that when n is even, both nx and ny must be even for derivatives to be nonvanishing. When n is odd, we have two possibilities. When nx is odd, we find that the nonvanishing derivatives have the following structure dx(yyXxx,(xx)...B=(-l)nrLnC.

(3.24)

The nonvanishing derivatives above have xx or yy pairs in sequence with an x at the extreme left. The order of xx and yy pairs is immaterial. If instead, ny is odd, we get Sy(xx)(xx)
(3.25)

In this case, we have xx or yy pairs with y at the extreme left. Again the order of xx and yy pairs makes no difference. All other derivatives are zero. Thus the Equations (3.18) have the virtues that all order derivatives can readily be written down. Commutation relations can be obtained by comparing a derivative with one obtained from it by interchanging two neighboring indices. We see that all nonvamshing commutators defined in this way are given by either the right hand side of Eqs. (3.23),(3.24) or (3.25) (up to sign). We next make use of Eqs. (2.8) with (2.9). We can interchange indices of the derivatives if we make use of the commutation relations. We can continually permute indices until all the y indices appear on the extreme right. Then Eq. (2.8) with Eq. (2.9) is made up of two parts. One gives the contribution from the path-5, and the second part is determined from the commutation relations, as in the last section. The latter term represents a correction at all points of the x,y plane to the path _ 1 A straight forward calculation shows the correction term is an infinite series with the structure £ a i ( a , - l ) a 2 (dx) 2 dy } [L3D] +±a2(a2-l)

a,(dy) 2 dxi[-2L 3 C]

+ ^ a , < a , - l ) £a 2 (a 2 -l)(dx) 2 (dy) 2 ±[-5L4B + L 4 B] +J f a,(a 1 -l)(a I -2Xai-3) a 2 (dx) 4 dy ±[-L5D] +±a2(a2-l)(a2-2)(a2-3)

a,(dy) 4 dx |[4L 5 C]

+ A a , ( a , - l X a i - 2 ) £ a 2 ( a 2 - l ) (dx)3 (dy)2 £ [9L5C-L5C]

Math Aesthetics/Nonintegrability

48

+±a2(a2-l)(a2-2) i Q ,(arl)(dx) 2 (dy) 3 ^[-2L5D] +J [ a l (a,-l)( Ql -2)(a 1 -3) ±a2(a2-\) (dx)4(dy)2 i[14L6B-2L6B] +jia2(a2-l)(a2-2)(a2-3)

£a,(ai-l) (dx)2(dy)4 ^[14L6B-2L6B]

+

(3.26)

The numbers 3,6,5,10,15,etc in the denominators in Eq. (3.26) represent the number of arrangements of indices for which nx and ny are both fixed. This is given by n!

(3.27)

nx!ny! In some of the terms in Eq. (3.26) we have written the bracketed part as a sum of two terms. This reflects the different origin of the two terms. In analyzing the origin of the contributions we find that the bracketed terms can be obtained from general formulae written below. We call the expressions in brackets in Eq. (3.26) a(nx,ny). We note there is no contribution from terms having nx=0 or ny=0. The formulae for a(nx,ny) are: (a) nx even, ny even (n=nx+ny> m = -,m x = —' ,my = y ) , a(n x ,n y )=t^- T -l]L n (-l) m B+[-^---l]L n (-l) m - , B, mx!my! nx!ny! (b) nx odd, ny odd a(nx,ny)= 0, n-1 nx-l ny-l n+1 (c) ny even, nx odd (p = — ,px = — ,py s - y - , q = — ) a(nx,ny)= [~-~ -1] Ln(-l)pC + [ - £ - -l]L»C(-iy , nx!ny! my!px! (d) ny odd, nx even a(nx,ny)= -&— Ln(-1)"D . (3.28) mx!py! We can readily see that the bracketed expressions of Eq. (3.26) are special cases of Eq. (3.28). For the two component lattice of section 1, the commutator method did not give any evidence of planar maxima and minima. The steady buildup of the field suggested singular behavior. The fact that the buildup was so regular can be taken as evidence that the errors were not a problem. We were able to integrate considerably farther from the origin as compared with the sum over path method. On the other hand, for the three component lattice studied here, we did find planar maxima and minima. If the pattern

49

The Commutator Method

Figure 3.1. Map of-B using the commutator method. This system gives a 3 component lattice when we specify the integration path. The system obeys Equation (3.18). Numbers are 100 times actual numbers.

found continues, we can expect an infinite number of such maxima and minima. An x,y map is given in Figure 3.1. Should we prescribe an integration path as previously, we see that the zero magnitude contour lines go off to ± oo in the y direction. In the present case, the zero contour lines are closed curves that resemble circles. In Figure 3.1 we see that the magnitudes of the maxima and minima all have the same value 0.87 ± 0.01. This is the same soliton magnitude that we get when we specify an integration path. Thus the property that the soliton magnitude is preserved by the integration scheme is present here as well as the more complicated lattice system of Chapter 2, section 6. In Figure 3.1 we map four quadrants, each of which has 24x24 points. The separation between points is .3. The grid used is of magnitude 10"'° in some instances. The number of segments, a\,oti,

Math Aesthetics/Nonintegrability

50

needed to get from the origin to a point was as high as the order of 10 .In programming for the computer we made sure that the grid was always multiplied by an a, so as not to allow extremely large or small numbers to enter the calculation from the grid or a. The order of the calculation n=nx+ny, was adjusted so we obtained convergence for the answer. The points farthest from the origin in Figure 3.1 made use of order 48. We found more and more terms in the series Eq. (3.26) are needed as we go farther from the origin. The regular looking pattern of Figure 3.1 is an indication that errors are not a problem in the region studied. Note regularities of the zero contour lines. We have also studied the same system using the sum over path technique shown in Fig 3.2. Using the summation over path method we were unable to integrate far from the origin. Also, we were obliged to use a .3 grid, which is very coarse, in order to include at least a few maxima and minima. Although, we qualitatively see a similar picture the quantitative agreement is not good. For example, the closest maximum to the origin using the sum over path method was 0.87, which is the same as the commutator method. The farthest minimum from the origin that we observed had a magnitude of .77 which did not agree that well with the commutator method of .87. It is clear that if we wish to use the sum over path method an approximation procedure is necessary. This problem will be dealt with in section 4 in some detail. The commutator method is accurate enough, as evidenced that the soliton magnitude was preserved throughout Fig 3.1. However, we need to remember that the commutator method is only useful if the basic equations have a simple enough character so that the expression for derivatives is tractable. 3. The Product Method We can also evaluate the field at all points directly from Eq. (2.8) with Eq. (2.9) without introducing any commutation relations. Instead we make use of Eqs. (3.23)(3.25) by themselves. In equation (2.8) with (2.9) this amounts to replacing the derivative terms by products of the fields. A derivative having nx indices labelled x, and ny indices labeled y is replaced by the factor b(nx,ny) as follows: (a) n even, nx even, ny even b(n x ,n y )=- m —-B(-l) m , m x !m y ! (b) n even, nx odd, ny odd b(nx,ny)= 0 , (c) n odd, nx even, ny odd

b(nx,ny)= -Bl—rX-l)" , m x !p y !

(d) n odd, nx odd, ny even b(nx,ny)= - H L _ c ( - l ) i . m y !p x !

(3.29)

51

The Commutator Method

In this method no integrations are needed along any paths. In Chapter 2 this method was called the product method. The results give the same picture as Figure 3.1. 4. Random Path Approximation The commutator method, when viable, is more accurate than the sum over path method. Hence we introduce an approximation scheme to the sum over path method by taking a subset of paths, where this subset is chosen in a random way. The random path approximation is then compared with the commutator method. We work with the same three component lattice of the previous two sections. The first question we ask is how many random paths should we take in any random path calculation? In Table 3.2 we give the results of calculating a representative component, -B, using a varied number of random paths. All entries in the table correspond to a single point. The segments in x and y are each 80. The grid is 0.0375. Table 3.2

Number of Random Paths 1500 1000 750 500 300 250

Value of -B Obtained from Summation Over Random Paths 0.39 0.39 0.39 0.39 0.39 0.38

Thus, we see that the number of random paths is not a critical parameter so long as we have a ''reasonable" large number of paths. The corresponding value of -B using the commutator method is .40. We still have another parameter at our disposal— the grid size. We studied the results at the same point as above, but now with a 0.01875 grid. The number of random paths was taken to be 1500. We obtained a value of .40 for -B, which agreed with the commutator method. Of the two parameters, the number of random paths and the grid size, we have found that the grid size is the more crucial parameter so far as obtaining agreement with the commutator method. In Table 3.3 we compare the results of the random path method with those of the commutator method for different points. The number of random paths in each case was chosen to be 1500. Comparison was made for the component -B. Selected but representative points are shown in the table. The table illustrates that the agreement between the random path approximation to the sum over path method vis a vis the commutator method can be considered rather good in that the two methods give the same results as shown in Figure 3.2. The discrepancy in the table is not more than .03.

Math Aesthetics/Nonintegrability

52 Table 3.3 — A comparison of the Sum Over Path Method with the Commutator Method

a

a2

grid size

12 16 20 24 28 32 36 88 96 104 112 120 128 136 288 304 320 336 352

12 16 20 24 28 32 36 88 96 104 112 120 128 136 288 304 320 336 352

0.075 0.075 0.075 0.075 0.075 0.075 0.075 0.0375 0.0375 0.0375 0.0375 0.0375 0.0375 0.0375 0.01875 0.01875 0.01875 0.01875 0.01875

Random Path Method for -B -0.52 -0.75 -0.85 -0.80 -0.60 -0.32 0.04 0.66 0.81 0.83 0.69 0.43 0.10 -0.25 -0.59 -0.80 -0.87 -0.79 -0.57

Commutator Method for -j -0.53 -0.75 -0.86 -0.81 -0.62 -0.31 0.04 0.68 0.84 0.84 0.71 0.43 0.09 -0.27 -0.57 -0.78 -0.84 -0.76 -0.55

This concludes our remarks concerning techniques to handle nonintegrable systems. We have seen that with the use of the computer such systems can be addressed. We make use of the 2 component and 3 component lattices of the aesthetic field equations for illustrative purposes here.

53

The Commutator Method

Figure 3.2. The sum over path method applied to the system of Figure 3.1. Numbers are 100 times actual numbers.

CHAPTER NO. 4 NONINTEGRABILITY AND THE ARROW OF TIME 1. The Arrow of Time Nonintegrability implies a superposition principle at each point, which represents an additional degree of freedom that may well be useful when rethinking some basic physical problems that are not presently understood. When we do not have a means to calculate the field along extended paths we turn to the integration scheme Eq. (2.26). We have already remarked at the end of section 5 of Chapter 2 that this approach has attractive features, so we shall make use of this approach in what follows. The notion of time is present regardless of which equations we are dealing with. As the form of equations is not critical, we will make use of the ABJL equations, with x —> t and y - > x , and see what nonintegrability can tell us about the notion of time. As the number of spatial dimensions are not important in our understanding of time we will work with one spatial dimension. Intuitively we recognise that time goes forward never backward. You would never know this by looking at Newton's equations. We can integrate these equations either forward or backward in time. If we integrate them forward we predict the future. If we integrate them backward we verify the past. The situation is different in quantum mechanics. It's not that you can't integrate the Schrodinger equation either backward or forward in time in a valid way. In quantum mechanics we have an additional ingredient—the measurement operation. The wavefunction ip collapses. We can then use ip to make predictions concerning the future (in a probabilistic sense) but we can't use this ip to verify the past. If we integrate ip backward in time we get meaningless numbers. Landau and Lifshitz1 attribute this distinction between past and future to the quantum measurement operation. This effect we may call the arrow of time. But we can question whether we have really obtained an understanding of the arrow of time in terms of the quantum measurement operation. After all, the measuring aparatus is still a quantum mechanics system, albeit a complicated one, satisfying the Schrodinger's equation, which can be integrated either forward or backward in time in a valid way. Perhaps some greater understanding of the measurement operation will enable us to get a better understanding of the arrow of time. We shall take another point of view. We propose that the arrow of time can be understood in terms of a classic theory that is nonintegrable. What we will do to start off is use quantum measurement theory as a guide to define precisely what we mean by the arrow of time. We define the arrow of time as follows: (1) From information at the present we can make predictions about the future (at least in a probabilistic sense). (2) From information at the present we cannot verify the past. This is precisely the effect that one has in quantum measurement theory.

55

Nonintegrability and the Arrow of Time

The results of integrating A in equation Eq. (2.1) (grid used is .0375 and spacing between points is .01875) and using Eq. (2.26) gives us: Table 4.1

4» -1.22 -1.20 t -1.12 -1.00

-0.90 -0.93 -0.91 -0.82

-0.58 -0.66 -0.67 -0.62

-0.29 -0.39 -0.41 -0.38

> x If we use the information in the top row and integrate backward in time we get: Table 4.2

^ -1.22 -1.20 t -1.12 -0.99

-0.90 -0.92 -0.90 -0.82

-0.58 -0.60 -0.62 -0.60

-0.29 -0.28 -0.31 -0.33 > x

It's not that we can't integrate the basic equations (ABJL equations) backward in time in a valid way, it is that there are other contributions that have to be taken into account. Integrating forward in time is shown by solid lines while integrating backwards in time is shown by wavy lines below. Y^W

X

R^ S

U

P

T

Q

In the above picture one can integrate from S to W as well as undoing it by integrating from W to S. However in integrating forward in time there is a contribution from V, while in integrating backward in time there is a contribution from R. Thus, the operation of integrating forward and integrating backward are different, and this accounts for the numbers being different in Table 4.2 as compared with Table 4.1. Thus the arrow of time has its origin in the integration scheme Eq. (2.26). We note that if the integrability equations k]=0 or k2=0 are satisfied, the effect of the arrow of time as defined above disappears. Thus, by just altering the origin point data the effect of the arrow of time is no longer present. Thus, we can say that the arrow of time has its origin, according to the above definition, in the notion of nonintegrability.

Math Aesthetics/Nonintegrability

56

We emphasize that the arrow of time as defined here is present regardless of whether the basic equations are time reversal invariant, as the effect follows from the integration scheme rather than the basic equations such as Eq. (2.1). There is another kind of asymmetry in time2 other than the arrow of time effect. We are talking of the concept of entropy as well as the expansion of the universe. If we take enough numbers we can say whether the numbers come from the future or the past. That is by comparing sets of numbers the more disordered set refers to the future, in the case of entropy. In talking about expansion of the universe the greater total amount of numbers refers to the future. These effects refer to an asymmetry in time, but have nothing to do with the arrow of time as we define it. This is clear as we can integrate the motion of particles backward in time in a valid way (from our definition above, we see there is no arrow of time). We can get a similar situation by integrating the ABJL equations using Equation (2.26). Note, the ABJL system is a simpler lattice system than what appears in Figure

Figure 4.1. Map of +t,+x quadrant for the quantity A in the ABJL system of Chapter 2.The grid used is 0.0375. The vertical line is the x axis and the horizontal line is the t axis. Numbers in the figure are ten times actual numbers.

57

Nonintegrability and the Arrow of Time

Figure 4.2. The situation in Fig 4.1 in the -t,+x quadrant. We see the solution has a totally different character in this quadrant.

2.2. The ABJL equations on useage of Eq. (2.26) gives rise to two dimensional maxima (minima) in the +t, +x quadrant but in the -t, x quandrant there are no maxima and minima. We see this in Figures 4.1 and 4.2. Thus if we are dealing with +x, by seeing enough numbers, we can tell whether the numbers came from the past or future. For example if we find a maxima this implies that the numbers came from the future. Thus the ABJL equations illustrate that we can have an asymmetry in time without it having to do with the arrow of time. Said another way, just because there is an asymmetry in time this does not mean that one cannot go backward in time. These are separate effects. What we see here would apply in the case of entropy or with the expansion of the universe. Thus we conclude: the present determines the future (at least in a probabilistic sense). But the present does not determine the past. We can know the past by keeping a record of the past. We can also have clues of the past from present information. However we can't obtain the past from the present in the case of the nonintegrable system described by Eq. (2.26). The effect described here is what we call the arrow of time.

Math Aesthetics/Nonintegrability

58

The superposition principle in the case of the simple two component lattice (where the two component lattice is obtained by specification of path), associated with the ABJL equations, does not give rise, from Figure 4.2, to a promising system. We see by comparing results with the 5 component lattice (section 6 of Chapter 2) that the effect of superposition is different for lattices of different character. The effect of the superposition principles is in general very much dependent on the system under study. References 1. L. Landau and E. Lifshitz, Quantum Mechanics Nonrelativistic Theory, 2 nd edition, Pergamon Press (1958), page 24. See also L. Landau and E. Lifshitz, Statistical Physics, Pergamon Press (1958), page 31. 2. P.C.W. Davies, The Physics of Time Asymmetry, Univ. of California Press, Berkeley California, (1977).

CHAPTER NO. 5 THE GAMMA EQUATIONS AS A SOURCE OF FUNDAMENTAL BUILDING BLOCKS 1. Introduction. As nonintegrability plays an important part in the aesthetic fields program, we may expect it to make its presence felt in fundamental physics. However, the effects of nonintegrability would be difficult to establish until one has a clear understanding of the solutions to the basic equations (in our case Eq. (1.13). Once we gain an understanding of the content of the basic equations, we could investigate how the superposition principle at each point (whether Eq. (2.21) or Eq. (2.26)), alters the results obtained when we specify an integration path. There are situations where the exact form of the basic equations is immaterial, as in the previous chapter. Withstanding such a situation where the form of equations is not important, the next problem would be to understand what sort of information is contained in the basic set of Equations (1.13). As we are not here involved in how the techniques of nonintegrable systems alter solutions (that is to say how a superposition principle at each point affects the solutions) we shall explore what sort of behavior Eq. (1.13) leads to along integration path segments in what follows. In dealing with two dimensional plots we will use the specification of path approach. As results of integration depend on path we can get to any point in a unique way by first integrating along x°, then z, then y, then x. As we cannot justify this integration path rather than another, this approach will be used only to see what kinds of solution Eq. (1.13) is capable of. It is another problem to see how the techniques of nonintegrable systems alter these results. This problem will not be attempted here in any large scale as we need increased computer capability to deal with this project, although some work on this problem has been done for simple systems and is discussed in previous chapters. We have found already that Eq. (1.13) has a close association with the simple sine curve in Chapters 1,2 and 3, and this point will be elaborated further in this chapter. The sine curve can be thought of as the basic building block of physics and this may encourage us to believe that Eq. (1.13) may well be relevant for physics. We consider in section 6 the set of origin point data that leads to sine within sine behavior. An aim of the program would be extending the data, in the way of greater complexity, to see if this can lead to realistic particle structure in the form of wave packets. In the early part of this century much information concerning fundamental physical processes was learned by studying basic building blocks — namely sine curves and superposition of sine curves in the form of wave packets. Later was added the concept of an agitated vacuum. We recall, the Heisenberg Uncertainty Principle as well as the Bohr atom can be understood using wave packets, without the need of the Schrodinger Equation. Having played this important role in the early development of quantum theory, the use of wave packets has diminished in the study of fundamental processes. To illustrate this point we note that Landau and Lifshitz1 hardly mention wave packets in their book on quantum mechanics. Nevertheless, the precise predictions of the

Math Aesthetics/Nonintegrability

60

wave packet approach in the Bohr atom, including the value of the Rydberg constant, suggest that there is value in such a mindset, and this approach of making use of basic building blocks to construct model mathematical universes will be pursued. A set of numbers at each point constitutes a model mathematical universe. The aesthetic fields program has sufficient content to describe interesting mathematical model universes, we shall see, based on fundamental building blocks of the type discussed above. In linear systems such as in quantum mechanics, one routinely obtains wave packets from a superposition of sinusoidal curves with different wave numbers. The linearity implies that the different contributions, having different wave numbers, act independently, so the spreading of the wave packet becomes a serious problem in attempts to associate the wave packet with particles. Goswami2 in his book considers an electron described by a Gaussian wave packet with width of one Compton wavelength. This wave packet then has a width of about 2x 109 meters after one second has elapsed. This suggests that wave packets associated with nonlinear, rather than linear, systems be studied. This will be pursued in section 7 and section 10. This uncontrolled spreading will not present a problem there. 2. Sinusoidal Behavior Along any Path Segment Arising from Mathematical Aesthetics Program Consider the following origin point data:

r1 = A rj 2 = A rj 3 = A r| 0 = B 1

n r1 = A

r22 = A r23 = A r20 = B

L

2]

1

r =A 1

I V A

31

1

r =A r x

1

01

1

=A A

0 2

r] 3 = A

r^0 = B

r03 = A

r

oo =

B

r 2ii = -A r22 = -A r23 = -A r^o = -B

1

r 2 = -A r22 = -A r23 = -A r 20 = -B r 2 = -A r12 = -A r»=-A

x

r

2

r 3ii = J

r

3

=J

r32 = L r33 = j ri

3

2 2

r r

3

1

01

oo = "B

=J r

3 2

r] 0 = J

=j

rL = J

=L r 3 =j

r^0 = J

=TL

r

3

21

3

r

= -A r^2 = -A rg3 = -A

1

i

r| 0 = -B

31

= J 1^2= L

3

3

r»=J

r3 = i oo J

1

61

Gamma Eq as Source of Building Blocks

r°,=-j r?2 = -L r°3 = -j

r° 0 = -j

r2I = -J r22 = -L r23 = -J

r20 = -J

MI

=

"J

^ 3 2 = "L

r33 = -j

r 3 0 = -j

r°,=-j r°2 = -L r»3 = -j

IV-J .

(5.1)

This structure is maintained at all points. We see this by integrating from the origin to any point using the fourth order Runge Kutta method. Although this conclusion is drawn from numerical work we believe this to be an exact result. This is true also for the 3 component lattice of Chapter 3. To illustrate this feature consider the following numerical results. We integrate the above data along x, then z, then x°, then y, then x, then z, then x°, and finally along y using (1.13). Each segment is of 100 points with a grid of .003. As an example we refer to the following table (note the numerical calculation was only done to 13 decimal places): Table I. Numerical Example to Illustrate that Structure of Eq. (5.1) is Maintained at All Points

Component r?,=J ri=J 1,=J rj,=J r^=J

Origin Value 1.0 1.0 1.0 1.0

Value at x=.6,y=.6,z=.6,x°=.6 0.4243123148468 0.4243123148468 0.4243123148468 0.4243123148468

In all instances we found that the structure Eq. (5.1) was maintained to 13 decimal places even though components, such as J, changed considerably from their origin point value. This indicates that the maintaining of the structure Eq. (5.1) is an exact rather than a numerical result. When we study the sine within sine system we shall find a similar situation as well. The gamma Equations (1.13) we find, making use of the algebraic capabilities of the computer via REDUCE, then collapse into the simple set of equations: dA

ax"

= "-I ki

dA

dy" dA

lh~

= *k,l v

= k,

dA

dx°

9 B

9 3

V

U T

dy

aT klJ dB

= ki

T

t

t

9 J

V

A

V

A

— =k2A ax 9 3

— =k 2 A dy dJ A — = kv2 A az 9} , „ ^ = k 2 B

9 L

^

A

V

A

— =k2A ax d L

— =k 2 A dy dL A — =kv2 A az <9L , „ a^=k2B

(5.2)

Math Aesthetics/Nonintegrability

62

with k,sA-B

k2 = J - L

(5.3)

These equations are maintained at all points since the origin point structure is the same at all points. Considering only changes along x and y, these equations are identical to the ABJL Eqs. (2.1) and (2.2). The integrability equations are k]=0, or k 2 =0 as in the two dimensional case. When we map the situation using the same parameters as in Chapter 2 we find a regular array for maxima and minima in three space dimensions for A,B,J,L. This arises since the system uncouples to yield, for example, for A

|?=k2k,A a?

= k2k|A

l ^ M A - k , ) .

(5.4)

Thus, A has sinusoidal behavior for ki k2 < 0 along x,y, and z. Along the x° axis A-ki is sinusoidal, so A itself is a displaced sine curve. The lattice particles do not preserve their magnitude in x° so they are not solitons. No matter which axis direction we integrate along we get sinusoidal behavior. As the equations are valid at all points we conclude that the ABJL equations lead to sinusoidal dependence along any path segment (path segments are parallel to x, y, z, x° axes). The ABJL equations represent a linearization of the system described by Eq. (1.13). If the integrability equations are satisfied we lose the lattice, and the equations are not interesting. For a different choice of origin point data the gamma equations, we shall see, collapse into a simple set, that are intrinsically nonlinear, and describe sine within sine behavior along any path segment. The equations are again nonintegrable. One would have a hard time overestimating the importance of the sine curve in the natural sciences. One might believe that everything one would want to know about sinusoidal behavior would already be well known. The subject of nonlinear equations, and in more than one spatial dimensions nonintegrable systems, gives evidence that this is not the case. In linear systems one can find a different way to obtain sine within sine behavior than the standard approach as well. As sinusoidal behavior is of such prominent significance in the sciences we shall make a digression from mathematical aesthetics to discuss sine within sine behavior. We will find that plots we obtain from the study of nonlinear systems, which are nonintegrable as well, when we have more than one spatial dimension, are indistinguishable from the plots obtained from traditionally obtained linear sine within sine systems.

63

Gamma Eq as Source of Building Blocks

3. Linear Sine within Sine System By sine within sine structure we mean, ip= msin(ki x +(j>]) + nsin(k2x +cf>2),

(5.5)

with ki=27r/Ai and k2=27r/A2. We shall consider the case that A2 is considerably greater than Aj. The system described by Eq. (5.5) is a six parameter system. An equivalent form for V> is i\>- a sin k]X +b cos k]X +c sin k2X + d cos k2x .

(5.6)

We start off with an elementary discussion. Consider the following equations: dN , „ — = -k, S dx

dP

dS — = k,N dx

d

Q

,

(5.7)

These equations have as solutions N=aN sin(k|X + 0,)

P=aP sin(k2x + 0 2 )

S=-aN cos(k)X +<j>i)

Q=-aP cos(k2x +(j>2)

(5.8)

We define A B C D

= = = =

N+P S+Q P-N S-Q.

(5.9)

Then A,B,C,D have the sine within sine structure ip. To many this information is sufficient to not pursue the situation any further. On the other hand, it is a simple matter to get differential equations for A,B,C,D by just adding and substracting the above equations. There are 6 quantities in Eq. (5.7), namely N,S,P,Q,ki,k2, so we get, using Eq. (5.9), 6 equations dA — = - L, B + L2 D dx dB — = L, A + L2 C dx

Math Aesthetics/Nonintegrability

64

dC = L, D dx '

UB

dD = -L, c dx"

-u A

dL, "dx"

=0

dL2 =0 "dx" ux

(5.10)

i

where, L, = (k, + k2)/2 U = (k 2 -k,)/2.

(5.11)

The solutions for A,B,C,D can be gotten from the definitions of the quantities appearing in Eq. (5.9) and take the form A = aN sin (k)X + 4>\) + ap sin (k2x + 2) B = -aN cos (kjx + x) - ap cos (k2x + 2) C = ap sin (k2x + <j>2) - 3N sin (k\X + x) D = -aN cos (k]X + \) + ap cos (k2x + 4>2).

(5.12)

We can also verify this by inserting Eq. (5.12) into the differential Equations (5.10) and setting the coefficients of sin k]X, cos k\\, sin k2x, cos k2x to be zero. A,B,C,D are given in terms of 6 parameters aN,ap,k|,k2) 4>\,<j>2 which can be expressed in terms of the 6 pieces of origin point data associated with the differential equations. We can also write Eq. (5.12) in alternative form. For example A = aA sin kjx + bA cos kjx + CA sin k 2 x + dA cos k 2 x.

(5.13)

Then we see aA = aN cos cf>)

CA - ap cos <j>2

bA = aN sin 0,

dA =ap sin 2.

(5.14)

65

Gamma Eq as Source of Building Blocks

Thus, we have equations like:

CA +
(5.15)

These equations are nonlinear relations between the coefficients aA, bA, We have integrated the differential Equations (5.12) along the x axis using the origin point data A=.5, B=.4, C=.3, D=.2, L,=3.17, L2=-3.11. The plot in Figure 5.1 is for A. 4. Another Linear Sine within Sine System Now that we have the set of Equations (5.10) it is a simple matter to allow the computer to alter the right hand side, so the basic structure is maintained -but the quantities A,B,H,M are interchanged, one with the other. In this way we come up with the following set of equations dA — =L, B + L 2 M dx dB — = -L, A - L2 H dx dM — = -Li M - U A dx — = L, H + L2 B dx ^i=0 dx ^ = 0 dx

(5.16)

Note that the M equation here involves M on the right hand side, and the H equation involves H on the right hand side. These features are not present in Eq. (5.10), as the quantities on the left side did not appear on the right side in this system of equations. We claim that the solution of Eq. (5.16) has the form: A = aA sin Iqx + bA cos kix + cA sink2x + dA cos :k2:x B = aB sin k|X + bB cos k]X + cB sin k2x + d1B cos k2x M = aM sin k]X + bM cos k]X + cM sin ....... k2x +dM dM cosk cos 2x H = aH sin kix + bH cos k[X + CH sin ck22xx++dn dHcos coskk; 2x.

(5.17)

Math Aesthetics/Nonintegrability

66

By inserting Eq. (5.17) into Eq. (5.16) and requiring that the coefficients of sin k]X, cos k)X, sin k2X, cos k2X be zero, we see that Eq. (5.16) is identically satisfied provided aA = (Li b B + L2 b M ) /ki

aM = (-L] b M - L2 bA)/ki

aB = (-L, b A - L2 b H ) /kt

aH = (Li b H + L 2 bB)/ki

b A = (-2 L] b M + b B L2)/L2

b H - -b;M

K i ~~ J_(i

' J_<2

cA = (L) dB + L2 dM)/k2

cM = (-Li dM - L2 dA)/k2

c B = (-L, dA - L 2 dH)/k2

cH = (Li dH + L2 dB)/k2

dM = (-L2 dH - 2 L] dA)/L2

dB = dA

k2=L2-L2.

(5.18)

Thus all the parameters in Eq. (5.17) can be expressed in terms of 6 pieces of information which we take to be L| ,L2,bM,bB,dA,dH which can be related to die origin point data A(0), B(0), H(0), M(0), L](0), L2(0). The 16 equations arising from the coefficients of sin kjx, cos k]X, sin k2X, cos k2X, being 0 lead to 12 restrictions on the a,b,c,d's and 2 conditions on k] and k2. The last 2 equations are then identically satisfied. Thus we see that we can have a linear sine within sine system in which the conditions between k),k2 and the constants appearing in the differential equations, are nonlinear. This is in contrast with the linear relations occuring in the previous set Eq. (5.11). The set Eq. (5.16) is obtained in the references3,4. We have not been successful in obtaining Eq. (5.16) or Eq. (5.10) from the gamma equations. Figure 5.2 shows a plot of the quantity A, using as origin point data A=.95, B=l.l, C=1.0, D=-1.05, L!=-.96, and L 2 =.97, as a function of x. 5. Nonlinear Sine within Sine System. Using the computer we may generalize Eq. (5.10) or Eq. (5.16) in a nonlinear way to get: dA — =-AM-BF ox dB —=AH+BM ax — = A2 - B 2 - C (F - H) ax

Figure 5.1. Plot of A vs. x for linear sine within sine system of section 3. 500 points are shown with spacing between points of 0.075 using DGEAR. ON

-J

a §

1

CO X>

g00 o -c1 ob o *n 03 C CL

5'

Figure 5.2. Plot of A vs. x for linear sine within sine system of section 4. 500 points are shown and spacing between points is .15 using integration scheme DGEAR.

05 03 0

— o 5

68

Math Aesthetics/Nonintegrability

<9H _ = -2AB -2CM dx ' dF _ 2AB+2CM dx~ dC _ 0. dx

(5.19)

We write the equations as partial differential equations as we will be extending these equations to three space dimensions later on. This extension to 3 space dimensions will be done without any increase in the number of parameters. As origin point data we take: C = -.96 H= -.55 F = -.85 .

A =.95 B = 1.1 M = .5

(5.20)

Integrating along x we obtain the results appearing in Figure 5.3 for the quantity M. From the plot, which is similar to what we saw in Figure 5.1, we infer the solution has the following structure as Figure 5.3 is characteristic of sine within sine behavior: 27TX

27TX

.

27TX

2TTX

M = aM sin —— + bM cos —— + CM sin —— + d\i cos —— Ai Ai A2 A2 27rx 2TTX 2irx lirx A= aA sin —— + bA cos —— + CA sin —— + dA cos —— A] Ai A3 A3 .

27TX

A]

.

27TX

27TX

.

27TX

27TX

Ai

A3

A3

27TX

27TX

27TX

B= ae sm — — + b^ COS — — + CB sm — — + de cos — — H= aH sin —— + bH cos —— + Ai A]

CH

sin —— + d H cos —— + L A2 A2

H + K C = constant.

(5.21)

Here L and K are constants. The numerical work also seems to indicate that A3 = 2A2

(5.22)

The small wavelength in Eq. (5.21) is A, and the long wavelengths are A2 and A3. The numerical work indicates that all components that vary have sine within sine behavior along x or a displaced sine within sine behavior in the case of the quantities H and F.

69

Gamma Eq as Source of Building Blocks

We emphasize that the conclusions are drawn from numerical integration using the 4th order Runge-Kutta method. These results were confirmed by Weyenberg3 from numerical integration using the IMSL routine DGEAR. To show that Eq. (5.21) with Eq. (5.22) is a solution of Eq. (5.19) analytically is a formidable algebraic problem even using the algebraic capabilities of the computer and has yet to be confirmed. However the numerical work is so clear cut, note Figure 5.3, that we will not concern ourselves with the analytic problem here. In other words, even if the solution to Eq. (5.19) is not exactly of the form Eq. (5.21), it simulates it very well and this is all that is needed for our purposes. In addition, in reference 3, we show as well, that the numerical integration can be fit numerically to a sine curve within a sine curve for the quantity M. When we insert Eqs. (5.21), (5.22) into Eq. (5.19) we note that the coefficients of sin lqx, cos k(x, sin k2X, cos k2X are no longer zero. This is because the system Eq. (5.19) is now nonlinear, and we must make use of the nonlinear trigonometric identities: sin2 e +cos20 = 1 cos 20 = cos2<9 - sin20 sin 29 = 2 sinO cosd.

(5.23)

6. Nonlinear Sine within Sine Behavior and the Aesthetic Fields Program. We now show mat we can obtain the nonlinear sine within sine Equations (5.19) as a special case of the gamma Equations (1.13). We consider the following intrinsically 3 dimensional origin point data

r], = 0.0 rj, = -A

rj 2 = o.o

r! 3 = o.o

r 22 = M

r 23 = -F

I*M = B

I12 = H

T\3=-M

2

r23 = F

H. = A

r

Ii = 0.0 rl, = -C rj, = -B it, = C

rf2 = o.o

rV-H

r? 3 = M

rL = B

r| 3 =A

iSi = 0.0

rf2 = o.o

iVo.o

2

= -M

r 2 3 = o.o

r 32 = - B r 2 3 = -A

(5.24)

This structure is maintained by the field Equations (1.13). We see this by integrating from the origin to an arbitrary point. The gamma equations then collapse into the set Eq. (5.19) together with the following:

Math Aesthetics/Nonintegrability

70

dA ~dy~

dA

= M 2 - B 2 + H (C = -A B - C M

~dz'

dB =AB+CM ~dy" dB = A 2 - M 2 + F (H ~dz~ <9M = -AM-BF 9y~ dM = AH + B M

Ifr. dC = -2AH-2BM dy'' DC = 2 A M + 2 B F ~dz'

an = 0

dy' <9H = - 2 A M - 2 B F ~dz~ dF = 2AH + 2BM dy" dF _ = 0.

~di~

The equations above imply F + H + C = K'

(5.26)

with K' constant. The equations are in general nonintegrable. We note we can force integrability to be satisfied by requiring for the origin point data A =1.0 B=1.0

M=1.0 C = -1.0 H = -1.0 F = -1.0

(5.27)

We then find that the quantities A, B, M, H, F, C, do not change from point to point (trivial solution). This result, again, illustrates the restrictive character of the integrabihty equations. As in the case of the ABJL equations and the three component lattice system, everything interesting is lost once the integrability equations are satisfied.

71

Gamma Eq as Source of Building Blocks

Numerical integration of the equations in y and z again shows the sine within sine structure similar to the case along x. From the numerical integration w e infer: . 27ry M = aM sin — 1A4

DM

27ry . 2-zry 27ry cos —— + CM sin — — i - d^ cos —— A4 A5 A5

27ry 27ry . 27ry 27ry A = a A sin —— + b A cos —— + c A sin —— + d A cos —— A4 A4 A6 A6 27ry B = a e sin —— + A4

DB COS

27ry . 27ry 27ry —— + CB sin —— + d B cos —— A4 A5 A5

. 27ry , 27ry . 27ry 27ry , F = ap s m + bp cos —— + cp sin —— + dp cos —— + L A4 A4 A6 A6 C=K -F-H H = constant,

(5.28)

with A5 = 2 A6 ,

(5.29)

and 27TZ

27TZ

.

27TZ

27TZ

M = a M sin —— + b M cos —— + c M sin —— + d M cos — A7 A7 A9 A9 27TZ

27TZ

27TZ

27TZ

.

27TZ

27TZ

A = a A sin —— + b A cos —— + c A sin —— + d A cos — A7 A7 A9 A9 .

27TZ

27TZ

B = a B sin —— + b B cos — - + c B sin — + d B cos —— A7 A7 As As C= ac sin

27TZ A7

, 27TZ . 27TZ 27TZ „ + b c cos -—- + c c sin —— + dc cos —— + L A7 As As

H = - F - C + K' (5.30)

F = constant, with A9 = 2A 8

(5.31)

Math Aesthetics/Nonintegrability

72

Here the small wavelengths are A4 and A7. The wavelengths involved in integrating along y are not the same as along x. Also the wavelengths associated with integration along z are different from the above. K',L'L" are constants. Note aA,bA,.— etc. are different in integrating along x, y and z. We see a similar behavior associated with x,y and z although there is a difference. We see C is constant along x, while H is constant along y, and F is constant along z. As Eq. (5.19) and Eq. (5.25) are maintained at all points (since the structure for the gammas Eq. (5.24) is preserved by the gamma Equations (1.13)), we conclude that the numerical integration leads us to believe that the system of equations, Eq. (5.19) and Eq. (5.25), and thereby the gamma equations themselves, describes sine within sine behavior, or displaced sine within sine behavior, along any path segment. Before we showed that the gamma equations, for a different choice of origin point data, describe sinusoidal, or displaced sinusoidal, behavior along any path segment. We see mat die gamma equations have a rather remarkable association with sinusoidal behavior. The gamma equations enable us to extend the sine within sine system Eq. (5.19), to three space dimensions without needing to introduce any new parameters. A,B,M,C,H,F are determined at all points in terms of six pieces of origin point data. All quantities that change show sine within sine behavior, or displaced sine within sine behavior. If we alter magnitudes for the origin point data in Eq. (5.24) we can still end up with sine within sine behavior. As an example consider the following data: A=6 C=-.96

B=7 H=-.55

M=5 F=-.97.

(5.32)

-M is plotted along x, here in Figure 5.4, showing the characteristic sine witiiin sine behavior. All the components that vary exhibit this behavior, or the displaced sine witiiin sine behavior along any path segment in x, y or z. An x,y map for -M, using the specification of path procedure, is shown in Figure 5.5. Not all values for the origin point data consistent with Eq. (5.24) show the sine within sine pattern. As an example the data Eq. (5.27) leads to a trivial solution. The data of Chapter 2, section 6, also has the structure Eq. (5.24), but leads to different looking pictures. Compare for example Figure 5.5 with the map appearing in Figure 2.2. This latter map describes a soliton lattice system. We called this lattice a 5 component lattice since the equations are the same as Eq. (5.19) and Eq. (5.25), and there is one linear relation between A,B,M,C,H,F, given by Eq. (5.26). Once one has sine within sine structure, one opens up the door to the notion of a "big" picture and a "small" picture. We can make use of computer graphics that emphasize the large scale variations, or "big" picture, although one does not get a good impression of what is going on in the "small". In order to get a big picture we compress (say) 4000 x

Figure 5.3. -M vs x for nonlinear sine within sine system of section 5. 1200 points are shown with spacing between points of .15 and grid of 0.0375.

o

I 1

w

J3

B CO

o c -1 o rt o M j 03 c

~ E Figure 5.4. -M vs x for nonlinear sine within sine system using data at end of section 6. 500 points shown with spacing between points of 0.028125.

5

TO

00

— .8

sr

Math Aesthetics/Nonintegrability

74

points onto a single computer page. Each x value in this plot then stands for 33 or 34 actual x values. In the vertical direction alphabet letters are printed for each of the coarse x values. The letter A at a certain value of, say, -M (vertical axis quantity) says there is one actual x value having the -M value shown; the letter B indicates two values of x had the same -M value; the letter C indicates three values of x have the same -M value, etc.. These letters A,B,C,... are not very important in our considerations since we are only interested in the nature of the large scale variation. An example of a big picture plot is given in Figure 5.6. Here we make use of the sine within sine nonlinear equations with origin point data given by Eq. (5.20). -M is plotted along x. This figure shows that the large wavelength oscillation appear sinusoidal, as expected from the small picture plot given in Figure 5.3. Whenever we have a problem involving a characteristic time in the sciences, we have a situation in which there is a small picture, as well as a big picture. A simple

Figure 5.5. x,y map of nonlinear sine within sine system using data at end of section 6. Grid is 0.0375, spacing between points is 0.3. Map involves all 4 quadrants with origin at center. Numbers are 100 times actual numbers. Map is obtained by specification of path technique. Contrast this map with Figure 2.2. Both systems have similar origin point structure Eq. (5.24), but magnitudes are sufficiently different to cause different looking patterns.

75

Gamma Eq as Source of Building Blocks

approximation to such a system could be treated as a sine curve within a sine curve. Just as we have perturbations of sine curves such as anharmonic oscillators, we could use the sine curve within a sine curve as a base for more complicated behavior. The way we do this in our program is to increase the complexity of the origin point data. By doing this we will be able to obtain a wave packet solution. Such a solution will be described in the next section. 7. Wave Packet Solution Consider the following set of gammas:

iV-r^.o? M2 = " r i 2 = .1

rfi=- r|, = -05 p i _ p 2 _ pO _ p i _ p2 1 02 L 10 - L 21 - ' l 20 " l 01

= -r°

p l _ p 3 _ pO _ pO _ p3 -1 30 ^ 01 - i 13 - " 1 3 1 ' l 10

= -r'

= 1.0

1

12

x

= 1.0

03

p 2 _ p 3 _ pO _. pO _. p3 1 03 ~ l 20 - l 32 ~ " l 23 " l 02

= -r 2

p l ___ Lp2 _ _ x p3 _ _l p l 1 10 20 - 30 01

-r 3 = 0.1

1

33

*22

Ml

-

_l p2 02

_

_

3

r = - r° = - r° = o 01 00

1

30

i

03

1

= 1.0

30

03

U 1

p 0 _ pO _ pO _ pO _. p l . 1 10 ~~ * 20 _ l 01 - x 02 _ " l 00

i

x

U U1

-

-r 2 = 0.01 i

oo (5.33)

-

with the other rj k zero. We next integrate this system using the 4th order Runge-Kutta method in conjunction with Eq. (1.13), to the point x=.6,y=.6,z=.6,x0=.6 as in section 2 of uiis chapter. The resulting Tjk are then used as origin point data. In Figure 5.7 we observe a big picture plot for a representative component, TQ0 = T^, (all plots in this section will be for this same component) along the y axis. Here the grid is .009375 and the spacing between points is .075. Figure 5.7 resembles beats of sound waves where one combines two sine curves of slightly different frequencies. The resemblance is not total as the amplitude of successive wave packets gets progressively larger as y increases. Whether this effect continues or there is a turnaround in magnitude eventually is not clear. The results were confirmed using one-half and one quarter the grid size. This picture shows that we are finding oscillations growing more complex, in a more or less systematic way, starting with the sinusoidal behavior along any path segment which occured when we

Math Aesthetics/Nonintegrability

76

.2

I

I I .2

1 X

I

I I •o

I !

i

i I

I

I if

I

77

Gamma Eq as Source of Building Blocks

consider the ABJL system Eq. (5.1). This gives an impression that one can build almost anything starting with the gamma equations. We will return to this feature in Chapter 6.

Figure 5.7. Big picture plot along y axis for the wave packet solution of section 7. 16000 points along +x and 16000 points along -x (arising by adjoining 4000 point plots) are combined. Grid is 0.009375 and spacing between points is 0.075. The picture has some similarities with beats occuring in sound waves. Plot is for T^.

In Figure 5.8 we show a small picture plot of the current system along the x axis. The grid is 0.028125 and spacing between points is .1125. 1500 points are shown in the figure. In Figure 5.9 we show a big picture plot of this system. One 4000 point plot along -x is adjoined to a 4000 point plot along +x. The grid is 0.01875 and spacing between points (in the 4000 point set) is 0.075. We again obtained a similar picture when the grid size was lowered. We lowered the grid to as small as 0.00029296875. We call such a system a wave packet solution. What is of interest besides the wave packet itself is what surrounds the wave packet. We call this region "vacuum" although the vacuum is not empty. There is important structure in the vacuum. Some gamma components can be very large in the vacuum, although T^ and some other components are small. For example, r^,=0.00012 at x=984.375 (actual x value) when we use the grid 0.00234375. At this point which is in the vacuum region, twelve components of rj k lie in the range .2 to .3 x 105 Oscillations of varying wavelengths are occuring for the different gamma components. Thus, the vacuum is anything but empty and uninteresting. It is not unreasonable to expect that a study of the vacuum for such a complex system could give us insights to some fundamental physics. We shall have more to say of this investigation in the next section. We specify a path in the manner previously described and study big picture plots for different y. In all these plots we see the wave packet structure. In Figure 3.9 the largest magnitude appearing in the plot is about 0.25. We find that for certain y this largest magnitude can be quite large. In these cases we call the system a two dimensional wave packet particle. As examples of this we see wave packet particles at y=-43 and y=-134.7 (units of y are .005). These are illustrated in Figures 5.10 and 5.11 respectively. The largest magnitudes here for T^ are in the range between 10 and 11 (contrast this with Figure 5.9). The spread of the wave packet at y=-43 is much larger than usual. Figure

s > CD CO

BCD

o"

Io

a a' Figure 5.8. Small picture plot for r } , for the wave packet system of section 7. Grid is 0.028125 with spacing between points of 0.075. 1500 points are shown.

»

I

00

Figure 5.9 Big picture plot of Tj, for wave packet solution of section 7, along x. Grid is 0.01875. Spacing between points is 0.075. 4000 point along +x and 4000 points along -x are combined in the plot.

79

Gamma Eq as Source of Building Blocks

5.10 involves 16000 points arising by adjoining 4 4000 point plots along x while fig 5.11 involves 4000 points in x. Our investigations indicate that the system Eq. (5.33) describes a multi wave packet two dimensional particle system. We next ask do we see additional wave packet structure as we increase x along these y=constant plots. The answer is we have not seen additional wave packet structures as a function of x. The indication is that magnitudes tend to increase without sign of bound as we increase the magnitude of x sufficiently. Whether this can be accounted for by numerical errors or not is not clear. Thus, the wave packet solution is not without problems. A more viable wave packet solution which does not appear to have this deficiency, at least in the domain studied, will be discussed in section 10. For now we will turn our attention to properties of the vacuum which are not without interest. 8. A Study of Properties of the Vacuum for the Wave Packet Solution We study y=constant big picture plots for the wave packet solution of the previous section. Fig 5.12 is at y=66. This shows a wave packet structure with vacuum appearing at the right of the plot. The peak magnitude here is less than in Figure 5.9. In Figure 5.13 at y=67 we see a continuity with Figure 5.12. At y=68 we see an abrupt change from Figure 5.13. There is then a range in y for which the plots do not resemble Figures 5.12 and 5.13. Again, at y=80 the situation is similar to Figures 5.12 and 5.13. Figure 5.14 is at y=68 where we see this abrupt change of the vacuum. The plot at y=69 is shown in Figure 5.15. Note this figure involves 4 4000 point plots adjoined to one another while the other figures in this series involve 4000 points. The largest magnitude in Figure 5.15 is comparable to that of Figure 5.9. The results of Figures 5.12, 5.13, 5.14, and 5.15 show that the wave packet has its presence felt considerably to the right without the appearence of a signal continuously moving from left to right. This example is not isolated. We see that the effect arises not because of a continuous displacement of peak magnitude as a function of y. Instead it is more as if "the floor rises (and falls)" to the right in a cooperative way (band structure of the vacuum). The effect appears as we have a multi component nonlinear theory, in which the vacuum can not be considered empty. It is not unreasonable to expect this feature to appear in time as well, as the coordinates in Eq. (1.13) behave like one another. Thus, we cannot expect to follow the wave packet motion in time as though it were a rigid body. We see it is not unreasonable to expect wave packets to make their presence "felt" over space in an "abrupt" and "noncontinuous" looking manner. We may say that the system gives the appearance of what is called "nonlocality". By nonlocality we mean that the wave packet has it's presence felt considerably to the right or left of it without evidence of a signal moving from the wave packet to the far away point under consideration.

2

» > VI

&

n a-. o o 3

3" n

1 •f

oo ©

Figure 5.10 Big picture plot at y=-43 using specification of path technique. Units of y are 0.005. Grid in y is 0.00005. 16000 points, arising by adjoining 4000 point plots along x, are shown in plot. Grid along x is 0.009375, and spacing between points along x is 0.075. Plot is for T^. We call the resulting structure a two dimensional wave packet particle.

81

Gamma Eq as Source of Building Blocks

Figure 5.11. Big picture plot using specification of path method at y=-134.7. The plot differs from Figure 5.10 in that only 4000 points along x appear. Also the grid along x is 0.01875, although the spacing between points is the same as in Figure 5.10. The plot again is for r},. y=-134.7 gives a second location of a two dimensional wave packet particle associated with the wave packet solution of section 7.

As we do not have an understanding of the mechanics obeyed by the wave packet system at this time it would be premature to speculate whether the nonlinear nonintegrable system discussed here can account for quantum phenomena. However the features already seen could have a bearing on such an issue. In quantum mechanics the wave function "collapses" when a measurement is performed. Thus it appears that the wave function can change in an abrupt way. We have seen that the function T^ can change in an abrupt manner (see Figures 5.13 and 5.14), so there is no reason that classical concepts cannot allow for abrupt looking changes. This can be accounted for by a system in which oscillations have different scales of oscillation wavelengths. However a conceptual understanding of quantum mechanics requires more than this. It is argued that in order to understand quantum mechanics from a classical base it is necessary that the classical theory be nonlocal. This goes under the name of Bell's Theorem5. For example, consider the case when conservation principles are at work. In measuring the spin of one particle, the spin of another particle would also have to abruptly change, even though the

Math Aesthetics/Nonintegrability

82

Figure 5.12. Big picture at y=66 using specification of path method. 4000 points are shown. Grid is 0.01875 and spacing between points is 0.075. Units for y are 0.005. The system is for the same wave packet solution of section 7. Plot is for T^.

second particle is far enough away from the first particle that there is insufficient time for a signal to reach the second particle from the first, to communicate the abrupt changes occurring in the first particle. Such considerations were raised by Einstein, Podolsky and Rosen6. Nonlocal classical theories have often been considered unreasonable. Here we see that the presence of an "agitated" vacuum leads to phenomena that appear nonlocal in a natural way. It is reasonable to expect that this feature would appear in reasonably complex wave packet theories. Thus the preliminary results from studying the above wave packet solution suggests that nonlocality should not be looked at as a stigma preventing classical concepts from ultimately accounting for quantum phenomena. 9. Assignment of Origin Point Data If we assign the origin point data according to Eq. (5.1) we see sinusoidal behavior along any path segment. If we assign the data according to Eq. (5.24) we observe sine within sine behavior along any path segment. One of the crucial problems in our studies is how to assign the origin point data. The highly symmetrical solutions of sections 2 and 6 (see Figures 2.1 and 5.3 for example) obey relations like T|k = 0 and T^ = 0. Tensor relations are maintained at all points by the field equations. This theorem was established in Chapter 1, section 8 and can be shown to be valid using numerical integration to a desired tolerance. Thus, the simple

83

Gamma Eq as Source of Building Blocks

Figure 5.13. The system of Figure 5.12 at y=67. We observe a wave packet structure at the left and vacuum at the right.

Figure 5.14. The system of Figures 5.12 and 5.13 nowat y=68. We see an abrupt change for the vacuum as compared with Figures 5.12 and 5.13.

Math Aesthetics/Nonintegrability

84

Figure 5.15. The system of Figure 5.12, 5.13, and 5.14 now at y=60. Here 16000 points are shown by adjoining 4 4000 point plots. The grid is 0.009375 along x and the spacing between points along x is 0.075. The wave packet has its presence felt to the right without the appearance of a signal moving to the right. Figure 5.14 and 5.15 illustrate what may be called nonlocality.

lattice system of section 2 and the sine within sine system (Figure 5.3) obey algebraic tensor relations at each point. We considered origin point data of a more complicated nature in section 7 of this chapter. We did encounter problems in that we were unable to establish in our numerical work a bounded solution even along the x axis (despite the use of extended precision) and the plot along the y axis, see Figure 5.7, has a symmetric character, and magnitudes for each wave packet structure appear to increase with no sign of abatement for large y. The origin point data is, again, not general as T^ = T^. This direction of more and more complicated origin point data then suggests the hypothesis that the origin point data should be free of all tensor restrictions which we know are maintained at all points. This can be considered as a new mathematically aesthetic principle as the alternative restricts the theory in an ad hoc way. Given a set of origin point data, it is not a simple matter to argue that there are no nonlinear restrictions on the data. It is an easier matter to seek a system that obeys no linear tensor restrictions on the origin point data, as the number of possible linear tensor restrictions is limited. Furthermore, we would be interested in a system that could be treated on the computer without unmanageable problems concerning numerical errors. There is, in principle, nothing wrong with a system which deals with magnitudes of all orders and for which oscillations have values for wavelengths of all orders. But numerically such systems would in general lead to problems having to do with numerical accuracy. The problem is not then in writing down a set of data free from tensor restrictions. The problem is one of finding, not only a system that is not restrictive, but one for which numerical errors are not a serious problem, at least for a large enough region such that we can establish interesting structure. With this practical restriction in mind we shall study a more general origin point data in the next section.

85

Gamma Eq as Source of Building Blocks

10. A More Viable Wave Packet Solution Consider the following set of gammas:

r!,=o.o

rj 2 = o.o

rj 3 = o.o

r| 0 = o.i

r21 = o.o

T\2 = 0.0

r23 = -o.o7

r 20 = -i.o

r 3 ,=o.o

r32 = o.i

r33 = o.o

r30 = o.o

To, =0.1

r02 = io

rJ3 = o.o

rj„ =-ooi

2

rf, = o.o

r n = o.o

r?3 = o.o7

r?0 = i.o

r|, = o.o

Y\2 = 0.0

Y\\ = 0.0

rf0 = o.o

r^, = -cos

T\2 = 0.0

r^3 = o.o

rJo=-i.o

r 01 =-i.o

rjj2 = o.o

r03 = i.o

r^o = o.o

r], = o.o

r] 2 = -o.i

r] 3 = o.o

r] 0 = o.o

ri, = o.o5

r ^ = o.o

r|, = o.o

r| 0 = i.o

r^, =o.o

rf2 = o.o

r ^ = o.o

r| 0 = -o.i

r^=o.o

r02 = -io

r03 = -o.i

r j , =-o.oi

r°,=-o.i

r°2 = -i.o

r?3 = o.o

r° 0 = o.oi

r°, = I.O

T°22 = 0.0

1^3 =-1.0

r° 0 = o.o

rg, = o.o

r° 2 = i.o

r° 3 = o.i

r° 0 = o.oi

r°,=o.oi

r° 2 = o.o

r°0i = o.on

rS„ = o.o.

(5.34)

The system Eq. (5.33) is characterized by

+ r! ± It, + 11. ± l

tk

0 0 AI* 4

Bri,

(5.34)

Math Aesthetics/Nonintegrability

86

with A, B, constant. Thus, we do not have linear tensor restrictions present at all points for the above system. We can get away from such simple numbers as appearing in Eq. (5.34) (like 1.0, 0.1. 0.001) by integrating the gamma equations away from the origin in the same manner as in section 7. The resulting gammas, after such a series of integrations, are then used as origin point data. We note all 64 VL are now nonzero by this operation, as is the case of the wave packet system of section 7. We are interested in what sort of content can be found in the gamma Equations (1.13), using the above set of data. Thus, which component we choose to study is not an important consideration either here or in section 7. Here, we choose to study the representative component r ] 3 . In two dimensional studies we make use of the path specification method where we integrate first along y and then x to get to any desired point in the x,y plane. We study the system by means of a series of plots along the x axis for different y. Our plots along x employ a grid size .0005859375 which is a smaller grid than we have been generally using. As this is not a symmetrical solution, our best hold on numerical errors is to check the results when the grid size is altered. We have redone the calculation at y=50 and y=100 (units of y are 0.025) with half as well as double the grid. The plots obtained were identical to the original plots. Agreement was found to six decimal places in these tests. In the plots the spacing between points along x is .075. 24,000 such points are compressed onto a single computer page. Thus, the plots go as far as x=1800 (actual value of x). These plots are referred to as "big picture" plots. Note in sections 6 and 7, we considered big picture plots in which only 4000 points are compressed onto a single computer page. In the plots that follow in this chapter we dispense with the letter system in the plots discussed at the end of section 6. The grid used for y is taken to be 0.00005. The spacing between points along y is 0.025 (which we note is less than in the case of x), and 100 such points in y are used to make a series of big picture plots along x. We label the plots as y=l,2,....,100 in the figures (it is understood that the units of y are 0.025). In making the plots we have made use of the newly acquired University of North Dakota Cray Supercomputer. When we specify a path, this problem is particularly amenable to supercomputer studies, as data associated with different y values can be simultaneously integrated along x. The big picture plots along x corresponding to the 100 different y values took 31 days of continuous running on the Cray. In order to study a . . (0.075) comparable region in y as in x, would require 24000x points in y, which would imply 720 such 3 \ day computer runs, which is clearly not practical for us, even without considering runs for different z and x° Despite limitations dictated from our computational resources, we can still learn interesting information about the wave packet system Eq. (5.34) by studying features present in the 100 plots associated with different values of y, so we shall focus on such a study here.

87

Gamma Eq as Source of Building Blocks

11. Features of the More Viable Wave Packet Solution Labels of the coordinate axes are at our discretion. Thus our plots for different y can be thought of as time plots if we wish. We list some of the features observed: (1) All plots are characterized by wave packets with "vacuum1' in between. No sign of potential unboundedness was observed in the region studied. We can then say that a multi wave packet system is a stable kind of arrangement to expect. (2) Unlike results of section 7, we saw multi wave packet structures as x was varied for fixed y. Because of this and feature number 1 we can consider the present system as a more viable wave packet solution than previously observed. In this paper we also consider a considerably larger domain than previously (about 6 times greater in x). The large number of points studied here and the small grid size used, requires large amounts of computer time and serves as a limitation to the present study. As computer time is such a factor here a clear picture using the integration scheme used in the arrow of time chapter (Chapter 4) was not feasible. (3) We see no evidence for the "uncontrollable" spreading of the wave packet discussed previously, at least as a function of y. (4) On comparing plots for succeeding y, one often sees a continuity from one plot to the next. For example, consider y=13-22 (figures 5.16-5.25). (5) However there are examples of abrupt looking transitions for successive y plots. Consider y=22 (Figure 5.25) and y=23 (Figure (5.26). Another example occurs when we compare y=3 (Figure 5.27) and y=4 (Figure 5.28). In these instances successive plots do not resemble one another very well. (6) Even if there appears to be a smooth transition from one plot to the next, we could be overlooking significant wave packet structure in the region between the plots. We may demonstrate this by considering plots obtained by forming a set comprised of every fifth plot. Then the plot at y=35 (Figure 5.29) and y=40 (Figure 5.30) would be adjacent plots. There appears to be a smooth transition between these two plots. Next, consider plots at y=36, y=37, y=38, and y=39 (Figures 5.31,5.32,5.33,5.34). We would have no idea of the existence of the wave packet structure that develops at the right by merely looking at Figures 5.29 and 5.30. Thus, what appears to be a smooth transition need not be. (7) If we do fine tuning by considering y values between plots which do not resemble one another very well the situation can be rendered more smooth. Consider feature number 6 above. Figure 5.34 at y=39 doesn't resemble Figure 5.29 at y=35. However Figure 5.34 (y=39) does resemble Figure 5.33 at y=38 and Figure 5.32 at y=37. We then see what appears at first glance to be a paradoxical situation. In some instances, the more we fine tune the situation the more regular the transition as in y=37,38,39 (Figures 5.32,5.33,5.34). On the other hand in other cases the more we fine tune (note feature number 6) as in the case of y=35 and y=40 (Figures 5.29 and 5.30) the more unpredictable the situation appears. Note again y=37,38,39 (Figures 5.32,5.33,5.34). Thus, the nonlinear nonintegrable theory based on mathematical aesthetics allows for complicated behavior patterns.

Math Aesthetics/Nonintegrability

88

(8) We pay particular attention next to the vacuum. As examples consider y=25 (Figure 5.35) and y=50 (Figure 5.36). The vacuum in these situations has a different character than, say, y=17 (Figure 5.20). The vacuum has a uniform character to the right in Figures 5.35,5.36, but maxima (minima) magnitudes are greater than in Figure 5.20. Remember the vacuum is described by small oscillations and is definitely not empty. The small oscillations are so close together that what we observe to the right in Figures 5.35 and 5.36 is a band appearance. These bands appearing in the vacuum allow wave packets to make their presence felt in far away regions without the need for a signal that leaves the wave packet and travels to the distant regions. We also observed such an effect in section 8, where it was referred to as the effect of nonlocality. The more "robust" vacuum in Figure 6.35 covers the region from x=900 to the full range of the figure at x= 1800. (9) Wave packet structures exhibit "tails" which are sometimes pronounced as at y=7 (Figure 5.37). This effect has similarities with what we discussed in feature number 8. (10) Wave packet structures can materialize from the vacuum and can disappear into the vacuum as in the case described by plots at y=35-40 (Figure 5.29,5.31,5.32,5.33, 5.34,5.30). The wave packet structure has also been observed in all our limited work when z is not zero. As an example the figure appearing at the begining of this book after the preface describes such a plot along x when z=-5, y=15 (in units of 0.005). Taking into account all the plots, we can say that spontaneity and unpredictability as well as more regular behavior are characteristic of the results. These features as well as nonlocality described above are present also in the study of fundamental processes (quantum theory). This suggests that further studies of the nonlinear nonintegrable system may well be useful in possibly gaining insights of fundamental processes, although this would require an extensive commitment of computer resources. To conclude, we see that with a small input based on mathematical aesthetics, we have been able to obtain basic building blocks. These are the same building blocks —sine curves, wave packets, vacuum regions—from which much information was learned about fundamental processes in the early part of the century. The system under study here is both nonlinear and nonintegrable. We may say that we are in the business, figuratively speaking, of achieving something out of almost nothing. It can be considered remarkable how much can be obtained based on a few simple ideas (note figures). A greater computer capability would enable us to study three dimensional big picture structures and a resulting mechanics. The present studies involve a big picture study in x but not in y or z orx°.

OQ

a p

ffl J3

o c o

o CD

Figure 5.16. Solution for r j 3 along +x at y=13 (units of y are 0.025) for the more viable wave packet solution of section 10. This is a big picture plot with 24000 points along x compressed onto one computer page. The parameters used are described in the texi:. Other plots shown are similar in all respects except they occur for different values of y. Values of x here and in subsequent plots are actual values rather than the number of points along x.

c 3' Cro CD

o

2

P PT

> o> V)

EX

a o w

1 o

3 3

f?

<* P

cr

5'

MD O

Figure 5.17. Plot at y=14.

91

Gamma Eq as Source of Building Blocks

1 cS o Cu oo

a

Math Aesthetics/Nonintegrability

92

I

I

93

Gamma Eq as Source of Building Blocks

1 s

I a IT)

1

s 0=

> a v>

Ba o" en

5 O

3 5'

1 o-

Figure 5.21. Plotaty=18.

95

Gamma Eq as Source of Building Blocks

OS

O Cu

op

E

Math Aesthetics/Nonintegrability

96

I

97

Gamma Eq as Source of Building Blocks

1

1

Math Aesthetics/Nonintegrability

98

I o 0.

I

99

Gamma Eq as Source of Building Blocks

CO

eS O

C4

i

s p

3*

> « o 3

a' 3g;

OQ

•5'

o o

Figure 5.27. Plot at y=3.

101

Gamma Eq as Source of Building Blocks

1 00

I

Math Aesthetics/Nonintegrability

102

I o

in

I

103

Gamma Eq as Source of Building Blocks

©

1 OH

© m U

a>

2

> ft Qa

r? o

I o

=3

5' OQ

3 a-

■5'

o5

Figure 5.31. Plot at y=36.

105

Gamma Eq as Source of Building Blocks

m

o CU c-j

rn

!

2

>

I

o 3

5"

o ON

Figure 5.33. Plot at y=38.

107

Gamma Eq as Source of Building Blocks

It.

<*

I

Math Aesthetics/Nonintegrability

108

in

I ea O CU

iri

m

I

109

Gamma Eq as Source of Building Blocks

! a

I

I

s > CO

5* a o'

I o

3

5'

3 a"

o

Figure 5.37. Plot ar y=7.

111

Gamma Eq as Source of Building Blocks

References 1. L. Landau and E. Lifshitz, Quantum Mechanics: Nonrelativistic Theory, 2nd Edition, Pergammon Press, (1958). 2. A. Goswami, Quantum Mechanics, William Brown Publishing Co., Dubuque Iowa, (1992), pages 39,40 3. M. Muraskin, Oscillations within Oscillations, Applied Math, and Computation 53 (1993)45. 4. M. Muraskin, Sine Curve within a Sine Curve, Physics Essays 5 (1992)331. 5. J. S. Bell, On the Einstein-Podolsky-Rosen Paradox, Physics 1 (1964) 195. 6. A. Einstein, B. Podolsky, N. Rosen, Can Quantum Mechanical Description of Physical Reality be Considered Complete, Physical Rev. 47 (1935) 777.

CHAPTER NO. 6 A STUDY OF SOME ADDITIONAL SOLUTIONS TO THE GAMMA EQUATIONS 1. A Lattice of Closed String Solitons We obtain different solutions to the gamma equations by altering the origin point data. We study, in this chapter, the results of different choices of origin point data with the aim of uncovering the wealth of information implied by the few mathematically aesthetic principles developed this far. First we study the three dimensional system:

rJ2 = i.o

rf2 = -i.o

rf, =-i.o

it, = i.o.

(6.i)

The other gammas are all zero at the origin. This system was studied in Chapter 2 section 6, and led in three space-time dimensions to a point soliton lattice. We now extend the data to 4 dimensions using Eq. (1.40) and Eq. (1.41). We specify an integration path by first integrating along x°, then z, then y, and finally x. Figure 6.1 shows an x,y map for this system. We see a more complicated lattice structure. Figure 6.2 and Figure 6.3 are sketches of the resulting 3 dimensional lattice at time zero. Maxima and minima lie on closed loops (closed strings), as we see in the figures. Everywhere on a loop the

Figure 6.1. Planar x,y map for r ] , for data Eqs.(6.1), (1.40), (1.41) showing a more complicated looking lattice than observed previously. Specification of path method is used. Numbers in the map are 10 times actual numbers.

113

Additional Solutions To Gamma Eq.

Figure 6.2. Sketch of system of Figure 6.1. Magnitude of maxima and minima are .64. The magnitude for the maxima (minima) lie on closed loops. In the figure the region above and below the plane at z=-1.4 is shown. Shaded regions within these loops lie below the plane. Dashed lines refer to portions of the loops hidden from the observer. Speckeled areas refer to areas hidden by another loop. The letter B refers to loops centered at the plane z=-1.4. Only ± 0.64 contour lines are shown. The map is for r j , .

magnitude of the maxima and minima is the same. In addition, this magnitude is unchanged as time evolves. Thus the string particles can be called solitons as well. The loops do not remain static as time evolves. There is a joining of loops at certain instants. After a joining the loops separate again. The separation of the loops does not occur at the same location (or even in the same vicinity) where the loops joined. After the separation we again have what appears to be an infinite system of closed loop particles arranged in a symmetric pattern. The system Eqs. (6.1), (1.40), (1.41) was studied using the sum over path procedure using the random path approximation. In Figure 6.4 we see an x,y plot (origin at the center of the map) for the component r},. We see as in previous soliton systems, that the magnitude 0.64 ± .02 of the maxima (minima) is preserved by the integration scheme. An interesting feature is similar to what was observed for the 3 component lattice (Fig. 3.1). The zero contour lines, for example, have a more circular type appearance than when we specify a path. Figure 6.4 shows a more complicated structure for contour lines than the specification of path result of Figure 6.1. In Figure 6.5 we observe a larger domain with the use of a coarse grid. The circulatory nature of the system shows up better when use is made of the larger region under observation. The density of maxima (minima) appears to diminish the farther we go from the origin. Evidence shows when we consider different z

Math Aesthetics/Nonintegrability

114

Figure 6.3. Loop particles centered at planes z=-7.4 and z=4.7 are added to the situation found in Figure 6.2. A,B,C represent loops centered at the 3 planes in the figure. For illustrative purposes, so that the loops do not appear excessively jumbled together, we have diminished the size of the axes of all the loops by onethird. Shaded areas and dashed lines have the same interpretation as in Figure 6.2. We also use dashed lines to indicate that the top of loop B (loop C) is below plane A (plane C ). Again only ±0.64 contour lines are drawn.

that the two regions labelled A in Figure 6.5 link up (or nearly link up) as we alter z (in the vicinity of z=-8 and at z=8) giving support that we still have closed loop structure for the maxima (minima) when we sum over paths. There is evidence this is also the case for other regions such as those marked B in the figure. Additional computer time would be useful here as conclusions are reached with coarse grids. In Figure 6.6 we plot a -0.40 contour line at different z's to show the link up (or possibly near link up) of the A regions. The loop particles studied appear to "loop" around the origin region rather the symmetric sites as in the case of when we specify a path. This same system was studied using the second approach to nonintegrable systems employing Eq. (2.26). Figure 6.7 represents an x,y map for r{, using a 0.009375 grid. The origin is at the center. We see evidence that the soliton magnitude of 0.64 is maintained by the integration scheme, although a deviation from this value shows up in

115

Additional Solutions To Gamma Eq.

Figure 6.4. The system of the previous 3 figures when we sum over paths making use of the random path approximation. Map is for r{, at z=0, x°=0. Grid is 0.375 and number of random paths is taken to be 500. Numbers in map are 100 times actual numbers.

the map. Finer grids indicate that this deviation is due to numerical errors. There are other magnitude maxima and minima appearing in the map as well. The situation is similar to what we saw in Figures 2.8 through 2.12. This suggests that if we integrate in z we will find evidence that these other magnitude maxima (minima) will increase to the soliton magnitude as z is altered. Figure 6.7 tells us that the superposition principle rearranges the solitons into a more disorderly arrangement. As far as three dimensional structure coarse computer runs in reference 1 suggest closed string structure, although we may have some open strings as well (no sign of closing was observed in the domain studied in some instances). These results are quite sketchy and serve to illustrate the need for greater computer capabilities.

Math Aesthetics/Nonintegrability

116

Figure 6.5. The system of Figure 6.4 is studied with a courser grid of 0.075 and a larger region. Contour lines shown are ± 0.40. Number of paths have values between 100 and 250.

2. Ladder Symmetry In the last section we studied a loop soliton lattice system which occurs when we specify a path. The question we pose is whether there exists some operation that maintains the form of the basic equations, preserves the soliton concept, but rearranges the solitons so they have a more disorderly looking arrangement as contrasted to the symmetric lattice of Figures 6.1-6.3. What we have ultimately in mind is as follows. Although we will not be able to make definitive statements regarding the foundation of quantum mechanics at this time, a step along the way would be whether we can induce a disorderliness in a natural way, in a classical system, by some mathematical operation that does not alter the form of the basic equations. It is well known that nonlinear systems can simulate chaotic systems. On the other hand, quantum mechanics is present in even the simplest free field situations when the equations used are very simple. Although this argument is not compelling in itself, the

117

Additional Solutions To Gamma Eq.

Figure 6.6. The regions labeled A in Figure 6.4 are studied for -8 < z < 8 (z is in units of 0.6). Contour lines are -0.40. We see that the A regions approach and/or join one another at z= ± 8, giving evidence for a closed string particle. A number in the figure adjacent to a contour line represents the value of z for which the contour line has the shape indicated.

possibility exists that the superposition principles associated with nonintegrable systems may yet play the above mentioned role. Figure 6.7 gives no evidence to contradict such a role for the superposition principle Eq. (2.26), so we will further explore this possibility. We have centered our efforts on the loop soliton lattice of the last section as this lattice has more structure than the two and three component lattices studied previously (where superposition principles did not induce a disorderliness of the type envisioned) and yet is still a simple enough system to assess the effect of the superposition principle in a straight forward manner. We thus continue to study the origin point data of the previous section, namely Eqs. (6.1),(1.40), (1.41). We saw from a coarse assessment, Figure 6.5, the sum over path method did not give one the feeling of "chaos" in that there was a definite circulatory character to the map obtained. However the second approach to nointegrable systems, which we already recognized in Chapters 2 and 4 has attractive features, leads to the disordered looking picture in Figure 6.7. With considerably greater computer time made available, and the technique of storing on tapes the values of the gammas along the last x=constant line, we reinvestigated the

Math Aesthetics/Nonintegrability

118

Figure 6.7. The data of Figure 6.1 using the second approach to nonintegrable systems. The grid is 0.009375 which is a course grid. Numbers are 100 times actual numbers. Evidence is consistent with the soliton magnitude being preserved by the integration scheme. The solitons have a more disorderly arrangement than when we specify an integration path as in Figures 6.1-6.3. The map is for r } , .

system. The stored gammas are used as input for the next computer run. We were then able to map a considerably greater region in the x,-y plane than observed previously. The results appear in Figure 6.8 for the quantity r[, using a much finer grid of 0.00234375. We see first of all that the soliton magnitude of 0.64 ± 0.04 is preserved by the integration scheme. With the additional computer time we have uncovered a rather rigid symmetry in each quadrant of the x,y plane. We call such a symmetry a ladder symmetry. When we pass to the right of the x=-y line, we find for fixed y, that maxima (minima) of the same

119

Additional Solutions To Gamma Eq.

Figure 6.8. The second approach to nonintegrable systems is studied in the x,-y quadrant over a considerably greater region than in Figure 6.7. The grid is now a finer 0.00234375, otherwise the system is the same as in Figure 6.7. This picture illustrates the ladder symmetry. Numbers in the map are 100 times actual numbers.

magnitude are repeated in x, for a fixed spacing Ax. The symmetry to the left of the x= -y line occurs in Figure 6.8, instead when we alter y for fixed x. Also, maxima and minima were found to lie on lines parallel to x=-y lines with an equidistant spacing along these lines. We did not observe the symmetry between quadrants when x —> -x, and positive numbers go into negative numbers as described in Chapter 2, section 6. However we do note that maxima having the value 0.64 occured only in the + - and — + quadrants, while minima with value -0.64 ocurred only in the ++ and — quadrants. The x= ± y lines (and lines parallel to these lines) appear in a basic way in the ladder symmetry when combined with lattice soliton systems. We note that the second approach to nonintegrable systems involves one step in x and one step in y before we sum contributions. Thus the x= ± y symmetry is likely to have it's roots in the nature of the integration scheme when combined with the lattice of Figure 6.1-6.3. Thus we conclude the evidence indicates that the integration scheme by itself is not sufficient to induce the disorderliness starting with the data of the loop soliton lattice. If we wish a more chaotic behavior the route would be to extend the origin data beyond the simple system Eq. (6.1). It should be kept in mind that altering the origin point data would be expected to alter the soliton character of the solution. The ladder symmetry is also present in x,y plots for different z. 3. Loop Lattice With a Doublet Basis Consider the following origin point data: 1 ^ 3 = ^31 ^20

= r

r

=

i0

02

^01

=

r " i 2 = -r 32 = -T 1 3 = -r 21 = 1.0

= r

=

2 1 = "ri2

^13

=

"r31

"ri0

=

=

"r30

"r01

=

=

"r03

=

l-° =

*0

r?o=rl 3 = r«2 = -r«3 = -rl 0 = -r302 = i.o,

(6.2)

120

Math Aesthetics/Nonintegrability

with the other Tjk zero. We specify a path in the manner described previously and map r 0 0 . The result is given in Figure 6.9. This set of data leads to a loop lattice but with two types of maxima (minima) having the magnitudes in the vicinity of 1.36 and .50. In our previous sets of data leading to a lattice there was but one magnitude for the maxima (minima) when we specify a path. In three dimensions we observe the loop structure previously seen in section 1. In Figure 6.10 we apply the second approach to nonintegrable systems. We again see the ladder symnmetry. 4. Concept of Imperfect Lattice We next discuss a set of data for which the notion of lattice appears when we specify a path, but in a coarse sense,

r4 = 8 r20=.4 r]2=i.o ri2=-o.5 T2u=-.6T20=-3

I* =-1.0 I * = .5

r ? 2 = - 6 r° 2 =-.3 i t , =.8

r°,=4,

(6.3)

with the other gammas zero. We use Eqs. (1.40) and (1.41) to then obtain our origin point data. When we specify a path we obtain Figure 6.11. This figure looks like a lattice. But on closer scrutiny we see, as an example, planar maxima of magnitudes 0.74 and 0.80 within the region enclosed by a single 0.35 contour line. For a perfect lattice these numbers would be the same. Figure 6.11 was obtained using a 0.0375 grid which is a finer grid than we generally use in obtaining such lattices. We also see from computer runs that the .10 contour lines (not shown in map) are not as symmetric as the 0.35 contour lines. Thus, for these reasons, we refer to the system as an "imperfect" lattice. We emphasize that from a numerical point of view we cannot say with certainty whether a lattice looking system is perfect or imperfect. We will then rely on visual observations of the maps in so characterizing a solution. Maps for different z show that the two maxima (minima) in close proximity eventually merge (both for increasing z and decreasing z) so that maxima (minima) lie on a loop. For an imperfect lattice the magnitudes of the maxima (minima) vary somewhat (in this case the range is of the order of ten percent). The magnitudes of the loop was of the order of 0.80. A loop was studied as the time x° was varied. We found at all instants studied a loop with magnitude varying in the 0.80 range. This result suggests that, in an imperfect sense, we have a soliton loop lattice. The results were then studied using the integration scheme Eg. (2.26). We used a grid of 0.009375. From Figure 6.12 we do not see the rigid ladder symmetry. Thus the imperfect lattice is a way to avoid the rigid symmetry seen in section 2. However, even though there is no longer a strict symmetry we can recognize that "type" structures do repeat (in a nonexact way). For example, the 0.70 magnitude surrounded by a 0.10

121

Additional Solutions To Gamma Eq.

Figure 6.9. Data of section 2 when we specify an integration path. Grid is 0.075. Map is for T^. We see two types of maxima (minima) when we specify a path. Magnitudes in figure are 100 times actual values. The system is again a loop lattice.

2 & sr

> re

3re o'

o 3

5' re

1 If

Figure 6.10. Data of Figure 6.9 when we use the integration scheme Eq.(2.26). Grid is 0.00234375. The map is for rJo in the +- quadrant. Again we see the ladder symmetry. Numbers are 100 times actual numbers.

123

Additional Solutions To Gamma Eq.

Figure 6.11. System based on Eq. (6.3). Path is specified. Grid is 0.0375. Spacing between points is 0.3. Results indicate what we call an "imperfect" lattice. The two maxima (minima) within a contour a contour line have slightly different magnitudes in general. r{, is mapped. Numbers are 100 times actual numbers.

contour line on the top left is mirrored by a 0.72 maximum surrounded by a 0.10 contour line close to the end on the top right. Even though the contour lines have somewhat different shape, there is a corresponding structure that appears as we increase x. Thus an imperfect lattice is reflected by an imperfect symmetry when we use the integration scheme Eq. (2.26).

Figure 6.12. System of Figure 6.11 when use is made of the integration scheme Eq. (2.26). No exact symmetry is observed. There is a repeat of similar type structures (for example, the 0.72 maximum surrounded by the contour line .10 on the right is in a coarse sense a repeat of the .79 maximum surrounded by the .10 contour line appearing on the far left. Grid used is 0.009375. Numbers are 100 times actual numbers. Map is for T j , . The repeat of structures continues beyond the range of the map.

s > w 3n>

o"

o a

3' a

l

§

Figure 6.13. System of section 5 when we use the integration scheme Eq. (2.26) associated with nonintegrable systems. Map is of r } , in the x,-y quadrant. Numbers in this figure as well as other figures associated with this system of data are 100 times actual numbers. The grid used is 0.00234375. The map shows an irregular appearance for the 0.10 contour lines.

125

Additional Solutions To Gamma Eq.

5. "Chaotic" Looking System By using a more involved set of origin data we can obtain a more disorderly system. We shall term the solution "chaotic", although qualifications are needed. A truly chaotic looking system of maxima and minima would have the magnitude of the maxima (minima) not only different, but magnitudes should vary over all values. Thus, different maxima (minima) would differ by many orders of magnitude in a truly random looking system. However, if magnitudes differ by many orders of magnitude, then numerical errors would become an important factor. Also from a numerical point of view, we could not distinguish between a large magnitude and a singularity. Thus, when we talk of chaos we shall only consider random looking locations for the maxima (minima). Our maxima and minima shall be required to vary over a small range in magnitude. In order to say we have we have a random looking solution, we will need to study a relatively large region of space. We will need to use a very small grid as we have no other hold on the numerical error problem. For lattice solutions, or for that matter symmetric solutions, we can tell if errors are a factor by the deviation from the symmetric pattern. Numerical errors would not be expected to maintain the symmetry. The only weapon at our disposal, then, in obtaining reliable results is to use tiny grids and then repeat the calculation with smaller grids. If the results do not agree well, this would be a sign that we should not have confidence in the results. Using tiny grids to study a relatively large region means we need a large amount of computer time which we were able to obtain for this project. We mapped a region of 22,882 point by 2,689 points in the x,y plane using the grid 0.00234375 together with the superposition principle Eq. (2.26). Then the calculation was entirely redone using the grid 0.001171875. For the maximum farthest from the origin we found an agreement in the two calculations up to 0.003, so to the accuracy of the maps we can say that the results are the same. The system under consideration is based on the following data for the gammas:

r1 = r 2 1

_ p i _ p2 _ — x 01 — i 02

x

pO _ pO . 2 2' = 0 1 11 " 1 22

10

r1 3 - r1 3 = -rf3 = -o.2 03

30

1

r = r2 1

02

1

10

r1 = r 3 1

30

X

01

r 2 - ri 3 i

03

20

_ pO _ p i _ _ 1 21 ~~ _ 1 20 ~ _ p0 _ — J- 13 _ p0 _ _ l 32 _

pi _ -1 03

p 2 _ pO r° 2 01 ~~ 'L 12

_1

p3 _ pO L x 10 — ~r?x 31 =

p2 _ p3 _ pO 30 _ "X 02 ~ _ i r° 233

_1

= i.o 1.0

= 1.0.

(6.4)

The other T ^ are zero. We then made use of Eqs. (1.40) and (1.41) to arrive at our origin point data. This set of data is not general as r[ k = 0 and 1^, = 0. The above data were then mapped in the x,-y quadrant using the gamma equations, Eq.(1.13), and the integration scheme, Eq.(2.26), for the representative component T\,. The results are given in Figures 6.13-6.15. Figure 6.14 fits to the right of Figure 6.13 and Figure 6.15 fits to the right of

Math Aesthetics/Nonintegrability

126

Figure 6.14. The contour lines of this map do not seem to form any simple pattern. We do not observe the ladder symmetry here, although there are some vestiges of it which appear in diagonal directions. In the context of the above, we may refer to the system as a "chaotic" system, although this is somewhat imprecise (see also below). Figures 6.136.15 are not very deep in y. In Figure 6.16 we present a picture of the +- quadrant close to the origin with a greater range of y, looking for any obvious pattern. We used a 0.0046875 grid here. We saw no simple pattern for the planar maxima (minima) although we did see some similar looking structures. In this sense we can say that the random appearance is more pronouncd in x than y. The set of data above does not give rise to an (obvious) lattice solution when we specify a path. We saw lattice solutions in so much of our previous work summarized in this chapter. In Figure 6.17, we show the map obtained when we specify a path, as usual by first integrating along y and then x to get to any desired point. The map is for the + quadrant. We see no symmetry pattern as we increase x. In Figure 6.18, we show a region in the ++ and + — quadrants which is deeper in y. We do see a similarity in structures here. Also we see evidence for a repeating of maxima (minima) in an oscillatory way. Here we focus attention on minima inside similar structures as we alter y (note Table 2 of reference 2). When we specify a path, we do not get a lattice; however, we do see evidence for a symmetric pattern as we alter y. The way this effect is reflected when we use the integration scheme Eq. (2.26) appears to be by way of appearance of similar looking structures as was discussed previously The results appear to be consistent with the notion that the less the results look like a lattice, or have a regular pattern, when we specify a path, the more erratic looking the contour maps when we use the superposition principle Eq. (2.26). 6. Instanton Solution In Chapter 2, section 6, we studied an intrinsically 3 space-time dimensional system that gave rise to a soliton lattice when we specify an integration path. By considering a slightly more involved set of origin data we can obtain a three space-time dimensional instanton system, again when we specify a path. Consider the following set of Vg : T\2 = -T\2 = \.Q

rli =-r|, =o.8 . We make use of Eq. (1.40) together with the 0.88 0.50 0.44

3 dimensional af -0.42 0.22 0.90 0.30 -0.60 1.00 .

(6.5)

(6.6)

127

Additional Solutions To Gamma Eq.

I

I

I

Ii

I 3

I

I

I

2

1

3

£

I i

I

s

I

I

% o'

o

I

1

to

Figure 6.16. The system of Figure 6.13 which is now deeper in y.

Figure 6.17 The system of Figure 6.13 when we specify an integration path. The system of section 5 does not lead to a lattice when we specify a path, although there are some regularities present. Map is for Tj, in the x,-y quadrant.

129

Additional Solutions To Gamma Eq.

The set Eq. (6.5) has the structure Eq. (5.24), but after using Eq. (1.40) and Eq. (6.6), this is no longer the case. In Figure 6.19 we see a map of T ^ when we specify a path. The grid used is 0.0375. We do not see a picture that is characteristic of loops as is Figure 6.1 or Figure 6.9. At first sight, Figure 6.19 looks like a reasonably complicated multi-maxima (minima) system. However, we have uncovered systematic patterns by studying the locations of two-dimensional maxima (minima). Consider a maximum at x=9, y=-8, in units of 0.15. The maximum has the magnitude 0.129. When we vary y in the range Ay=36.5 to 39, we come across a succession, or family, of maxima whose locations in x are graphed in Figure 6.20 (there is an uncertainty of ± 1 unit in x in the plot). We see that the x locations of the maxima in the family suggest in a striking way a (discrete) sine curve. The relative spacing in y between maxima in this family has an uncertainty of two units (as there are two maxima involved in obtaining a relative spacing). Thus, we can say that the relative spacings in y do not change very much for the entries in Figure 6.20. As for the magnitudes of the maxima within the family, we find they oscillate in a way that is more complicated than a sine curve. The largest magnitude observed for a maxima is 0.130 ± 0.002 and occurs for 4 out of 26 entries in making the plot for Figure 6.20. The smallest magnitude in this plot had the value 0.077. As we move to the right we uncover other families of maxima (minima). For example, consider the family of successive minima obtained by starting with a minimum at x=132, y=-9. The x locations of succesive minima in this family are again given by a discrete sine looking curve. However as we move to the right the value Ay between successive members of a family becomes greater. As we go farther to the right the situation becomes more jumbled. When the range Ay increases it is more difficult to say that a maximum (minimum) belongs to one family or another. We follow the motion of a particular minimum in time, when we specify a path. We focus our attention on a minimum having the value Of-0.126 at x=20, y= -11, z=0 (units are 0.15). We followed this minimum from z= -20 to z=52. We first recognize that we are not dealing with solitons, as the magnitude of the minimum does change with time. The minimum under study waxes and wanes in time. Eventually, it disappears at about z=52 (the magnitude before it disappeared was 0.066). At this time it is "swamped" by another minimum close by. The maximum magnitude that we ever see as we follow the motion of the minimum is 0.130 ± .002 which is attained for short intervals at different times. We note that in all of our studies of this system, we never observed a magnitude greater than 0.130 ± 0.002. The number 0.030 ± 0.002 appears as well in our z=0 maps. It also appears when we follow the minimum above in time. In addition it appears when we follow other maxima (rninima) in time. We shall call the number 0.130 the instanton value. By thinking of the particle system as a collection of instantons, we have an ensemble of particles (the suggestion is that we have an infinite number of them) having identical properties.

2 B-

> a v>

$■

o' o 3

3

a

1

O

Figure 6.18. Map of the system of Figure 6.17 occurring when we specify a path. This figure is deeper in y than Figure 6.17 and includes portions from the ++ and +- quadrants. We see more similarity in structures as we alter y as contrasted with x (note Figure 6.17).

131

Additional Solutions To Gamma Eq.

If g i

i!

H

u

i II I! il i is ill pi

!|l ill

fi'l if!

ill

Math Aesthetics/Nonintegrability

132

Figure 6.20. x location for maxima within a family. The family is obtained by varying y. The system is the same as in Figure 6.19. The results are consistent with a (discrete) sine curve.

The instanton picture contrasts with a picture where particles are represented by maxima and minima. In this latter situation, we have an unfortunate situation where some particles have less magnitude than is found in regions nearby where there are no maxima or minima. For this region we can question the use of maxima (minima) to represent particles. We thus can see an advantage in representing particles by either solitons or instantons. Another very attractive approach to representing particles is by wave packets which was discussed in Chapter 5. We see that solitons and instantons occur from simple patterns for origin point data. On the other hand wave packets occurred using more general looking origin point data. We observed 13 instantons in the x,-y quadrant going 55 points in x and 45 points in y and from z=0 to z=20 (again units are 0.15). The study was not extensive enough to determine whether the locations of instantons fall within a regular pattern. Thus, it is not clear if we have a lattice of instantons or not. However, we did note that planar maxima (minima) at z=0, having the instanton value did have a regular appearance within families. The minimum magnitude for maxima (minima) is 0.065 ± 0.001. We saw a profusion of these as well. After attaining this value, the maxima (minima) can disappear in time or the magnitude can grow in time reaching the instanton value. In addition we found on occasions some maxima (minima) not falling into the pattern above. For example, we found a three dimensional maximum of magnitude 0.118 at z=.9,10,ll, x=28,29, y—33, -34, -35. and a three dimensional minimum value Of-0.123 at z=9,10, x=40,y=-32, -33. We did not see many of these, although we cannot expect such an effect to be isolated. These examples did not stay around very long as they disappeared when we integrated both forward and backward in time.

133

Additional Solutions To Gamma Eq.

The evidence then suggests an infinite instanton system having magnitude 0.130 ± 0.002, although other 3 dimensional magnitudes for maxima and minima (other than 0.065 ± 0.001) do appear on occasion. We next studied the system using the integration scheme Eq. (2.26). The results are given in Figure 6.21 and Figure 6.22. The grid is 0.00234375 and the maps are in the x,-y quadrant. The component mapped is T] 3 . Figure 6.22 fits to the right of Figure 6.21. Again we find that the maximum magnitude found anywhere in the maps is 0.130 ± 0.002. This result includes coarse runs in time. This suggests that the instanton value is preserved by the superposition principle. We saw no evidence for a quantitative repetition of numbers to the right of the x=-y line as we increase x, as was the case in the ladder symmetry. What we do see are similar type shapes for the top set of -0.01 contour lines and for the bottom set of-0.01 contour lines although the -0.01 contour line on the left of Figure 6.22 is not of the same pattern. There is a tendency of maxima (minima) to fall on lines parallel to x=-y lines, although there are exceptions. We have not seen the regularities as in Figure 6.20 which resulted from a specification of path. We would like to investigate, in particular, the x= -y parallel directions to see if there are regularities for locations of maxima and minima. Thus, considerably greater computer time would be useful towards a better understanding of this system. The instanton system represents another type of particle system admitted by the set of mathematically aesthetic principles. 7. A Dynamical Lattice We can think of an infinite system of mass points connected by strings as a lattice of mass points. When we have small oscillations, causing the mass points to deviate from the lattice positions in a sinusoidal way, we may call this a dynamical lattice for the mass points. We can have a similar situation in classical field theory. When all the maxima (minima) are located in an equidistant manner we call this a lattice solution. We have studied lattice systems of different types since Chapter 1, within the nonintegrable aesthetic field theory. These lattices arise when we specify an integration path. We now study small deviations from the lattice structure such that the deviations have a sinusoidal character. We call such a system a dynamical lattice. We shall study an intrinsically three space-time dimensional system. In this way we don't have to deal with the added complication of string systems. In section 6, we did find a solution where deviations from the lattice for the x coordinate, for maxima (minima) within a family, did have a sinusoidal character (see Figure 6.20). We shall find here in addition to the sinusoidal variation of the x location of members of a family, that motion in "time" is that of simple harmomc motion (in a numerical sense). In addition the magnitude of the maxima (minima) within a family at time zero will vary in a sinusoidal way as we alter y. Also, for a representative maximum (minimum) the magnitude will vary sinusoidally as a function of time (again, these effects, outside the first kind of Figure 6.20, were not found previously). Thus, with all these sine curves, we can say that a dynamic lattice exists within the aesthetic fields

2 3-

> CD CO

Bo'

o g 5'

I ■?

Figure 6.21 The system of Figure 6.19 when we use the superposition principle (2.26). The grid is 0.00234375. Numbers are 1000 actual numbers. Map is in x,-y quadrant for the quantity r\3. We do not see the ladder symmetry, but there are some regularities. The results are consistent with the notion that the instanton value of 0.130 ± 0.002 is preserved by the integration scheme but the instantons are rearranged in a way that would require considerably greater computer time to unravel.

Figure 6.22. This figure fits to the right of Figure 6.21. Similarities in contour structures can be observed.

135

Additional Solutions To Gamma Eq.

program. This is not to say that we have found the most general dynamical lattice we can think of as there are curves associated with the location of maxima (minima) that we cannot confirm to be of sinusoidal character. For example, the y location within the family closest to the y axis appear to be uniformly spaced. It is a tedious job sifting through a large number of solutions looking at deviations from the lattice structure. Thus we will be content with the solution found. We choose as origin point data the following 3 dimensional system: r 21 =0.03

T\2 =0.03

T23 =-0.95

r 31 =-o.o3

r 3 2 =i.o

r ] 3 =-o.o3

T2U =-0.03

r2n= -o.o3

r2u= 0.95

rf, =-0.98

T\2 =0.03

T2n =0.03

rf, =0.03

T]2 =-1.0

r]3 =0.03

I ^ =0.98

Y\2 =-0.03

rV-0.03,

(6.7)

with the other gammas zero. This set of origin point data has the structure given by Eq. (5.24), which we recall led to the nonlinear sine within sine system. However, the magnitudes used here are very different from the magnitudes used in Chapter 5 (namely Eqs. (5.20) and (5.32)) and magnitudes are of importance in the kind of maps observed. Compare Figure 5.5 with the map found in Figure 6.23. This latter map arises when we specify a path and make use of the data Eq. (6.7). All maps in this section are for the representative component T\3. The grid size is 0.075. We see a symmetric looking pattern of contour lines. The maxima and minima vary between 0.051 and 0.055 magnitude within the region mapped in Figure 6.23 (note as mentioned before, all magnitudes appearing throughout the text are truncated rather than rounded off). On first sight it looks like a two dimensional lattice system. However, closer scrutiny reveals deviations from a strict lattice, and it is these deviations that we shall concentrate on. Consider the minimum -0.052 at the far left in the second row of loops from the top in Figure 6.23. Its location is x=8, y=25, in units of .15. When we proceed 41 ± 1 units above and below this minimum, we find, as in section 6, a whole family of minima. The x location of members of this family are plotted in Figure 6.24. An uncertainty of one unit, either to the right or left is allowed in the x location. The grid used here is 0.0375 and units are again taken to be 0.15. The results of Figure 6.24 show that the x location within a family is consistent with sinusoidal behavior. Such an effect for x location was first observed in section 6, although the system studied there was not a dynamical lattice, as the motion of maxima (minima) was not simple harmonic motion.

s

I

> V

3o'

o 2. 3 re

0\

Figure 6.23. Map using the data of section 7 when a path is specified. Map for this system as well as Figures 6.24-6.29 are for T\3 and are for the same system described by equation (6.7). The grid used is 0.075. Numbers are 1000 times actual numbers. The map includes portions of the ++ and + - quadrants. The deviation of locations of maxima (minima) has a sinusoidal character in many respects and is called a dynamical lattice.

137

Additional Solutions To Gamma Eq.

We pointed out that the magnitude of maxima (minima) in Figure 6.23 are not all the same, although they vary within a small range. In Figure 6.25 we plot the value of the magnitude of the minima for successive members of a family using the same parameters as in Figure 6.24. The results resemble a sine curve although there is some greater weighting at the bottom of the curve as compared to the top. A plot similar to this was not found in section 6. We next studied the motion of a maximum of magnitude 0.046 at x=19, y=-14.5 (units are 0.15) in time (z coordinate). The grid used was 0.075. In Figure 6.26 we show the x location of the maximum as a function of time, and in Figure 6.27 we show the y location of the maximum as a function of time. Numerically the motion appears to be consistent with simple harmonic motion in both instances leading to elliptic motion. We did not obtain such a motion in section 6. In Figure 6.28 we plot the magnitude of the maximum at x=19, y=-14.5 as a function of z. The magnitude varies from 0.041 ± 0.001 to 0.054 ± 0.001, in again what appears consistent with a sine curve. We do not have a soliton system as in Chapter 2, section 6, as the magnitude of the maxima (minima) does vary in time. Instead we have an instanton type system as the maximum of all the maxima in Figure 6.28 takes on the same value 0.054 ± 0.001. We note also fom Figure 6.25 that the minimum of all the minima within the family is also 0.054 ± 0.001. The results are consistent with a multi instanton system having instanton value 0.054 ± 0.001. However, since the magnitude of the maxima in Figure 6.28 do not vary much as time evolves, we can think of the system as a quasisoliton system with average magnitude 0.048 db 0.007. We can follow the motion of quasi-solitons for as long as we wish due to the sinusoidal character of x(z) and -y(z) as evidenced by Figures 6.26 and 6.27. The results are consistent with the y location between successive members of the family studied in Figure 6.24 being constant rather than sinusoidal. In addition when we start with the minimum of magnitude 0.054 at x=8, y=-17 and add 11 units in x, we end up with another minimum of magnitude 0.054. By adding 10 or 11 units in x, for the same y plus or minus one unit, we end up with a succession of minima all having magnitude 0.054 ± 0.001. Thus, we can again say that we do not have what would appear to be a general dynamical lattice as we do not see sinusoidal character here. Also Ay for members in a family increases as we move towards theright,rather than being sinusoidal. We next studied the effect of the superposition principle Eq. (2.26) on this system of data. The results are shown in Figure 6.29. The map is of a portion of the x=-y quadrant with grid size 0.00234375. Location of maxima (minima) appear to lie on lines parallel to x=-y lines with equal spacing, although greater computer time would be useful here. This feature is present in what we call ladder symmetry. Coarse computer runs also indicate that quasi-solitons can no longer be followed in time for as long as we wish (as is the case when we specify a path) as quasi-solitons appear and disappear. The dynamical lattice again points up the rather remarkable relationship between the mathematically aesthetic principles and the sine curve.

s a-

>

io

2. a CP

«

I

00

Figure 6.24. x location of minima within a family for successive members of a family. Grid used is 0.0375. Units for x are 0.15.

139

Additional Solutions To Gamma Eq.

Figure 6.25 Magnitude of minima within a family for successive members of a family. Grid is 0.0375.

Figure 6.26. x(z) for a maximum. Grid is 0.075. Units for x and z are 0.15. The result is consistent with simple harmonic motion.

Figure 6.27. -y(z) for the maximum in Figure 6.26. Grid is 0.075. Units for y and z are 0.15. Results are again consistent with simple harmonic motion.

Math Aesthetics/Nonintegrability

140

Figure 6.28. Magnitude of the maximum in Figures 6.26 and 6.27 as a function of z.

Figure 6.29 The system of section 7 using the integration scheme Eq. (2.26). Grid is 0.00234375. Map is a portion of the + - quadrant. Numbers are 1000 times actual numbers. Maxima (minima) lie on lines parallel to x=-y lines (ladder symmetry).

141

Additional Solutions To Gamma Eq.

8. Irregular Oscillations Along All Path Segments In Chapter 5, section 2 we showed for a particular choice of origin point data the aesthetic field equations collapse into a simpler set, that describe sinusoidal behavior along any path segment. Such a choice of origin point data amounts to a linearization of the gamma equations. For a different choice of origin data we showed, in Chapter 5, section 5, that the aesthetic equations collapse again into a different set, that describes sine within sine behavior along any path segment. This system is nonlinear. Here we show that for a more general set of origin point data, the equations describe irregular oscillations along the coordinate axes for both the small picture and the big picture (Chapter 5, sections 5 and 6). From this we infer that the aesthetic field Eq. (1.13) describes irregular oscillations along any path segment (at least close to the origin). The solution under study is a wave packet solution. The origin point data is free of linear tensor restrictions as is discussed in Chapter 5, section 9. Consider the following set of gammas:

r? 3 = - r 2 3 = o.o7 r} 2 = -r? 2 = o.i r i , = - r i , =0.05 -pl -p2 _ -pO _ -pi _ 1 02 ~ l 10 ~ l 21 ~ ~l 20 ~

"p2 __ -pO 01 ~ _ i 12

= 1.0

p3 _ 10

pl 03

= 1.0

p2 _ p 3 _ pO _ pO _ p3 __ p 2 1 03 - L 20 - L 32 ~ _ 1 23 - ± 02 _ 1 30

= 1.0

p l _ lp3 _ lpO _ 1 30 ~ 01 ~ 13 ~

pl _ p2 _ 1 10 ~ L 20 ~

-i

-i

pO _ 31

x

x

p3 _ p l _ p2 _ -p3 _ = 30 ~ l 01 ~ L 02 _ 1 03 "

-i

0.1

r°3i = -T°u=-T022 = o.i pO _ p 0 _ pO _ pO _ 10 — l 20 ~" l 01 "* l 02

1

P3

pl _ 00

_1

= _r° =r° =c\ nn<;

p2 .= 00 "

-1

0.01 (6.8)

with the other gammas zero. We integrate this data to x=.6,y=.6,z=.6, x°=.6 in the same manner as in Chapter 5, section 2. The resulting gammas are then used as origin point data. We are interested again in what sort of content can be found in Eq. (1.13) using a set of origin point data. Thus, which component we choose to study is not of importance here. We shall choose to study a representative component which we take to be T\3. In Figure 6.30 we observe a small picture plot along the y axis. Figure 6.31 fits to the right of Figure 6.30. Figure 6.32 fits to the right of Figure 6.31 and Figure 6.33 fits to the right of Figure 6.32. The grid used in each plot is 0.003515625 and the spacing between points is 0.1125. Each figure contains a thousand such points.

Math Aesthetics/Nonintegrability

142

Figure 6.30. Small picture plot for T^ along the y axis for the system studied in section 8. Grid is 0.003515625. Spacing between points is 0.1125. The figure contains 1000 such points.

Figure 6.31. This figure should be adjoined to the right of Fig. 6.30. The parameters are the same as in Figure 6.30.

Figure 6.32. This figure shold be adjoined to the right of Fig. 6.31.

Figure 6.33. This figure should be adjoined to the right of Fig. 6.32. Figures 6.30-6.33 illustrate the irregular oscillations along the y axis. Small picture plots show irregular oscillations along all the coordinate axes.

143

Additional Solutions To Gamma Eq.

Wefindas well, irregular oscillations in small picture plots along the other coordinate axes. The distance between successive maxima and minima were tabulated along the x axis as we move farther from the origin. For the first 208 entries these values varied from 2 to 26 units (unit is 0.1125). The numbers in the range 2 to 26 are not random, as the number 26 does not appear as often as the number 8, for example. Still, one can say that the pattern of these values has something of an irregular character for the region studied, although we noted large numbers tend to occur more often farther from the origin. We next observe big picture plots along the coordinate axes. The big picture plots are obtained by compressing 4000 points onto a single computer page. The spacing between points is 0.1125, and the grid size is 0.003515625. There are in total 16000 points in each plot. Figure 6.34 is along x, Figure 6.35 is along y, Figure 6.36 is along z, and Figure 6.37 is along x° These plots show the irregular oscillations along the axes, and by inference irregular oscillations along integration path segments (at least close to the origin). These results appear common in many other wave packet type solutions, although on the contrary we noted big picture symmetries in Figure 5.7. We may see the wave packet character at say y=-38 (a unit of y is 0.005) when we specify a path. This is shown in Figure 6.38. The grid in y is 0.00005. In x, the grid used was 0.01875, and the spacing between points is 0.075. At y=-214, for example, the magnitude becomes so large that we were unable to obtain a bounded solution in our computer studies. Although we have a problem with boundedness at certain y=constant plots, the problem is not as widespread as encounterd with the system Eq. (5.33). On the other hand we did not have a problem with boundedness in the domain studied for the wave packet system associated with Eq. (5.34). We did not have sufficient computer capabilities to do justice to the problem of the effect of the superposition principles Eqs.(2.21) and (2.26) on the current system. The pictures in this chapter illustrate the wealth of information contained within a few mathematically aesthetic principles. As the work here only scratches the surface, one can only wonder what else can be learned from a study of these few simple ideas.

Math Aesthetics/Nonintegrability

144

Figure 6.34. Big picture plot for the system of section 8, along the x axis. The spacing between points is 0.1125, and the grid is 0.003515625. The big picture plot has 4000 points compressed onto a single computer page. Four such computer pages are joined together, so the figure involves 16000 points. The plot isforT^.

Figure 6.35. Big picture plot along the y axis. Parameters are similar to Fig.6.34.

Figure 6.36. Big picture plot along the z axis. Parameters are similar to Fig. 6.34.

145

Additional Solutions To Gamma Eq.

Figure 6.37. Big picture plot along the x° axis. Parameters are similar to Fig. 6.34. Figures 6.34-6.37 show that big picture plots along the coordinate axes lead to irregular oscillations in the domain studied. From the big and small picture plots we infer that the system of section 8 leads to irregular oscillations along any path segment (at least when we are close to the origin).

Figure 6.38. Big picture plot for the system of section 8 when we specify a path at y=-38. Units Of y are 0.005. Grid in y is 0.00005. Grid in x is 0.01875. Spacing in x between points is 0.075. 4000 such points along x appear in the figure. This plot shows the wave packet character for the solution.

Math Aesthetics/Nonintegrability

146

References 1. M. Muraskin, Three dimensional particle structure using the new approach to nonintegrable aesthetic field theory, preprint 2. M. Muraskin, Chaos and Aesthetics, Computers Math. Applic. 26 (1993) 93.

CHAPTER NO. 7 MATHEMATICAL AESTHETICS: ADDITIONAL TOPICS 1. Component to Represent the Particle System If we rotate the Cartesian coordinate system, as an example, this should not alter the structure of a particle. We have therefore suggested that the component r[J0 represent the particle system as it is unchanged by three dimensional rotations. Another possibility is g, as this quantity is unchanged by four dimensional rotations, g is the determinant of a second rank tensor gjj. The problem is that it is possible to define gy in an infinite number of ways. In this section we show that the way that gy is introduced is irrelevant. We can introduce an independent gjj by specifying origin point data for such a quantity without reference to r] k . We could also construct from TL an infinite number of second rank tensors. For example, we can write down ( D = Lr t1 ps Bij a sj (2) = r t &1J

ps

(3) _ pt 6ij

1

' i s 1

tj

ps

tl

L

JS

„(4) _ x-pk xp t xp m x p s _(l)np 6ij im ks jn pt 6

etc..

(7.1)

g® n p g£ = *sn

(7-2)

Here g(l)np is defined by

The superscript in g[]) in equation (7.1), tells us which member of the infinite set of gy, constructed from r^, we are talking about. In any of the cases mentioned above, the change of gy is determined from our uniform prescription (chapter 1, sections 6 and 7) to be dgy S k = r U + rj kgit . dx

(7.3)

As a result, we obtain (appendix D) :

dxk dxk etc. .

2gr[k

VsT\k (7-4)

Math Aesthetics/Nonintegrability

148

Equation (7.4) holds whether gy is symmetric, as is the case of gf. , or nonsymmetric. It holds whether gy is specified independently at the origin or not. Furthermore, no matter how gij is constructed from r ] k (as in Eq. (7.1)) the map of g is the same, provided its origin point data is the same. In addition, if we multiply the origin value of g by some amount A, then all the entries in the map of g are multiplied by A. Thus, up to a scale, there is a single map for g no matter what structure gy has (or has not) in terms of other fields, . / g has the same property. Thus far, we have argued that the map of g is independent of how gy- is formed. We next ask whether we should represent the particle sysem by ^/g, g, g2, etc. We note there is but one independent map, since all the maps involving functions of g can be obtained from, say, the y/g map. If we get a lattice for yfg, we will also get a lattice when we square all the entries in the map. We have a similar situation previously. We may choose to map T ^ rather than T°m T^,, on grounds of simplicity. Simplicity is then a motivating factor in these considerations. Thus far, we have argued that TQQ and g are attractive candidates for representing the particle system. We shall not draw any definitive conclusions here. In reference 1, we have studied maps for , / g , although we could consider maps for g instead, as the quantity g is unchanged upon inversions, which can be considered desirable. If the origin point data is positive for g or y/g, it remains so at all points from Eq. (7.4). The positive definite property is then similar to the energy density, for example, in electromagnetic theory. In our earlier considerations as in Chapters 1-4, the origin point data obeyed T ^ O and so g did not change from point to point. It would be necessary to obtain multi-particle solutions in order to have an interesting model universe. We shall next show that with the following origin point data we can obtain a two dimensional lattice type solution for the quantity , / g . Consider the following origin point data: J-02

A

21

x

12

1

01

LM

r?3 = rL=-r 2 3 = -r?2 = i.o

(7 .5)

with the other T^y zero. We use Eq. (1.40), with

/ . . 0 l\ 0, 0 1\ 0 0 1 1)

V

1 0/

(7.6)

149

Additional Topics

This gives a r] k , which obeys 1^=0. We next alter r°k so that the values 0.5 are replaced by 0.6, and the values -0.5 are replaced by -0.6. The resulting Tjk are used as origin point data. We see that T^ is not zero here. In Figure 7.1 we map ^/g in ++ quadrant in x,y plane for ^ = 1 at the origin. This represents a lattice, although not as simple as we have seen before. Although we see multi-maxima and minima, here we do not see maxima (minima) in all quadrants when we use the integration scheme Eq.(2.26) (the results resemble Figures 4.1 and 4.2) so we have not pursued this system further.

Figure 7.1. Map of , / g at z=0, x°=0, in the ++ quadrant for the system described in section 1. The grid is 0.15 and the separation between points is 0.60. Specification of path approach is used, ^ g at the origin is 1.0. Map shows a complicated lattice structure. Numbers are 100 times actual numbers.

Math Aesthetics/Nonintegrability

150

2. Lorentz Invariance of the Gamma Equations A Minkowski space is characterized by the length of a vector obeying (the summation runs from 1 to 4 on both sides of the equation) £ ^

A;

= £ A; Ai ,

(7.7)

which when written out is given by (A',)2+ (A'2)2+ (A'3)2-(A'Qf = (A,) 2 + (A 2 ) 2 + (A3)2-(Ao)2.

(7.8)

The "length" of the vector here is contrasted with the Euclidean length, in which all the terms on each side of Eq. (7.8) have positive signs. Transformations for which Eq. (7.8) is satisfied are called Lorentz Transformations. We can obtain a Minkowski space from the requirement that all vector components with the index zero are pure imaginary, and tensor components with an odd number of zero indices are accordingly pure imaginary. That is, for example, T^, is (i) 3 times a real field component. The basic equations are still Eq. (1.11), as these equations were obtained on general grounds, regardless of the Minkowski or Euclidean property of the space. As the equations are tensor equations they have the same form when a Lorentz Transformation is made. When the space is Minkowskian, and we expand the gamma equations we do not get the same system as before. For example, consider Eq. (1.11) when i=l, j=l, k=l, 1=1. This now gives

dr'^rj.rj.+r^iVr^r 3 , r!0r°,)dx.

(7.9)

This differs from the Euclidean space result for which the last term is positive. However the system of equations becomes identical with the Euclidean equations if we alter the origin point data so T° —> - rjj and also let dx° —» -dx°. As the choice of origin point data is arbitrary and what we call +x° and -x° is not of importance so far as Eq. (1.11) is concerned, we conclude that the equations in Minkowski space have the same content as the equations in Euclidean space. Thus, there is no physical content, in itself, in that the equations are Lorentz invariant, as there is no content difference in the solutions if the space is Minkowskian or Euclidean. 3. Higher Dimensions We have been working throughout with 4 space-time dimensions (although for calculational ease we have considered systems with dimension less than 4). We have never furnished arguments for such a restriction. If there were indications that greater dimensions would be useful we would consider deviating from the present program. Here we present some preliminary work done in higher dimensions even though we can offer

151

Additional Topics

no argument to the effect that 4 dimensions is inadequate. This calculation will be done with an eye towards probing what possible role higher dimensions could play. One feature of higher dimensions is that it enables us to get away fom the notion that there exists a single universe. Instead we may have the possibility of an infinite number of universes of the type@+(D f @ + 0 We should point out that we have yet to furnish reasons why the number 4 appears so special. To illustrate the kind of thing we have in mind, we first consider the case that 4 is replaced by 2. We can define a universe by means of ^nXnJ'-uJ'i2'^u^U'^2\'^22Another universe can be defined from 1^3,130^03, Too.rjj^jo.roj.roo. The independence of the universes is maintained by the gamma equations. In a similar manner we can have an infinite number of 4 dimensional universes existing ''side by side" Next we could allow for a small coupling between these universes. We may call the 4 dimensional structures subuniverses, rather than universes, due to the lack of independence now. Provided the coupling, which appears through components like F\5, remains small, the 4 dimensional character of the subuniverses would be maintained. This coupling furnishes a possible means by which our subuniverse can be affected by other subuniverses. We shall explore the notion that subuniverses can affect or "fuel" one another below. We chose to study a complex six dimensional space. This, has been studied in the literature (see for example references 24), although we emphasize that this is meant as an example, as we have no convincing argument for six dimensions. x',x2,x3 are taken to be real and x4,x5,x6 are taken to be pure imaginary. This then represents a symmetric generalization of the Minkowski hypothesis. Components of TL are taken to be real if the number of times the indices 4, 5, or 6 appearing in r"k is even. rj k is pure imaginary if the number of times the indices 4, 5, or 6 appear in T]k is odd. A similar prescription is taken for a" (of equation Eq. (1.40)). The first thing we recognize is that these reality properties are maintained by the field equations. Real fields can be said to comprise the subuniverse Ui, while imaginary fields comprise the subuniverse U2. The integrity of Uj and U2 are maintained, as mentioned previously, by the field equations, although Ui affects U2 and vice versa. Although other authors interpret x4, x5, x6 as a three dimensional "time" we shall think of the situation in terms of two subuniverses that effect one another. In this respect we could just as well extend the present model to 8 dimensions in which the subuniverses Ui and U2 are both 4 dimensional, although for calculational ease we will consider the six dimensional system here. We could call Ui a universe, and U2 an anti-universe as we have only two subuniverses under discussion here. We decompose T ^ into real and imaginary parts ( A | r B ^ are real) a

= A^ + i B ^

(7.10)

Origin point data is chosen as follows. For nonzero A^7, we take: A23 = 8

Af^-1.0

AJ2=1.0 Afa = -.6 A? 3 =-.6 A|, = .8 .

(7.11)

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In addition our nonzero B l are taken to be: B«6 = .10

B|4 = - 1 5

B*5 = .I5

Bf4 = .10

Bf6 = --13

B« 5 =-.13

(7.12)

We also write (f" and g? are real) a« = f« + i g f ,

(7.13)

and use r Q -- aa'Q aa^ a7 TQ j \ l Pi'

(7.14)

We choose alsct f,1 =.88

f2' =-.42

f3' =-.32

fi=o

f5'=0

f'=0

f?=.5

ff=.9

ff=-.425

f|=o

f| = 0

f|=o

f?=.2

ff=-.55

f 3 =89

f43=o

f53=0

f3=0

f?=0

f? = 0

f*=0

^J =1.01

£=.3

^=-2

ff=0

f|=0

ff=0

f45=.46

f|=.88

f65=-.35

ff=0

f»=0

f|=0

f|=.18

f*=-.3

ff=.94

g!=o

g2=°

■S-o

g{= -22

gj=-41

g^-26

g?=o

gH

g?=o

^=-3

83--2

g^=-12

g?=o

4=0

g^=o

g^=.6

g^-3

g^=-35

gt=.44

g<=-.16

g^=-39

gJ=o

g?=o

g^=o

g^=-48

gf=-24

g|=-.16

gi=o

g^=o

gf=o

g«=-.38

g^-2

g|«.5

g^=o

g^=o

g6=0.

(7.15)

To assess the effect of U2 on U], we first study the system when B ^ and g? are zero so we have a three dimensional system. We then make use of Eq. (7.14), but with f3= 1.0, to get the origin point data. When we specify an integration path as before, we get the map of Figure 7.2. In this map z=-.6. We see a soliton lattice, with soliton magnitude .13 ± .02. This small uncertainty in soliton magnitude may indicate that the lattice is an

Additional Topics

Figure 7.2. Data obtained by studying the 3 dimensional system obtained by requiring that imaginary components are all zero in section 3. The map arises by using the specification of path method. Grid is 0.15. z has value of -.6. Map is for r ] , . System is consistent with soliton loop lattice that is stationary in x4, with magnitude 0.13 ± 0.02. With this uncertainty, system could be an imperfect lattice. Numbers are 100 times actual numbers here and in Figures 7.3 and 7.4.

imperfect one. In Figure 7.3 we apply the integration scheme Eq. (2.26). We use a grid of 0.00234375 and the map is in the x,-y quadrant. The nonsymmetric region is enlarged compared to, say, that of Figure 6.8. However contour lines on the far right look like contour lines on the left, telling us that we have a ladder type symmetry (at least in an approximate sense). Maxima (minima) appear evenly spaced on each of a set of lines parallel to the x=-y line, although this effect is not seen for all maxima (minima), at least for the grid size used. Next we study the full six dimensional system Eqs. (7.10)-(7.15). In Figure 7.4 we map Aj | in the + - quadrant of the x,y plane when we specify a path. The results look like a loop lattice, except the magnitudes of the planar maxima (minima), which are preserved when we integrate in z, vary from loop to loop. Maxima (minima) lie on certain y= constant lines and have magnitudes that vary in an oscillatory way, which appears to be sinusoidal in a discrete sense (note Table 3 of reference 5). The influence of U2 then is to lead to maxima (minima) having different magnitudes (compare Figure 7.2 and Figure 7.4). When we make use of the superposition principle Eq. (2.26) we obtain Figures 7.5-7.8. Figure 7.6 fits to the right of Figure 7.5, etc. The plots are in the + - quadrant. We no

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154

Figure 7.3. System of Figure 7.2 studied using the integration scheme Eq. (2.26). Grid is 0.00234375. Map is in x,-y quadrant. Although the lattice of Figure 7.2 is altered in a significant way, we still see a repetition when x is increased sufficiently. That is even though the nonsymmetric region has been increased we still see a ladder type symmetry, at least approximately.

Figure 7.4. Data based on the full 6 dimensional system of section 3. Map is in the x,y plane. Path is specified. Grid is 0.15. This is a loop lattice, however the magnitudes of the maxima, as a function of x, vary in an oscillatory way that may be be that of a discrete sine curve. Thus we see an effect arising from U2 by comparing this map with Figure 7.2. Map is for A j , .

155

Additional Topics

Figure 7.5. The system of Figure 7.4 using the superposition principle Eq. (2.26). Grid is 0.0046875. Numbers are 1000 times actual numbers. Map is for A|, in x,-y quadrant. The effect of U2 shows up in a greater disorderliness than in Figure 7.3, although regularities are still observed with respect to locations of maxima (minima) along lines parallel to x=-y lines.

Figure 7.6. This figure fits to the right of Figure 7.5.

Math Aesthetics/Nonintegrability

Figure 7.7. This figure fits to the right of Figure 7.6.

Figure 7.8. This figure fits to the right of Figure 7.7.

156

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Additional Topics

longer see the repetition when x is increased sufficiently as was the case of the 3 dimensional system of Figure 7.3. Thus, the effect here from U2 is to lead to a more irregular looking system. Although the system appears more disorderly, there still are some regularities, as we see evidence for locations of maxima (minima) along lines parallel to x=-y lines as was also observed in Figure 7.3. We point out that for other more symmetric origin point data we do not see additional disorderliness despite the higher dimension. The present calculation tells us that higher dimension can ultimately have a useful role as it leads to a more complicated system here. However, as there is so much to be done in four dimensions, and as we have yet to show any inadequateness in a 4 dimensional description, we therefore have not pursued the notion of higher dimensions further at this time. In any case, we can think of higher dimensions as a means of widening one's horizons. 4. A Brief Summary High energy experimental research increasingly implies high financial costs. Thus it remains a question whether new generation accelerators will continue to be built. This then suggests that new potential directions be explored. If we can somehow come up with the correct set of field equations, then coupled with advances in computer technology, it may be possible to explore fundamental physical processes in an alternate fashion to the accelerator approach. Coming up with the correct set of equations may not be as difficult a program as it might at first glance appear. The underlying hypothesis we make is that the foundation of physics lies in mathematical aesthetics. This is the only criterion that appears to make much sense, in view of limits to empiricism. The theory would involve entities and the way these entities change. With such a simple base we would expect that the number of candidates for the basic equations would be quite small. This is then the reason for the mathematical study of the discipline of mathematical aesthetics. We have shown in this work, that there exist principles which can be categorized as "mathematically aesthetic", and that these aesthetic mathematical principles can be cast into a set of nonlinear equations that describe change. In particular we require that all Cartesian tensors, regardless of rank, and the way these tensors change, are treated according to a uniform prescription. In contrast, the approach to fundamental equations in the past has relied on generalization of equations that have been shown to be valid in some domain. Thus, so far as we know there are no other mathematically aesthetic equations that compete for attention with Equation (1.13). Furthermore this author has been unable to come up with an alternative system despite much effort in this regard. Thus, the next pressing problem is what sort of information is contained in the mathematically aesthetic equations. In studying this problem much is needed in the way of patience. After all, the equations depend solely on mathematical aesthetics, and are not formulated in terms of patently "physical" quantities such as force, energy, etc.. Instead the emphasis has been to seek solutions of the fundamental equations which describe basic building blocks, such as sine curves and wave packets immersed in an unstable vacuum that has considerable structure. Thus much like a tinker-toy we can then hope to construct complex model universes from the basic building blocks. It is only after we

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158

study the motion of these wave packets, in particular in the presence of other such wave packets, that can we think of introducing physical concepts such as force, energy, etc. This would involve considerably greater computer resources than has been at the command of the author, so it is a problem for future generations. In our work we have put our energy in the study of simple solutions to the mathematically aesthetic equations, such as lattice solutions and generalizations thereof. This modest study has already had important consequences with the recognition of the importance of nonintegrable systems. By nonintegrable we mean that results of integration depend upon path. The simple lattice system arising by a linearization of the mathematically aesthetic equations, called the ABJL equations, serve as a model to develop the theory of nonintegrable systems. The theory of nonintegrable systems can be developed in its own right and this is the subject matter of Chapters 2 and 3. We have argued that nonintegrable systems are more natural than integrable systems, and are thus apropos to a fundamental type theory. The simplest way one may procede in a nonintegrable theory is to specify an integration path. As we have no reason to favor one path over any other, we have developed the theory of nonintegrable systems. We have formulated two separate approaches to nonintegrable systems. In the first approach we have formulated three equivalent methods called the Product Method, the Sum over Path Method and the Commutator Method. In addition we have discussed an approximation scheme to the Sum over Path Method, called the Random Path Approximation which has been successfully tested with the Commutator Method. A second approach to nonintegrable systems, with many attractive features, has also been developed. We have seen from the second approach to nonintegrable systems that the way we integrate a nonintegrable system of equations can transform a system where soliton particles have well defined trajectories when we specify a path, to a system of solitons not having well defined trajectories as solitons now appear and disappear. Also, the second approach to nonintegrable systems can also offer an explanation for the phenomenon that one cannot go back in time (called the arrow of time in Chapter 4). This latter result does not depend on the specific equations under consideration, so long as derivatives are given by products, so we used the ABJL equations as an example. By merely altering the origin point data, so the integrability equations are now satisfied, the effect of the arrow of time disappears, so we can argue that this effect may be tied directly to the notion of nonintegrability. Nonintegrability opens up new vistas. In the case of the ABJL equations once we require integrability we lose the lattice and the equations are rendered totally uninteresting. For the 3 component lattice of Chapter 3, section 2, again once we require integrability we lose the lattice and the equations become trivial. For the sine within sine system of Chapter 5, section 6, once we require integrability we lose the sine within sine structure, and the equations again are trivial. All the multi-particle solutions in 4 dimensions discussed throughout the text, including the wave packet solutions, are characterized by the integrability equations not being satisfied. This is grounds enough for studying nonintegrable systems. This study has led to the generalization of the concept of derivative. This enables data at the origin to be truly arbitrary, and enables results of integration to be dependent on path (it is unrealistic for results of integration to be independent of path since different paths traverse different environments), and still we can have consistency with equations such as Eq. (2.5). Once one deals with maps, as in the case of the many figures displayed in the text, one is

159

Additional Topics

observing functions, and mixed derivatives of these functions are symmetric. Thus, the effect of nointegrability can be thought of as hidden. Nonintegrability is important in constructing the field from contributions from different path segments. However once one employs the superposition principle at each point, which arises from the theory of nonintegrable systems, we are men dealing with functions for which the standard rules of the calculus are applicable. It is in the construction of the maps themselves that the additional degree of freedom associated with nonintegrability makes its presence felt (Chapter 2, section 6). For this reason we would expect nonintegrable theory to be useful in the development of a fundamental theory of physics. In particular quantum mechanics as it stands remains an enigma as it stands apart from classical descriptions of nature. In quantum theory one has sum over path techniques6 and a commutator approach, and particles do not have well defined trajectories. Thus, we have similarities with nonintegrable systems, but this identification remains only suggestive at this time. Experiments can be thought of as indicating that we have a stochastic substructure appearing in fundamental processes. The Henon-Heiles Equations (1.2), exhibit chaotic looking properties in phase space. We have seen examples of random looking behavior asociated with the mathematical aesthetics system, Eq. (1.13). This would already indicate to us that classical nonlinear equations are capable of describing a stochastic substructure. This has prompted other authors as in reference 7, to propose that nonlinearity may ultimately account for quantum theory. However the problem with an understanding of quantum mechanics is deeper than this. It has been argued by Bell, and others and goes back to the Einstein, Rosen, Podolsky studies (see references 5 and 6 of Chapter 5), that if any classical nonlinear theory accounts for quantum mechanics, the classical theory must be nonlocal. As nonlocal theories have been suspect, this has been the major obstacle in trying to understand quantum effects using a classical base. By nonlocal we mean that a system can effect a system that is far away, in such a short time span, that a signal from the original system doesn't have the time to reach the far away system. Thus, in trying to gain an understanding of quantum theory, the problem of nonlocality needs to be addressed. We have shown in Chapter 5, sections 7,8,9,10 that the nonlinear, nonintegrable theory based on mathematical aesthetic allows for wave packet solutions. The wave packet system is immersed in a vacuum that is unstable and has considerable structure. We then went on to show that the appearence of nonlocality is quite natural in such systems. Thus, we would say that the notion of nonlocality should not, in itself, stand in the way in the ultimate understanding of quantum phenomena in terms of classical concepts. 5. Outstanding Problems The mix of nonlinearity and nonintegrability based on mathematical aesthetics developed in this text, has barely been explored. We have reached a point that the computer facilities that have been available to us this far (coupled with a downsizing of the computer facilities at Univ. of North Dakota), are insufficient for us to progress in a straightforward manner along the track we are presently on. We here list some of the more pressing problems at this juncture:

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160

(1) We have obtained multi wave packet solutions in Chapter 5, section 10. We would like to study the structure of three dimensional wave packet particle systems, with an aim of understanding the physical concept of "particle". (2) We would then like to study the motion of wave packet particles in the presence of other wave packets with an idea of exploring the mechanics of these systems. What sort of force laws are obeyed by such systems? Can we introduce the concept of energy? What sort of phenomenological laws can be obtained from the maps? Can we make contact with equations such as Maxwell's equations and the Schrodinger Equation, as examples? (3) We would like to study the vacuum in considerably greater detail, as there is considerable physics in the notion of vacuum. We have already mentioned the notion of nonlocality. Can we see effects common to quantum phenomenon here? (4) We have obtained a more viable wave packet solution in Chapter 5, section 10. Are there other sets of origin point data that are more advantagous for study? The possibilities for origin point data is immense. Interesting origin point data lead to systems which should be studied in detail, which implies considerable computer time. (5) Although wave packet solutions have commanded greater attention in the lectures we have given, soliton, instanton, and closed string particle systems, are in need of greater attention. By altering the origin point data and/or the dimension of the system can we get useful particle systems? Again the evolution in time of such perturbed solutions would be of interest. (6) The role of superposition principles associated with the concept of nonintegrability is a paramount problem. Although we have studied this effect in simple lattice systems and have argued that in this way the arrow of time may be understood, we know very little on how the superposition principles Eqs. (2.21) and (2.26) alter the specification of path results for complicated systems such as the more viable wave packet solution of Chapter 2, section 10. The computer time required is considerably greater than when we specify a path. We note, when we use the specification of path approach we have not had sufficient computer time to study pressing problems 1-3 above, much less deal with the notion of the role of the superposition principles. (7) Even though we have no compelling argument for higher dimensions, the rudimentary work of section 3, suggests possible useful features here indicating future study may be helpful. 6. Mathematical Aesthetics and Epistemology We are not that far into the computer age yet we can seriously study mathematical model universes using the computer. The figures illustrate just how much has been done already. When we first started we were pleased to obtain just an extremum for a component. This told us that the mathematical aesthetic ideas, at least, said something, as contrasted with being totally empty. In order to study model universes here, one need only supply a set of origin point data and let the computer do the rest. The situation can be looked at as analagous to a microscope. With a microscope, once one supplies a slide, one can observe different "worlds". In a similar way with a "computerscope" one can also observe different worlds, in the form of model mathematical universes.

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Additional Topics

In so doing one sometimes gets a feeling as an observer, akin to an 18th century, Age of Reason, clockmaker God, who observes a model universe unfold, without interfering with the goings on in the model universe. Such a line of thought tends to encourage philosophical reflections. Even though such ruminations are generally considered more appropriate to a philosophical treatise, in view of the subject matter covered in this work, we shall pursue such points here. The material addressed here can be found in an article by the author in the philosophy journal Darshana International8. We may ask, can we in principle (not concerning ourselves with the problem of numerical errors) know everything about the universe we live in, by studying solutions to a set of basic equations? The answer to this question is categorically no, as we will see below, and furthermore no amount of future study can ever change this conclusion. The question then, is whether knowledge of numbers at all points is sufficient to understand everything about our universe. We have alluded to before the Phythagorean idea that "all things are numbers" This in fact, is the most optimum situation so far as knowledge is concerned. That is, if units were also needed, then there would, in addition, be entities which could not be understood in terms of something more basic. Thus, so far as knowledge is concerned, there would be more in the way of unanswered questions if units were needed in the description of nature. Let us get back to the question stated above, in the most optimum situation in which all things are numbers. Consider the basic Equations (1.13), which were obtained with the use of logic (called logical self-implication by Schwalm in reference 5 of Chapter 1). We may seek a general set of origin point data, as in Chapter 5 section 10, when we obtained wave packet solutions. If, in principle, we knew the values of the gammas at all points of space by solving the basic equations (in reality a totally impractical situation), would we know everything about the universe? The problem is this. There is more than one solution to the basic equations. We can see from Eq. (1.13) that there exists a trivial solution as well. By trivial, we mean that the gammas are all equal to zero at all points and that this is consistent with the equations. By means of logic we cannot say that the nontrivial solution has any greater reason to be chosen rather than the trivial solution. Said in another way, an empty universe is perfectly logical. We cannot choose between an empty universe and a nontrivial universe on grounds of logic. This is true whether the basic equations have the form Eq. (1.13) or some other (undetermined) form. Thus, we may say that science is capable, in principle, of answering all questions about the world we live in (in the most favorable situation in which all things are numbers), except for one thing. The fact that we are here, as contrasted with an empty universe, is totally incomprehensible and will always remain so. That is, by means of logic (reason), one has no grounds to favor the nontrivial solution to the basic equations as constrasted with the trivial solution. We may ask what role does the notion of God play in all this. The nature of God is complicated because there are several characteristics commonly attributed to God. This will lead us to define a type A God and a type B God. We shall point out that there is a mathematical way of introducing a type A God, but there is no proof of existence. On the other hand, a type B God will be shown unambiguously to exist.

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Many people believe that God has the ability to affect the course of events in our world. God can perform "miracles" and is the object of prayers for help. There is a mathematical way that our 4 dimensional subuniverse can be "affected" by external influences. We saw this in section 3, when we talked of higher dimensions. A type A God is defined as being able to affect our four dimensional world from the outside. We see that higher dimensions offer a vehicle for this. At present it is not clear that there is a need for more than four dimensions. Thus, we cannot say that type A God exists, although the possibility cannot be ruled out. It is possible to define God in more abstract terms. We may define type B God as knowledge that is beyond the reasoning process. Such knowledge, thus, cannot be deduced by means of logic. We have already argued that an empty universe is not inconsistent with any logical principle. Hence, we conclude that there exists knowledge beyond the power of reason, and hence a type B God exists. Even if type A God exists, there is still a type B God. That is, even in the case, say, of a higher dimensional universe the empty trivial solution is also possible. Once having said that a type B God exists, there is nothing more that can be said, in a logical way, about such a God. It is analagous to a great painting. The only place that we know about the artist himself, is from a signature in the corner of the picture. The only way that we know about type B God is from the bottom line, i.e. that type B God exists. Is it possible to describe the Creation within a basic field theory? There is no logical way of constructing something out of nothing at some time (that is there is no reason why nothing shouldn't remain nothing). This implies that the universe existed for all timesthat is, an infinite universe. This is perfectly consistent from a mathematical point of view. However, an empirical physical universe appears at odds with the notion of an infinite universe. There is no problem of principle here. That is, an "empirical physical universe" would be based on particles. We have already seen in computer examples, for example, that solitons can appear spontaneously— in effect being "created" out of the vacuum. Also wave packet particles have been shown to spontaneously appear, at least as a function of y. There is nothing illogical about a finite particle universe emerging from an infinite mathematical universe at some stage. We note that a finite particle universe coming out of an infinite mathematical universe can be argued without invoking a need for a type A or type B God. The 18th century "clockmaker" God, by definition, affected our world at some instant but no longer does so. This would be a special case of a type A God. No characterization of type B God is possible using the process of reason. All that we can do making use of logic is to say that type B God exists. This is the beginning and the end of what logic can tell us. We can proceed one more step. We, as thinking beings are capable of knowing of type B God. After all we are here (even if this point cannot be deduced), and this tells us something—namely, that the nontrivial solution has been "chosen".

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References 1. M. Muraskin, The Determinant of gy in Aesthetic Field Theory, Computers Math. Applic, 22(1991)43. 2. Y. Murai, On the Group of Transformations in 6-Dimensional Space, Prog. Theoretical Phys., 9 (1953)147. 3. E. A. B. Cole, Particle Decay in 6-Dimensional Relativity, J. Phys. A. Math. Gen., 13 (1980) 109. 4. C. Patty and L. Smalley, Dirac Equation in a Six-Dimensional Spacetime; Temporal Polarization for Sublimal Interactions, Phys. Rev. D, 32(1985)891. 5. M. Muraskin, Rearrangement of Lattice Particles, International Journal of Mathematics and Mathematical Sciences 16 (1993) 593. 6. R.P. Feynman, The Space-Time Approach to Non-relativistic Quantum Mechanics, Rev. Mod. Phys., 20 (1948) 367. 7. J. Ford, How Random is a Coin Toss, Physics Today, 36(1983)40. 8. M. Muraskin, The Physics of God, Darshana International, 21 (1981) 47. Note on page 51, the fourth line from the bottom should read "beyond the power of reason"

APPENDIX A ELEMENTS OF THE CALCULUS We review the basics of the calculus with an eye towards generalizations to nonintegrable systems. Derivatives are defined in terms of finite differences. At the last step the limiting case of infinitesimal grid size is taken. We are dealing throughout with regular functions such that the results of this limiting process yields a convergent answer. In our computer work, of course, we do not go to the limit of infinitesimal grid size, so we check to see that a reduction of the grid appears consistent with convergence. Consider the following points: T U X R S W P Q V Then in terms of finite differences derivatives are defined at a point P as follows (F is any field component and dx,dy refer to partial derivatives respect to x and y respectively.): (d x F)dx = F(Q)-F(P) (5yF)dy = F(R)-F(P) (3xxF)dx dx s dx((dxF)dx)dx= 3x(F(Q)-F(P))dx = F(V)-F(Q)-(F(Q)-F(P))=F(V)-2 F(Q)+ F(P) (9yyF)dy dy = F(T)-2 F(R)+F(P) (axyF)dx dy = dx(dyF dy)dx= ax(F(R)-F(P))dx = F(S)-F(R)-(F(Q)-F(P))= F(S)-F(R)-F(Q)+F(P) (c^F) dx dy = dy(dxF dx)dy =3y(F(Q)-F(P)) dy = F(S)-F(Q)-((F(R)-F(P))= F(S)-F(Q)-F(R)+F(P) etc

(Al)

Thus, we see that second mixed partials are symmetric, ^ F = 5 yx F

(A2)

In a similar way we see that all mixed partial derivatives are symmetric in all indices. This is true as dx and dy approach zero as well. As usual regular functions are assumed. From the definition of derivatives we have: F(x+Qidx, y+a2
c ^ F dx dx

165 + +

02(02-1)

Elements of the Calculus

_ . . , ai(ai-l)(a!-2) a w F dy dy + d ^ F dx dx dx

02(02-1X02-2) a^yF dy dy dy

Qi(oi-l) a2(Q2-l) , + — — 02 "xxyF dx dx dy + ai dxyyF dx dy dy 2

+

'

(A3)

That is, by inserting Eq. (Al) into Eq. (A3) we see that Eq. (A3) is identically satisfied for all a i and all 02. Qi is the number of segments of extent dx parallel to x and 02 is the number of segments of extent dy parallel to y. Thus, in our picture above, to get to the point U we need a i = l and 02=2. The formula Eq. (A3) is easily extended to any number of spatial dimensions Eq (A3) says we can calculate F at any point of space in terms of the derivatives of F evaluated at the origin point. To get F(S) the formula Eq. (A3) simplifies since a\=l, 02=1, so F(S)= F(P)+ d x F dx+ dyF dy+ d^F dx dy

(A4)

We can retrieve many of the standard formulas of the calculus starting with Eq. (A3). For example, we consider any two points that are near to each other. We can then approximate Eq. (A3) as follows. We subdivide the grid so dx and dy approach zero. This means that o i and 02 approach infinity, so that ojdx approaches a finite but small number a, and Q2dy approaches a finite but small number b. Then Eq. (A3) becomes a2 a3 F(x+a, y+b)= F(x,y)+ a d x F+ b dyF+ - dmF+ - d^F b2 b3 a2b + dyyF+ dyyyF+ a b 9 x y F + i\. v. "21" 5xxyF 2 b a + — 9WF+

(A5)

In Eq. (A5), (oj-m) with m an integer, are approximated as a, for m small. When m is large, so (oj-m) is no longer large, terms involving these factors multiply higher order terms in the differentials of displacement and are neglected. As a and b are small, a few terms in this expansion normally suffice. Eq. (A5) is just the Taylor series approximation. If the point S in Eq. (A4) is so close to P that dx and dy approach zero then Eq. (A4) becomes F(S)= F(P)+ d x F dx+ <9yF dy .

(A6)

dF= d x F dx+ dyF dy

(A7)

or which is another often used formula in the calculus. It is not necessary that x,y characterize locations in space. The above formulas are valid provided F is a function of these variables, regardless of their significance.

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For nonintegrable systems those formulas above not involving mixed derivatives are still valid. In the first approach to nonintegrable systems all the formulas in this section are valid if we replace mixed derivatives by symmetrized derivatives. In the second approach to nonintegrable systems the only mixed derivatives consistent with the basic equations are the second. Higher derivatives are not defined here. Thus, only formulas Eq. (A4), with second mixed derivative replaced by the symmetrized derivative, and Eq. (A6) and Eq. (A7) are valid in this case. We emphasize that for nonintegrable systems derivatives defined away from the origin (the origin is where the arbitrary data is given), as in Eq. 2.23, are more clumsy to deal with.

APPENDIX B THEOREM OF THE CALCULUS REQUIRING PATH INDEPENDENCE FOR INTEGRABLE SYSTEMS The system under investigation is Equation (1.13). This system, which is integrated using the computer, has the structure Eq. (Bl). Since the number of indices is not of importance here, we have suppressed indices. Also the number of dimensions is again not significant here so we will largely use two dimensions for illustrative purposes. Then the structure of the equations to be studied is given by Si(x,y)=

af(x,y)

~d^r-

(B1)

We define M s S, N s S2.

(B2)

Then the system (Bl) becomes M(x,y)=

af(x,y) ax

df(x,y) dy ' ;t 3 b) we gt dM d2f dxdy " a7 N(x,y)=

d d If we take —- of (B3a) and — dy ox

(B3a) (B3b)

d2f _ aN (B4) ax3y dx As a consequence of the definition of derivatives (Appendix A) mixed derivatives are symmetric for regular functions. Thus we have ^ =^ . (B5) ay dx This is called the integrability equation. The integrability equation says that M and N can not be any function of x and y. The calculus does not allow flexibility on this point, M and N must satisfy Eq. (B5). In 3 space dimensions we define in addition P = S3.

(B6)

VxS=0.

(B7)

The integrability equations then become

Math Aesthetics/Nonintegrability

168

Here the curl is defined in the usual way. Going back to our two dimensional system the calculus goes on to say that results of integrating f f df df

J (Mdx+ Ndy) = J (— dx+ — dy)

(B8)

are independent of path. Use was made of Eq. (B3) to obtain the right hand side. In three df dimensions we would add the term Pdz on the left and — dz on the right. The oz independence of path is a consequence of the integrability conditions as we shall see below. From the definition of Sj Eq. (B2) we may write Eq. (B8) as /S-dl.

(B9)

If we integrate this around a closed curve, then Stokes theorem (where a is a unit vector perpendicular to the infinitesimal area element and sense given by a right hand rule) is written,

fSAX=ll

VxS • a dA .

(BIO)

The line integral is over any closed curve and the integration on the right is over an area bounded by the closed curve. The integrability equations, Eq. (B7), tell us that the line integral around a closed curve is zero. This in turn implies that Eq. (B9) is path independent. Stokes theorem can be obtained from an alternate definition of the curl in terms of a line integral,

*if

a ■ VxS= limA

Sdl

OH)

A represents an area bounded by the closed curve. The two definitions of the curl can be shown to be equivalent. We shall work with a closed curve in the x,y plane which we consider for illustrative purposes, dl which represents an infinitesimal tangent to the closed path, has components dx and dy. a has length one and is perpendicular to the x,y plane. We take the circulation to be counter-clockwise. Then by definition a points in the positive z direction. Since the area A enclosed by the closed curve goes to zero, there is no loss in generality in taking the closed curve to be a rectangle in the x,y plane as shown below: 3

Ay

2

x , y

4

A x

1 X

169

Therom of the Calculus

In this case Eq. (Bll) becomes: (VxS)3 Ax Ay= / S, dx+/ S2 dy-/ S, dx- / S2 dy 1

3

2

(B12)

4

Using a Taylor Series expansion Eq. (A5), keeping terms of first order in smallness becomes (VxS)3 Ax Ay=y > (S 1 (x ) yo)-^ ^ ) d x + |(S 2 (x 0> y)+ ^ f,„

9Si Ay }

J (Si(x,yo)+— -y

f

J ^^^-^

^)dy

dS2 Ax

-y)dy-

(B13)

Here the derivatives in the first and third integrals are evaluated at x,y0 and the derivatives in the second and forth integrals are evaluated at x0,y. We make use of cancellations, and use the fact that in lowest order all the derivatives can be evaluated at xo,yo, so the derivatives can be pulled outside the integrals. Therefore Eq. (B13) becomes (VxS)3 Ax Ay= (-^-

—I) Ax Ay,

(B14)

and we get from Eq. (B2) ,„ s <9N dM (VxS)3= — - — , B15) ay ax which estabUshes the equivalence of the two definitions of the curl in the two dimensional case. The condition (VxS) 3 =0,

(B16)

we remember is identical to the integrability equation. The integral definition of the curl makes it easy to obtain Stokes theorem Eq. (BIO). In two dimensions the closed curve appearing on the left hand side of Eq. (BIO) is any finite closed curve in the x,y plane. We can obtain Eq. (BIO) by adding up the contributions from many infinitesimal rectangles using Eq. (Bll). Taking all circulations to be counter-clockwise we get cancellation for line integrals in the interior, thus leading to Eq. (BIO). On account of the integrability equations the right hand side of Eq. (BIO) is zero so we have #S-dl=0, for any closed curve. Consider any two points A,B on a closed curve as follows: B

£7

(B17)

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170

Then we have

B

A

#S dl= / S ■ dl + / S • dl. A

B

(B18)

Interchanging limits on the second integral we get B

B

/S

d l = / S - dl

A

A

path 1

path 2

(B19)

on using Eq. (B17). As the closed curve containing the points A and B can be any closed curve it follows, using Eq. (B2), that the integral

B

f (M dx + Ndy) A

(B20)

is independent of path between A and B as a consequence of the integrability Eq. (B16), which follows from the symmetry of the mixed derivative Eq. (B4). In three dimensions B

f (M dx+ N dy+ P dz) is independent of path by extending the above arguments. When the above integration depends on path the system is said to be nonintegrable. In Chapter 2 we argue that nonintegrable systems are natural, can indeed be handled using the computer, and have considerable content. We there introduced generalized derivatives which agree with the usual derivatives in the case that the integrability equations, arising from symmetry of mixed derivatives, are satisfied. That is to say, by generalization of the notion of derivative it is possible to consider Eq. (Bl) without restricting M and N in any way.

APPENDIX C ELEMENTS OF TENSORS 1. Tensors and Coordinate Transformations In order to locate a point we introduce a set of coordinates, which we label in 4 dimensions as x , x , x 3 , x°. The number of dimensions is not of importance in our considerations. These coordinates are not unique. Another set of coordinates is given by x ' , x' , x ' \ x'° The new coordinates are related to the old ones since the point which we are locating is the same in both instances. This can be written as x'' = x'\x\x2, x\x°).

(Cl)

From the notion of derivative we have from Eq. (A8), with F replaced by x" dx'' dx

" = axl d x J

(C2)

The set of components dx' refers to a difference of displacements. The inverse transformation is written xi = x i ( x ' l , x , 2 , x ' 3 , x ' 0 ) ,

(C3)

from which we get again from the notion of derivative dx

= £ * <**'

Thus, there exists two types of partial derivatives associated with the coordinate transformation, namely dx" dx1 (C5)

ft? ^

aF

Tensors are objects having certain transformation properties when a coordinate change is made. The set of components constitute the tensor quantity. There are objects, A', that transform similar to Eq. (C2), and are written with index as a superscript. Their transformation properties are then given by f)\'' A"(x",x' 2 ,x'\x 0 ) = — A J (x',x 2 ,x 3 ,x°). (C6) ox* The dependence of objects such as A" and A> on the coordinates is similar to the above and will in general be suppressed in what follows unless we are interested in emphasis. There are objects that transform making use of the partials with the prime on the bottom. These objects are written with subscripts for the indices. The quantities A; then are taken to transform according to 8x> *

%

>

*

>

■

(

C

7

>

An example of this is the gradient of a scalar. A scalar is an object that can be obtained by the contraction operation (see section 2) and is unaffected by the coordinate transformations. Let s be a scalar field. Then Eq. (A7) becomes

Math Aesthetics/Nonintegrability

172 #s

,

, i

ds=^-dx'. (C8) Since s is taken to be a scalar, we can replace the unprime by primes on the left hand side to get A >, 'i

'2 '3 ,o,

ds(x',x 2 ,x 3 ,x°)

j

ds (x ,x ,x ,x )= —. dx . (C9) k We divide by the change Ax' , corresponding to a small change Ax , and take the limit of Ax' becoming small, so it can be written dx' , and denote derivatives as partial derivatives as the derivatives involve several independent variables. We then get 9s' 3s dx' • = -^ ■ (C10) dx" dx' dx" Eq. (C10) is called the chain rule, and we see it can be obtained (note Appendix A) starting with the concept of derivative. It is assumed we are dealing as always with well behaved functions. We note that both Eq. (C2) and Eq. (C10) are valid regardless of the type of coordinate transformation. The transformation of the differentials of'the displacement and the gradient of s obey different laws in general (contrast Eq. (C2) with Eq. (C10)) although they are both one index objects, and refer to different types of vector. We note at this point we have not introduced any mechanism for relating A1 and A; (raising and lowering indices). It is not necessary to introduce any such relationship between upper and lower index vectors in obtaining the desirable attributes of tensors as outlined in the next section. Thus, at this stage, one should not think of A1 as representing components obtained by parallel projection of a vector, and Aj as components arising from perpendicular projection. That is to say, we are not requiring Aj and A1 be different kind of components of the same vector. We may call the upper index objects contravariant and the lower index objects covariant. Vectors whether with index appearing as a superscript or as a subscript, are called first rank tensor tensors. Second rank upper index tensors have the property of behaving like a product of two upper index vectors when we make a coordinate transformation. Thus we have „•, dx" dx" „ TJ =

^axT

T

(Clla)

A second rank lower index tensor is then required in a similar way to behave like a product of lower index vectors , dxs dx1 T1fJ= —r r T st , K(Cllb) dx" dx" and a second rank mixed tensor obeys A dx" dx1 S. T',J = - s :T . . (Cllc) dx

dx"

'

Higher rank tensors are introduced by the same procedure. An n"1 rank tensor transforms as a product of n vectors. An n"1 rank tensor can be upper index, lower index or mixed, with n indices which are either subscripts or superscripts. The order of indices is dx" dxm important and any ordering is permissible.The objects —r , are not tensors. oxJ dx'

173

Elements of the Tensors

Entities may be tensors with respect to some coordinate transformations but not to others. For example the electric field is a vector with respect to rotations, but not a vector under Lorentz transformations. An important result can be obtained from Eq. (C2) by dividing by Ax'k and taking the limit of this quantity being small as we did in obtaining the chain rule Eq. (ClO). We then get

dx" _ dxn dxm dx" ~d^d*)

2

(C12)

3

Since x' ,x' ,x' ,x'° are all independent variables it follows that the left hand side is zero or one. The left hand side is one when i=j and zero otherwise. The Kronecker delta, 6, is <9x' defined to have these properties. This feature is also present for the objects —r Then Eq. (C12) can be written as ^ xJ { s , _ 5 x " dx< cs dx' dx

^

=

^ ^

5

'

=

^ ^ T '

(C13)

which gives the transformation property for a second rank mixed tensor. The Kronecker delta is called an isotropic tensor since it has the same components in the primed as well as unprimed system. 2. The Importance of Tensors The basic equations of the last, section Eqs. (C2), (ClO), (C13) have their basis in the definition of derivative (note Appendix A). Thus the results we obtain are general regardless of the transformation of the coordinates and regardless of the relationship between upper and lower index objects. In the text we have taken tensors to be the basic entities to be used in constructing a field theory based on mathematical aesthetics. The desirability of tensors arises from the theorem that tensor equations have the same structure after a coordinate transformation. We shall demonstrate next that tensor equations do in fact have such a property. We will continue to work with general coordinate transformations introduced in the previous section. By tensor equation we mean any equation in which the only objects appearing are components of tensors and for which certain rules are obeyed. As an example consider the following equation ^ = Bjj C'j

(C14)

The index i is called the free index, and in a tensor equation, it must match up on both sides of the equation. The free index can be labelled by any letter so long as it matches on both sides. The index j is called a dummy index. The dummy index appears two times, once as a superscript and once as a subscript. The dummy indices are summed over. In 4 dimensions the summation is from one to four. How we label the dummy index is immaterial. We can change the letters j to, say, k in (C14) without altering the equation. A,= Dikk with sum over k, by the above rules is then not a tensor equation.

174

Math Aesthetics/Nonintegrability

The input for the theorem is a tensor equation, such as Eq. (C14), together with the statement that the objects appearing in (C14) are tensors, obeying the transformation laws ,_dxm 1

dx" , _ dxm

dx1

B i i

~ W W dx" C" = — C

Bmt

(C15

Then inserting this into the tensor equation Eq. (C14) gives dxm dxm dx' dx" Am = ^ T 7 XT? T 7 B™ C P (C16) [ dx" dx' dx" dxP Using the relationship Eq. (C13) and from the properties of the Kronecker delta, we have dxm — ( A m - B m t C ' ) = 0. ox' Since the coordinate transformation can be any transformation, it follows Am = B m t C

(C17)

(C18)

Thus, the same equation appears in the primed system, Eq. (C14), as well as in the unprimed, Eq. (C18). This completes the theorem for the Eq. (C14). This result can be established for any tensor equation using a similar procedure. As to the aesthetic field system introduced in Chapter 1, one set of Cartesian coordinates should be indistinguishable from any other Cartesian system so far as the form of the equations is concerned. Thus, all the equations appearing in Chapter 1 should be tensor equations with respect to 4 dimensional rotations. Next we illustrate an important theorem called the quotient theorem, which is basically similar to the above theorem, except with different features taken as given information. Consider, as an example, the (Cartesian) tensor equation dAi = r j k Aj dxk

(C19)

By tensor equation, we mean that such an equation is valid in the primed as well as unprimed systems, so Eq. (C19) holds with all the objects being primed. Also, the quantities dA;, Aj, dxk are given as tensor quantities with Aj and dxk being arbitrary. dAj is a vector when we make rotations from a Cartesian system. Such a situation is a special case of general coordinate transformations so all the previous formulas are still valid. Thus, we have dxm dAj = ,dA m 1 dx" s . dx A = r As ' dx"

175

dx = —

Elements of the Tensors

dx< .

(C20)

Then, it follows from the theorem that 1 ^ is a third rank mixed tensor. The idea is that all quantities but one in the tensor equation are given tensors. Then, so long as the tensors multiplying the unknown quantity are arbitrary, it follows that this remaining object is a tensor. To see this, insert Eq. (C20) into the primed version of Eq. (C19). Then multiply dx" both sides of the equation by - — and use the general formula Eq. (C13). Then use Eq. (C19) without the prime. This gives _ A J . dx" dx" dx'k „ r ^ A " d X = ^ ^ ^ A " d

X

. ' -

< C21 >

Since An and dx' have never been specified it follows from the general relation Eq. (C13) that _,i dx" dx{ dx1 „, r k = T; ■ —z r, , (C22) Jk

ax s dx" dx,k "

which implies that T'-k is a third rank mixed tensor. We can see in a similar manner that this is a general result, not depending on any relationship between upper and lower index objects or on the type of coordinate transformation under consideration. We can also see from this theorem that the quantity A1 of equation Eq. (1.9) is a vector. An operation that is useful involving tensors is that of contraction. Here, we take one upper index and one lower index, and we set them equal and then sum over the allowed values of the index. This operation reduces the rank of the tensor by two. As an example consider the second rank tensor A' J B k . We set j=k and sum to get A' J B'. Then from the transformation properties of A" and Bj we get dx" dxs Am — B s . (C23) A"B| = — ' dxm dx11 Using Eq. (C13) this gives A' j B] = AJBj (C24) Thus, we have an object with no free indices having one dummy index as superscript and one dummy index as subscript (this arrangement for the indices is of importance) for which the primed result is the same as the unprimed result. The quantity in Eq. (C24) is then defined to be a scalar which is a zero rank tensor. So we see that the operation of contraction lowers the rank of the tensor by two. Again we see in a similar way that this result is general and does not depend on any relationship between upper and lower index quantities and regardless of the nature of the coordinate transformation. Discussion of tensors can be found in many sources. As examples, consider references 1,2,3.

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Math Aesthetics/Nonintegrability

3. Rotations of a Cartesian System As a simple example of a rotation, consider the case of a rotation about the z axis. Then (x1 = x, x2 = y, x3 = z) we have x' = x cos# + y sin9 y = -x sin9 + y cos9 z'=z .

(C25)

We then take differentials of Eq. (C25). Since we are requiring the transformation to be a constant transformation, this means 9 is constant. We then have dx' = dx cos# + dy sin# dy7 = - dx sin# + dy cos0 dz/ = dz

(C26)

Equation (A7) here takes the form . dx' dx1 J dx' A dx = dx + —— dy + —— dz ox ay dz

, oy

ay

ay

dy = —— dx + —— dy + —— dz ox ay oz , dz' dz1 dz? dz' = — dx + — dy + — dz . (C27) ox oy oz Thus, comparing Eq. (C26) with Eq. (C27) and use the fact that dx, dy, and dz are independent. This implies ox' „ ox' . „ ox' — = COSP -—— = smo —— = 0 ox oy oz —— = - sin^ ox oz'

^-=0 ox

—— = cos^ —— = 0 Oy oz dz1

— =0 oy

dz!

— =1 oz

(C28)

Ox'j These are the partial derivatives, ——r, introduced in section 1 in the particular case of constant rotations about the z axis. We can obtain the inverse transformation by replacing primes by unprimes in Eq. (C25) and letting 9 -> -9. This gives x = x' cos9 - y sin9 y = x' sin9 + y cos9 z = z/ .

(C29)

177

Elements of the Tensors

Using the same procedure as before, taking differentials etc., we obtain the partial dx' derivatives — r , dx" dx _ <9X <9x dy dz'~° ^ = 0 dz'

dy

^ = 0 ^ = 0 (C30) dx' dz' ay By comparing Eq. (C28) and Eq. (C30) we see dx'{ _ dx> (C31) ~dx^~~dx7i ' This result is valid for any constant transformation that preserves the Cartesian character of the coordinates. We see this as follows. We define a four dimensional rotation by the condition that the Pythagorean length of a vector be unchanged by the transformation. Then we have 4

4

V. A" A" = J2 A' A1 , i=l

(C32)

i=1

and thus (all summations here go from 1 to 4)

yAiAi

=Y i,m,s

£2L^LAmAs

3xm dxs

(C33)

In order for this to be valid for any A1 it follows fl-v'' die1'

with 6ms being 1 if m=s, and zero otherwise. Then comparing with Eq. (C13) gives the result Eq. (C31). As a result of Eq. (C31) the transformation law for A' and A; become the same. With no apparent need to distinguish between upper and lower indices the simplest thing to do is to require A1 = A, A3 = A3

A2 = A2 A0 = A0 ,

(C35)

which can be written as Ai = gij AJ, with

(C36)

Math Aesthetics/Nonintegrability

178

*■" V o ' , )

(C37)

*- (o1-,

(C38)

if the space is Euclidean, or

),

if the space is Minkowskian, in which case A 0 = -AQ. The formulas of the previous sections are valid for any coordinate transformations and are not dependent on how we raise and lower indices, so they are also valid in the present situation. 4. Raising and Lowering of Indices and Mathematical Aesthetics Even though there is no distinction between A1 and A; with respect to transformation laws in the situation of section 3, this does not imply that Eq. (C35) is satisfied. After all Aj and B; in Chapter 1 (Equations 1.5 and 1.6) have the same transformation laws, but this does not require that they be equal. Aj could represent Pjj, for example, while Bj could represent t\t. By requiring Eq. (C35), we have assumed a specific way to relate upper and lower indices, namely the simplest way. We recognize that a more general way to relate upper and lower indices will not prevent us from preserving the desirable features of the tensors. But why would one wish to have a more involved way to raise and lower indices other than Eq. (C35)? We look for such reasons from the notion of mathematical aesthetics. We emphasize when we talk tensors below, we mean dynamical tensors. A dynamical tensor is one that changes from point to point in the manner of Chapter 1. Examples of nondynamic tensors are the Cartesian unit vectors i,j,k. By requiring that all tensors be treated in a uniform way with respect to change we see that even if gy had the form Eq. (C37) or Eq. (C38) at a single point, it would not keep this structure at all points in view of the Equations (1.8), in which gy changes from point to point by a uniform set of rules. Said in another way, an important reason for a need for both upper and lower indices is that it enables us to introduce the notion of a dynamical scalar product as in Chapter 1 in a nonrestrictive way. In Chapter 1, we saw that scalars, in addition to being unaffected by coordinate transformations, do not change from point to point and are thus constants. The property that all scalars (remember we are talking about dynamical scalars here) are constant can be considered a desirable feature. Otherwise tensors multiplied by a scalar would behave differently from the original tensor with respect to change. Multiplying the tensor by the scalar squared, say, would in general produce still another law of change for the original tensor. If there were no distinction between upper and lower indices, we could if we choose write all indices either as upper or as lower indices. Then the requirement that all scalars be constant, which is important as it enables all tensors to be treated in a uniform way with regards to change, would imply

179

£d(A,Ai) = 0.

Elements of the Tensors

(C39)

i

That is, the Pythagorean length of the vector would be constant. Howevever in a basic theory involving a vector, A, to require its length be the same at all points, at the outset, would imply a restrictive theory. To see how restrictive, we may rewrite the formalism of Chapter 1, with all indices as subscripts. We shall here treat all repeated indices as being summed over as this is equivalent to working with a gy qiven by Eq. (C36) or Eq. (C37). Then for the change of Aj we have analagous to Eq. (1.5), dA, = r ijk Aj dxk

(C40)

We then see that Eq. (C39) implies right away that the change function is antisymmetric in the first two indices. Requiring that Tu& behaves like a vector Dj, and Tidk behaves like another vector Fj, further restrict the change function. We can satisfy these conditions by demanding that T^ be antisymmetric in all indices which leads to a theory which our computer studies indicate as trivial. It should be said that antisymmetry in all indices may not be the only way to satisfy these conditions. In Chapter 5 we obtain from a few mathematical aesthetic principles such things as the wave packet systems. We can see in a practical way that we would be throwing away all these results by restricting the change function unnecessarily. An interesting feature of the system requiring a general relationship between upper and lower indices is that the Pythagorean length (Ai)2 +(A2)2 +(A3)2 + (A0) is unchanged by any rotation in view of Eq. (C31), so it is a scalar, but it is not a dynamical scalar. The dynamical scalar length is defined in terms of a dynamical gy by means of AA^A'AJgy,

(C41)

and is constant as is the case for all dynamical scalars in the theory. Thus, we have a theory in which Aj is not restricted by having its Pythagorean length the same at all points, as would be the situation if we adopted Eq. (C35) and required all scalars be constants. In addition, when we think of dynamical tensors we recognize that all the desirable features of the tensors of section 2 are still present, as these features are not dependent on the way we raise and lower indices. Thus, the aesthetic fields program furnishes us with a reason to introduce upper as well as lower indices even in a Cartesian system. In summary, we preserve all the tensor formulas of section 1 and 2. We differ from common procedure, by allowing for a general relation between upper and lower indices. gij is not even required to be symmetric in Eq. (C36). This is to be the case even when the coordinate transformation is that of a rotation from one Cartesian system to another as in section 3. This enables us to formulate a theory where all dynamical scalars are constant in a nonrestrictive way, which in turn allows us to treat all dynamical tensors in a uniform way with respect to change, again without restrictions.

Math Aesthetics/Nonintegrability

180

We shall discuss the mathematical aesthetics program in the case of general coordinates in Appendix E. References 1. L. Landau and E. Lifshitz, Classical Theory of Fields, 2nd edition, Pergamon Press, 1962. 2. J.L. Synge and A. Schild, Tensor Calculus, Univ. of Toronto Press, 1949. 3. B. Spain, Tensor Calculus, Wiley (Interscience), New York, 1953.

APPENDIX D ELEMENTS OF DETERMINANT THEORY 1. Transformation Law for a Determinant of a 2nd Rank Tensor An array of coefficients or numbers with certain properties with respect to addition and multiplication, when we are dealing with more than one array, is called a matrix. We shall write a matrix as a capital letter, such as P, or as the collection of its elements (p'). The matrix elements can be labelled any way we wish, so we may call them p' or py or pij or PjJ. In what follows, when we have a repeated upper and lower index, this will imply an understood summation over the range of this index. If all the indices are subscripts (superscripts) then any summation will be indicated by a summation sign. In any case the first index will refer to which row we are talking about, and the second index will refer to the column. If we have two matrices P and Q, with matrix elements p'. and q1., we represent matrix multiplication by R=PQ, defined by

ij'Ml'

(D1)

for all i and j (it will be understood that free indices vary over the entire range of values here and in what follows). Note the sum is over adjacent indices. Matrix multiplication is non commutative as the order PQ is not the same as QP in general. Matrix addition is defined by rij=pUqiJ

(D2)

Addition is only meaningful for matrices having an identical number of rows and an identical number of columns. Multiplication of a matrix by a number, means multiplication of each element by that number. Two matrices P and Q are equal if p'j = q'j for all the elements. The antisymmetric symbol e;jk...is defined to be +1,0, or -1. It is +1 if ijk.. needs an even number of transpositions of neighboring indices to end up with the sequence 123... . It is -1 if the number of transpositions is odd, and zero if any two (or more) indices are the same. The antisymmetric symbol is defined to be the same in all coordinate systems. Thus, in dealing with the example of three subscript indices we have ei23 = +1, em = -1. e32i —1, ein = 0, etc. . eijk- is defined to be +1, -1, 0 according to the same prescription as ejjk... . Associated with a square matrix of order n, is a quantity or number called the determinant of P and denoted by | p | . It is defined by (there are n subscripts associated with the antisymmetric symbol) eijk.... | p | = eabc.... pai pbj pck...

(D3)

Math Aesthetics/Nonintegrability

182

A determinant of order 3 is written

P1.

P\

P'S

P2,

P\

P23

P3,

P\

P33

(D4)

e,jk

1 Pi

= e abc p a , p b p c k

(D5)

In this case Eq. (D3) becomes

We evaluate this for i=l, j=2, k=3, to get P I = P' 1 P22 P33 + P*2 P23 P31 + P*3 P21 P32

P'SP^PVP'IP^PVP^IPV

(D6)

From the definition of the antisymmetric symbol we obtain the same answer for | p | regardless of the choice for ij,k provided these indices are all different. If the indices are not all different then Eq. (D5) leads to zero equals zero. We see from the above that if any two rows (columns) of the determinant are the same then the determinant is zero. We see also if we interchange any two rows (columns) of the determinant, then the determinant changes sign. | p | = 0 if entries comprising a row or column are zero. If we multiply a row or column by a quantity k, then the determinant itself is multiplied by k. We state the following theorem. If the square matrices R,P,R satisfy R=PQ then the corresponding determinants obey | r | = | p | | q | We illustrate this theorem for order 3. By the same argument it is true regardless of order. We use Eq. (D5) but with the symbol p replaced by r everywhere. Then using the rule for multiplication of matrix elements Eq. (Dl) we get eijk I r | = eabc pas p b t pcu q s ; q1, qu k , = | p | estu q\ q'j quk

(D7)

In order to get this we used Eq. (D5). Using Eq. (D5) again gives e,jk

= IP

eijk

(D8)

This equation implies, what we set out to demonstrate, that when R = PQ, then the determinant of R is equal to the determinant of P times the determinant of Q. Let us next introduce any second rank tensor gjj . We work in 4 dimensional space here. These components comprise an array and thus can be thought of as a matrix. The determinant of this matrix is denoted by | g | , or more simply with the bars off, as g (see chapter 7, section 1). The transformation of this second rank tensor under a coordinate transformation is given by Eq. (Cl lb). We reexpress Eq. (Cl lb) as gij = asi a'j gst

(D9)

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Elements of Determinant Theory

where dxs

a

i -

^7

ax'/t

.,

A

j -

ax

The chain rule is written

TT

(D10)

ox

aSjAjt = 6\

(Dll)

A s ,a , m = <5sm

(D12)

or

The determinant of g in the primed system in a similar way as before, takes the form (remember the antisymmetric symbol is defined to be the same in all coordinate systems) g'eijk, = e»bed ^ g^ g^ g^, .

(D13)

Then, using Eq. (D9), and the same procedures used in establishing the above theorem (Eq. (D8)), gives us g' = | a |

|a | g

(D14)

We note the set asi although not a tensor, is an array, for which we can obtain its determinant from an expression similar to Eq. (D13). Since the Kronecker delta has determinant 1, the theorem Eq. (D8) implies |A|

= -ri-p

I al

(D15)

and, thus from Eq. (D14) we get g = | A |2 i

(D16)

Taking the positive square root, we get the transformation law for what is called a pseudoscalar. The antisymmetric symbol is not a tensor quantity. We show next that we can obtain a tensor quantity from the antisymmetric symbol by multiplying it by ^/g. That is we claim that eijk... =

y/ge&...

(D17)

is a tensor. The way we will show this, is to assume it is true, and then find we end up with an identity. Thus, we assume (we work as previously in 4 dimensions, although as before the argument is valid for any dimension), ,-' — a „b eUk. = aV b j a c„ck a d„d 1 £.bcd n

(D18)

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Inserting Eq. (Dl 7) on the right gives 4ki = (a a i a b J a c k a d 1 e a b c d ) A /g = | a | eijk, y/g ,

(D19)

when use is made of Eq. (D3) to eliminate the bracketed term. In addition, Eq. (D17) evaluated in the primed system is
>

(D20)

as the antisymmetric symbol has the same values in the primed system. Taking the positive square root of Eq. (D14), we have v / g 7 = I a] y/i,

(D21)

so Eq. (D20) becomes 4k] = I a I v/g eijkl .

(D22)

Inserting this on the left hand side of Eq. (D19) then leads to an identity, indicating that Eq. (D18) is correct, and telling us that e p is indeed a tensor quantity. We shall see how this fits into the aesthetic fields program in section 4. 2. Expansion of a Determinant in Terms of Cofactors We discuss here an alternative way of expanding a determinant, other than Eq. (D3), called the cofactor method. As we will have repeated indices that are not meant to be summed over, we shall dispense with the summation convention, and indicate sums by the summation sign. We shall write matrix elements with only subscripted indices here. A cofactor, called Cy, represents a determinant obtained by deleting the i"1 row and j * column of a determinant | p | , with the results then multiplied by (-1)1+J. The set of cofactors, for all i and j represents a matrix array, called the cofactor matrix. The determinant | p | can then be written I P I = £ Pik Cac = £ pik Cik . i

(D23)

k

In the first expression the summation is over i, and k is fixed, while in the second expression the reverse is true. The second expression reflects the fact that rows are not favored over columns in talking about determinants. We illustrate the situation with a 3x3 determinant | p | . Then, from the first expression on die right, in Eq. (D23), we have for k=l, | p | = PnC 1 ,+p2iC 2 l +P3iC3i . (D24)

185

Elements of Determinant Theory

with P22 P23

c 21 =-

CII =

P32 P33

)

Pl2

Pl3

Pl2

Pl3

C 3 i= P32 P33

)

P22 P23

(D25)

Thus, Eq. (D24) gives the same results as Eq. (D6). In a similar way we can see that Eq. (D23) gives the same answers for the determinant as Eq. (D3), regardless of the order of the determinant and regardless of the choice of unsummed index in Eq. (D23). In addition we see by direct expansion that / , Prk Qk - 0 (D26) and 7] Pkr Ciq - 0 ■

(D27)

This situation in Eq. (D26) is tantamount to replacing the i"1 row by the r"1 row in | p | leaving two rows the same in the determinant | p | , which gives zero. In Eq. (D27), we have a similar situation but with columns rather than rows. These expressions can be combined by writing

P I h - z J Pk j Cki

(D28)

and

P I h~

Pik c■Jk ^2 ; k

(D29) We next introduce the inverse matrix P"1 , with matrix elements a$j, according to the relations y ^ pik «kj=<5y • k

(D30) By comparison with Eq. (D29) we see that the inverse matrix elements are given by

a»= T ^ V •

(D31)

As a simple example consider the matrix P given by

A

2

V-i

A

2 1 0 0 1/

(D32)

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Then the cofactor matrix, called cof(py), is:

2

/I

f-

COf(pij) =

\-l

-2 2 2

1\

-- 23 /1

(D33)

Then to get the inverse of the original matrix, we interchange rows and columns and divide by the determinant of the original matrix, to get: -1/2 1 1 -1 -1/2 1

1/2 -1 3/2

(D34)

3. The Equations for the change of g We again discuss a representative second rank tensor gjj. These components comprise an array and thus can be represented by a square matrix. This matrix, then, has a determinant associated with it called g. We introduce the inverse matrix, labelling the matrix elements g'k. From the notion of inverse these matrix elements then obey gijgjk = <5f

(D35)

This is matrix multiplication as the adjacent indices are summed over. Multiply both sides by gmi This then yields (g n u g u )g j k = gmk ■

(D36)

For this to hold independent of the choice for gy, it follows that g^gij^f,

(D37)

as well. In matrix notation this just says G G"1 = 1, where 1 represents the unit matrix (1 in the diagonal elements and zeros elsewhere), implies G"1 G = 1. We next show dg = g glk dgk,

(D38)

Together with

dg=Hdx m

dglk = g £ d x m ,

Eq. (D38) gives

H=-'^

(D39)

187

Elements of Determinant Theory

We still need to obtain Eq.(D38). We shall illustrate the situation with order three. Then taking differentials of Eq. (D6), with p'j replaced by gy, we get dg - dg,,(g22g33 - g23g32) + dg| 2 (g 2 3g31 " g2lg33) + dgl 3 (g2lg32 " g22g3l) + dg 2 l(gl 3 g32 " gl2g33)

+ ....

(D41)

Thus Eq. (D41) can be written dg = E d&i C'J '

IJ (D42) where Cy is the cofactor associated with deleting the ith row and j " 1 column in gy. From the rule of taking differentials of a product, we see, for example, that the term in brackets multiplying dgn, does not have any contributions from elements in the first row or first column. This is just the criterion for the cofactor. In this way we see that the results Eq. (D42) is not just true for a third order determinant, but is true regardless of order. From Eq. (D31) we get, with ay replaced by the inverse g'J, and | p | replaced by g,

Cy=ggJ'

(D43)

Then Eq. (D42) becomes (with upper and lower indices now present the summation convention is implied), dg = dgygj'g,

(D44)

which is what we wished to show, namely Eq. (D38),fromwhich Eq. (D40) follows. 4. Relationship with the Aesthetic Fields Program In order to evaluate Eq. (D40) we make use of the results obtained from the aesthetic fields program, in particular Eq. (1.14), which we restate

g=r; k gy + r j k g i t .

(D45)

ThenfromEq. (D40), using Eqs. (D35) and (D37) we get

|Hgr:k,

< D46)

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which appeared in the text as Eq. (7.4). The equation for y/g, also appearing as Eq. (7.4),

-^r

=

7i r lk.

(D47)

then follows from Eq. (D46). In section 2, we established that e^u obtained from the antisymmetric symbol is a fourth rank tensor. Then from the aestheticfieldsprescription, it behaves like a product of vectors, AjBjCkDi, with regards to change. Thus from Eq. (1.5) we get '' ,kl = r&n esjkI + r? eiM + r ^ eijs, + I*ta eijks. (D48) ax" Inserting Eq. (D17) which is the definition of e^i into Eq. (D48), and evaluating the system for, say, i=l j=2,k=3,l=4, (1=4 is the same as what we have been calling 1=0 in the text), then gives us back equation Eq. (D47). The choice of ij,k,l makes no difference provided these indices are all different, otherwise Eq. (D48) just becomes zero equals zero. In the aesthetic field theory all tensors, regardless of rank, are treated in a uniform way. The fact that we are dealing with an infinite number of tensors should not lead to additional restrictions on the theory. We have seen here that the introduction of the tensor epi, obtained from the antisymmetric symbol, fits into the aforementioned pattern. That is, the results appearing in the text (Chapter 1, section 7), are present when the antisymmetric tensor is introduced, as we have here discussed.

APPENDIX E CURVILINEAR COORDINATES 1. Parallel Transport and the Connection The way one chooses to describe the universe, i.e. the coordinate system one introduces, should not affect how the universe runs (that is, the form of the basic equations). We can consider this a mathematically aesthetic principle. We will find that this basic principle is in harmony with regards to the equations used in the text. We shall study the principle to see what insights can be obtained from it. Said in another way, the basic laws (which are objective) should be independent of the means used to describe the system (which is subjective). The field components do change when we impose a new set of coordinates, but according to the above principle the form of the equations should not. In the text we worked in a Cartesian system as this is the simplest coordinate system to work with and does not simulate any dynamics. However, in principle, nothing stops us from working with a more complicated coordinate system, even if it makes the computation more difficult. A general coordinate system is described by

x'WW.xW)

(El)

We call such a coordinate system curvilinear. Tensor equations have the desirable property (Appendix C) of having the same form regardless of the coordinate system, whether simple or not. Thus, we require that the basic equations should be tensor equations, with respect to general coordinate transformations. Consider a vector field A' (which behaves like the differentials of displacement so far as coordinate transformations are concerned), having the transformation law Eq. (C6). By making use of the chain rule dx" dxm ^ — = 5 ™ , (E2) dx> dx" ' we get dxm Am = — A " (E3) dx" Taking differentials of this we obtain d A " = ^ d A ' + A ' d ^ =^ d A ' + A ' ^ d x " (E4) dx" dx" dx" dx'sdx" Thus, as a result of the presence of the last term dAm does not have the transformation property of a vector, and therefore is not a vector quantity. The reason for this is not only does A' change from point to point by virtue of it being a vector field, these components also change just because the coordinate system is not the same as it was at the original point. Consider two neighboring points P and Q. In order to obtain a differential vector

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190

quantity, we must first parallel transport the vector from P to Q. We then compare this parallelly transported vector with the vector already at Q. We are then comparing two vectors at the same point, where the coordinate system is now identical for the two vectors, and so the result of taking differences is itself a vector. To be concrete, we define the following quantities: dA* = A ( Q ) - A ( P ) <5A = A(P.T. from P -> Q) - Aj(P) DA1 = Al(Q) - Aj(P.T. from P -> Q) ,

(E5)

where A'(PT. from P —» Q) gives the components of the vector at Q after it has been parallelly transported to Q from P. We then see an identity arising from the above definitions DA1 = d A - <5A'

(E6)

Of the quantities appearing in Eq. (E6), DA1 is the vector quantity. We next need a formula for 6A'. 6A1 should depend on dx k , the differential displacement between the two points, and the vector under consideration A' We neglect higher order terms in dxk as the two points are infinitesimally close. The dependence on A1 should be linear. This would guarantee that if we sum two vectors and then parallel transport the result, we would get the same answer if we parallel transport the two vectors and then add. We shall illustrate this below. From the proceeding considerations we have (the minus sign is present to have agreement with other authors) <5A = -Ajk A j dxk

(E7)

The coefficients Ajk are called the connection. They are also called Christoffel Symbols. The above result is valid for any vector under consideration. Thus, we have also <5Bj = - Ajk B j dxk

(E8)

If we sum the two vectors, we get A + B1 = C.

(E9)

We may then parallel transport the results to obtain (as Eq. (E7) holds for any vector), <5C = - Ajk Cj dxk

(E10)

On the other hand, by adding Eq. (E7) and Eq. (E8) first, we get S(A' + &) = - A]k (Aj + Bj) dxk

(El 1)

] 91

Curvi linear Coordinates

By then adding the two vectors as in Eq. (E9), we obtain the same as Eq. (ElO). Thus with the form Eq. (E7), it does not matter if we add the two vectors first and then parallel transport or vice versa. This would not be the case, if we chose instead of Eq. (E7), as an example •5A1 = -Ajkl A> Ak dx1

(El2)

In the Cartesian system there is no change of components of a vector as a result of parallel transport, so A.'k = 0 at all points. Since a scalar is unaffected by the change of coordinate axes at P and Q, we have *(A i B i ) = 0 .

(El 3)

As a result of Eq. (E7), and since A1 is any vector, we see <5Bj = A\k Bj dx k ,

(El4)

and this holds for any Bj. Inserting Eq. (E7) into Eq. (E6), together with the elementary formula from the calculus Eq. (A7), we get <9A' DA' = — dxk + Ajk AJ dxk

(El5)

We define A'.., called the covariant derivative, by means of DA' s A'^dxJ

(E16)

Since the difference between vectors at the same point is itself a vector, it follows from the Quotient Theorem (Appendix C) that A'., is a second rank mixed tensor. Comparing Eq. (E15) and Eq. (E16) yields dA'

A B

* aF + A*AJ

(E17)

Starting off with A,, rather than A1, gives the identity DAi=dAi-<5Ai

(El 8)

DA, = A i ; k dx k

(E19)

We define

Then, by using similar arguments as previously, we find Ajj is a tensor quantity, and is given by the expression 8A Ai;k = ^ -A^ k Aj(E20>

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Math Aesthetics/Nonintegrability

An example of a second rank upper index tensor is Au = A'BJ. Then from Eq. (E7), we get <5Aij = -Aj,,, Amj dx1 - A^ Aim dx1.

(E21)

We require that all second rank upper index tensors behave in the same way under parallel transport. This maintains the linearity property, so that the procedures of parallel transport and addition remain commutative. Starting instead with Eq. (El4) by a similar procedure, we find 6Aij = AS,Amjaxt + Ajj,Aimdxl

(E22)

As DAij and DAjj are tensor quantities, we again use the Quotient Theorem to establish that A1J.k, Aij;)c are also tensors, where DA« = Aij.kdxk DAy = Aij;kdxk .

(E23)

Then, in the manner established previously, we obtain flA'J

A

gAjj

*' = 19x' T T " AiT A™J - AJT A ™ ■

(E24)

In this way, we can obtain the covariant derivative of any tensor quantity. As an example, the covariant derivative of the fourth rank tensor AL is given by 9A A' A

jkl;m

!— + A'

=

Q

T A m

A'

- A'

tm Ajkl

- A L Ajkt .

A'

■'^jm -^tkl

-A' /v

A'

km ^ t l

(E25)

We see a simple pattern emerges, by inspection of Eq. (E25), so covariant derivatives of any rank tensor can be written down readily. It is a simple matter to establish that covariant derivatives obey a product rule similar to that of ordinary derivatives. That is we get formulas, such as (AiB^AyBj+AiBy,

(E26)

We see this by using Eq. (E20) and Eq. (E24). Covariant derivatives can be thought of as tensor derivatives, as covariant derivatives of any rank tensor are themselves tensors. On the other hand, Ajk is not a tensor quantity. We next obtain the transformation properties for Ajk under general coordinate transformations. We start off with the transformation properties of A^

193 A

Curvilinear Coordinates

k =^ T ^ dx

a

■

(E27)

"

We next take —- of both sides, and use the chain rule Eq. (E28),

dK _ dA^ ax" ax! ax'1 axj '

(E28)

to get

_ a2x,m ax1 axjaxk

m 5A4 a x '

dAy

. d2

< + axj axk

/m

i

k A'. ,n 4

ax,m ^ A ; ax"

' axk ax" ax-

dx &c "

(E29)

Now, from Eq. (E20) and the fact that a covariant derivative is a tensor, and transforms accordingly, we have

aAk

A A„

A,i = ^ r - A

n

ax" ax,m .

A ^ ^ — A [

; m

.

(E30)

aAk We eliminate —— from Eq. (E30) using Eq. (E29). We make use of the fact that Eq. (E20) is a tensor equation, so it is valid in the prime system. This enables us to substitute for A|.m in terms of A'| m . In addition we make use of Eq. (E27) to substitute for An in Eq. (E30). With these operations Eq. (E30) becomes: (9A[ m

W

'

ax" ax'ro _ a V ^ -'ax k a x j d ^ d x * ^ 1

n lm ,

chT aK, a / axk a x " a x j

ax"1 - ^ A ; A

S

.

(E31)

At this point we notice a cancellation between the first term on the left and the second term on the right. Transposing all the terms to one side, and extracting a common factor Aj, leaves us with (labels on dummy indices have been routinely changed along the way),

ax" ax/m

aVn

ax'n axs

The bracketed part must be zero since AJ, is arbitrary. Multiplying through by — - , using the chain rule, and rearranging gives ki

=

aV" <**_ a / ih*_ &r axjaxk ax/n

axk ax'n ax'

„ ta

l

;

The system which we call prime and the system which we call no prime is up to us. Thus, in addition, the following formula is also true

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194

,s _ ki

<92x" ,k

"~ dx"dx

dx^_+d^_dx^_d^ a

dx

+

dx'k dx" dx,{

ta

'

This is the transformation law for a connection. It is not a tensor due to the presence of the first term on the right. We can, if we wish, choose the unprimed system to be Cartesian, which is characterized by AjJ,, = 0. Then the formula for the connection, from Eq. (E34), becomes d2xa dx's : z -=—■ (E35) dx"dx'k dxn ' This formula enables us to calculate the connection in any coordinate system. We note, since x" is a regular function of xJ, mixed derivatives are symmetric here (as contrasted with fields in which one is obtaining the field in terms of contributions along path segments as in Chapter 2). Thus, we obtain the symmetry property ,„ A ki =

A]k = 4 j -

(E36)

The first step in determining the connection, is to specify the coordinate transformation which we wish to consider. A simple choice was given in Appendix C (Equation C25) where we were considering rotations. We next compute partial derivatives in the manner done in Appendix C, section 3. In this latter situation second derivatives were all zero, since the angle 9, did not vary from one location to another, so there is no change of tensors under parallel transport in the case of constant rotations. That is, in the simple example of Appendix C, section 3, the connection is zero. 2. Mathematical Aesthetics and General Coordinates The basic postulate, of this appendix, is that the form of the equations is the same regardless of the coordinate system chosen. Thus, the basic equations should be tensor equations with respect to general coordinate transformations. The procedures of Chapter 1, section 6, can then be reproduced using the symbol D rather than d. Thus, we introduce a change tensor r j k , such that DAj = r j k Aj dxk .

(E37)

As DAj is a vector, the Quotient Theorem tells us that T\k is a third rank tensor. Eq. (E37) holds for all vectors Bj, as one (dynamical) vector is not favored over any other (dynamical) vector. Aj on the right of Eq. (E37) implies that the sum of two vectors has the same kind of change equation as Eq. (E37). For the product gy = Aj Bj, we have from above °gij = (T-k gtj + T]k git) dxk .

(E38)

As one kind of second rank tensor is not favored over any other second rank tensor, we require all second rank tensors change by the same set of rules.

195

Curvilinear Coordinates

j

From Aj, gy we can introduce A as follows Ai=gijAJ . All equations used are to be tensor principle set down in this appendix. A1 is a vector. Provided gy has an necessary in Chapter 1), we get from

(E39)

equations as a result of the mathematical aesthetic We then see that the Quotient Theorem implies that inverse (a hypothesis which we have shown is not Eqs. (E37), (E38) and (E39) D A = - r j k A j dxk .

(E40)

The change function, being a third rank tensor, is required to behave like A' Bj C k . We conclude that the change function must determine its own change in the same manner that it determines the change of other tensor functions. Thus, we have

Drjk = (r^r]|' + rimr--r]];rinl)dxl

(E 4i)

Introducing the covariant derivative, which is a tensor quantity, in the same way as in the previous section, we have D^k = rj k;1 dx1 .

(E42)

Then the basic equations for the change function becomes dV Jk r< it r 1 Q

\

l

mk

T^i

Am

Am r 1

A

jl

l

l » i

T^m

A

Am + r m A '

jm A kl

+ l

i

jk A ml

= r ^ r ; + rj m r™-r]jr^,

(E43)

which can be written

rk^r-

re+^rs-rsm,,.

(E44)

The simple rule emerges, ordinary derivatives are replaced by covariant derivatives. Then the equations are valid in any coordinate system, rather than just the Cartesian system used in the text. This rule is valid in other fields of physics, as for example, Maxwell's Equations, where it is commonly used. Thus, for example Eq. (1.14) of the text becomes | |

- A'k gtj - Ajk git = T'k gtJ + T]k g, t .

(E45)

Ajk is then calculated for the particular coordinate system under consideration, by means of Eq. (E35). In the case of computations, the Cartesian system is the simplest, and for this reason has been adopted in all our numerical work. Even though the computation is simplest in the Cartesian system, there are advantages from a conceptual point of view, in proceeding as we have done here, as there is a fundamental mathematical aesthetic principle involved. The equations should be

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196

unchanged in form when a general coordinate transformation is performed. There are two types of partial derivatives for the transformation, namely dx ;i

dx'

8*

^

« *

(E46)

and thus at the outset there are two types of vector introduced using the two types of transformations, which we have labeled with an upper index as well as a lower index. As Eqs. (E43) and (E45) are valid regardless of the coordinate system, they are therefore valid in a Cartesian system. Thus, there is a need for upper and lower indices in the aesthetic fields program even when working in a Cartesian system. Even if gy had the form Eq. (C37) or Eq. (C38) at the origin, it would not keep this structure elsewhere in view of Eq. (E45), even when A'k= 0, and thus Eq. (E39) would not imply Aj = ± A' at all points. If, on the other hand, we started in a Cartesian system, so Eq. (C31) is valid, so that we have a\>

5x"

one might be tempted to introduce a single type of vector and then keep all indices as subscripts. This would lead to a mathematical aesthetics program based on Eq. (C40), which we have remarked implies a restrictive theory. We would not then have been able to obtain the results displayed in this book. Thus, the mathematical aesthetic principle, leading to curvilinear coordinates, plays a role in justifying the need for two types of vectors even when the system is Cartesian. Upper indices have a different character with regard to dynamics than lower indices. We have a similar situation when we talk of the determinant of gy, called g, on the one hand and dynamical scalars on the other hand. Dynamical scalars are constant, and thus have different dynamical properties from g which obeys Eq. (D40) together with Eq. (E45) for any coordinate system. This is so, despite the fact that in Cartesian systems dynamical scalars have the same transformation properties as g since | a | | a | is 1, and use is made of Eq. (D21). Thus, by solely looking at Cartesian systems, just as with the case of upper and lower indices, one could easily be misled and opt for the restrictive theory having T[k=0 (see Eq. (D46)). 3. A Non Dynamic Tensor Although we have been concerned with the change of dynamical tensors, there exist non dynamical tensor quantities in the theory. We mentioned the Cartesian unit vectors i, j , k previously in the text. Although these objects do not affect the change equations, we shall discuss a particular one here, the metric tensor, in order to tie in with what appears in standard texts. When we consider curvilinear coordinates, we have already noted that components change in the operation of parallel transport because the coordinate system is no longer the same at the points involved. There is another feature that comes into play when we have curvilinear coordinates. For illustrative purposes, consider the simple case of two dimensions (the results can easily be extended to any number of dimensions) where we

197

Curvilinear Coordinates

have an oblique set of coordinate axes. We introduce independent basis vectors aj along the coordinate axes. Note here the index i tells us which basis vector we are dealing with. Then a vector A can be expanded in terms of the basis vectors according to (the summation convention for upper and lower repeated indices is understood) A = Aj a{ .

(E48)

A refers to the vector. Equation Eq. (E48) amounts to the parallelogram rule for vector addition as shown in the diagram below, and A' refers then to components of A obtained from parallel projection. This represents an alternate approach to the introduction of vectors from the transformation approach of Appendix C.

A1 a 1

The set of basis vectors is not unique. We may introduce a second set bj, which can be used to define a second set of coordinate axes directions. We can expand A in terms of this set as well A = A'1 bj.

(E49)

Now, any vector can be expanded in terms of the original basis vectors. This is true for the set bj, so we get for each i bi = a k iak .

(E50)

We could also express the aj in terms of the bj, so we have aj=A k ,b k .

(E51)

Then from Eqs. (E48), (E49) and (E50), considering we are dealing with the same vector A here, we get A^A'"^.

(E52)

Math Aesthetics/Nonintegrability

198

We will be able to establish that this transformation is the same as what was introduced in Appendix C, once we bring in partial derivatives associated with the transformation, later on. Inserting Eq. (E51) into Eq. (E50), and using the fact that bj are arbitrary, we obtain (we may also insert Eq. (E50) into Eq. (E51)) a' k A J . = «Jk

A' ja *, = ^

(E53)

We next define Aj = a; • A,

(E54)

and correspondingly A[ = b, A . Then inserting Eq. (E51) into Eq. (E54) and using Eq.(E55) gives A' k A; = A k .

(E55 (E56)

The transformation law for A' and A are seen to be different by comparing Eq. (E52) and Eq. (E56). The definition Eq. (E54) has a simple geometric interpretation, in terms of perpendicular projection of the vector A onto the coordinate axes, shown below in the diagram. A2 /

^ / -» ^^ A^/ i >^ I ^ 1

A / a2•a2

A, V

Q\- Q

We may introduce reciprocal basis vectors, a' by means of a(

aj = <5^

b; • bJ = <5,J

(E57)

In 3 dimensions this implies - e*a; x at a'= '2V

v(E58)

199

Curvilinear Coordinates

where V= aj • aj x ak. In V the indices are all different and cyclic with 123. There is no summation between indices in the numerator with those of the denominator. Eq. (E58) can be established by inserting into Eq. (E57) and observing that Eq. (E57) reduces to an identity for all choices of the free indices. It is a straightforward problem to demonstrate that parallel projection with respect to the original basis vectors aj, is the same as perpendicular projection with respect to reciprocal basis vectors a1, and vice versa. (For example, write A = A'k'ak, defining a parallel projection with respect to reciprocal basis vectors, take the dot product with aj, and use Eq. (E54).) We can introduce tensors of any rank by introducing irreducible tensor products of basis vectors below. This generalizes the way a vector was introduced in Eq. (E48). The tensor is given by T, and its components are given Tuk"' This represents another way to introduce the concept of tensors as contrasted with the transformation approach, although we do not obtain a geometric picture as was the case with a vector. The tensor is then defined by T = T'Jk a iaj a k .... = T'mnp" b m b n b p ...

(E59)

The second part of this expression follows when the tensor is expanded instead in terms of the new basis vectors bj. From the relationship of the new basis vectors in terms of the old ones, Eq. (E50), we get the transformation law TiJk=T'mnp-aimaJnakp...

(E60)

We can as well expand T in terms of reciprocal basis vectors T = Tljk a' aJ ak ... = T^ p .. b m b" bP...

(E61)

The second part follows since it is the same tensor T we are talking about. The new and old reciprocal basis vectors are related by means of b^A'j-aJ a1 = a i j b i

(E62)

We see this is valid by inserting Eq. (E62) into Eq. (E57) and observing that we end up with identities when use is made of Eqs. (E50), (E51) and (E53). Then substituting Eq. (E62) into Eq. (E61) gives the transformation law T, j ,.. = T L p . . . A - A " j A P k

(E63)

We can also introduce mixed components by means of T = Tijk"-

a,ajak...amanap...

(E64)

Math Aesthetics/Nonintegiability

200

The indices can appear in any ordering. It is not necessary that upper indices are ordered first or together. We can also expand in terms of the new basis vectors b1 and bk- Since it is the same T that is involved, we get by the same procedures the transformation law for mixed tensors Tijk-„mp... = T ' a b c - x y z

a ' . a V A*m AV

(E65)

Reciprocal basis vectors, like any other vector, can be expanded in terms of the original basis vectors, so we have ak=Gkiaj .

(E66)

Taking the dot product with a1 on both sides, and using Eq. (E57) yields G^ = ak • a 1 .

(E67)

As the dot product is symmetric in the ordering of the vectors, it follows that Gkl = Glk .

(E68)

In a similar way, we expand a, in terms of a', to define Gjj as = Gy aJ

(E69)

Taking the dot product with a\ gives Gy = a i

aj

(E70)

Inserting Eq. (E66) into Eq. (E69), and using aj is arbitrary, we get Gjk G t a = 6>m .

(E71)

In the new coordinate system defined by the basis vectors bj, Eq. (E67) becomes G'k' = b k

b1 .

(E72)

Then, starting with Eq. (E72) and using Eq. (E62) and Eq. (E67), we get G'ij=AmAjtGmt .

(E73)

When we introduce the curvilinear system defined by Eq. (El), we shall obtain the formulas Eq. (D10), so G'-' has the transformation law of a second rank tensor, Eq. (Cl la). The set of components G1', is called the metric tensor. In a similar way, we obtain G^ = a"i a": G™

(E74)

201

Curvilinear Coordinates

Now A can be expressed in terms of reciprocal basis vectors (parallel projection in terms of reciprocal basis vectors is the same as perpendicular projection with respect to the original basis vectors, so the coefficients in the expansion are A,), so we have A= Aj aj

(E75)

Then using Eq. (E48) and Eq. (E66) gives AiG ik = Ak .

(E76)

AJ Gjj= Aj .

(E77)

In a similar way we get

Thus, the metric tensor has a significance of relating components describing parallel (perpendicular) projection of a vector to the components describing perpendicular (parallel) projection. We may introduce natural basis vector according to the definition dr = aj dx1

(E78)

dr is the vector describing infinitesimal displacements between a point and a neighboring point, while dx1 are infinitesimal increments along the coordinate axes. From the notion of derivative, we have dr=|^dx: dx' Thus, comparing Eq. (E78) and Eq. (E79), we get, since the dx1 are all arbitrary dr ■i = 7T- • dx' The line element (distance between neighboring points) is then using Eq. (E70) ds 2 = dr

dr = a* dx'

a; dxj = Gy dx1 dxJ

(E79)

(E80)

(E81)

We illustrate the notion of natural basis vectors with a couple of examples. For a Cartesian system, the differential displacement vector is dr = i dx +j dy +k dz

(E82)

Then, from Eq. (E79), we see dr . — =i dx

dr . — =i dy J

dr — =k , dz

(E83)

202

Math Aesthetics/Nonintegrability and thus from Eq. (E80) a,=i

a2=j

a3=k

(E84)

Thus, for a Cartesian system, the natural basis vectors are the usual unit vectors i,j,k. From Eq. (E70), Gy is 1 if i=j, and 0 if i ^ j , which is the same as Eq. (C37). For polar coordinates the differential displacement vector is dr = r» dr + rd
(E85)

where ro and OQ, are unit vectors along the coordinate axes. Then from Eq. (E79) we get

g=r.

*-,* ,

aj = r 0

a2 = r 0O-

(E86)

and from Eq. (E80) we see (E87)

The natural basis vectors have dimension length, and are not necessarily unit vectors. Also, dx' are not necessarily lengths (note for example d#). From Eq. (E70) and Eq. (E87) we get G„=l

G12 = 0

G2i=0

G 2 2=r 2 .

(E88)

The metric tensor does not have any nonvanishing elements for i ^ j if the system is orthogonal, as is the case of polar coordinates, since the line element Eq. (E81) is a sum of squares. A curvilinear coordinate system is described by x" = x " ( x \ x 2 , x 3 , x°),

(E89)

in 4 dimensions. The natural basis vectors in the primed system are dr b, = - .

(E90)

Then, from the chain rule we get dr 6x> bj = — — r . dxJ dx" Using Eq. (E80), this gives

(E91)

203

Curvilinear Coordinates

».-,— .

(E92)

dx" a^b,— .

(E93)

and in a similar way

Thus the transformation laws Eqs. (E52), (E56), (E60), (E63), (E65), (E73), (E74) are the same transformation laws as appearing in Appendix C (in some cases we need to multiply through by the inverse transformation and make use of the chain rule), when we use the identification .

dx'

a1; =

'

r

ax"

A

r^

(E94)

Thus, the two descriptions of tensors, one making use of an expansion in terms of products of basis vectors, and the other which makes use of transformation laws when we change coordinates, are hearby linked up. By similar procedures, we see the metric tensor can be used to raise and lower indices for any tensor quantity. For example, we raise the index m in TJ m by multiplying by G™ to get, Tji = G im T j m .

(E95)

We can raise an index in the metric tensor itself to get Gjj G>m = G,m

(E96)

We do not stagger indices here since Gjj is symmetric in the order of its indices. Comparing with Eq. (E71) we see that the mixed components of the metric tensor are just the Kronecker delta. The metric tensor, as we have seen, relates reciprocal basis vectors to ordinary basis vectors (Eq. (E66)). We can if we wish use the metric tensor to raise and lower indices, although we emphasize this is not the only way to raise and lower indices, as we see from Eq. (E39). Next we obtain equations relating the metric tensor at different points. G1' relates parallel and perpendicular projection for both DA' and A'. Thus we have DAj = Gik DAk ,

(E97)

A = Gik A k .

(E98)

as well as

204

Math Aesthetics/Nonintegrability

Inserting this on the left hand side of Eq. (E97) gives D(G ik A k ) = G i k D A k .

(E99)

As DAj is proportional to the infinitesimal quantity dx k , we find that D of a product obeys a product rule the same as an ordinary differential. Thus Eq. (E99) leads to 0 = DGik E Gik;m dx m ,

(E100)

as A1 is arbitrary. Since dx m is also arbitrary we get the equations for the change of the metric tensor Gik;m = 0 ,

(E101)

which when expanded out (see Eq. (E20)), yield ^ - A | k G t J - A ] k G i t = 0.

(E102)

Ak,i, = AJ Gn* ,

(E103)

Defining Akiji as

we see Eq. (El02) becomes <9Gik Changing the labels of the free indices according to i —> k, k —> 1,1 —>■ i gives <9Gki -5-T -Aui-A k ,,i = 0 .

(E105)

If instead we let i —► 1, k —> i, 1 —> k in Eq. (El04), we get • ~ - A i J k - A u k = 0.

(E106)

Multiplying Eq. (El05) by minus one, and then adding the three above equations, and using the symmetry property Eq. (E36), and the definition Eq. (E103), gives _ n Am 1 dGik <9Gii 3G kl A Ailk, - G,mAk, - - ( — + — - — ) .

(E107)

Multiplying through by G" and using Eq. (E96) then gives At

_ i r,,^Gik

dGH

dG kl

which relates the connection (Christoffel Symbols) to the metric tensor.

(E108)

205

Curvilinear Coordinates

Note Gy is a tensor, but it is a nondynamic tensor, and it is fixed once the coordinate system is specified. From Eqs. (E73) and (E94) we have

°

.y = <9x" dx"G ..

^^

(E109

)

st

We may start with a Cartesian system so G is 1 if s=t and 0 if s ^ t. Then from the partial derivatives of the transformation, obtained in the same manner as in Appendix C, section 3, G"J may be calculated at all points. Curvilinear coordinates implies we have components of a vector obtained by parallel projection as well as by perpendicular projection. The relationship between these different type components is fixed once the coordinate system is specified and can be calculated using the metric tensor (see forthcoming Eq. (El 11). To obtain perpendicular projection components from parallel projection components (and vice versa), we see the operation is independent of any dynamics. Another effect appearing due to curvilinear coordinates is the change of components of a vector under parallel transport. This effect is described by Ajk, which is related to derivatives of the metric tensor and appears as Eq. (El08). We recognize that the quantities Glj , Ajk describe the effects of curvilinear coordinates and are not dynamical quantities. We see explicitly that Gy is not a dynamical quantity by noting that the change of Gy, Eq. (E102), is independent of the change tensor rj k . To develop the aesthetic field equations, we require that all (dynamical) tensors are treated in a uniform way with respect to change. When we relate upper and lower indices, in formulating such a theory, we do it by means of a dynamic gy Ai=gijAJ.

(El 10)

This defines A-" in a different way than if we use Ai = GyAj

(El 11)

A' in the former is a dynamical vector, while Aj in the latter situation refers to parallel projection components with respect to basis vectors along the coordinate axes, as contrasted with perpendicular projection components with regards to the same vector. We could always obtain this latter quantity (if we are so interested), if we use Gy obtained from partial derivatives of the transformation as in Eq. (El09). In Appendix C we point out that the tensor calculus is extremely general. The desirable properties of tensors appear regardless of the coordinate system under investigation and regardless of the way we raise and lower indices. This point was emphasized and illustrated in Appendix C. We recognize from Eqs. (El 10) and (El 11) that the operation of raising and lowering indices is not unique. Again we stress, to obtain the aesthetic field equations most directly, we only concern ourselves with dynamical tensors, rather than tensors whose purpose is to describe the coordinate system. The effect of the coordinate system does appear in the aesthetic field equations Eq. (E43) via the connection, which is calculated for the particular coordinate system using Eq. (E35). Gy does not explicitly appear in this change equation.

Math Aesthetics/Nonintegrability

206

One could, if we wish, form new tensors using the metric tensor such as G'J rjj. The change of this quantity would then be obtained using Eq. (E43) and Eq. (E102). As a result, we do not obtain the unified system of equations written as Eq. (1.30) (but with ordinary derivatives replaced by covariant derivatives to take into account the general set of coordinates), but we could calculate the change of these non solely dynamic tensors nevertheless. Thus, in this chapter we observe the role of the metric tensor in the aesthetic fields program. The equations involving the metric tensor that we have discussed, remain valid. However, in developing the aesthetic fields program, we have focused on dynamical tensors, rather than tensors that are descriptive of the coordinate system. We obtain dynamical quantities by choosing to raise and lower indices in a dynamic way, rather than by use of the metric tensor. The aesthetic field equations then take on a simple, and what can be considered, elegant structure. In calculations we stick to the Cartesian system where Ajk = 0, and Gy is 1 if i=j, and 0 if i / j , as this makes the calculation the simplest. 4. A Restatement of the Mathematical Aesthetic Principles When we first considered a set of mathematical aesthetic principles in Chapter 1, we did not have an appreciation of the importance of nonintegrability, nor of the value of general origin point data. Also, we worked exclusively with Cartesian coordinates, even though we recognize in this Appendix, that a basic principle is involved in the use of general coordinates. Thus, at this point we shall restate the mathematically aesthetic principles, made use of, in this book. Principle #1. The basic laws for change, which are objective, should be independent of the coordinate system used to describe the system (which is subjective). Thus, we require that the basic equations should be tensor equations with respect to general coordinate transformations. Principle #2. The supplied information, or data, associated with the equations, is taken (to minimize arbitrariness as compared to hyperbolic equations) to be arbitrary at a single point. Furthermore, this origin point data should be chosen in a general way. That is, the data should not be subjected to conditions that are maintained at all points by the change equations. Principle #3. All (dynamic) tensors are treated in a uniform way with respect to change, regardless of rank. We may say that the principle on uniform treatment of tensors represents the heart of the aesthetic fields program. Nondynamic tensors are tensors that describe the coordinate system. Principle #4. It is understood that all changes to be computed are continuous and singularity free. The superposition at each point (principle #6) should not allow backtracking in forming a path contribution as this implies an infinite number of paths for a finite grid size. Furthermore the results of superposition should lead to well behaved functions. In practice, this postulate means that as we decrease the grid size the result of all numerical

207

Curvilinear Coordinates

integrations and summations should give the same answer to some numerical tolerance, when the grid is reduced. Principle #5. We require the system be self-consistent, adhering to the rules of Aristotelian logic. This principle enables us to obtain the explicit structure of the basic equations. Principle #6. The basic equations, obtained using the previous principle, are expressed as a set of total differential equations that describe change along any path segment. Contributions to the field from segments along different integration paths are treated by a uniform democratic procedure. Results of integration between the origin and any other point are not restricted by requiring integration be independent of path (nonintegrability). The effect of this postulate is a superposition principle at each point. We should continue to keep this list of principles flexible as there remain unanswered qustions. For example the dimension of space remains an unresolved parameter in the theory. We have made suggestions as to which component should represent the particle system, but this has yet to be resolved. We have not ruled which superposition principle at each point should be employed. We have introduced complex fields in Chapter 7, but again we have not introduced any principles to indicate whether the gammas be real or complex. In addition we allow for the possibility of questions not heretofore presented.

References to More Recent Research Articles by the Author 1. M. Muraskin, Update of Mathematical Aesthetic Principles with Discussion of a More Viable Wavepacket Solution, to be published Physics Essays. 2. M. Muraskin, Techniques for Nonintegrable Systems, to be published Hadronic Journal. 3. M. Muraskin, Dynamical Lattice Systems, Computers Math. Applic. 2 (1994) 77. 4. M. Muraskin, Oscillations Within Oscillations, Applied Math, and Computation, 53 (1993)45. 5. M. Muraskin, Chaos and Aesthetics, Computers Math. Applic, 26 (1993) 93. 6. M. Muraskin, Proceedings of the 8th International on Mathematical and Computer Modelling held at the Univ. of Maryland, The Calculus of Aesthetics, Math. Modelling andSci. Computing, 2 ( 1 9 9 3 ) 5 3 . 7. M. Muraskin, Instantons and Nonintegrable Aesmetic Field Theory, Computers Math. Applic, 26(1993)81. 8. M. Muraskin, Proceedings of the Ninth International Conference on Math, and Computer Modelling held at Univ. of Calif, at Berkeley held in July 1993, Mathematical Aesthetics, Nonintegrability and Construction of Mathematical Model Universes using the Computer, Math. Modelling and Sci. Computing, to be published. 9. M. Muraskin, Rearrangement of Lattice Particles, International Journal of Math, and Math. Sciences, 16 (1993) 593. 10. M.Muraskin, Sine Curve Within a Sine Curve, Physics Essays, 5 (1992) 331. 11. M.Muraskin, Beyond Newton and Leibniz, Applied Math, and Computation, 52 (1992)417. 12. M.Muraskin, Computer Two Dimensional Maps of Loop Soliton Lattice Particles Using the New Approach to No Integrability Aesthetic Field Theory, International Journal of Math, and Math. Sciences, 15 (1992) 563. 13. M. Muraskin, Determinant of g in Aesthetic Field Theory, Computers Math. Applic, 22(1991)43. 14. M.Muraskin, Study of Three Component Lattice System, Math, and Computer Modelling, 15(1991)63. 15. M. Muraskin, Summation over Path Degree of Freedom and the Loop Lattice, Applied Math, and Computation, 42 (1991) 197. 16. M. Muraskin and R. Mohnen, Approximation to Summation over Path Approach to Aesthetic Field Theory, Math. Modelling and Sci. Computing, to be published. 17. M. Muraskin, Thw Arrow of Time, Physics Essays, 3 (1990) 448. 18. M. Muraskin, Proceeding of the Seventh International Conference on Math, and Computer Modelling held at Chi. 111., Math, and Computer Modelling, 14 (1990) 64. 19. M. Muraskin, Irregular Oscillations and Aesthetic Field Theory, Applied Math, and Computation, 16(1993)593. 20. M. Muraskin, Wave Packet Solutions and Mathematical Aesthetics, Physics Essays, 6(1993)409. 21. M. Muraskin, On the Nature of Time, International Journal of Math, and Math. Sciences, 13(1990) 179. 22. M. Muraskin, Use of Commutation Relations in No Integrability Aesthetic Field Theory, Applied Math, and Cofrpuimtsfr Sfif W&tB) till.

209

References

23. M. Muraskin, Introd. to Derivatives Which are Not Necessarily Symmetric and a Possible Role for them in Physics, Physics Essays, 2 (1989) 375. 24. M. Muraskin, Study of Different Lattice Solutions in Aesthetic Field Theory, Applied Math, and Computation, 30(1989)73. 25. M. Muraskin, Alternative Approach to No Integrability Aesthetic Field Theory, Math. and Computer Modelling, 12 (1989) 721, Errata 16 (1992) 149. 26. M. Muraskin, Trajectories of Lattice Particles Using the New Approach to No Integrability, Math, and Computer Modelling, 12 (1989) 1545. 27. M. Muraskin, Nonintegrable Aesthetic Field Theory, Math and Computer Modelling, 10(1988)571. 28. M. Muraskin, A Two Dimensional Lattice Solution Satisfying Integrability, Math. and Computer Modelling, 9 (1987) 883. 29. M. Muraskin, Sinusoidal Decomposition of the Lattice, Hadronic Journal Supp., 2 (1986) 600. 30. M. Muraskin, Trajectories of Lattice Particle, Hadronic Journal Supp., 2 (1986) 620. 31. M. Muraskin, Aesthetic Fields Without Integrability, Hadronic Journal, 8 (1985) 279. 32. M. Muraskin, More Aesthetic Field Theory, Hadronic Journal, 8 (1985) 287. 33. M. Muraskin, Aesthetic Fields: A Lattice of Particles, Hadronic Journal, 7 (1984) 296. 34. M. Muraskin, Further Studies in Aesthetic Field Theory Hadronic Journal, 7 (1984) 540. 35. M. Muraskin, Sinusoidal Solutions to the Aesthetic Field Equations, Foundations of Physics, 10(1980)237. 36. M.Muraskin and B. Ring, Increased Complexity in Aesthetic Field Theory, Foundations of Physics, 7(1977)451. 37. M. Muraskin, Particle Behavior in Aesthetic Field Theory, International Journal of Theoretical Physics, 13 (1975) 303.

Index A

ABJL Equations 17,28,158 -Arrow of Time and these equations 54-57 -Commutator method 40-44 -Four dimensional situation 60-62 -Lattice solution 18,19 -Nonintegrability of 17 -Second approach to nonintegrable systems 55-57 -Special case of gamma equations 60-61 -Specification of path method 18-19 -Structure maintained at all points 61 ABMCHF Equations 67,68,70 Accelerator Physics 157 Action principle 3 Aesthetics 1-15,206-207 -Field Equations 5,194-196 -Motivation for 1 -Principles 3-5,15,20-21,24,82-84,157-159,178-180,189,194-196,205 -Summary of program 157-159 Antisymmetric symbol 181 -184 Arbitrary origin point data 4,15,82-84 Aristotelian logic 4,5,195 Asymmetry in time 56-57 Arrow of time 37,54-58,158 B

Backtracking 24,27 Basis vectors 197-203 -Natural basis vectors 201 -203 -Reciprocal basis vectors 198-201 Beats 75,77 Bell's theorem 81,159 Big picture plots 72,74,76,77,78,80,81,82,83,84,86,89-110,144,145 110,144,145 Bohr atom 59,60 Building blocks 59,60,157 C

Calculus, elements of 164-166 -Path independence for integrable systems 167-170 Cartesian Coordinates 2,4,15 Chain rule 172 Change function 4,194 -Equations for 5,10,195 Chaotic looking system 125,126 Characteristic time 74 Christoffel symbols (connection) 190-194,204 -Calculation of 194 -Symmetry properties 194 -Transformation law of 192-194 Circulatory pattern of contour lines 49,116,117 Cofactor 184,185,186,187

211 Cofactor matrix 186 Colon derivative 9 Component to represent particle 12,147-149 Computer project 10 Computerscope 160 Commutator method 40-50 -ABJL equations 40-44 -3 Component lattice 45-50 -Comparison with Random Path Approximation 51,52 Contractions 171,175 Contravariant index 172 Coordinate Transformations 171 -173,182-183,189,192-194,197-200,202-203,205-206 -Rotations 176-178 Covariant derivatives 191 -192,195 Covariant index 172 Creation 88,162 Curved space.absence of need for 1-2 Curvilinear coordinates 189-207 -And mathematical aesthetics 194-196,206 D Derivatives 164 Derivatives (Generalized) 19,21-23,158 -All orders consistent with basic equations 21-23 -At points other than origin 26 Determinants 181-188 -Of second rank tensor 147-149,182-184,186-188 DGEAR (IMSL Routine) 10,69 Dirac equation 3 Dimension (higher than 4) 2,13,46,150-157,162 Dynamical lattice 133-140 E Einstein-Podolsky-Rosen Paradox 82,159 Entropy 56 Epistomology 160-162 Error Problem 10,7584,125 Expansion of the Universe 57 F Feynman R.P. 34 Five component lattice 28-38,72,112-119 Foundations of physics 1,81,161 Functions 33 G Gamma Equations (see Aesthetic Principles under field equations) -Exact Solutions 11-12 -General coordinates 195 -Solutions of 28-38,90-110,112-145,150-157 General coordinates and mathematical aesthetics 189,194-196,206-207 GGUD (IMSL routine) 32 Gradient 171

Index

Math Aesthetics/Nonintegrability

212

Goswami A. 60 Gupta S. 1 H Heisenberg Uncertainty Realations 59 Henon-Heiles Equations 2,4,6,12,15,20,25,159 Hidden effects associated with superposition principles 34,158,159 Hyperbolic equations 4,20,27 I Imperfect lattice 120-124,153 Indices, raising and lowering 175,177-179,201,203,204,205 Infinite number of equations 3,9 Instanton solution 126-133,160 Integrability equations 7-10,11 -Restrictive character of 17 (ABJL system),45 (3 component lattice system), 70,158 (sine within sine system) Inverse of a matrix 185-186 Irregular oscillations along path segments 141-145 J K Knee-Jerk operation 25,26 L Lagrangian 32 Landau and Lifshitz 54,59 Ladder symmetry 116-119,120,122,126,133,137,153 Lattice solution 12-14,19,29 -Doublet basis 119-120 -Dynamical lattice 133-140 -Five component 28-38,72 -Imperfect 120-124,153 -Soliton 28-38,46-50,112-115,116-119,137,152 -Soliton string (loop) 112-116,119-120,160 -Three component 45-50 Logical self-implication 6,161 Lorentz invariance of gamma equations 150 M Mathematical aesthetics (see Aesthetics) Mathematical model universes 60,157,160 Matrix 181 Matrix, inverse of 185-186 Metric tensor 200-206 -Definition 200 -Raising and lowering of indices with 201,203,205 -Relationship with Christoffel symbols 204 -Symmetry property 200 -Transformation property 200,205

213 Minkowski hypothesis 151 Multiparticle solutions (see also Lattice solutions) 12-14 N Newburgh R. 20 Newtonian mechanics 28,54 -Arrow of time 54 -Favoring first and second derivatives 28 Nondynamical tensors 4,196-206 Nonintegrability 13-15,17-38,158,207 -Arrow of time 54-57 -Commutator method 40-50 -Product method 21,50-51 -Random path approximation 51 -523 -Second approach 26-28 -Sum over path method 23-25 Nonlinear Equations, other types 2,3 Nonlinear sine within sine system 69-75 Nonlocality 79-82,87-88,159 Nonsymmetric mixed derivatives 15,17,19,21-24,28,166 O

Origin point data 4,15,82,84,206 Outstanding problems 159-160 P

Parallel projection of a vector 172,197,199 Parallal transport 189-192 Particle solutions -Component to represent particle system 12,147-149 -Instanton 126-133,137 -Maxima (minima) for field component 12,31,132 -Quasi soliton 137 -Soliton 10,28-38,46,112-115,116-119,120,132 -Soliton string 112-115,116,119,120 -Wave packet 59-60,75-79,81,85-88,89-110,132,141-143,157-158,160 Path dependent quantities 25-26,34,158-159 Path independence for integrable systems 167-170 Perpendicular projection of a vector 198,199 Phenomenological theories 34 PhippsT.EJr. 18,20 Phythagoreans (all things are numbers) 1,161 Product method 21,50-51 Proliferation of paths 32 Q Quadrant symmetry 35,119 Quantum Measurement Theory 54 Quantum Mechanics 36,60,81,82,116,159,160 Quasi-solitons 137 Quotient theorem 174-175

Index

Math Aesthetics/Nonintegrability

214

R Ramanathan M. 10 Raising and lowering of indices 175,177-179,201,203-205 Random path approximation 32-34,51 -52 REDUCE 61 Relativity and nonintegrability 20 Restrictive character of integrability (see integrability) Rotatations of a Cartesian system 176-178 Runge-Kutta approximation scheme 10 Rydberg constant 60 S Scalar (dynamic) 7,178-179,196 Schrodinger Equation and arrow of time 54 SchwalmW. 6,161 Second approach to nonintegrable systems 26-28,55-58,118-119,120,125-126,127-128,133,134,137138,143,149,153-157 Self-consistency 4,5,6,195,207 Simple harmonic motion of lattice particles 31 -32,46,133,137,139 Side by side universes 151,162 Sinusoidal behavior along any path segment 18,45,60-62 Sinusoidal wave solution 11 Sine within sine behavior (linear) 63-65,67 -Another linear sine within sine system 65-66,67 Sine within sine behavior (nonlinear) 66,68-75 -Along any path segment 71 -72 Six dimensional system 13,14,150-157 Small picture plots 72,73,78,141-143 Soliton solution 10,28-38,46-50,112-115,116-118,120,132,152 Spatial coordinates 26,55 Specification of path method 13,15,18-19,29-31,74,112-114,121,123,128,130,131,134,136,149,153 Spreading of wave packet 60,87 Stochastic substructure 88,159 Strings (open) 115 String solitons, lattice of 112-116,119-120,160 Stokes Theorem 168-169 Structure of ABJL system maintained at all points 61 -ABMCHF equations 69 Sum over path method 23-25,33,34,49,53,115-116,117,160 Superposition principle at each point 24-25 (see also sum over path method and second approach to nonintegrable systems) Symmetric mixed derivatives and path independent integration 25,167-170,194 T Tensors (dynamic) 4-6,171 -180,182-184,188,189-192,194-196 -Constructed from antisymmetric symbol 183-184,187-188 -Contraction operation 171,175 -Nondynamic 4,179,196-206 -Importance of 173-175 -Quotient theorem 174-175,194,195 -Raising and lowering of indices 135,177-179,201,203-204,205 -Second rank (dynamic) 4-5,6,7,8,172,179,182-184,186,187,191,194-195,200,205 -Tensor equation 173-174 -Tensor equations maintained at all points 10

215

-Vectors 4,6,9,171,197-203 Three component lattice 45-53 Time 26,29,31,55,87,113 -Arrow of 54-579 -Time reversal invariance 56 Tinker-toy concept 157 Trajectories of lattice particles 31 -32,35-38,46,129,137,139 -Non well-defined trajectories 36,129,158 Transformation law of tensors 171-173,203 -Determinant of second rank tensor 183 -Metric tensor 200,205 Two component lattice (see ABJL equations) U Units, lack of need for 1,161 V

Vacuum 59,77,79,88,89-110 Volterra Predator-Prey relations 2,4,6,20 Vectors (see tensors) -parallel and perpendicular projection 197,198,203,205 W

Wave packets 59-60,75-79,81,85-88,89-110,132,141-143,157-158,160 -More viable wave packets 85-110 -Wave packet particles (2 dimensions) 77,80,81 -Wave solutions satisfying integrability 4,11 WeyenbergC. 10,69 X,Y,Z

Index