Gravitation And Cosmology - From The Hubble Radius To The Planck Scale

I1 Application To Gyro Experiments

4. Motivation Previous researchers, ~ o z ~ r e vand " Hayasaka and ~akeuchi,'~ have observed decreases in the weights of small gyroscopes as a linear function of frequency of rotation. They operated their gyroscopes using 400 Hz three-phase power, and measured the weight decreases on chemical balances. The weight decrease had a magnitude as high as 10 mg. Polarity - spin-vector down (clockwise rotation viewed from above) operation resulted in weight decrease, while spin-vector up operation was a null result, spin-vector horizontal resulted in one-half the maximum decrease". The results were strongly criticized in the literature, not on an experimental basis, but on the inability of some researchers to replicate the results. An unsuccessful replication of the results13 used compressed-air jets to accelerate the gyroscopes. and ignored the electromagnetic aspect of the experiments. The opinion that those researchers may have left out some fundamental features motivated the present projects. Two experiments were carried out - one at NASA Goddard Space Flight Center (GSFC), and one in-house at Research Support Instruments, Inc. (RSI).

THE GEM THEORY OF FIELD UNIFICATION

5. Materials and Methods The investigations used gyroscopes from Sperry Flight Systems - Gyro-Horizon Indicator Model # 608588-28. The rotors had a weight of 289.6 g, and a radius of 2.25 cm. The investigations did not seal the gyroscope in vacuum; based on the experimental work of Kosyrev, hermetic sealing rather than vacuum sealing was deemed important. The gyro was mounted on a small aluminum bracket that could itselfbe mounted in two positions to give the gyro spin vector either an up-down orientation (parallel or anti-parallel to gravity vector) or a horizontal orientation (perpendicular to gravity vector). The whole assembly of the gyroscope, photocell rotation speed monitor, and mount had a mass of approximately 580 g. A Jack and Hientz model F20-5 inverter supplied three-phase power with frequency proportional to voltage (at full 115V, frequency was 400 Hz). The wires to power the gyroscope were run along the fulcrum ofthe balances to minimize their effect on the experiments. A CdS photocell and a red LED were added to the gyroscope to observe the modulation of reflected light from a black stripe painted on the rotor. A Tektronics model TDS 210 digital oscilloscope provided a measurement of rotor frequency. The NASA GSFC experiment used a Mettler mechanical balance with a 1 kg capacity and 0.1 mg resolution. It was difficult to use - the readout on a Mettler balance is mechanical, and as such, has a slow response time. The weights had to be adjusted by hand (using front panel knob controls), so it took some time after turning on the gyroscopes to get a reading. The in-house RSI experiment, on the other hand, used an electronic balance, carefully separated from the gyroscope by building a mechanical beam balance that rested on the electronic scale and a knife-edge fulcrum. By counterbalancing the gyroscope with almost equal weight on the other end of the beam, and by locating the scale at 1/4 the distance from the fulcrum as the gyroscope, a 4x resolution magnification was accomplished. A photograph of the balance is shown in Figure 1. herestingly enough, this follows a similar setup by Podkletnov and ~ i e m i n e nwhere '~ weight changes above a superconducting disk were observed. The similarities are intriguing; the superconductor was levitated by a toroidal magnet, rather than bearings, the disk was spun using 400 Hz 3-phase power, much l i e the gyro experiments, and the balance was separated from the electronic scale. The significant difference is that gravitational shielding was observed; that is to say, it was a sample located above the superconductor that lost weight. The Podkletnov apparatus is shown in Figure 2 to illustrate the similarities.

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Figure I: Experimental Apparatus

Two approaches were used for recording weight and frequency. For the GSFC experiment, due to problems with measuring weight changes on a Mettler balance in any quick fashion, the gyroscope frequency was measured during each experimental run, and an average frequency and scale weight change were recorded. In RSI experiment, the frequency and weight were recorded in real-time as the gyroscope spun up and spun 5 second time resolution. This allowed the measurement of frequency down, with dependence ofweight change in real-time, rather than accumulated average.

-

THE GEM THEORY OF FIELD UNIFICATION

rnagcwta with

rntartlng

firtjd

C w

S U ~ O C ~

Figrre 2: Podkletnov and Nieminen Apparatus (1992)

6. Results The results for the GSFC experiment are given in Figure 3. The results were a weight loss approximately 115 of the values reported by Hayasaka and Takeuchi. The best explanation at this time is that the effect of apparatus design on the gravity modifications is poorly understood. If the local gravity field is indeed modified, various parts of the equipment will lose or gain weight, and this means that a slightly different experimental configuration may produce significantly different results. When the experiment was replicated at RSI, using the mechanical balance addition to an electronic scale, the results shown in Figure 4 were obtained.

J.E. BRANDENBURG. J.F. KLINE, AND V. DIPIETRO

276

am-41Emmews OSFC DATA i A # ?

Spin Vector Up

Figure 3: Results of NASA GSFC Experiment Spin Vector Up Hequerrcy b p n d m e d RotOr meet Both AcceletMng and DecskraHngRotor Shown

0

2

4

6

8

tO

Rotor Frequency tto3 p)

Figure 4: RSI Experimental Results

Ln this case, there was excellent magnitude agreement with Hayasaka and Takeuchi, though the spin vector orientation was opposite. Their results are shown in Figure 5.

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Figure 5: Results of Hayasaka and Takeuchi

7. Discussion From the results above, it is clear that the Hayasaka and Takeuchi experiment is repeatable if their experimental methods are followed carefully, and in other circumstances also. These experiments appear to support the prediction of the GEM theory that gravity fields can be modified directly by EM fields. To avoid uncertainty in the interpretation of balance results, a balloon-based neutral buoyancy experiment has been attempted with intriguing results. Improved balloon are planned for the future. Further development of the GEM theory will also be pursued. Thus it appears possible that the GEM theory, unifying the Planck and Cosmic scales, may allow gravity modification and ultimately human flight technology.

Acknowledgements The authors wish to thank Joe Firrnage for his kind support of our research along with Creon Levitte, Fausten, Cheryl my muse, and the rest of the ISSO merry band. Special thanks goes also to Michael Corson of RSI and Robert Feddes of Aerospace for their interest and encouragement in this research. Special thanks also to Richard F. Post and Edward P. Lee for their early encouragement of this research.

J.E. BRANDENBURG, J.F. KLINE, AND V. DIPIETRO

References 1. Brandenburg, J . E. (1992) Unification of Gravity and Electromagnetism in the Plasma Universe, IEEE Transaction in Plasma Science Vol. 20. 6. pp 9-U 2 Brandenburg, J. E. (1995) A Model cosmology based on Gravity-Electromagnetism Unification, Astrophysics and Space Science. 227, pp133 3.Klein, Oskar, (1926) "Quantum Theory and Five-Dimensional Relativity Zeitschrift fur Physik, 37, 895. 4. Sakharov A.D. (1968) "Vacuum quantum fluctuations in curved space and the theory of gravitation" Soviet 5. Zel'dovich. Ya. B. (1967). Cosmological constant and elementary particles. Sov. Phys.--JETP Lett. 6. 316317. 6. Puthoff, H.E, (1989) "Gravity as a zero-point fluctuation force", Phys. Rev. A 39,2333 7. Haisch, B. , Rueda, A, and Puthoff, H.E., "Advances in the Proposed Electromagentic Zero-Point Field Theory of Inertia", AAIA 98-3143. 8. Vallee, Rene-Lwise,(l970) Gravitational and Material Electromagnetic Energy Synergetic Model, (unpublished) 9. F-EClose (1979)"An Introduction to Quarks and Partons" Academic Press, New York, p411 10. Brandenburg, J.E. (2000) "The GEMS theoryn Manuscript in preparation. 1 l.Kozyrev, N., A., (1968) Joint Publications Research Service # 45238 12.Hayasaka, Hideo, and Takeuchi, Sakae, (1989) Anomalous Weight Reduction on a Gyroscope's Righ~ Rotations around the vertical Axis on the Earth, Phys. Rev. Letters, Vol. 63, 25,pp 2701 13.Faller J.E-. et al. (1990) Gyroscope Weighing Experiment With a Null Result, Phys Rev. Lett -64, 8, pp 825 14. E. Podkletnov and R. Nieminen, Physica C 203 (1992) 4 4 1 4 4

CAN GRAVITY BE INCLUDED IN GRAND UNIFICATION?

PETER ROWLANDS I Q Group m d Science Communication Unit, Depurtment of Phyics, University qf Liverpool, Ogord Street, Liverpool, L69 7215, UK.

JOHN P. CULLERNE I Q Group, Department of Computer Science, University of Liverpool, Peach Street, Livepool, L69 72F, UK.

Key words: Grand Unification. Planck mass. U(5) Abstract:

A model of quarks, based on integral charges, suggests a value of sin20w= 0.25, with a consequent prediction of Grand Unification at the Planck mass, and a value of l / a = 11 8 at 14 TeV.

1. A quark model with integral charges Grand unification of the electromagnetic, weak and strong interactions is at present GeV, about four orders below the Planck believed to occur at an energy of order 1 0 ' ~ at which quantum gravity becomes significant. However, there are serious mass (Mp), problems with the current minimal SU(5) model, which fails to predict a full convergence of the three interactions. Also, the assumed electmweak mixing parameter, sin2ow = 0.375, is at total variance with the experimental value of 0.231. A 'renormalization' procedure has been devised to reduce the GU value to about 0.21 at the Z boson mass (MZ).However. applying the renormalized value to the equations for the coupling constants leads to sin2& = 0.6 at GU, in complete contradiction to the 0.375 assumed. In addition. though the weak and strong coupling constants are assumed to be exactly unified at GU, the incorporation of the SU(2) x U(1)electroweak model appears to require a modified value of the electromagnetic coupling, and so GU is not exact between the interactions but occurs only through an assumed group structure. The problem, we believe, lies in the current, rather 'mechanistic , understanding of the fractional values observed for quark charges. Experiments, such as the ratio of hadron / muon production in electron-positron annihilation events, and the rate of decay of neutral pions to two photons. have. of course. repeatedly shown that quarks carry charge units that are fractions of those of the charge on the electron e, either 213 in the 7

279 R.L. Arnoroso et a1 (eds.), Gruvitation and Cosmology: From the Hubble Radius to the Phnck Scale, 279-286 O 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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case of u, c, t, or -113 in the case of d, s, b. Though experiments do not access individual quarks, they show, quite clearly, that only a fractionally-charged quark model can be constructed to explain the phenomenology of the interactions of baryons and mesons. There are, however, logical problems with the position that the experimental results require us to take. One is that it either makes charge imperfectly quantized or it suggests that quarks (or leptons) are not truly fundamental. Of course, we can redefine the fundamental unit of charge to be e13, rather than e, but this would mean that a particle with two units of charge (u) would be as fundamental as one with a single unit (4; it would also mean that electrons, which show no sign of' being composite, would nevertheless somehow acquire three units of the new fundamental charge. A second problem is that grand unified theories imply that quarks and leptons can be combined in a single overall unification scheme. with possible quark-lepton transitions; different units of charge, however, would make it difficult to see how this could be accomplished. In developing a model for an SU(5) group structure, and its breakdown to SU(3) x SU(2) x U(l), based on a symmetry with the Dirac algebra, we have found that fractional quark charges are best treated as an 'emergent' phenomenon, as suggested by Laughlin (1983, 1999). In the case of the fractional quantum Hall effect, in condensed matter, ensembles of particles with only exact units of e somehow acquire the characteristics of perfect fractions of this unit. Laughlin (1999) has explained this, with reference to Anderson (1972), as 'a low-energy collective effect of huge numbers of particles that cannot be deduced from the microscopic equations of motion in a rigorous way and that disappears completely when the system is taken apart'. In his 1998 Nobel Lecture, he states the following opinion: 'The fractional quantum Hall effect is fascinating for a long list of reasons, but it is important in my view primarily for one: It establishes experimentally that both particles carrying an exact fraction of the electron charge e and powerful gauge forces between these particles, two central postulates of the standard model of elementary particles, can arise spontaneously as emergent phenomena. Other important aspects of the standard model, such as free fermions, relativity, renormalizability, spontaneous symmetry breaking, and the Higgs mechanism, already have apt solid-state analogues and in some cases were even modeled after them (Peskin, 1995), but Fractional quantum numbers and gauge fields were thought to be fundamental, meaning that one had to postulate them. This is evidently not true.' The difficulty of establishing a universal charge quantization suggests that there is something not quite fundamental about the present status of the quark theory - just as the widely different masses for the negatively-charged electron and the positivelycharged proton suggested that there was something not quite fundamental about the theory of matter at an earlier epoch. Laughlin's proposal indicates that the redefinition of quark fractional charges as an emergent phenomenon might be a way forward, but still leaves us without a mechanism for achieving this. There is. however, an older theory which might be of some use in this context. This theory has often been seen as an 'alternative' to the standard theory with fractional charges, and, although not completely discarded as a possible model, has gradually fallen into disuse, because of its apparent

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disagreement with accepted experimental results. If, however. we regard it, not as an alternative theory, but as a possible description of a m e c h i s m For the standard theory, then it might again be seen to have value. In introducing the concept of colour to the quark theory in their well-known paper of 1965, Han and Nambu proposed that the differently coloured quarks carried either unit or zero values of electric charge, in preference to the fractional charges assumed in the version of the quark theory proposed in the previous year by Gell-Mann and Zweig. In the Han-Nambu theory, the charge assignments were of the form: Blue

up down

e

I

Green

Red

e

0

0

0

-e

Elue 243

Green 243 - el3

Red 243

I

rather than the more familiar

up down

- el3

-63

required by the Gell-Mann-Zweig version. One of the advantages of this theory was that it made 'colour' differences a natural result of the existence of different charge structures and not an arbitrarily added extra property. It is widely assumed, and this may have been Han and Nambu's original intention, that such a theory implies that, at some sufficiently high energy, the integral nature of the charges will become manifest and colours directly reveal themselves. However, if we assume pegect gauge invariance for the strong interaction, and hence perfect infrared slavery, then no such transition will occur; there will be no finite energy range at which integral charges or colour properties will emerge. The charges will be exactly distributed between the quark components of baryons and mesons, and will be exactly fractional, in every way identical to the fractional charges in the standard theory. The Han-Nambu model will become just another way of representing the theory of fractional charges. Close (1979) expresses it in the following way: 'Imagine what would happen if the colour nonsinglets were pushed up to infinite masses. Clearly only colour 1 [singlets] would exist as physically observable states and quarks would in consequence be permanently confined. At any finite energy we would only see the "average" quark changes and phenomenonologically we could not distinguish this from the Gell-Mann model where the quarks form three identical triplets.' The fractional charges would not, in fact, even be 'averages'; they would be exact because of the effectively infinite rate of 'rotation' between the coloured states or phases. On such a model, the 'intrinsic' charges would be integral, but the observed charges 'fractional'. Although this would not make fractional charges 'emergent' in quite the same way as Laughlin has demonstrated for the fractional quantum Hall effect, involving the collective action of a large ensemble of particles, they could be described as such on the grounds that they would be derived from the collective action of a multiplicity of possible states.

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In earlier work (Rowlands, 1998, Rowlands and Cullerne, 2000), we have found that, assuming that the ultimate charges are integral, but become fractional through perfecl gauge invariance that cannot be broken down at finite energies, we are able to generate both the overall SU(5) and its breakdown to SU(3) x SU(2) x U(1). In principle. we derive five representations of the electric, strong and weak charge states (A-E), which map onto the charge units (e, s, w), and the five quantities (m, p, E) involved in the Dirac equation. The 24 SU(5) generators can be represented in terms of any of these units, for example, in the form:

The Standard Model disregards what would be the 25th generator in a U(5) representation on the grounds that it is not observed; it may be significant that, if such a particle existed, it would couple to all matter in proportion to the amount, and, as a colour singlet, would be ubiquitous.

2 Grand Unification calculations Of course, a model with integral charges would appear to be identical in all effects to the standard representation, and, though perhaps more satisfying from a logical point of view, would contribute no new physics and no new testable predictions. There is, however, another possibility, and its effects would be perfectly testable. and relatively easy to calculate. This is that. although all charge-related phenomenolog~would remain exactly as assumed at present, the gauge relations between interactions, being of a more fundamental nature, would reflect the newly-assumed underlying structure producing the observed fractional charges. Such processes would most likely be evident in the GSW electroweak theory or in Grand Unification, where we would be comparing the contributions of different interactions at a fundamental level on the basis of group parameters independent of the phenomenology. One possibility would be the weak mixing angle in the GSW SU(2) x U(1) theory, calculated from sin 2 0 -

w-x~2.

GRAVITY IN GRAND UNlFICATION?

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For the weak component, with only left-handed contributions to weak isospin, from 3 colours of u, 3 colours of d, and the leptons e and v,

This should be unchanged for intrinsic integral charges. For the electromagnetic component, however, this is not the case. Intrinsic fractional charges, with both left- and right-handed contributions, would lead to

from which sin2ow= 0.375 . With intrinsic integral charges, we have ~ ~ ~ = 2 x ( 1 + 1 + 0 + 0 + 0 +l +1O+) = 8 , leading to

sin2 ow= 0.25 .

Now, it has been observed (Weinberg, 1996) that the value 0.375 for sin2& is in 'gross disagreement' with the experimental value of 0.231. However, 0.25 is much closer, and second order corrections, as we shall see. may suggest a reason for the relatively small discrepancy. In standard theory (Weinberg, 1996, Georgi and Glashow, 1974, Georgi et al., 1974, Masiero, 1984), to obtain anything like the 'correct' value for sin2&, we take the renormalization equations for weak and strong couplings:

and

_1

I - i n q . a3(ru) - ai; 4 P

(3)

and combine these with a gauge-related modification of the equation for the electromagnetic coupling (1 / a), based on an assumed grand unified gauge group structure: where

From (2), (3) and (4), we obtain M x(the Grand Unified Mass scale) of order 10Ir GeV, and then apply (2) and sin2 Ow=

eCu1'

P. ROWLANDS AND J.P. CULLERNE

284

to give 'renormalized'values of sin2ewof order 0.19 to 0.21. This procedure, in fact, not only assumes a particular value for the Clebsch-Gordan coefficient, (= 5 / 3). but also proposes that the final unification between the three interactions is not an absolute equalization, but is achieved only through the mechanism of a group structure. The coupling constants for the strong and weak interactions are assumed to achieve exact equalization with each other, but not with that for the electromagnetic interaction. The calculation also proposes that the exact value for sin2@, at Grand Unification becomes renormalized at lower energies, but does not suggest a mechanism for this renormalization or an independent way of calculating it. It should be stressed here that, while (2), (3) and (6) are well-established results, (4) and (5) are speculative, and are not supported by the experimental evidence. Though the calculation clearly has several rather ad hoc aspects, and its assumptions, especially those referring to the Grand Unified gauge group, are still only plausible conjectures, it is regarded as giving the best available fit to the data. It is not, however, a very good fit, for when the Grand Unification constants are substituted into the equations for the individual couplings (2), (3), (4), these manifestly fail to converge to a single value for the Grand Unified coupling ( a ) . In addition. recalculation of the value of sin2@, at y = 10" GeV gives 0.6 rather than the 0.375 initially assumed in setting up the equations! The equations not only fail to provide answers consistent with experiment, but are also massively inconsistent with each other. The discrepancy is so serious that a supersymmetric solution has been suggested (Amaldi et al., 1991). If, however, we have an indepazdent value for sin2& of the right order, we can perform a much simpler calculation for Mx, which makes no assumptions about the group structure. We combine (2), (3) and (6) to give:

Using typical values for p = Mz = 91.1867(21) GeV, a(~:) = 1 / 128 (or 11129)' @(M:) = 0.1 18 (or 0.12) (Weinberg, 19%, Novikov et al., 1999), and sin2ow= 0.25, we obtain 2.8 x l o i 9G ~ V for the Grand Unified mass scale (Mx). Intriguingly, this is of the order of the Planck mass (1.22 x loi9). If we a s s m that Mx is the Planck mass (on the basis that purely first-order calculations of Mx tend to overestimate its value (Kounnas, 1984)), we obtain a (the Grand Unified value for all interactions) = 1 / 52.4, and = 1 / 31.5, which is, of course, the kind of value we would expect for the weak coupling with sin2@, = 0.25. We also obtain unit strength for the strong interaction (g= 1) at the energy level of baryonic and mesonic structure (approximately, y = m,). There is, as we have implied, some justification in using such precise values for Mx and sin2ewas 1.22 x 1 0 ' % e ~ and 0.25, in the results of the higher order calculations which have been performed for the intrinsic fractional charge model. Kounnas' two-loop approximation (1984) reduces the value of Mx by a factor of about 0.64, while the presence of massive gauge bosons depresses the effective values of 1 / and sin2@, in the energy range Mw- M,, where they are normally measured. Kounnas' plots for sin2& against p! show a distinct dip at Mw- M,, against an overall upward trend. The actual

@(MZ~)

GRAVITY IN GRAND UNIFICATION?

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decrease is from about 0.22 to 0.21, representing a possible decrease in what would be otherwise expected of up ro about 0.02. Of course, these calculations are based on a model assuming intrinsic fractional charges, but a similar depression of the measured values of 1 1 a2and sin2& might be expected on any related model. The renormalization equations for weak and strong couplings require no modification as a result of changes in the quark model, but the U(1) electromagnetic coupling requires alteration in the hypercharge numbers. In particular. from 1 / 6 to I / 2, while goes from -2 / 3 to -1, -1 or 0, depending on hechanges colour, and (dr)Lfrom 1 / 3 to 0, 0 or 1. The fermionic contribution to vacuum polarization is, conventionally (Masiero, 1984),

(:)L

where n, = 3 is the number of fermion generations. However, when modified for integral charges, this becomes

corresponding to the change from c2= 5 / 3 to c2= 3, when sin2eW= 1 / (1 + c2) changes from 0.375 to 0.25. By our understanding, 0.25 is specifically the value for a broken symmetry, produced by asymmetric values of charge, whether or not it is contained within a larger group structure such as SU(5), and would be the value expected at the mass scale appropriate to the electroweak coupling, that is at p = Mw- M,. We do not expect it to be valid at Grand Unification. An analysis of the behaviour of the electromagnetic coupling may indicate the value we should expect. Here, we find that the addition of the term (3 / $I In (M~'/ ,d)to % leads directly to the coupling a for the electrornagnetic interaction, and not to the modified coupling a,,normalized to fit an overall gauge group, assumed in most Grand Unification schemes, for when M, = 1.22 x 1019GeV, p = MZ= 91.1867 GeV, and 0% = 1 / 52.4,

This is ofparticular interest, since it suggests. a little unexpectedly, but quite plausibly, that the 'unification' atMx might involve a direct numerical equalization of the strengths of the three, or even four, physical force manifestations, without reference to the exact unification structure, unless, at Grandunification, c2= 0 and sin2& = 1 in a group structure such as U(5), involving gravitation and a 25'h generator. We may, therefore, propose that the actual symmetry group, incorporating gravity, is U(5), with the additional generator, coupling to all others, representing the gravitational interaction (at least in numerical terms), and SU(5) being the first stage of the symmetry breakdown. Such a unification would be far more exact than one dependent on the constants of a particular group structure, and would confirm the interpretation of sin2ow as the electroweak constant for a specifically broken symmetry, taking the value of 0.25 at the energy range (Mw-M,) where the symmetry breaking occurs.

P. ROWLANDS AND J .P. CULLERNE The particular value of this theory is that it is eminently testable and the testing procedure is quite straightfonvard, with the virtue of not requiring any additional experimentation beyond the programme already laid out for experimental high energy physics. We simply take the equations (2), (3) and (8) (of which the last is new), and the values for sin2ow,Mx,and R, as calculated in this paper, and determine the three interaction strengths at increasing energies. As the energy of our experiments increases, the divergences between the values found from this model and those from the version based on intrinsic fractional charges will become increasingly apparent. A particularly striking case occurs with a, which, on this model would be 1/118 at 14 TeV (the maximum energy of the proposed LHC), in comparison with the 1/125 predicted by minimal SU(5).

3. References Amaldi, U., Boer, W. de and Fiirstenau, H. (1 991 ) Phys. Lett. B, 260,447. Anderson, P. W. (1972) Science, 177, 393. Close, F. E. (1979) An Introduction to Quurks and Partons (Academic Press), p 167. Georgi, H and Glashow,.S. L. (1974) Phys. Rev. Lett., 32,438. Georgi, H., Quinn, H. R. and Weinberg, S. (1974) Phys. Rev. LRtt., 33,451. Han, M. Y. and Nambu, Y. (1965) Phys. Rev. B, 139, 1006. Kounnas, C. (1984) 'Calculational schemes in GUTS', in C. Kounnas, A. Masiero, D. V. Nanopoulos and K. A. Olive, Grand Unljkation with and without Super.yvmmtletry and Cosmological Implication (World Scientific), pp. 145-281. 188-227. Laughlin, R. B. (1983) Phys. Rev. Lett., 50, 1395. Laughlin, R. B. (1995) Rev. Mod. Phys., 71 (1999), 1395. Masiero, A. (1984) 'Introduction to Grand Unified theories', in Kounnas et al. (op. cit.), pp. 1-143, 20-25. Peskin, M. E. and Schroeder, D. V. (1995) Introduction to Quanturtz Field Theory (Addison-Wesley). Rowlands, P. (1998) 'The physical consequences of a new version of the Dirac equation', in G . Hunter, S. Jeffers, and J-P. Vigier (eds.), Causality and L o c a l i ~in Modern Physics and Astronomy: Open Questions and Possible Solutions (Kluwer Academic Publishers), 397-402 Rowlands, P. and Cullerne. J. P. (2000) 'The connection between the Han-Nambu quark theory. the Dirac equation and fundamental symmetries', Nuclear Physics A (in press). Weinberg, S. (1996) The Quantum Theoy ofFields, 2 vols. (Cambridge University Press), Vol. 11, pp. 32732. Novikov, V. A., Okun, L. B., Rozanov, A. Z. andvysotsky, I. (1999)Rep. Prog. Phys., 62, 1275.

GRAVITATIONAL ENERGY-MOMENTUM IN THE TETRAD AND QUADRATIC SPINOR REPRESENTATIONS OF GENERAL RELATIVITY

ROH S. TUNG California Institute for Phyics and Astrophysics (CIPA) 366 Cambridge Avenue Palo Alto, California 94306, USA ar~d Enrico Fermi Institute, University of Chicago 5640 South Ellis Avenue, Chicago, Illinois 60637, USA Email: tung @calphysics.org JAMES M. NESTER Deparfrrzent of Physics and Center for Corrzplex Systewls National Central Universih Chungli 320, Taiwan, ROC Email [email protected]

Abstract In the Tetrad Representation of General Relativity, the energy-momentum expression, found by Moller in 1961, is a tensor wrt coordinate transformations but is not a tensor wrt local Lorentz frame rotations. This local Lorentz freedom is shown to be the same as the six parameter normalized spinor degrees of freedom in the Quadratic Spinor Representation of General Relativity. From the viewpoint of a gravitational field theory in flat space-time, these extra spinor degrees of freedom allow us to obtain a local energy-momentum density which is a true tensor over both coordinate and local Lorentz frame rotations.

1. Introduction Conservation of energy-momentum, which is associated with spacetime symmetry, plays an importanl role in physics. When we trace back the history, we find that a new physics has usually been born with a violation of conservation of energy-momentum. Perhaps the only exception was Einstein's radical idea: general relativity --- a theory of spacetime itself. 287 R.L. Anmroso eta2 (eds.), Giuvitation cind Cosmnlogy: From the Hubble Rcrdius to the Plcinck Scale, 287-294. O 2002 Kluwer Acader~licPublishers. Printed in the Netherlartds.

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The problem of determining the gravitational energy-momentum arose immediately after Einstein's formulation in 1915; attempts looking for a local energymomentum only resulted in a set of pseudotensors. After much effort, people concluded that there is no proper physical local energy-momentum density for the gravitational field. The situation was gradually clarified to the following conclusions: (i) The energy-momentum concept in a gravitational field can be introduced if we replace the spacetime symmetry in ordinary relativistic field theory by the concept of asymptotic flatness, i.e. total energy is well-defined, (ii) Because of the equivalence principle, gravitational energy-momentum is not localized. As the famous textbook of Misner, Thorne and Wheeler teaches [I]: "anybody who looks for a magic formula for local gravitational energyn~onrentumis looking for the right answer to the wrong question

".'

Newtonian gravity theory is based on action-at-a-distance. so in that case we expect only a total energy for a gravitating system. But for relativistic gravity theories we expect meaningful local quantities since the interactions are local and they exchange energy-momentum locally. Roger Penrose was not satisfied with only the total energy for gravitation being defined; he stated 'lt is perhaps ironic that energy conservation, a paradigmatic physical concept arising initially from Galileo's (1638) studies of the motion of bodies under gravity, and which now has found expression in the (covariant) equation V,T"~= 0 - a cornerstone of Einstein's (1915) general relativity - should nevertheless have found no universally applicable formulation, within Einstein's theory, incorporating the energy of gravity itself." and then proposed the idea of quasilocal (i.e. associated with a closed 2-surface) energy-momentum [2]. There have been several proposals (an extensive literature was given in Ref.1 of [3] ) for quasi-local energy-momentum. They need either a reference background [3, 41 or a globally defined spinor field [5]. The meaning for a reference background was in fact pointed out by Poincark [6, 71-that the physical description is often based on a prion' conventions. For spacetime geometry, two points of view are possible, (i) According to general relativity, the line element between neighboring events is measured by using rods or clocks with the same length or rate which are independent of the field present. The resulting spacetime is curved in general, (ii) On the other hand, one can define the line element to be of Minkowskian form. Accordingly, the rods and clocks are affected by the gravitational field. Apart from global topological questions the two complementary points of view are equivalent. The origin of Weyl's gauge idea is in fact to abandon the idea of adding lengths in general relativity, in Weyl's opinion, keeping a rod to have the same length is a concept which involves actionat-a-distance [8]. The same applies to clocks. '1n order to be a good student, we should probably not ask such a "wrong question". But in this paper, we are being naughty students.

GRAVITATIONAL ENERGY-MOMENTUM

289

Microscopically, it is very difficult to have classical concepts such as rods and clocks giving a simple microscopic understanding of gravitation, Hence the main effort of current quantum gravity is to find new concepts for the space-time at the Planck scale, there have been many stimulating ideas, e.g., strings (p-branes), twisters, or loops. These concepts provide a rich structure for space-time at the Planck scale, From the field theory point of view, microscopically space-time geometry enters only as a background concept necessary to defining a field theory. The search for a good expression for local energy-momentum is thus especially important since the energy concept is associated with the fundamental structure of space-time, Since for Einstein's general relativity, there is no space-time symmetry in general, we do not expect a local conserved energy-momentum. In this case, we expect at most a quasi-local definition. However, considering Einstein gravity as an ordinary field theory in flat Minkowski background space-time, we do have a Minkowski space-time symmetry. Can we then obtain an energy-momentum density tensor in this case? In this paper we will obtain such a quantity by a change of variables and by adding extra spinor gauge variables.

2. Metric Representation The Hilbert Lagrangian density for General Relativity is LH = -t/--TjR. The traditional approach uses the metric coefficients in a coordinate basis as the dynamic variables, so CH = CH(g, dg, adg). Because of the second derivatives, this is not suitable for getting an energy-momentum density. However a certain (noncovariant) divergence can be removed (without affecting the equations of motion) leading to Einstein's Lagrangian CE = CE(g, ag) = CN - div. One can now apply the standard procedure and get the canonical energy-momentum density. It is known as the Einstein pseudotensor; its value depends to a large extent on the coordinate ("gauge") choice. No satisfying technique has been found to separate the "physics" from the coordinate gauge.

3. Tetrad Representation An alternative is to use an orthonormal frame (tetrad), a pioneer of this approach was Mgller [9]. Let ~ P U= gabea~ebv,with gab = diag(+l, -1, -1, -l),and regard the Einstein-Hilbert Lagrangian as a function Le (e, ae, dae) of the tetrad eaP. A suitable total divergence can again be removed yielding a Lagrangian density which is first order in the derivatives of the frame. Now there is an associated energy-momentum density which is a tensor (density) under coordinate transformations, but it depends on the choice of orthonormal frame (Lorentz gauge) [9, 10, 111. 4. Quadratic Spinor Representation

Let $ be any Dirac spinor field and let

Q

= &q, where .9 = fluya = eapyadzfi

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R. S. TUNG AND J. M. NESTER

+

. ~ covariant differential, DQ := dQ wQ, includes is a Clifford algebra l - f ~ r m The the Clifford algebra valued connection one-form w := i X b w a b . NOW consider the second covariant differential on Q, using Dfia = 0 , (i.e., the torsion 2-form vanishes for the Levi-Civita connection), we obtain

Here we have introduced the convenient (Hodge) dual basis qa-. := * ( f i a A . . .) and used the identity ymyab = 2gmIax1- €mabc7C75.The first term vanishes by the first Bianchi identity and the second term.

is proportional to the Einstein tensor. This provides a succinct representation of the Einstein equation. The Quadratic Spinor Lagrangian (QSL) [15, 16, 17, 18, 191 is given by

This QSL satisfies the spinor-curvature identity [20]

where $2 = :flabxb = dw + ww, is the Clifford algebra valued curvature 2-form. The field equations 0 2 8 = 0 and D ( % ~ , ~ ~=~0 Qare ) equivalent to the Einstein equation and torsion free equation, respectively. The metric is given by gp, = Q(,Q,) . The rhs of (4) expands to

Since fl a b A qab = -R

* I, for a spinor field

$J, normalized according to

we find that this QSL differs from the standard Hilbert scalar curvature Lagrangian only by an exact differential,

In the action this corresponds to a boundary term which does not affect the local equations of motion. *Our Dirac matrix conventions are '~(,ya) = h b , ~ ~:= t , ' ~ := 5 7 0 7 l 7 2 ~ .We often omit the wedge A; for discussions of such "clifform" notation see [12, 13, 141.

GRAVITATIONAL ENERGY-MOMENTUM

5. Spinor Gauge Invariance of the QSL

From the form of the Lagrangian (7), the QSL action for an extended region actually depends on the (normalized) spinor field only through the boundary term, not locally. A change of the spinor field within the interior of the region will leave the action unchanged. Consequently the Dirac spinor field $ has complete local gauge invariance subject to the two restrictions (6). This six real parameter spinor gauge freedom can be represented in the form $ = U$,-, where t,bo is a normalized Dirac spinor with constant components and U is the Dirac spinor representation of a Lorentz transformation. Thus the gauge freedom of the normalized spinor field is a kind of local Lorentz gauge freedom. Considering the scalar curvature term in the Lagrangian (7), it can be recognized that the theory also has the usual local Lorentz gauge freedom associated with transformations of the orthonormal frame. Hence there appears to be two Lorentz gauge Freedoms here. But are they really independent? The boundary term is

Let us consider a gauge transformed spinor field l(r' = U ~ Then . = $U-l, D$' = UD$ and = D($)U-l. The gauge transformed boundary term then becomes

DV

The unitary transformations on the gammas induce Lorentz transformations, U-'ya ycLCa,on the orthonormal frame indices. Thus. the six parameter spinor gauge freedom $ (with normalization condition) is entirely equivalent to applying the transformation $IC = LCaflato the orthonormal frame alone. Hence the boundary term really has one physically independent Lorentz gauge freedom. We showed that the QSL is dynamically equivalent to the tetrad (teleparallel) representation [21]. In the QSL we have a spinor field which has a six parameter local gauge Freedom which effectively replaces the local Lorentz frame gauge freedom of the tetrad representation.

6. Gravitational Energy-Momentum Density From Noether's theorem, with&, = Cup* t h o = a,] d+ A (aL/ad+) - a,] we obtain a canonical energy-momentum 3-form.

C(#),

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R. S. TUNG AND J. M. NESTER

satisfying the conservation of energy-momentum dEp = 0. Here the covariant differential D only operates on the spinor but not on the spacetime index. Expression (10) is still another pseudotensor, however, if we consider as a field in Minkowski spacetime, then this is a gravitational energy momentum tensor. Therefore classically, whether we need the concept of an energy-momentum density or not, depends on our viewpoint: (i) When treating the spacetime to be curved, we have no spacetime symmetry in general. According to the equivalence principle, there is no well defined energy-momentum density, (ii) With a Minkowski line element, associated with the spacetime symmetry we have a covariant energy-momentum density. This is useful in microscopic physics, where space and time lose its operational meaning.

*

7. Discussion

By changing the ten parameter metric variables gp,, to the sixteen parameter tetrad variables eup in the variational principle, Mgller [9, 101 obtained a local energy-momentum density which is a tensor wrt coordinate transformations but is not a tensor wrt local Lorentz frame rotations. Recently de Andrade, Guillen and Pereira [ l l ] gave a refined version of Mgller's expression, but it still depends on a local Lorentz frame rotation. In this paper, we showed that by adding an auxiliary Dirac spinor field to the action, we can obtain an expression which is a tensor wrt both coordinate and local Lorentz frame rotations. This extra six parameter spinor gauge freedom (eight parameter Dirac spinor with two normalization conditions) was shown to be equivalent to the Lorentz transformation for the associated orthonormal frame. By comparing the formulation with Yang-Mills gauge theory (Table I ) , we find that we can define a gravitational field strength FG = DY. Table 1: Comparison between Quadratic Spinor Representation of General Relativity and Yang-Mills Gauge Theory

Potential

A=AI~~X'T~

Field Strength

FyM= D A

Field equations

D*F=O

3 ! = el,dxfiyrplt

FG = Drl!

D y s F ~= 0

Presently it is not so clear here what is the corresponding gauge group in this formulation. Several related approaches may clarify the situation: (1) using a semi-direct sum of the group SL(2,C) and c4 [lg], (2) the Teleparallel approach [8, 21, 22, 231, (3) considering the spinor one-form as an anticommuting field

*

GRAVITATIONAL ENERGY-MOMENTUM

293

[24, 25, 26, 271. We also note that there is an interesting generalization of the quadratic spinor Lagrangian to the Einstein-Maxwell system [28]. We close by noting that the approach discussed here also suggests that we might be able to find an expression for a covariant gravitational "Lorentz force law". References [l] C. W. Misner, K. S. Thome and J. A. Wheeler, Gravitation, (Freeman, San Francisco, 1973), p 467. [2] R. Penrose, Proc. R. Soc. London A381.53 (1982). [3] J. D. Brown and J. W. York, Phys. Rev. D 47, 1407 (1993), gr-qc/9209012. [4] C.-M. Chen, J. M. Nester and R. S. Tung, Phys. Lett. A 203, 5 (1995), grqc/941 1048. [5] A. J. Dougan and L. J. Mason, P h y ~ .Rev. Lett. 67, 2119 (1991). [6] H. Poincare, La science et lfhypothPse, (Flamrnarion, Paris, 1904). [7] C. Wiesendanger, Class. Quanturn Grav. 13, 681 (19%). [8] F. Gronwald and F. W. Hehl, in Proc. of the 14th Course of the School of Cosrnology and Gravitation on Quanturn Gravie, P. G. Bergmann et a1 ed. (World Scientific, Singapore, 1996), gr-qd9602013. [9] C. Mflller, Mat. Fys. Dan. Vid. Selsk. 1, No.10, 1 (1961); Ann. Phys. 12, 118 (1961). [lo] J. N. Goldberg, in General Relativig and Gravitation - one Hundred Years After the Birth of Albert Einstein Vol.1, A. Held Ed. (Plenum, New York, 1980), p.478. [I 11 V. C. de Andrade, L. C. T. Guillen and J. G. Pereira, Phys. Rev. Lett. 84, 4533 (2000), gr-qc10003100.

1121 A. Dimakis and F. Miiller-Hoissen. Class. Quantum Grav. 8. 2093 (1991). [13] E. W. Mielke, Geornetrodynarnics of Gauge Fields, (Akademie, Berlin, 1987). 1141 F. Estabrook, Class. Quantum Grav. 8, L151 (1991). 1151 J. M. Nester and R. S. Tung, Gen. Rel. Grav. 27 . 115 (1995), gr-qd9407004 [16] R. S. Tung and T. Jacobson, Class. Quanturn Grav. 12, L51 (1995), grqc/9502037.

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R. S. TUNG AND J. M. NESTER

[17] D. C. Robinson, Class. Quantmz Grav. 12, 307 (1996). [18] D. C. Robinson, Int. J. Theor. Phys. 37, 2067 (1998). [19] R. P. Wallner, J. Math. Phys. 36, 6937 (1995). [20] J. M. Nester, R. S. Tung and V. V. Zhytnikov, Class. Quantum Grav. 11, 983 (1994), gr-qcl9403026. [21] R. S. Tung and J. M. Nester. P h p Rev. D 60. 021501 (1999), gr-qcl9809030. [22] Y. M. Cho, Phys. Rev. D 14,2521 (1976). [23] V. C. de Andrade and J. G. Pereira, Phys. Rev. D 56, 4689 (1997), grqcl9703059. [24] L Bars and S. W. MacDowell, Phys. Lett. B 7l, 111 (1977); Gen. Re1 Grav. 10, 205 (1979). [25] S. W. MacDowell and F. Mansouri. Phys. Rev. Lett. 38. 739 (1977). [26] F. Mansouri, Phys. Rev. D 16, 2456 (1977). [27] R. S. Tung, Phys. Lett. A 264, 341 (2000), gr-qcl9904008. [28] L. McCulloch and D. C. Robinson, Class. Quantum Grav. 17, 903 (2000).

SPINORS IN AFFINE THEORY OF GRAVITY

HORST-HEINO V.BORZESZKOWSKI Technical University Berlin, 1nstitute.for Theoretical Physics Hardenbergstrasse 36 0 -10623 Berlin. Gernlany HANS-JURGEN TREDER Rosa-Luxemburg-Str. 17a 0 -14482 Potsdam, Germany

We consider some mathematical aspects of purely affine theories of gravity. In particular, we show that in affine spaces one can establish a truncated spinor formalism reducing in metric-affine spaces to the standard one. As a consequence, one can formulate gravitational equations with matter sources to which there can exist solutions with a Riemann-Cartan geometry.

1.

Geometrical fundamentals

1.1 AFFINE GEOMETRY To begin with, let us make some general remarks on the linear connection field from the point of view of the tangent bundle.' We consider the connection defined in the tangent bundle T(V4) over V.. Then to the transformation ( x i ,x A i ) -,( x r,xAJ7) of the bundle coordinates there corresponds the following transformation of the local components AABi of the Lorentz connection r

Here dxk/dxr corresponds to the transformation xi W A ' ~ = O 8 ~A ' w ~ ) to the transformation xAi + X A ' ~ . Using in T(V4 ) natural bundle coordinates xki onehas

+xi'

and

~ A ' B

(with

and (1) reduces to the (Riemann-Christoffel) transformation law of a linear connection under coordinate transformations (it acts only on space-time indices): For the fibre bundle theory see, for example, 111; for the above discussion, see also [2].

295 RL. Amoroso et a1 (eds.), Gravitation arzd Cosmology: From the Hubble Radius to the Planck Scale, 295-302 O 2002 Kluwer Academic Publishers. Printed in the Netherlands.

H-H V. BORZESZKOWSKI AND H-J TREDER

On the other hand, for xi + 6i1x1 (i.e., dxi/axi' = 6:,, from (1) one obtains the (RicciLevi-Civita) transformation law of the connection under Lorentz transformations (it acts only on the anholonomic indices)

+ COA'cCOKC

AA'~ti'=

(4)

,j.

Considering now the transformation xAi = hAkxki (with hAi E GL(4,R ) ) from anholonomic to holonomic bundle coordinates, then, according to (1) the following (EinsteinCartan) transformation ri1+ of the connection is associated with it:

rtl

AAgi = hA rn hskrmki+ hAmhsm,i= hAmhsm;i .

(5)

This relation is equivalent to the so-called Einstein lemma (I/denotes here the covariant derivative with respect to the connection r i k l and /// the derivative acting simultaneously on holonomic and anholonomic indices) hBiiiil:= hBj,l - hBrrrif+ hCiABci= hBiiil + hCiABcl=0. (6) The local Lorentz transformations and the coordinate transformations which are above considered as different transformations of the coordinates in a Lorentz bundle can also be regarded as the homogeneous and inhomogeneous parts of the local Poincark group2 hi'

dxr = ~ ~ k ( x l4--dXk ) d ~ ~

(7) axk acting in an affine space. Then, relation (6) can be considered as an implication of the transition from one item of the Poincark transformation to the other.3 The mathematically equivalent relations (3)-(7)can also be justified by the physical requirement that a physical theory has to enable one to define measurement values. That this requirement is really satisfied by (6). and thus by the related formulas (3), (4). (3, can be seen as follows. The so-called Einstein coefficients Aiki:= hgihBk,l= -hig,lhBk are the components of the connection defining the teleparallelism of Einstein and Cartan: The measurement values VB= ~ B ' X and V B= h B;V i of vectors and coi vectors V do not change when they are transported by means of this connection: =, - A =0 6 1 / 1 1 = @::,I - VrArii = 0 1

1

f-)

++

V B ,=~0 VB,;= 0.

(84 (8b)

2 ~ ngeneral, instead of the local Poincar6 group (semiproduct of translations and pseudo-orthogonal transformations), one can consider the affine group (semiproduct of translations and general linear transformations) [cf., e-g., [3], [4], [5], [6]]. Here we confine ourselves to the Poincark group since this group can be interpreted most simply as an requirement following from the principle of general relativity. Following Schouten [7], we here consider the two parts of the Poincarb transformation as assotiated to anholonomic and holonomic transformations. For another understanding, see the Poincari gauge field theory (Hehl et al. 1995 [6]). In this approach working in the framework of metric-affine space, the homogeneous

rL

(linear) item of the Poincark group is ascribed to a tensorial connection part which has to be added to the inhomogeneous (translational) part T".This procedure exploits the fact that the sum of a connection

+ can conflict with and a tensor is again a connection. To our mind, the generalized connection " r L the Weyl lemma establishing the connection between the holonomic and anholonomic representations.

rT"

SPINORS IN AFFINE GRAVITY

297

That the coefficients A i k l really provide a teleparallelism can be seen from the fact that the transport is transitive and integrable and thus independent of the way along which it is done. This is reflected by the vanishing of the curvature tensor formed from the Einstein coefficients: Aiklm

= -Aiki,rn

+ Aikm,l - AirnrA'kl

-tAir*Arh.

(9) With the A n s a t z given in (5),this provides the Cartan-Schouten representation of the Riemann tensor in anholonomic coordinates,

In order to couple gravity to matter one has to introduce spinorial quantities in affine spaces. To do so, it is helpful to remind of the fact that spinors vV,yz" (with V , V = 1,2) are "scalars that satisfy a particular rule of differentiation" [8], [9]. In other words. we employ the one-to-two correspondence of the Lorentz group 0(3,1) to the unimodular group SL(2,C) and require that the differentiation has to be covariant with respect to unimodular transformations a/: with det a , p = det aofi= 1,

(12)

and where the avpand avfi are connected by the relations

The coefficients of the spinor connection, variant derivatives [lo],

&,I

and Ab+j,which allow to built the co-

are specified by demanding Einstein's rule of teleparallelism (6) which now takes the form of "Weyl's lemma" demanding the covariant constancy of the spin-vectors4 o p v k :

+ =C T P ' ~, ~ d ' " r ~ ' ~=/ 0.

+

b p v k ] l ] l = G P ~ ~ , Ib p v r r ~ ~ ~ a v k ~ p ~a pi f i k ~ v

d

(15)

This provides the following expressions for the coefficients of the connection, Aikl

=c

fikl

=oip&okpP~'fil

i . c pv I pv k $ 9

z.

(1 6 4

+ biapokP' nap,

Asp = 1 ~ a ~. k b p p k l l i ,

(16b) (1 6c)

This is the notation introduced by Infeld and van der Waerden [lo]. In modern literature, sometimes it is also called "soldering form".

H-H V. BORZESZKOWSKI AND H-J TREDER

298

1.2 RIEMANN-CARTAN GEOMETRY AS A SPECIAL CASE OF AFFINE GEOMETRY In special cases, this purely affine framework can turn into a metric-afJine framework, namely when the field equations, (SLf ST)ST = 0 (for them se 5 2), have a solution riklthat allows one to introduce a covariantly constant (symmetric) metric tensor satisfylng giklll = 0 -

(17)

The solution of (17) is given by

={:I+

K;[

where the contorsion KLl is anti-symmetric in the first two indices, Kikl =-KkilOtherwise, the Weyl lemma translating the connection from holonomic (space-time) to anholnomic (internal) coordinates provides for the anholonomic representation of the connection r the expression

AAs1 = Y yikl

+K A ~ l

~ B I

= -ykil ,

Kikl

=-Kkil,

(19) (20)

hAkhjll1,and the last expression in (19) is defined as KAsr = h A jhBk K L ; ~ both are antisymmetric in the first two indices. As a consequence, the anhlonomic components of the internal connection is antisymmetric in the first two indices, too: Ajkl = -&if. (21)

where

~ A B are I

the Ricci rotation coefficients,

YABI =

Exploiting the validity of the lemma of Ricci together with the relation finally one finds gikllf

= g i k l l l f = ( h A i h B k q ~ B ) /= t ~0l

gik = hAihAk,

(22)

such that and, thus, is satisfied. This recovers the usual spinor f~rmalism:~

5 ~ means y of another line of arguments, this was also shown by Hayashi [ I l l . Even more, it was demonstrated there that this formalism can only be introduced iff the nonmetricity is of the Weyl form or vanishes.

SPINORS IN AFFINE GRAVITY

2. Affine Theory of Gravity In the afline franlework based on the symmetries given by the Poincark group, geometry is completely determined by the affine connection r i k l . Thus. a field theory has to start from a purely affine Lagrangian L = L ( r *y) (26) Its variation with respect to rikl and the matter field provides the following purely a f fine field equations

A classical example of an affine theory is the Einstein-Schrodinger theory stemming from the Lagrangian density [12], [13] (for its canonical structure, see [14])

According to the intention of their founders, it should represent a unified geometric description of gravitational and matter fields so that an additional matter Lagrangian need and must not be introduced. For we, however, want to regard (28) as the Lagrangian of vacuum gravity, here a matter Lagranpian has to be supplemented. It should be stressed that, as long as we aim at field equations of second order, (28) could only be supplemented by the square root of the determinant of the homothetic curvature. But, for in our framework we shall be interested in special solutions of the field equations which enable us to introduce a metric, this term is equal to zero. Using the Einstein affine tensor

as field coordinates the field equations stemming from (28) read [14]

where

and the Ricci tensor is given as

These equations satisfy the differential identity gpklk,i =0

300

H-H V. BORZESZKOWSKI AND H-J TREDER

which first time was derived by Einstein and Kaufmann [12]. Eq. (34) holds due to the invariance of (29) under the transformation,

In more detail, it holds:

As was shown by Bergmann [15], the principles of general relativity (i.e., of coordinate covariance) and of equivalence (i.e., of minimal coupling), implying a total Lagrangian of the form6 L = LG[g, rl + [g,r, VI (37) always lead to the following identity

where

8 the change of a quantity with fixed values of the coordinates,

6xi = 5' is the

infinitesimal change of the coordinates coming in via the relation 8Q3 = -(G{i),i satisfied for a scalar density G , and 6L/m and SL/Sy the Euler variations of the Lagrangian with respect to the connection and the matter fields. If the gravitational and matter field equations are satisfied the last three terms on Eq. (38) will vanish. In that case there exists a covariant vector density

satisfying the equation (Noether identity) 6',i = 0 . The divergence (40) is a scalar density, too. In paper [15], it is also shown that, if we consider the case of a rigid displacement along one coordinate direction, = ai = const., then the expressions (39) and (40) reduce to the following 6' = 8jkak, 8 i k , ' = 0

ti

The coefficients 29;k are no covariant tensors, but only affine tensors and can be considered to be the components of the energy-momentum complex of the total system including gravitational and matter parts.7 In our case of the pure vacuum Lagrangian H , the relation (38) takes the form

For the sake of generality, in this passage, we shall assume that there exists also a metric which is independent of the connection, and we include the the matter term which comes into the game only in Sec. 2 . To be more precice, the points summarized in this passage are in [15] partly formulated for the case of GRT. But they are evidently true for all Lagrangians of the form (37).

SPINORS IN AFFINE GRAVITY

301

so that, in the case that the field equations (30) are satisfied, Einstein's conservation law results:

Thus, the corresponding Noether identity reads ti,i = 0,

where

Therefore, the Hamiltonian density corresponding to the Lagrangian H is given by the expression

This shows that the Einstein-Schrodinger theory with the field equations (30) can be rewritten in a canonical form. where the field coordinates are given by the components of the Einstein affme tensor U k and the field momenta by the components of the Ricci tensor R,,. Interestingly, as relation (36) shows, the momenta are invariant with respect to the gauge condition (35). As far as the canonical form is concerned, one finds here the same situation as in Maxwell's electrodynamics. Starting in the Maxwell case from the Lagrangian

one obtains the canonical energy-momentum tensor

revealing that the field momenta that are canonically conjugate to the field coordinates are given by the field tensor which are invariants of the gauge group of the electromagnetic theory, & = A,- + @,i . This gauge invariance can be preserved in the case of matter coupling considered in [16]. Adding the Dirac Lagrangian to (28) then the total Lagrangian is invariant with respect to gauge transformations of the second kind,

r&4

I-;, -+ 6; qSl

pV + ei@tpv, xi' + eci@xv, where (50) corresponds to thetransformation ly + ei#ty of the the Dirac spinor

(49)

(so)

In vitue of the Noether theorem, this invariance leads to the vanishung of the covariant divergence of the current density. ly.

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3. References Kobayshi, L., and Nomizu, K.: Foundations of DifSerential Geometly, Interscience, London, 1963. Treder, H.-J., and v. Borzeszkowski, H.-H.: Int. J. Theor. Phys. 8, (1973), 219. Hayashi, K., and Nakano, T.: Progr. Theor. Phys. 38 (1%7), 49. Treder, H.-J.: Znt. J. Theor. Phys. 3 (1970), 23. Kopczyliski,W.. and Trautrnann, A.: Sprrcetime and Gravitation, Wiley, Chisester and PWN Polish Scientific Publishers, Warsaw, 1992. Hehl, F. W., McCrea, D., Mielke, E. W. and Ne'man, Y.: Metric-affine theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance, Phys. Reports 258 (1995), 1-171. Schouten, J. A.: Ricci-Ccrlculus, Springer, Berlin etc., 1954. Einstein, A., and Mayer, W.: Semivektoren und Spinoren, Sitz. Ber. Preuss. Akrrd. Wiss. Berlin, 5D-550 (1932). Weyl, H.: Zs. Physik 56 (1929), 530. Infeld, L., and van der Waerden, B. L.: Die Wellengleichung des Elektrons in der allgemeinen Relativitatstheorie, Sitz. Ber. Preuss. Akcrd. Wiss. Berlin, 380-401 (1933). Hayashi, K.: Phys. Lett. 65B (1976), 437. Einstein, A.: Zur allgemeinen Relativitatstheorie, Sitz. Ber. Preuss. Akcrd. Wiss. Berlin, 32 (1923). Schriidinger, E.: Spcrce - Time - Structure, Cambridge University Press, Cambridge, 1950. Treder, H.-J.: Astron. Nachr. 315 (1994). 1. Bergrnann, P. G.: General relativity of theory, in Encyclopedicr ofphysics, vol. 11, S. Fliigge (ed.), Berlin etc., Springer, 1962 v. Borzeszkowski, H.-H., and Treder, H.-J.: Spinorial matter in affine theory (submitted for publication).

A NEW APPROACH TO QUANTUM GRAVITY An Overview

SARAH B.M. BELL JOHN P. CULLERNE BERNARD M. DIAZ Department of Computer Science 1.Q. Group The University of Liverpool Liverpool, L69 7ZF, England

Abstract. We quantize General Relativity for a class of energy-momentum-stress tensors. Key words: quantum gravity, Bohr atom, Sornmerfeld atom.

1. The New Method Of Quantization

1.1 THE GENESIS OF A CURVED SPACETIME It has been generally believed that QED does not allow a classical spin for the particle described. However, it has now been shown that QED permits the spin of the particle to behave l i e a four-vector (Bell et al. 2000a&b). This was done by mapping the Dirac and photon equation into a new form. the versatile form,

where e is the charge on the electron. This permits the application of QED as a quantum version of General Relativity. We will study one particular sort of curved spacetime here which we will use as the building block for all the others, In this section we discuss its extrinsic description and relationship to the versatile Dirac and photon equations. We define the curvature in terms of c o d h a t e systems. We shall call the original flat spacetime that applies to equation (1) the spacetime ofthe Large observer. We may co-ordinatethe x, ,x, and x, , x, planes of the 303 R.L. Amoroso et a1 (eds.), Gravitation arul Cosnzology: From the Hubble Radius to the Planck Scale, 303-312. O 2002 Kluwer Academic Publishers. Printed in the Netherlands.

S. B.M. BELL. J. P. CULLERNE AND B. M. DIAZ

304

Large observer with polar co-ordinates, XI

r, ,el and r2,8,, where we have

= rl sin(@,),X, = r, cos(8,), xo = ro sin(@,), x, = ro cos(8,) and

and

s:, si are the arcs corresponding to rl,r, and 8,,8, . We introduce two further

observers who we shall call the Mediurrz observer and the Srnall observer. We give the first the co-ordinate system ( x , ,sl,r,,x, ), while the second is given the co-oidinate system ( so,sI,rl,r0 ). The spacetime ofthese observers is curved in such a way that equation (2) is replaced by

where

RI,Ro are constant. s l yso are arcs as before. Both of equations (3) apply for the

Small observer, while the first of equations (3) and the last of equations (2) apply to the Medium observer. The spacetime of the Medium observer is associated with the circular orbits of the Bohr atom while that of the Small observer is associated with the elliptical orbits of the Sommerfeld atom. The Dirac equation (1) and the photon equations are transformed into the curved spacetimes of the Medium and Small observers using variables. ( x, ,sl,rl, x , ) for the Medium observer and ( s,,s,, r,,r, ) for the Small observer. The quaternion matrices are transformed to lie along the arc and radius. They are then also transformed to the curved spacetime of the Medium and Small observers so that they appear constant for every location i n t h e i r s p a c e t i m e s . The r e s u l t for t h e M e d i u m o b s e r v e r

kDI

+ 0 2

= i-s* 8 / 8 s s , +lr18/dr,,12, + D 3 =i[08/dXo+ ~ 3 8 1 d X 3 , ~ithDo

and 4, unchanged, while for the Small observer Dl

-

+ e2= [,q18 14,+ i,,8 1dr,,

Do + g3 = ii,d / as, + l,,d / dr,. . We have. and for the reflector matrices. Similar expressions apply to M and A. However, while the spacetime of the Medium and Small observers is locally flat. it is globally curved and this will affect any variable that depends on volume. The potential depends on a probability current per unit volume and therefore A does not refer to the potential the Medium or Small observer will see. Equations (2) and (3) are used to find the volume seen by the Medium or Small observer and the photon equation to find the correspondingpotential. This calculation shows that an inverse distance potential seen by the Large observer becomes a constant potential in the spacetimes of both the Medium and Small observers. More details are provided by Bell et al. (2000f).

QUANTUM GRAVITY

1.2 THE BOHR-SOMMERFELD ATOM We may now solve the Dirac equation amended as described in section 1.1 for the spacetime ofthe Medium observer to show the derivation ofthe Bohr atom as a consequence of QED. Our version of Bohr's equations. unlike his original treatment. will be relativistically correct. We suppose that the Bohr electron making up the oneelectron atom has a velocity in the s l direction, that is, the electron h'ds a circular orbit round the proton. We use the rest frame of the proton. We discover that

= exp(i(v" x i t ps, ))

- ii,,p - ieA') M" ) x exp{i(v-xi t p s , ) ) m"' = ( v - - e ~ " )t, p 2 , = ((if

where we have set electron,

h / 2n to 1, m" = (me / i )

with

x i = x, / i and we have a plane wave solution ,(dl,

(4)

me the mass of the

#,

), , where iv - is the

frequency and p the wave number. A- / i is the potential due to the proton in the Medium observer's spacetime. We have

where e is the charge on the proton.

R1 is the Bohr radius at which both the Large and

Medium observers see the same potential. R, is a constant. We see that the interaction between the proton and electron has introduced no more than a change in the vacuum potential and swap where co-ordinates ( x,,x2 ) are exchanged for co-ordinates ( s,,rl ). We may derive the Bohr equations from the solution to the D i c equation, equation (4). We see from the third of equations (4) that there is an equivalent free electron with wave number p and frequency

This is the Bohr electron. We require that the wave function to be single-valued from the point of view of the Large observer. De Broglie's relation between /I and the momentum of the Bohr electron then provides

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S. B.M. BELL, J. P. CULLERNE AND B. M. DIAZ

where ( - iv " ) is the velocity of the electron, we have restoredh / 2n and nois an integer? This is the first of Bohr's equations. Equations (5) and (6),and de Broglie's relation between and the energy of the Bohr electron provides

91

However, there is another way of obtaining the total energy and that is to consider the geometry ofthe path ofthe Bohr electron. From the Bohr electron's point of view the mass

im- and the electron is in a potential field. However, from the point of view of the atom as whole there is no potential field and im-is the frequency associated with the mass of the atom, iv- , traveling at velocity v" with respect to the atom's rest frame. From de Broglie's relation for the atom, we must therefore have m- = vb- / .&I + v b-2 ). This leads to

is

and to

which is Bohr's second equation. The total angular momentum ofthe Bohr atom is made up of an integer contribution from the orbital angular momentum, the half integer spin of the electron and a half integer contribution from the Berry phase (Anandan 19Sn)the electron gains each orbit because the orbit can be considered as a closed path containing the nucleus. We next derive the Sommerfeld model as a consequence of QED. This time we want to replace co-ordinates ( xo,x, ) with ( so,ro ) and take on the point of view of the Small observer. We adopt sl as our temporal co-ordinate with ( xo,x3)spatial so that the situation becomes exactly analogous to the our derivation of the Bohr atom above. We can do this because the Dirac equation (1) can be shown to have a suitable tachyon form. We suppose a tachyon particle and anti-particle emerge, if the atom is suitably excited, in the form a bound state. We call the new particles the Green electron and Greenpositron. We shall call the bound state they form the Green atom. The emergence of the Green atom causes a transition from the spacetime of the Medium observer to that of the Small observer. The wavefunctions of both the Bohr electron and Green electron must be single-valued along the new loop so and we may deduce two similar equations to (7). This is sufficient to enable us to calculate the tachyon equivalent of rest mass ofthe Green electron and positron, img" =

- (m-/ vg")(nr / no),where nr is an integer called the azimuthal quantum number. We may also reproduce Bohr's equations for the Green atom assuming s,is the temporal

QUANTUM GRAVITY

307

co-ordinate and the Green positron stationary. The velocity of the Green electron as it

.

appears in Bows equations is - ivg" We next swap sl for so as the temporal axis by swapping energy and momentum eigenvalues for the Green electron and positron and discover the total contribution the Green atom makes to the frequency of the complete system.

We add the contributions of the Bohr electron and Green atom. This gives us the total frequency and momentum of the whole stucture in the spacetime of the Small observer. Repeated use of analogues to equation (9) allow us to transform these to the spacetirne and frame of interest, that of the Large observer and ( xo,x,,x2,x3 ), where we retrieve the Sommerfeld expression for the energy levels ofthe atom. More details are provided by Bell et al. (2000e&f).

1.3 THE NEW QUANTIZATION In section 12 we sketched the derivation of the Bohr and Sommerfeld models of the atom from QED. Here we do the inverse, derive QED from the Bohr and Sommerfeld models. We do not need to do more than derive QED from the Bohr model because the extra Sommerfeld energy levels appear when the Bohr model is applied a second time, using the tachyonic Green atom. It is well known that the inverse square of distance law for the strength of the electrical field gives rise to the photon equation in differential form. We extend this using a similar method to the interaction of another particle with the potential field, A, described by the photon equation. We assume the Bohr equations, (7) and (lo), apply to an electron interacting with an spherical volume of constant charge density responsible for the potential field, A. at point b in the spacetime ofthe Medium observer. The Bohr electron orbits round the surface ofthe spherical volume at the Bohr radius and we may arrange for the electron to make as many orbits as may be required to consider the electron bound whatever the total interaction may describe. We do this by letting the spherical volume decrease in size. As the volume becomes smaller the mass of the electron, me,will increase along with the charge, e, as in renormalisation. If we add equation (8) describing energy conservation to the Bohr equations we may assemble a wave function that describes the Bohr electron in a manner similar to equation (4). As the spherical volume becomes infinitesimal this wavefunction will satisfy the Dirac equation at point b from the point of view of the Medium observer. However, the steps we took to get from the Dirac equation in the spacetime of the Large observer to the Dirac equation in the spacetime of the Medium observer are reversible. We may return to a Diac equation and solution with the same frequency term, v- ,and equivalent angular momentum

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S. B.M. BELL, J. P. CULLERNE AND B. M. DlAZ

at point b in the spacetime of the Large observer. If Bohr's equations are valid for infinitesimal volumes at all points in the field then the Dirac equation must hold everywhere too and we therefore find that the full quantum theory is QED. We note that we have not mentioned the global boundary conditions or constrained the magnitude of the wavefunction in our discussion. The magnitude of the wave function would vary as required by the global boundary conditions. We find that if we negate both charges on our interacting bodies and replace the potential, A by (A-1) QED continues to apply. More details are provided by Bell et al. (2000e).

2. Application Of The New Method To Gravity 2.1 BOHR'S FRST EQUATION We now derive Bohr's two equations for quantum gravity. With electromagnetism we are in the position of the Large observer and the spatial loop, s,, provides the initial bound state, with the temporal loop.

so,giving extra fine structure. For gravity we are in the position of

the Medium observer. and the first loop we see is temporal. Since we want to distinguish the gravitational hydrogen atom from the electromagnetic equivalent we will call thefoimer the Thalesium atom. We will call the equivalent of the Green electron for a Thalesium atom a Geotron.. We will identify the point ofview ofthe Green positron with that ofthe Thalesium atom. As we did above, we will assume the point ofview ofthe Thalesium atom initially and assume our temporal co-ordinate is spatial. We therefore have the fi-arne and spacetime ofthe Thalesium atom as being equivalent to that of the Medium observer except that a spatial and temporal co-ordinate are swapped, and the frame and spacetime of the Geotron as being equivalent to that of the Small observer except that a spatial and temporal co-ordinate are swapped and that the Geotron is a moving frame relative to the Thalesium atom. Spdmtiie appears flat to the Medium observer with his co-ordinates ( xo,sl,r,,x3) Cartesian since all the curvature produces is a variation in the vacuum potential. We will change our nomenclature and take the metric to be,

where we have assumed the point of view of the Thalesium atom and are treating x l as temporal. We shall sometimes need the variant

where ( r, $ ) are polar co-ordinates. The metric we shall use to describe the curved spacetime of the Geotron is

QUANTUM GRAVrrY

where we have set the gravitational constant to unity, t is the temporal co-ordinate and the analogue of S, in section 1. However, we are treating x, as temporal. a is twice the mass of the source responsible for the gravitational attraction, which takes on the role the Green positron played in the electromagnetic case. We suppose that equation (14) is an intrinsic description ofthe curvature we now link to QED. We want to consider a temporal loop where the Geotron is bound to the Thalesium atom and in a circular orbit. We therefore use metric (13) and set dt = rd8, where d8 is the change in angle according to the Thalesium atom. Equation (14) becomes

We consider a circuit of the Geotron round the centre of attraction in the plane r, 8 in the metric given by equation (15). From the point of view of the Geotron position four-vector of the Geotron does not change direction, but the co-ordinate frame of the Thalesium atom does. So we can consider the position four-vector parallel transported as for the Berry phase.

Let the two-vector giving the position of the Geotron be V = (V r ,V 8 ) T . Following Martin (1995) and assuming the Geotron orbits at a fixed radius r, we then find

8, is also the angle the position vector of the Geotron makes relative to the Thalesium atom. We require a steady state in which the system returns to start position after a cycle. In that case after n circuits, when from the point ofview ofthe Thalesium atom z = 4nrn, 8, must be an integer multiple of 2n, 0, = 2nm, with both n and m integer. Although the spin is

integer from the point of view of the Thalesiurn atom because the Berry phase is added, from the point of view of the Geotron the spin is halfinteger as is that ofthe electron which is why we have added an extra factor of 2 for z From the Thalesium atom's viewpoint the path

.

length z corresponds to an angle 20, = 4nn. We have

Substituting for

r and 8,

R is a constant and

in equations (16) we see this imposes a condition on r, r = R where

S. B.M. BELL, J. P. CULLERNE AND B. M. DMZ

&a/ R) = m / n .

(18) R is the equivalent of the Bohr radius and we shall call it that. defining it exactly in section 2.2. We would like to know the interval as the Thalesium atom sees it. Using metric (13) we obtain for the interval

dt, = rde,.

(19)

We would also l i e to know the interval, dt, ,as the Geotron sees it. Substituting from equation (19) into metric ( 14) and using equation ( 17) and ( 18) we obtain

dt, = dz = rd8,.

(20)

We see that the Geotron is also entitled to see his spacetime as flat. Equation (18) provides information which will allow us to derive Bohr's first equation.

2.2 BOHR'S SECOND EQUATION The Lagrangian associated with equation (14) gives us the acceleration of the Geotron for the orbit of the Geotron we discussed in section 2.1. The acceleration is

Next we calculate the acceleration seen by an observer with the frame ofthe Thalesium atom in his flat spacetime. We apply General Relativity to the problem (Kenyon 1 W ) and consider the covariant derivative for metric (12) and the equation for the geodesic in f l a ~ spacetime. From these we obtain

where vg is equivalent to the velocity ofthe Geotron and we have used equations (19) and (20) for the last two equations of(22). We see from equations (22) that it does not matter whether we are in frame and spacetime ofthe Thalesium atom or in the frame and spacetime ofthe Geotron. We obtain from equations (22) and (21)

From equations (18) and (23) we may formulate the Bohr equations in exactly the same

QUANTUM GRAVITY form as they took for QED as described by Bell et al. (2000c&d). 2.3

QUANTUM GRAVITATIONAL DYNAMICS

We have been taking the point of view ofthe Thalesium atom or Geotron for our calculations above. We now wish to assume the point of view of the Small observer for whom the circular loop is temporal. W e calculated the energy contribution for such a temporal loop in section 1.2, equation (11). For the current purpose we re-state the equation as En = m v We may g

g

.

also calculate the gravitational energy in the standard way (Martin 1995) from the metric (14) and Lagrangian. The two methods agree on the form of the result. On taking up the position where the loop is temporal our metric for the Medium and Small observer become dt2 =.

-dx: +dr: +dx: +dx:,anddr2 = - ( a / r ) d t 2+ ( r / a ) d r 2+dr:

+hi.

We consider the state of the Thalesium atom. For QED there is no preferred plane for the angular momentum when we are not in an eigenstate. All possible orbits exist as a superposition of states and the atom appears spherically symmetrical. We assume the same for the Thalesium atom. The metric of the Small observer becomes

dr ' = - (a / r)dt' + (r / a)dr2+ r 'd5 '+ r sin2 (5)d5',where rlabels a sphere of area 4m2and 5 and 5 describe colatitude and longitude. The Bohr equations can be derived in an identical manner from this metric as well, provided that the first two terms are set positive. We apply the method of quantization we used in section 1.3. We find that QED is the full quantum theory. Moreover, we can find the equivalent ofthe electromagnetic potential from the Bohr equations. In the frame of the Geotron it is A = al(2r). Since we are discussing gravity. while keeping the equation itselfthe same. we will rename the photon equation the graviton equation. If we take the global point ofview the potential will be a full four-vector. For the metric, we must set

g,,= 2 4 x , ),g, = W ( x , )I-' 7gGG = r(x,I2 g, = r(x, l 2 sin2(517 (24) 9

for the local infinitesimal volume and then Lorentz transform into the global frame by some Z(X,U). Equation (24) covers Schwarzschild metric for the spacetime applying to a spherically symmetric source ofgravitation (Martin 1995). We may therefore move to an interpretation ofour general metric (24) from the point of view ofGenera1Relativity. We impose a change the potential of+l for each infinitesimal spherical volume in the local rest frame and negate its charge. We negate the charge of the body interacting with the field. QED continues to apply, but each spherical volume has, in the local rest frame, the Schwarzschild metric. This means that, if we have the redefintion ofmetric (24) applying, at each location and time we have an infinitesimal system composed oftwo bodies interacting according to the theory of General Relativity. The quantum condition discussed in section 1.3 is still in place. At this

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S. B.M. BELL, J. P. CULLERNE AND B. M. DIAZ

point it can be shown that there is a one-to-one correspondence between the current density of QED, J p , and the energy-momentum-stresstensor, TpV ,of the Einstein equation for the permissible metrics of type (24) plus local Lorentz transformation

Z(x, ).

Acknowledgment One of us (Bell) would like to acknowledge the assistance of E.A.E. Bell.

References Anandan, J. (1992) The geometric phase, Nature 360, 307-313. Bell, S. B. M., Cullerne, J. P. and Diaz, B. M. (2000id Classical behaviour ofthe Dirac bispinor, Foundations of Physics 30, 35-57. Bell, S. B. M., Cullerne, J. P. and Diaz, B. M. (2000b) The Dirac bispinor: new properties and location, Presented at the 16th International Conference on Few-Body Problems in Physics, Taipei, Taiwan, March 6-10. Bell, S. B. M., Cullerne, J. P. and Diaz, B. M. (200012) A new approach to quantum gravity, Presented at the 16th International Conference on Few-Body Problems in Physics Taipei, Taiwan, March 6- 10. Bell, S. B. M., Cullane, J. P. and Diaz, B. M. (2000d) A new approach to quantum gravity: a summary, to appear in the Proceedings offhysical Interpretations of Relativity Theory VII" at Imperial College, London September 15-18. Bell, S. B. M., Cullerne, J. P. and Diaz, B. M. (2000e) QED and the Bohr model ofthe atom, To be published. Bell, S. B. M., Cullerne, J. P. and Diaz B. M. (2000f) QED and the Sommerfeld model ofthe atom. To be published Kenyon. I. R. (1990) General Relativity, Oxford University Press. Oxford. Martin, J. L. (1995) General Relativity, a first course for physicists, Revised edition, Prentice Hall, London.

MULTIDIMENSIONAL GRAVITY AND COSMOLOGY AND PROBLEMS OF G

M.A. GREBENIUK Institute of Gravitation a n d Cosmology, PFUR, MichlukhoMaklaya Str. 6, Moscow 117198, Russia AND V.N. MELNLKOV Center for Gravitation a n d Fundamental Metrology, VNIIMS, 3 - 1 M. U l y a m v o y Str., Moscow 11731.3, Russia Institute of Gravitation a n d Cosmology, PFFUR, M i c h l u k h Maklaya Str. 6, Moscow 117198, Russia Depto. de Fis ica, CINVESTA V-IPN, Apartado Postal 14-740, Mexico 07000, D. F.

Abstract. Main trends in modern gravitation and cosmology and their relation to unified models are analysed. The role of multidimensional gravitational models with different matter sources in solving the basic problems of cosmology is stressed. Exact solutions in multidimensional cosmology with p-branes for some special cases are considered. The Riemann tensor squared is calculated in a general case and investigated for a number of special solutions. Singularities of the Riemann tensor squared are also disscussed. Also the case of static internal spaces is considered. It is shown that external space for this case is de Sitter or anti-de Sitter one. Behaviour of cosmological constant and its generation by p-branes is demonstrated.

I. Introduction The necessity of studying multidimensional models of gravitation and cosmology [I, 21 is motivated by several reasons. First. the main trend of modern physics is the unification of all known fundamental physical interactions: 313 R.L. Amoroso eta1 feds.). Gravitationand Cosmology: From the Hubble Radius to the PlanckScale, 313-320 O 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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M.A. GREBENIUK AND V.N. MELNIKOV

electromagnetic, weak, strong and gravitational ones. During last decades there was a significant progress in unifying weak and electromagnetic interactions, some more modest achievements in GUT, supersymmetric, string and superstring theories. Now theories with membranes, p-brines and more vague M- and Ftheories are being created and studied. Having no any definite successful theory of unification now, it is desirable to study the common features of these theories and their applications to solving basic problems of modern gravity and cosmology. Moreover, if we really believe in unified theories, the early stages of the Universe evolution and black hole physics, as unique superhigh energy regions, are the most proper and natural arena for them. Second, multidimensional gravitational models, as well as scalar-tensor theories of gravity, are the theoretical frameworks for describing possible temporal and range variations of fundamental physical constants [3,4, 5, 61. These ideas originated from earlier papers of E-Miln (1935) and P-Dirac (1937) on relations between fenomena of micro and macro worlds and up till now they are under a thorough study both theoretically and experimentally (see review on G in [42]). At last, applying multidimensional gravitational models to basic problems of modem cosmology and black hole physics we hope to find answers to such long standing problems as singular or nonsingular initial states, creation of the Universe, creation of matter and entropy in it, acceleration, cosmological constant, origin of inflation and specific scalar fields which are necessary for their realization, isotropization and graceful exit problems, stability and nature of fundamental constants 141, possible number of extra dimensions, their stable compactification etc. Bearing in mind that multidimensional gravitational models are certain generalizations of general relativity which is tested reliably for weak fields up to 0,001 and partially in strong fields (binary pulsars), it is quite natural to inquire about their possible observational or experimental windows. From what we already know, among these windows are: - possible deviations from the Newton and Coulomb laws, - possible variations of the effective gravitational constant with a time rate less than the Hubble one, - possible existence of monopole modes in gravitational waves, - different behaviour of strong field objects, such as multidimensional black holes, wormholes and p-branes, - PPN parameters, - cosmological models behaviour and standard cosmological tests etc. As modern cosmology already became a unique laboratory for testing standard unified models of physical interactions at energies that are far beyond the level of existing and future man-made accelerators and other

MULTIDIMENSIONAL GRAVITY

3 15

installations on Earth, there exists a possibility of using cosmological and astrophysical data for discriminating between future unified schemes.

As no accepted unified model exists, in our approach we adopt simple, but general from the point of view of number of dimensions, models based on multidimensional Einstein equations with or without sources of different nature: - cosmological constant, - perfect and viscous fluids, - scalar and electromagnetic fields, - plus their interactions, - dilaton and moduli fields, - fields of antisymmetric forms (related to p-branes) etc. Our main objective was and is to obtain exact solutions (integrable models) for these model self-consistent systems and then to analyze them in cosmological, spherically and axially symmetric cases. In our view this is a natural and most reliable way to study highly nonlinear systems. It is done mainly within the Riemannian geometry. Some simple models in integrable Weyle geometry and with torsion were studied also. 2.

The model

Here we consider the model governed by the action [33, 341

where g = gMNdzM@ dz N is a metric,

cPIis

a dilatonic scalar field,

is a nr-form (nr 2 2) on D-dimensional manifold M , A is the cosmological constant and X J I E R, I , J E Q. We consider the manifold M = R x Mo x . . . x M,, with the metric

where w = f1, uis a time variable and g%s a metric on Mi satisfying the ~ ] Ji = const, i = 0,.. . ,n. equation R ~ c [=~ &gi, For any I = {il,. . . , i k ) E Q, il < . . . < ikr we put in (2)

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M.A. GREBENIUK AND V.N. MELNIKOV

+ +

where r - is thevolume &-form, d ( I ) = 6, . . . di, . For dilatonic scalar fields we put (d = v1(u), I E 0. Let

It is not difficult to verify that the field equations for the action (1) with the field configurations from (3), (4) may be obtained as equations of motion corresponding to the action

where ( a A )= ( @ , v ' , ~ P p )=, l / @ ,N = exp(yo- y) > Ois the lapse function, GAB ( a )is the matrix of the minisupermetric of the model and

is the potential. 3. The case with Ricci-flat internal spaces

Let us consider the harmonic time gmge y = yo($) and also put A = 0, ti = 0 for i = 1, . . .,n;So # 0. Here we also assume that the orthogonality conditions

aresatisfiedforall I # J , I , J E O , with the A=O, Si=Ofor i = 1, ...,n and to# 0. We also put 0 $ I , VI E 52. Then the exact solutions for the logarithms of scale fields and harmonic gauge in the considered case read

n

y= where

do d(I) Cdi$i =1 - do In If01 + C -d i=o I E ~ -

ln If11

+ aOu+ pO,

(9)

MULTDLMENSXONALGRAVITY

317

i = O ,...,n, I E R . Now let us investigate the obtained solutions for possible pbraneconfigurations. The Riernann tensor squared for the metric (3) takes the following form

We consider the case D = 11, XI = 0, I E 0. Due to the orthogonality conditions (8) and the relation (10) we have the following possibilities for the p-branes configuration

if d ( I n J ) = 1 , 2 , 4 , respectively,and 2v; = 1, I , J E R. Using the rules (12), we can list all possible sets R, or collections of overlapping 2-brane and 5-brane. Here we study the case with n = 4, do = 3, dl = d2 = d3 = 2, d4 = 1 , ~ ( i= ) f1, i = 1 , . . . , 4 . We also assume that Mo = sd0, w = -1. In this case the possible set R reads

The investigation of this model shows, that the behaviour of the scale factors and thensor Reimann squared is depends on the signatures of the internal spaces. For example in the case with ~ ( 1= ) ~ ( 2= ) 4 3 ) = -e(4) = - 1, to= -(& - 1) we have the expanding external space with compactified internal spaces. The Riemann tensor squared for the multidimensional metric in this case takes the following form

where l[gO]and RlgO] are the Riemann tensor squared and scalar curvature for the external space Mo respectively. Here sign(0) = 0. Thus, if we take the appropriate external space, we can get a nonsingular Riemann tensor squared for the multidimensional metric for this case. 4. The case of static internal spaces

Now let us consider the special case with the #k = const, k = 1 , . . . ,n; d = const, I E a. In this case we have one differential equation for 4 O , the set of algebraic equations for 4-d ~YI.

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M.A. GREBENIUK AND V.N. MELNIKOV

It should be noted that there are two possibilities in this case: 0 E I, V I E 0 or 0 4 I , V I E Q, i.e. when forms are present in our external space or they are completely defined on internal subspaces. Now Let us investigate the solutions in the first case. Let t,: dt, = erdu, be a synchronous time. Here we put w = - 1. Using the "fine-tuning" of a cosmological constant A

i = 1, ...,n , we get

5

g = -dts @I dt, + ch2H2 [~tsl+

gk,

k=l

where

Thus. for the new metric

go = -dt,

€3 dt,

2 [ ~ t , I0 + c hH2 9

1

we have ~ i c [ g O=] cog0. As we can see, for (Mo,go) = (sd0, g[sdO] = dflz,,) we get the de Sitter space with the relations

k = 1,...,n , J ~ 0 . Example. Now let us consider some more special cases. Here we put n = 2 and = ((01, ( O , l ) ,(0,211-

(19)

Thus, resolving the First equation in (1 8) we get the following formulas for the charges of forms

where Q2 is some arbitrary constant. Here, as we can see, one must put ~ ( 1=) ~ ( 2=) - 1 . Thus, the effective cosmological term of the $ space in this case reads

MULTIDIMENSIONAL GRAVITY

319

so, for the case with (Mo,go) = (srb,g[SdO]= dQ&) (ijOis de Sitter space) = +1 and we may get the extremely small effective cosmological constant A while the multidimensional cosmological bare constant Ahas the Planck scale. The case with A = 0. If we put A = 0 we come to the case with Ricci-flat internal spaces. And inflationary solutions are generated in this case only due to the fields of forms @-branes).

~(9

5.

Conclusion

In the paper were investigated some special classical solutions in the framework of the multidimensional cosmological model with additional scalar fields and antisymmetric forms (multidimensionalcosmology withp-branes) obtained in [34]. The Riemann tensor squared of the whole multidimensional metric was constructed and investigated for some special solutions with different p-branes configurations. The singularity of the Riemann tensor squared was also considered. Exact solutions for the model were also obtained, when scale factors of internal spaces are constant. It was shown that external space for this case is de Sitter or anti-de Sitter one. Behaviour of cosmological constant and its generation by p-branes was also demonstrated. References 1. V.N. Melnikov, Multidimensional Classical and Quantum Cosmology and Gravitation.Exact Solutions and Variations of Constants, CBPF-NF-051193, Rio de Janeiro, 1993. V .N. Melnikov. In: Cosmology and Gravitation, ed. M . Novello (Editions Frontieres, Singapore, 1994) p. 147. 2. V .N. Melnikov, Multidimensional Cosmology and Gravitation, CBPF-MO-002195, Rio de Janeiro, 1995, 210 p. V .N. Melnikov. In Cosmology and Gravitation.11ed. M. Novello (Editions Frontieres, Singapore, 1%) p. 465. 3. K.P. Staniukovich and V.N. Melnikov, "Hydrodynamics, Fields and Constants in the Theory of Gravitation", Energoatomizdat, Moscow, 1983, (in Russian). 4. V.N. Melnikov, Int. J. Theor. Php., 33, No 7 , 1569 (1994). 5. V. de Sabbata, V.N. Melnikov and P.I. Pronin, Prog. Theor. Phys., 88, 623 (1992). 6. V.N. Melnikov. In: Ciravitational Measure~i~ents, Fundamental Metrology and Constants. Eds. V. de Sabbata and V.N. Melnikov (Kluwer Academic Publ.) Dordtrecht, 1988, p.283. 7. V.D. Ivashchuk and V.N. Melnikov, Nuovo Cimento, B102, 131 (1988). 8. K.A. Bronnikov, V.D. Ivashchuk and V.N. Melnikov, Nuovo Cimento, B102, 209 (1988). 9. V.D. Ivashchuk and V.N. Melnikov, Phys. Lett., A135, 465 (1989). 10. V.D. Ivashchuk, V.N. Melnikov and A.I. Zhuk. Nuovo Cimento. B104, 575 (1989). 11. S.B. Fadeev, V.D. Ivashchuk and V.N. Melnikov. In Gravitation and Modern Cosmobgv (Plenum, N.-Y., 1991) p. 37. 12. V.D. Ivashchuk, Phys. Lett., A170, 16 (1992).

320

M.A. GREBENIUK AND V.N. MELNIKOV

13. V.D. Ivashchuk and V.N. Melnikov, Teor. Mat. Fiz., 98, 312 (1994) (in Russian). 14. V.D. Ivashchuk and V.N. Melnikov, Int. J. Mod. Phys., D3, 795 (1994), grqd9403063. 15. V.R. Gavrilov, V.D. Ivashchuk and V.N. Melnikov, J. Math. Phys., 36,5829 (1995). 16. V.D. Ivashchuk A.A. Kirillov and V.N. Melnikov, Pis'ma ZhETF, 60 No 4 (1994). 225 (in Russian). 17. V.D. Ivashchuk and V.N. Melnikov, Class. Quantum Grav., 12, 809 (1995). 18. A.A. Kirillov and V.N. Melnikov, P h p Rev., D52. 723 (1995). 19. K. A. Bronnikov, V.D. Ivashchuk in Abstr. VIII Soviet Grav. Con$, (Erevan, EGU, 1988) p. 156. 20. S.B. Fadeev, V.D. Ivashchuk and V.N. Melnikov, Phys. Lett, A161, 98 (1991). 21. K.A. Bronnikov and V.N. Melnikov, A m l s ofphysics (MY.), 239, 40 (1995). 22. V.D. 1vashchukandV.N. Melnikov, Class. Quantum Grav., 11. 1793 (1994). 23. V.D. Ivashchuk and V.N. Melnikov, Grav. & C m o L , 1, No 3, 204 (1995). 24. V.D. Ivashchuk and V.N. Melnikov, Phys. Lett., B384, 58 (1996). 25. U. Bleyer, V.D. Ivashchuk, V.N. Melnikov and A.I. Zhuk, Nucl. Phys., B429 117 (1994). gr-qc/9405020. 26. V.R. Gavrilov, V.D. Ivashchuk and V.N. Melnikov. Class. Quant. Grav.. 13. 3039 (1996)27. V.R. Gavrilov and V.N. Melnikov, Theor. Math. Phys., 114, No 3,454 (1998). 28. V.R. Gavrilov, V.N. Melnikov and R. Triay, Class. Quant. Grav., 14, 2203 (1997). V.R. Gavrilov, V.N. Melnikov and M. Novello, Grav. & Cosml., I, No 2, 149 (1995). V.R. Gavrilov, V.N. Melnikov and M. Novello, "Bulk Viscosity and Entropy Production in Multidimensional Integrable Cosmology", Grav. and Cosmol., 2, No 4, 325 (1996). 29. V.D. Ivashchuk and V.N. Melnikov, "Multidimensional Gravity with Einstein Internal Spaces", Grav. & Cosml., 2, No 3, 177 (1996). hep-thl9612054. 30. V.D. Ivashchuk and V.N. Melnikov, "Intersecting p-Brane Solutions in Multidimensional Gravity and M-Theory", Grav. & Cosmol., 2, No 4,297 (1996). heptW9612089. 31. V.D. lvashchuk and V.N. Melnikov, Phys. Lett. B, 403, 23 (1997). 32. V.D. Ivashchuk and V.N. Melnikov, "Sigma-Model for Generalized Composite pBranes", Class. and Quant. Grav., 14, No 11, 3001 (1997). hep-thf9705036. 33. K A . Bronnikov, M.A. Grebeniuk, V.D. Ivashchuk and V.N. Melnikov, "Integrable Multidimensional Cosmology for Intersecting pBranes", Grav. & Cosmol., 3, No 2, 105 (1997). 34. M.A. Grebeniuk. V.D. Ivashchuk and V.N. Melnikov, "Integrable Multidimensional Quantum Cosmology for Intersecting pBranes", Grav. & C m o l . , 3, No 3,243 (1997). grqcl970803 1. 35. M.A. Grebeniuk, V.D. Ivashchuk and V.N. Melnikov, "Multidimensional Cosmology for lntersecting p-Branes with Static Internal Spaces", Grav. & Cosmol., 4, No 2 (1998). 36. V.D. Ivashchuk and V.N. Melnikov, Int. J. Mod. Phys. D, 4, 167 (1995). 37. V.D. Ivashchuk and V.N. Melnikov, "On Singular Solutions in Multidimensional Gravity", Grav. & Cosmol. 1, No 3, 204 (1996). hep-thl9612089. 38. V.D. Ivashchuk and V.N. Melnikov, "Multidimensional Classical and Quantum Cosmology with Intersecting p-Branes", J. Math. Phys.. 39. 2866 (1998). hepthl9708157. 39. V.D. Ivashchuk and V.N. Melnikov, "Cosmological and Spherically Symmetric SCF lutions with Intersecting p-Branes", J. Math. Phys., 40, No 12, 6568 (1999). 40. M.A. Grebeniuk, V.D. Ivashchuk, V.N. Melnikov and R. Triay, "On Some Cosmological Models with Fields of Forms", Grav. & Cosvnol., 5, No 3, 229 (1999). 41. V.D. Ivashchuk and V.N. Melnikov, Proc. 2nd ICRA Workshop "The Chaotic Universe", WS, 2000, p.509. 42. V.N. Melnikov, "Fundamental Physical Constants and Multidimensional Black Holes", Grav. & Cosmol., 6, No 2, 81 (2000).

QUANTUM GRAVITY OPERATO

AND NASCENT COSMOLOGIES

LAWRENCE B. CROWELL ALpha Institute ofAdvanced Study I I Rutafa Street, H Budapest. H-11 65, Hungary

1. Introduction The Planck length is a fundamental scale of physics. Suppose that we have a black hole with

a ma& M

whose Scbwarzshild radius isRS known to be r = ZGWc, . Now let this black hole

have a de Broglie wdve length 2 = h 1 p , wherep = E/c = Mc the spacetime momentum of the black hole. Assume this wave length is equal to its diameter defined by the Schwarschildradius, further, the mass ofthe black hole is M = h / c/Z The analyst finds that the de Broglie wavelength ofthis blackhole is,

.

This length converts into energy units as scale is 10''

GeV.

2. Operators And Corrections To Gravity Andre Sakharov suggested foundations of gravitation as a metric elasticity [3]. This metric elasticity was based upon a microscopic structure analogous to atomic structure in material elasticity. This lead Wheeler to proposed pregeometry as something more fundamental than geometry. Then the gravitational coupling parameter is a grand sum over many pregeometric modes. So gravitation relies on the quantum physics ofthe vacuum. A proposed Lagrangian that describes the zero-point energy of the geometrodynamic vacuum advanced by Ruzmaikina and Ruzmaikin is a power series in spacetime curvature [4]

321 R.L. Antoroso et al (eds.). Gravitation m~clCosmology: From the Hubble Raclius to the Planck Scale, 321-330. O 2002 Kluwer Actulemic Publishers. Printed in the Netherlands.

LAWRENCE B. CROWELL

322 Here

a,P,Y ,6are coupling coefficients

of order unity, and k are the modes of the

pregeometric entities. This corrections to Einstein's general relativity has departures at high energy due to the breakdown of the spacetime picture. The first terni is expected to be removed by renormalination arguments. From the second term we can see

Ashtekarpresented the nature ofthe limits as G -+ 0 and h 3 0 as they recover classical gravitation and standard quantum mechanics [fl. This illustrates that quantum gravity possesses two limits, or that C and h are variable or renorrnalizable. In a similar manner this author has discovered that quantum radiance by horizons in spacetime will generalize the nature of quantum uncertainty [I] Here temperature cc gravity evaluated by the Killing field normal to the horizon

This demonstrates that the vacuum state is a thermal or mixed state. Now perform these calculations with boson creation and annihilation operators that are thermally distributed,

where these then act on a pure vacuum state

,t a,

a

and

are the creation and annihilation operators for a boson field, and equation (8)

is analogous to the Heisenberg picture, The commutator of these genemlized operators is then found to be

and uncertainty between position and momentum is

([

(Ap, )(Axk ) 2 T ( a,, ak I

I) . )

The uncmainty of one conjugate variable under a squeeze state operator diverges from the standard Heisenberg result. For large z uncertainty of the squeezed variable increases along with the uncertainty ofthe "desqueezed" conjugate variable. 11the squeeze state operator

QUANTUM GRAVITY OPERATORS

is applied to the operator ad we find that it is transformed according to flj

= s(z)o?~s'(z)

+e

27rw, (%aJ P2

+

There is then observed to be a gravitational squeezed state operator defined as

that enters into the commutation ofthe creation and annihilation operators as

The boson field operators examined in this example may be considered to be those for the gravity field. They may be considered as Sen connections written according to Ashtekar variables. Thus quantum gravity involves the renormalization of the Planck scale and a redefinition of quantum uncertainty.

3. Metric Fluctuations And Quantum Fields

Ashtekar [6, 71 provided a demonstration of semiclassical theory according to A -+ 0,and G -+ 0 [4], which holds here. The quantum expectation then has a maximum that determines the classical h + 0 limit. Consider a metric fluctuations dual to a term

'

8rr 'j, that

is a fluctuation in the

momentum metric rr conjugate to the metric g g for a spacial surface ofevolution. This fluctuation is then an acceleration that is associated with nongeodesic motion ofa test particle on the spacial surface ofevolution. Equation 6 shows that quantum fluctuations in the metric involves a renormalization of the Planck constant. We considermomentum metric fluctuations with the ADM Hamiltonianconstraint,

324

LAWRENCE B. CROWELL

that defines the operator in the Wheeler-DeWitt equation,

We may represent the momentum operator with the fluctuations in the momentum according to

where the momentum metric fluctuation is

where A[g] is the amplitude ofthe cosmological wave function !?! [g]. For low energies these fluctuations are small in comparison to the expectation values. From here on we set c = 1 where

d ~ t 2is the Plancklength. These momentummetric fluctuations arerelatedto the

metric fluctuations according to [*ij

1 ( r ) , ikl (r')] = [6%ij( r ) ,6i3' I ( rI)] = di- sjd(r - r').

Given that there is an "barrowof time" t expressed as

= NnP

+N

(14)

the momentum operator may be

This is computed with the Poisson bracket with the classical ADM hamiltonian as

The corresponding term with fluctuations in the shift parameter vanish. The fluctuation is evaluated to be

The Killing vector is a vector that is constant along the momentum of a particle:

QUANTUM GRAVITY OPERATORS

The fluctuation in momentum of a particle. as it is carried along with a fluctuation in the metric momentum, is then seen to be in components

which means that the fluctuation part ofthe momentum operator is seen to be

When this momentum fluctuation is projected onto the Killing field this results in the constant vector

N ow the projection ofthe Killing field onto the momentum fluctuation satisfies

where the first right hand side term vanishes, For a fluctuation mode with frequency ar

and so the fluctuation propagates on anull direction. We then can construct the Killing vector as

where we have used the Heisenberg uncertainty principle and have included the Planck constant expliiitly. Let us consider the fluctuation on a single spacial direction. This Killing vector on null coordinates is

LAWRENCE B. CROWELL

We have that V is the inertial time that is defined for the classical spacetime. The quantum fluctuation determines a Killing time V defined on the horizon h+ as

From the Killing vector we have that this equals

or that

We now consider the field expanded into normal modes as

The Killing time is defined by the fluctuations around the classical horizons. As such the phases are restricted to the horizon so that

fiwv

f ' = {

L~(Y, z)e 0,

9

v
(29)

A Fourier transform of this phase is Fw(iA

Y, z ) =

1

[00 (v, , 00

eiflVfw

y z)dV.

For V > 0we have that the phase flinction has the generator V . We then perform an analytical extension into the complex plane with V = iB to obtain

This means that the Fourier transform is ofthe form

QUANTUM GRAVITY OPERATORS

It is then possible to examine the same Fourier integral for to

V -+ -V and we have that

a + -Q,

which corresponds

As this corresponds for V < 0 we have an extension ofthe wave function across the horizon. We may then demonstrate that,

From this it can be demonstrated that the vacuum state is then defined as

Here it is evident that the modes associated with these fluctuations play the role of a temperature. From this the field assumes the form

4=

The field operators are then generalized accor ng to

which also converge to the zero temperature result for o + 0. These generalized operators are consistent with the results of Feinstein and Sebastian when the phase is averaged, or equivalently a coarse graining of the phase space. Then the scattering and tunneling of the vacuum has an entropy change S = In tush (2v) where r is the squeezing parameter [2]. The squeezing parameter then is a measure of the tunneling. Here the squeeze parameter may be evaluated as

so the generation of matter is associated with the initial entropy of the cosmology. Here p-,

328

LAWRENCE B. CROWELL

is the energy density ofthe vacuum. The generalized operators in equation 7 leads to the uncertainty spread

which are consistent with the result of Feinstein and Sebastian. Feinstein and Sebastian illustrate that the squeezing of states that are solutions to Wheeler-DeWitt equation will exhibit tunneling and that the entropy measure of this the process is the generation ofmass-energy within a cosmology [2]. Above it is illustrated that quant-operators in curved spacetime exhibit a form of gravitationally induced squeezing [I]. These two pieces ofthe puzzle indicate how the universe spontaneously tunneled out ofthe vacuum. Further, as the total Gibbs free energy created is zero, the mass-energy of the universe is the temperature times the entropy ofthe initial state of the universe.

4. Phenomenology With Strings And Membranes Now it is illustrated how the above theory is related to virtual fluctuations of strings and membranes. What follows is a phenomenological discussion of the problem. Strings and membranes are spaces with fields parameterized on them. The tension A of a membrane is the determinant of the cosmological constant. We consider the pbrane as embedded in n dimensions. The stress-energy for a pbrane is then

determined by the Wess -Zumin -Witten action [8,9], and Here thevielbeins n"e the dynamics as a a model. The membrane in this model is a Zbrane that sweeps out a

u-'~u

3-volnine. We then assume that the membrane is determined by gauge potentials on this volume. The WZW action for this system is then determined by these fields on the boundary of this volume dV and within it, -

s-

This action pertains to the dynamics ofa membrane whose boundaries are a two dimensional of spacetime with spacial and temporal directions. The first term can be rewritten with the Stokes theorem

I,@

= I , d @ to

329

QUANTUM GRAVITY OPERATORS

produce the Chern-Simons form as the integrand for the action. If we define the gauge potential in one direction as 't-' and apply a gauge covariant Euler-Lagrange equation we find the wave equation

For brevity we consider the gauge potential as being along one spacial direction and restrict it to just # . Here g is the coupling constant for the field. This differential equation requires that the gauge potential contain both Yang-Mills and Lorentz Chem-Simons potentials with different cohomologies [lo]. This cubic Klein-Gordon equation has been examined in gauge theories and QCD. This differential equation determines the motion of a soliton wave [ll]. We consider this soliton as moving in one direction. The solution to this equation is then

$ ( x , t ) = kib0sech(kx - w t ) ,

.

where the coupling constant is related to g2 = k - o, 2 We then consider this soliton as propagating along the membrane, and where the boundaries of this membrane are a virtual fluctuation of a 2-d spacetime. We may think of this soliton as propagating on this membrane. The field

Cf,b+ f *,b ') ) then acts upon the vacuum State

y = a,

We naively consider the virtual suing as a perturbing field that interacts with the membrane. The open ends of the string are tied to the membrane and satisfy D i c h l e ~ Boundary conditions. We then simply consider this interaction as the addition ofthe field # to

,. If we consider this perturbation to only 0(#) we then have that the string field

obeys the differential equation

An analysis may be approached with the raising operators for the scattered field. Let the incident field possess the lowering and raising operators a

and at. Then consider the

scattered and reflected wave have the lowering and raising operators b

2 ', where b ' s b ' and A

'

and

6' and c and

c # c^' We then posit the asymptotic wave functions

The analytic solution to the differential equation is easily seen to be

LAWRENCE B. CROWELL

which describes a topological soliton. Here A gives the reflection coefficient and B determines the transmission coefficient. The wave ftinction this may be written as U ( r , t )

+

~ s i n (?lo h z)e-qkX - w t ) ) , where for comparatively large squeezing parameter r = y oL sech($+-,z) ( ~ c o s(h~ / ~ z ) e q-( w"t )

The transmission coefficient

T2

=

(47)

1 ~ 1 ' when expressed according to the raising and

lowering operators for the field leads to

sechr =

t

C I= atrz '

(49)

which further leads to the result consistent with

These results are consistent with the derivation of a cosmology that tunnels out of the quantum vacuum [2]. This is effectively "creation without creation," and creation from nothing. Thomas Aquinas wrote in his Summa contra Gentiles III "Materia artificialiurrz est a natura, naturaliurn vero per creationem a Deo." Yet it appears as ifnothingness, or the quantum gravity vacuum pennits the creation of something from nothing "Creati Ex Nihilo,"

References [l] L. Crowell, Found. Phys. Left 12,6 (1999) [2[ A. Feinstein, M. Sebastian, Found. Phys. Lett. 13,2 (2000) [3] A. D. Sakharov, Dokiady Akad. Nank. S. SR. 177,7({71 (1967),lenglish) Soviet Phys. Dokiady 12 1040-1041 [4] Ruzinaikina and Ruzmaikin Somet Physics JETP 30.2. pp.372-374l19701 [5] A. Ashtekar Phys. Rev Lett 77,24 (1996) [q A. Ashtekar, Phys. Rev. Lett., 57,2244-7, (1986). 171 C. Rovelli, Class. Qnant Grav., 8,1613-1675, (1991) [El E. Witten, Comm. Math. Phw9. 92,455, (1986). [9[ 5. P. Novikov, Usp. Mat Nauk. 37,3, (1982). [lo] M. B. Green, 3. H. Schwartz. E. Witten Superstring Theory vol2, p 381-383, Cam- bridge, (1987) [l 11 H. B. Nielson, P. Olesen, Nucl. Phys. B61,45 (1973)

GRAVITATIONAL MAGNETISM: AN UPDATE SAUL-PAUL SIRAG International Space Sciences Organization (ZSSO) 3220 Sacramento Street, San Francisco, CA 94115

Abstract. Gravitational magnetism (or the Blackett effect) is the generation of a magnetic field by an electrically neutral rotating mass, whose magnitude is determined by analogy with the magnetic field generated by a rotating electric charge. Since 1947, there is increasing evidence for this effect by the measurements of the magnetic fields of the solar planets, the sun, other stars, and even pulsars, as well as the galactic magnetic field. However, the attempt to measure this effect in the laboratory depends on the ability to measure extremely weak magnetic fields and the shielding of extraneous magnetic fields. Early attempts to measure this effect in the laboratory depended on ad hoc extensions of the simple rotational version of gravitational magnetism. Recently there have been more sophisticated laboratory approaches. Also the extended observational evidence has generated a plethora of theoretical attempts to derive the Blackett equation in a larger context Of particular interest is the work of R.I. Gray, who performed an advanced version of Blackett's static experiment, and also related the Blackett effect to several other theoretical and empirical relations particularly the Wesson effect--the constancy of the ratio of spin to mass-squared for planetary, stellar, and galactic bodies. Pauli's anomalous magnetic moment (as a Blackett effect) is also considered as a bridge to the gravitomagnetic field generated by superconductors. 1. The Early Work: 1912 - 1979

In 1912 Arthur schuster1, in discussing "the possible causes of terrestrial magnetism," made the very tentative hypothesis: "If magnetisation and rotation go together, the sun and the planets would all be magnetic." Here presumably mass in rotation would play the role analogous to that of charge in rotation. Schuster's speculation was tested experimentally by wilson2 and by Swann and ~ o n ~ a c r eHowever. ~. these tests were not tests of the straightforward analogy suggested by Schuster. This situation was reviewed by P.M.S. ~ l a c k e t tin~ 1947. when he showed that the ratio of magnetic moment P to angular momentum U for the earth. the sun and the newly measured star 78-Virginis was a close fit to the simple formula:

P JE -=pU

2c (in emu);

p & --' 2 4

,

units).

(1)

33 1 R.L Amoroso et a1 (eds.),Gravitation utld Cosmology: From the Hubble Rculi~~s to the Plunck Scale, 331-336. O 2002 Kluwer Acadetnic P~~blishers. Printed in the Netherlcitlds.

332

S-P SIRAG

Where G is Newton's constant, c is the speed of light, and P is a "form factor" which should be close t o l . Note that this equation is usually written in Gaussian electromagnetic units, where the Coulomb constant is c2. But I have rewritten it in SI units where k is the Coulomb constant, to make the parallelism more clear. The consequences of this equation are called the Blackett effect, although for historical reasons it is also called the Schuster law, or the Schuster-Blackett law, or the WilsonBlackett law. [Note well: in my 1979 pap&, I called the consequences ofthe Blackett equation the Blackett effect and also the gravi-magnetic effect. And other papers have followed this "gravi-magnetic" nomenclature. However, in 1984 a paper was published6, which used the term "gravitomagnetic field" to describe (by analogy with Maxwell's equations) the purely gravitational field, one of whose consequences is the well known LenseThirring ("frame dragging") effect. Since then, the term "gravitomagnetism" has become standard. (See especially reference 7: Ciufolini & Wheeler.) Therefore, I will use only s the term Blackett effect; and will drop the use of "gravi-magnetism" to avoid confusion 9 with the term "gravitomagnetism."] q It is widely believed that Blackett did a laboratory experiment which ruled out this equation. However, his huge experimental paper10 described a static test in which a 15-kg gold cylinder at rest was presumed to pick up an induced current (producing a small, but measurable, magnetic field) from the rotation ofthe earth. In my 1979 "Gravitational Magnetism" I pointed out that the Blackett effect had yet to be definitively tested in the laboratory. Also 1 compiled the data from several newly measured astronomical bodies, which was a near fit to the Blackett equation. Thus I proposed that a fresh attempt be made using the latest magnetic field detectors (SQUID-magnetometers) to test the simple rotational version of this equation. The trend Line of the data for P/U (magnetic moment versus angular momentum) was fairly systematically offset from the prediction line. where is set to 1. so that the offset is indicated by the average . Of course, there would also be electrical-magnetic effects in many of the astronomical bodies. so that these electrical "dynamo" and other effects serve to dampen somewhat the primary Blackett effect. This idea is supported by the fact that Mercury provides the datum point closest to the prediction line. Because of Mercury's slow rotation rate and small size, it was presumed that Mercury had no magnetic field. When Mariner 10, which was equipped with a magnetometer to measure the magnetic field of Venus, also measured a magnetic dipole field on Mercury, it was a great surprise1'. The only datum point to fall far short ofthe Blackett prediction line is Mars. This may indicate that we are seeing Mars when the interaction between its primary (Blackett) field is being pumped down by its interaction with the electrical "dynamo" field. Here I should mention that surdin12 has proposed a formula (based on stochastic electrodynamics, SED) similar to the Blackett effect in which polar flipping is to be expected. He has suggested that Mars is presently being viewed in process of changing polarity. surdin13 has also done an experiment in which he claims to have measured the rapid changes of polarity of an electrically neutral rotating body. The measurement depends on signal autocorrelation, and various possible "parasitic" effects have to be ruled out This very intriguing experiment needs to be repeated.

GRAVITATIONAL MAGNETISM 2. Post 1979 Work

The most straightforward test of the Blackett equation would be to measure directly the magnetic field of a rotating neutral body (which is not also a ferromagnetic substance). ~lackett~"'suggested that a 1-meter bronze sphere spun at 100 Hz would do nicely. except that this is the maximum safe speed. and there are severe problems in nulling out extraneous magnetic fields. With modern SQUIDS and mu-metal shielded rooms. such an experiment can be attempted. Exactly such an experimental design14 was described at the SQUID '85 conference in Berlin. However. the results of this experiment have not been published. Another experimental result is not widely known because it is described in the book. Unified Physics, by R.I. ~ r a ~ " He . found the simple rotational version of the Blackett effect experiment too difficult, so he carried out an improved version of Blackett's static experiment with positive results. Gray also found an intriguing relationship between the Blackett effect and the somewhat analogous effect described by wesson16, in which the ratio of angular momentum to the square of the mass of astronomical bodies remains fairly constant over the vast range of planetary bodies to galactic clusters. If "fairly constant" can be idealized to "constant", these two effects can be compared, in this pristine form, as follows (and note that Blackett's symbols for magnetic moment P and angular momentum U have been replaced by and J, to make the relationships clear): The Blackett constant can be written as:

The Wesson constant can be written as:

(where Gaussian units have been used; and where G is the gravitational constant, c is the speed of light and 137 is the inverse of the fine structure constant). This suggests various relationships between b and w, such as:

a -b --

w =2c -

b

2

a

and

w

-fi

The fundamental nature of such relationships hints at deep comections between the microscopic world and the macroscopic world of astronomical objects, on which both the Blackett effect and the Wesson effect are based. Blackett himselp was motivated by the possibility of finding a comection between macroscopic physics and microscopic physics. ~ r a ~has" pushed this relationship much further. Jack Sarfatti (in these proceedings) has extended Gray's ideas to black-hole physics and the Planck-scale world, in which there is a mass-scale duality (as in M-theory) between the Wesson and Blackett effects.

334

S-P SIRAG

M-theory is a generalization of superstring theory, which is based on a fundamental Wesson-like parameter. As Abdus SalamI7 puts it: "A closed string is a loop which replaces a spacetime point. Its quantum oscillations correspond to particles of higher spins and higher masses, which may be arranged on a linear trajectory in a spin-versus-mass2 (Regge) plot. If the slope parameter of this trajectory - the only parameter in the theory - is adjusted to equal the Newtonian gravitational constant, one can show, quite miraculously, that in the zeroth order of the closed bosonic string there emerges fiom the string theory Einstein's gravity in its fullness! (The higher orders give modifications to Einstein's theory with corrections which have a range ofPlanck length = 10 -33 an.)=

Note that here, Salam is using units in which c is set to 1. so that the J/m2 Regge plot is being "adjusted" to G/c. which is essentially the Wesson constant 137G12c for the J/m2of astronomical objects. Peter ~rosche" has also analyzed the "mass-angular momentum diagram of astronomical objects'? and his work (as well as Wesson's) was compared to the Blackett ' ~ . analysis was continued in the book by effect by V. De Sabbata and M. ~ a s ~ e r i n i This De Sabbata and C. sivaram20, Spin and Torsion in Gravitation. Here torsion is described in the standard way of Cartan as "the antisymmetric part of an asymmetric affine connection." However, in this context, they push torsion beyond the formalism of the Einstein-Cartan theory, where (usually) torsion is minimally coupled to spin and cannot propagate. Nonminimal coupling of electromagnetism and gravitation is implicit in the Blackett effect, as is argued by James F. woodward21, who devotes most of his paper to an attempt to use 100 pulsars as data points to provide evidence for the Blackett effect. He finds that the Blackett "form factor'? P must evolve somewhat over the lifetimes of pulsars. This may indicate that there is some peculiar aspect of pulsars not yet understood. Perhaps the most sophisticated approach to the Blackett effect is that of A.O. Barut and Thomas ~omitz?' Here the concept of magnetic moment is analyzed in the context of 5-dimensional Kaluza-Klein theories of unification of gravity with electromagnetism. In particular, Pauli's ( 1 9 3 3 ) ~papers on spin in 4-d projective space (with 5 homogenous coordinates) are reviewed. In addition to the usual K-K type minimal coupling of gravity and electromagnetism, Pauli found an anomalous coupling to the electromagnetic field, in which the ratio of the anomalous magnetic moment to the spin of an elementary particle is

Thus Pauli says: "from the extra term it can be concluded that electrically neutral masses with a nonzero spin moment must have a small magnetic moment (which is with respect to the problem of the earth's magnetism not without interest)." Barut and Gomitz suggest that in a macroscopic body each elementary particle will have an anomalous magnetic moment equal to the spin S multiplied by

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Thus in a rotating macroscopic object. magnetic moment will accumulate and rise to the Blackett magnetic moment:

In this picture, the accumulation of elementary spin magnetic moments is due to the fact that elementary spins are really orbital angular momenta in the 5th dimension. Thus they say: "In conventional gravitational theories it seems to be difficult to understand a relationship of the type [Blackett's equation] without going to the fifth dimension." If the idea that spins (and therefore the anomalous Pauli magnetic moments) can accumulate via rotation of macroscopic bodies is correct, then there may be a relationship between the Blackett effect and the gravitomagnetic field. Readers who have heeded the "Note well" caveat near the beginning of this paper, will remember that the gravitomagnetic field is quite different from the Blacketteffect magnetic field, which I called the gravi-magnetic field in 1979. The gravitomagnetic field is not a magnetic field but a gravitational analog by way of writing the equations of general relativity in the form of Maxwell-lie vector equations. The bridge between the Blackett-effect magnetic field and the gravitomagnetic field might be built by way of the work of Ning Li and David TO^?,*^,'^. The key idea seems to be that, although ordinarily, the gravitomagnetic (and electrogravitic) fields are too small to measure, the alignment of spins in the lattice ions of superconductors, make the gravitomagnetic field much larger (by 11 orders of magnitude) than the magnetic field. As we have seen, according to the Barut-Gornitz picture, the rotation of a macroscopic body will accumulate (and thus magnify) the tiny anomalous magnetic moments associated with elementary spins. It would now seem that superconductivity is another way for anomalous magnetic moments to accumulate by the alignment of spins. This accumulation of anomalous magnetic moment would have the same form as the Blackett effect, According to Torr and ~ i "It ~is shown ~ ,that the coherent alignment of lattice ion spins will generate a detectable gravitomagnetic field, and in the presence of a timedependent applied magnetic vector potential field, a detectable gravitoelectric field." It would seem that the Blackett effect may well be entailed in these superconductors via the lattice ion spin analog to the Pauli-Barut-Gornitz rotational mechanism. Some possible technological consequences ofwork of Li and Torr are described in references 27 & 28. I have listed additional references, 29-35, to the Blackett effect. See especially reference 35, where the Blackett effect is (like the Wesson effect) extended to galactic "As far as we and intergalactic structures. As Opher and Wichoski say (in Ref. 35 know, cosmic magnetic fields pervade the Universe."

Acknowledgments It was Hal Puthoff who (at Vigier 111) mentioned Paul Wesson's paper (ref. 16) and later sent it to me. And it was Creon Levit who discovered R.I. Gray's book (ref. 15), and loaned it to me shortly after I received the Wesson paper.

S-P SIRAG References 1. Schuster, A. (1912) A Critical Examination of the Possible Causes ofTerrestria1 Magnetietism, Prac Lond. Phys. SOC.24, 121-137(1911-1912). 2. Wilson, H.A. (1923) An Experiment on the Origin of the Earth's Magnetic Field, Proc. Roy-Soc. A, 104, 451-455 3. Swann. W.F.G. and Longacre. A. (1928) J. Franklin Inst. 220.421. 4. Blackett, P.M.S. (1947) The Magnetic Field of Massive Rotating Bodies, Nature 159, 658-666. 5. Sirag, S.-P. (1979) Gravitational Magnetism, Nature 278, 535-538. 6. Braginsky, V., Polnarev, A. and Thome, K. (1984) Foucalt Pendulum at the South Pole, Phys. Rev. Left. 53, 863. 7. Ciufolini, I. and Wheeler, J.A. (1995) Gravitation andlnertia, Princeton U.P., Princeton. 8. Morrison, P. (1976) The Scientific and Public Life of P.M.S. Blackett, Scientific American 2354, 138-139. 9. Smith. P.J. (1981) The Earth as a Magnet. in D. Smith (ed), The Car~tbridgeEncyclopedia Of Earth Sciences, Cambridge U.P., New York. 10. Blackett, P.M.S. (1952) A Negative Experiment Relating to Magnetism and the Earth's Rotation, Phil. Tmns R Soc. Lond. Series A, 897, 309-370. 1 1. Sagan, C. (1975) The Solar System, Scientific American, 233:3, 23-31. 1 2 Surdin. M. (1979) The Magnetic Field of the Planets, I1 Nuovo Cimento,2C:5. 527-53. 13. Surdin, M. (1977) Magnetic Field of the Planets, J. Franklin Inst., 303:6,493-510. 14. Harasim, A., v. Ludwiger, I., b y , W. and Auerbach, H. T. (1988) Laboratory Experiment for Testing Gravi- Magnetic Hypothesis with Squid-Magnetometers, in H.D. Hahlbohm and H. Luhbig (eds.), SQUID '85, Superconcuctiong W n h u n Interference Devices and their Applications, de Gruyter, Berlin. 15. Gray, R.I. (1988) UnifirdPhy.~ics,NavalSurface Warfare Center, Dahlgren, Virginia. 16. Wesson, PS. (1981) Clue to the Unification of Gravitation and Particle Physics, Phys. Rev. D,23:8, 17301734. 17. Salam, A. (1989) Overview of Paaicle Physics, in P. Davies (ed) The New Phy.sics,Camb.U.P., New York. 18. Brosche, P. (1980) The Mass-Angular Momentum Diagram ofAstronomica1Objects, in P.G. Bergmann and V. De Sabbata (eds.) Cosmology and Gravitation, Plenum, New York. 19. De Sabbata, V. and Gasperini, M. (1983) The Angular Momentum of Celestial Bodies and the Fundamental Dimensionless Constants ofNature, Lett. Al Nuovo Cimento 38, 93-95. 20 De Sabbata. V. and S i v a m C. (1994) Spin and Torsion in Gruvitatiorz, World Scientific. Singapore. 21. Woodward, J.F. (1989) On Nonminimal Coupling of the Electromagneticand Gravitational Fields, Foundations ofphyhsics 19:11, 1345- 1361. 22. Barut, A.O. and Gomitz, T. (1985) On the Gyromagnetic Ratio in the Kaluza-Klein Theories and the Schuster- Blackett Law, Foundations 4Physics 154, 433-437. 23. Pauli, W. (1933) Anmlen der Phyusik 18,305 & 337. 24. L i N. and Torr, D. (1991) Effects of a Gravitomagnetic Field on Pure Superconductors, Phys. Rev D 4 3 2 457. 25. Li. N. and Torr, D. (1992) Gravitational Effects on the Magnetic Attenuation of Superconductors. Phys. Rev. B, 649, 5489. 26. Torr, D. and Li, N. (1993) Gravitoelectric-Electric Coupling via Superconductivity, Found-of Phys. Lett 6:4,7 1. 27. Stimiman, R. (1999) The Wallace Inventions, Spin Aligned Nuclei, the Gravitomagnetic Field, and the Tampere "Gravity-Shielding" Experiment: 1s There a Connection? Frontier Perspectives 8:l. 20-25. 28. Wilson, J. (2000) Taming Gravity, Popular Mechanics 177:10, 40-42. 29. Bennett, J.G., Brown, R.L. and Thring, M.W. (1949) Unified Field Theory in a Curvature-Free Five Dimensional Manifold, Proc. R. Soc. London Ser. A 198, 3961. 30. Ahluwalia, D.V. and Wu, T.-Y. (1978) On the Magnetic Field of Cosmological Bodies, Lett. Nuovo Cimento 23:11, 406-408. 31. McCrea, W.H. (1978) Magnetism and Rotation: Blackett's Speculation of 1947, Speculations in Science and Technology, 1:4 329-338. 32. Massa C. (1989) On the Generalized Gravi-Magnetic Hypothesis, Annalen der Physik 7:&2 159-160. 33. Muller-Hoissen, F. (1990) Gravity Actions, Boundary Terms and Second-Order Field Equahons, Nucl. Phys B 337, 709-736. 34. Srivastava, Y. and Widom, A. (1992) Gravitational Diamagentism. Phys. Lerr. B 280.52-54. 35. Opher, R. and Wichoski, U.F. (1997) Origin of Magnetic Fields in the Universe due to Nonminimal Gravitational-ElectromagneticCoupling, Phys. Rev. Letters 78, 787-790.

QUANTUM HALL ENIGMAS MALCOLM H. MAC GREGOR Lawrence Livennore National Laboratory Livennure, California, USA, 94550

1. Introduction to the enigmas The quantum Hall effect (QHE) is one of the most striking, and at the same time one of the most enigmatic, physical phenomena to be discovered in the last few decades. Experimentally, the results are simple, straightforward, and of almost unparalleled accuracy. Quantum Hall research facilities have sprung up all over the world, and the "von Klitzing constant" R = 25812.807 has become a metrological standard. Theoretically, however. this same simplicity has not emerged. Although a great many papers have been written on the quantum Hall effect, very few QHE books have appeared, and those that have emphasize the tentative nature of the theories. An introductory text, published in 1994, describes this situation very succinctly [I]: "A somewhat disturbing fact one faces when writing a book on the theory of the QHE is that, unfortunately, for the time being no conclusive theory of this fundamental phenomenon is available - in spite of the numerous attempts that have been undertaken to explain it from the time of its discovery. All that has been achieved so far is the working out of certain aspects which will be part of the ultimate theory that is still to be achieved. ... In general, publishing an introduction to a non-existing ... theory is not considered to be advisable." In the present paper we list and briefly discuss a series of enigmatic quantum Hall topics. Hopefully these results will lead to further investigations in this important field.

2. The experimental "classical" and "quantum" Hall effects The Hall effect was discovered by Edwin Hall in 1879. Using a flat metal foil, he measured its longitudinal voltage V and current 1, and hence its longitudinal resistance, R=VII. When he placed the foil horizontally in a vertical magnetic field B, he also measured a transverse voltage VH,proportional to B, which is caused by the displacement of the current slightly to one side of the foil under the action of the Lorentz force. The ratio of VHto the current I defines the transverse "Hall resistance," RH =VH / I , which is linearly proportional to B. We can write the Hall current Ias I = n,ve, where n, and v are the electron density and velocity, and e is the charge on the electron. We can similarly write the Hall voltage as VH= BV = ngvh/e (Faraday's law), where the magnetic flux quantum is 41=hie and the flux density is q.The Hall resistance RHthus becomes RH= B/enp = n,j Ine hle2. The Hall conductance is CH= l/RH= ve2/h, where v E n, /n+ is denoted as the Hall "filling fraction". This is the '6classical" Hall effect.

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The "quantum" Hall effect was discovered by Klaus von Klitzing in 1980. Carrying out a Hall experiment in a two-dimensional (2D) semiconductor (a Hall bar) at a very low temperature and high magnetic field [2], he found that the plot of RH versus B (or, in his case, gate voltage) exhibits a series of plateaus which are centered around the linear RH versus B curve. These plateaus occur at integer values of the filling fraction, v = k (k = 1. 2. .. .lo. ... ). They have extremely flat slopes. and the measured integer v values have the astonishing accuracy of one part in lo8!Furthermore. the longitudinal resistance (magnetoresistance) essentially vanishes on the plateau, which signals the formation of a single coherent quantum state. Thus these integer Hall plateaus are denoted as the "integer quantum Hall effect'' (IQHE). Additional quantum Hall results were obtained in 1982, when Horst Stormer and Daniel Tsui extended the quantum Hall measurements to higher 5-values [3,4], and unexpectedly discovered Hall plateaus that havefractional filling fractions, !V = k lm (k= 1, 2. 3, ...: m = 3, 5. 7, ...). The k values include all integers, but the nz values include only odd integers. These fractional Hall plateaus have the same experimental accuracy as the integer plateaus, and are denoted as the "fractional quantum Hall effect" FQHE)From a strictly experimental point of view, the integer and fractional quantum Hall effects appear to be identical to one another. In particular, integer and fractional Hall plateaus occur interleaved together in the same set of measurements. Quantum Hall experiments can be carried out either by holding the electron density n, constant and varying the external magnetic field B = n+. $ (semiconductor heterojunctions), or else by holding B constant and varying n, by means of a "gate" voltage (MOSFETs). In the present paper, we concentrate on the former case. The extreme accuracy of the plateau quantization indicates that the quantum Hall effect is a microscopic phenomenon, in the sense that the Hall plateau basis states all carry the value v.

3. The "Fermi sea" (FS) model for the integer quantum Hall plateaus The theoretical efforts to explain the Hall plateaus have followed the pioneering work The 2D quantum mechanical motion of electrons in a magnetic of Robert Laughlin field B is described by the Landau equations [1,6]. This motion is essentially the same in a semiconductor as it is for free electrons, except that in the semiconductor the electron mass me is replaced by an effective mass m*, which in GaAs is equal to 0.07 me. The conduction electrons in the semiconductor, being fermions, each occupy a distinct quantum state, with the continuum of filled states forming the "Fermi sea". The Landau equations group these electron states into a series of discrete energy bands, which are denoted as "Landau levels". The electron density of states in each Landau level is n, = e Blh [6].Thus for a B-value BI such that the first Landau level is exactly populated by the available conduction electrons, we have the Hall filling haction v = 1. If the field B is now decreased to the value B2= lL2 B1, with the electron density n, remaining constant, then two Landau levels are exactly populated. and we have v = 2.And so on. The Landau electron density on each Landau level corresponds to the close-packing of single-wavelength Landau cyclotron orbitals. The existence of a Hall plateau, wherein the microscopic Hall conductivity OH= ene IB remains constant as B is varied, requires that

[a.

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n, a B on the plateau. This follows from the Landau level density cited above, and

from localization-delocalization & semiconductor I)) theory [1,6], which specifies that only the energy states lying at the center of each Landau level are delocalized. and thus contribute to the Hall current. The other states are pinned by impurities in the semiconductor substrate. Hence, when the surface of the Fenni sea is located bemeen two Landau levels, variations in B affect only the localized electron states, and not the Hall conductance, so that we have plateau formation. As the field B is decreased from its v = 1. value, the rising Fermi surface sweeps over the Landau levels, successively filling them and generating the v = 2 . 3, spectrum of integer Hall plateaus. This Fermi sea (FS) model is the generally accepted explanation for these integer plateaus. but it cannot account for the fractional plateaus, as we now discuss.

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4. The "collective electron excitation" (CEE) model for the fractional plateaus In a quantum Hall experiment, increasing the magnetic field B above the value BI that corresponds to the v = 1 integer Hall plateau means that the conduction electrons all lie in the lower portion of the first Landau energy level. Thus it was not anticipated that Hall plateau formation could occur above B,. This is why the Stormer and Tsui discovery [3,4] of a Hall plateau at the field B = 3BI, and with afractional filling fraction, v = 1 13, was such a surprise. This plateau cannot be accounted-for by the FS model with its LD systematics. Hence a new type of quantum Hall theory was devised for the fractional Hall plateaus, wherein the basic quantum states consist of collective electron excitations (CEE) that feature fractional electric charges Electron coulomb interactions, which are unimportant in the FS model for the integer quantum Hall plateaus. play a dominant role in the CEE models for the fractional Hall plateaus [7-91.

[a.

5. The fundamental quantum Hall enigma: one experiment and two half-theories The FS model discussed above has played a dominant role in semiconductor physics, and the discrete ferrnionic Landau energy levels are manifested, for example, as Shubnikov-de Haas oscillations in many experiments [6]. Thus, to associate this model with the IQHE seems a logical procedure. But the FS model cannot accommodate the FQHE. Similarly. the CEE model has led to accurate calculations of many salient features of the FQHE [7-91, but it does not accommodate the IQHE From an overall physics point of view, we seem to be in the awkward position of having one simple and extremely accurate physics experiment, with two quite different and non-overlapping theories required to account for it. This at least suggests that both of these quantum Hall half-theories may be flawed, and that the search should be continued to find a more unified solution to this problem. In the present paper we discuss a number of enigmas that must be addressed in this search. These include questions about external magnetic flux quantization, trapping, transport, and release; a fermion-to-boson superconducting phase transition and its effect on effective masses; the statistics of 2D multilayer composite particles; and some eigenvalue problems associated with gauge invariance. We also offer an example of a more unified quantum Hall theory, the close-packed composite bason (CPCB) model [lo].

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6. The microscopic accuracy of the Hall filling fraction v

The Hall plateau tilling fraction v = n /n+ = k l m is the ratio of electrons e to magnetic flux lines $ = hle in the Hall current I, where k and m are integers. There are two key points to be noted here. (1) The measured accuracy of v is one part in 10'. (2) The boundaries of the Hall current region are very raggedly defined: there is a Hall "bulk current," in which the entire plateau quantum state (or at least the part not pinned down by impurities) drifts along; there are Hall "edge currents," in which the orbiting electrons bounce along the "walls" of the Hall bar; there are sometimes regions in a Hall bar where two different filling fractions co-exist side by side; and there are combinations of these, as can be ascertained by differential voltage measurements across the Hall bar. The only way that points (1) and (2) can be reconciled is if each individual Hall "current element" carries the value v , so that v is a microscopic quantity. This conclusion is confirmed by the Hall "shot noise" experiments, in which a constriction or bypass is created wherein the Hall current elements pass (tunnel) through one at a time. In the theory of these experiments, the Hall conductance CH= ve2/his characterized as a "topological invariant...reflecting the binding of charge and flux" [ll]. The task now is to determine the precise nature of these Hall plateau "current elements". 7. Magnetic flux quantization and the magnetic Reynolds number R M

Classically. a magnetic field B is treated as a continuum. At the quantum level, discrete flux lines $ can occur, but the quantization of these lines is not (as far as we can detect) innate to the field B, but depends on the quantum states with which the field B interacts. In superconductivity, electrons combine together to form bosonic "Cooper pairs". When Cooper pairs flow in a superconducting current loop, the magnetic quantization produced is in units of $ = h/2e, where 2e is the charge of the Cooper pair. In the quantum Hall system, where we have single orbiting electrons, the quantization is in units of 4 = hle. The point here is that this quantization is created by the Hall electron orbitals, so that the binding together of the orbitals with the external field lines is a natural result. Measurements made with a high-conductivity fluid such as mercury or sodium demonstrate that for transport times which are short as compared to the diffusion time, magnetic lines are frozen into the fluid and carried along with it [12]. The condition for this to occur is given by the magnetic Reynolds number RM = zvdrift/L,[I I] where zis the difhsion time, Vdrift is the fluid velocity, and L is the flow length: if RM <>1 it does. Applying these results to a typical Hall bar, we have Vdrift lo4cmlsec and L 0.1 cm. Off a Hall plateau z-lo-'' sec, which gives RM whereas on the plateau z 1 sec, which gives RM lo+'. Thus plateau flux trapping is the expected result. When we apply this systematics to the observed flux trapping on the quantum Hall plateaus, we are led to the following result: (7.1) An m h Landau electron orbital traps m external flux quanta = h/ e Since a superconducting electron orbital that contains rn de Broglie wavelengths h generates m induced (diamagnetic) flux quanta bnd = h 1 e , we obtain another result: (7.2) In a trapping electron orbital, the number of $ext and (oppositelydirected) bnd magnetic flux lines hle are equal to one another.

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8. Composite bosons (CBs) and composite fermions (CFs) A "composite particle" (CP) is the bound state of an electron and m magnetic flux quanta $ = h l e . We can envision this process as the direct trapping of external flux lines in a Landau electron cyclotron orbital (current loop), as described in (7.1). Alternately, we can picture the external flux lines as creating "vortices" in the 2D "'electron charge fluid" of the Hall bar, which then attract the electrons [3]. These vortices act as "ktitious" flux lines that are directed oppositely to the external flux lines. and are thus roughly analogous to the $ind flux lines delineated in (7.2) However. the latter are more localized. In the CEE models, the electron states are not cyclotron orbitals, but rather collective excitations which arise from the mutual coulomb interactions of all of the conduction electrons. In the case of the v =1 13, fractional Hall plateau, for example, each electron occupies three CEE "sites," so that each site carries an effective fractional charge of 113, and each site combines with a magnetic vortex line hle. A crucial feature of the CPs is that their 2D statistics depends on the number m of trapped flux lines: if m is even, the CP is a composite fennion (CF); and if ~n is odd, the CP is a composite boson (CB). The Hall plateaus are composed of odd-112 CBs, and thus resemble boson condensates. The non-plateau region near v = 1/2 contains even-111 CFs, which cannot combine bosonically, but which give rise to interesting experimental fluxtrapping results, as we describe in the next section.

9. The v = 1/2 CF Hall non-plateau region and magnetic flux-trapping Apart from the striking results that occur on the quantum Hall plateaus, important results also occur in the non-plateau region near v = 1/2, where plateau formation does not occur. In the v = 1/2 region, there are two available flux quanta for each conduction electron, so that CP formation leads to CFs rather than CBs. These CFs, being fermions, do not combine together as a bosonic quantum macrostate [13], but instead operate as individual entities. This gives us an opportunity to investigate two essential CF features: ( I ) the flux-trapping process; (2) the nature of the CF bound states. The concept of CFs was introduced by Jain [14]. and CFs play a prominent role in CEE theories. where they account for many systematic Hall results 19,151. Formally, the CF is defined by a Slater-determinant wave function that contains no mass parameter [14]. Thus the experimental determination of the CF properties is an important task. The trajectories of individual CFs in a magnetic field B have been measured in twoaperture experiments in the ~-112region [14]. For B-values below the v = 1/2 value Bm, there are more electrons than pairs of magnetic field lines, and the field lines are all captured, so that no CF trajectories through the two apertures are observed. Slightly above the BLQvalue, there are more flux lines than available electrons to capture them, and CF trajectories are observed that reflect the residual magnetic field B* =B-Bin. When this experiment is repeated with electrons at magnetic field values B=B*, the peaks in the electron two-aperture spectra match the CF peaks [14]. These experiments show two important things: (a) the CF composite state, which was originally introduced as a CEE collective state involving many electrons [14], exists as a localizable entity that has a particle-like trajectory in a magnetic field; (b) the trapping process that binds

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two magnetic field lines to each CF electron actually removes these field lines from the field experienced by the CFs. Thus this flux-trapping mechanism is more that just a theoretical artifact: the reality of these even-m CF states indicates that the odd-m CB states on the Hall plateaus, and their associated flux-trappings, are just as real. The final information about CFs comes from the experimental measurements of their properties [14]. Experimentally, the CF has a negative electric charge of -e, a spin equal to that of the electron, and an electron-like gyromagnetic ratio. Furthermore, whereas GaAs semiconductor electrons have effective masses m* = 0.07 me,the CF state has a mass roughly equal to the electron mass me. There is an old vaudeville saying that if something looks like a duck and quacks like a duck. it probably is a duck. Application of this show-business wisdom to the CF states suggests that the electron in the CF composite state is a real orbiting electron, and not a CEE many-electron collective state.

10. The fermion-to-boson phase transition The conduction electrons in a semiconductor live in a Fermi sea, and their FS motion in a magnetic field is defined by the Landau equations. Thus, when the integer quantum Hall plateaus were first discovered, it seemed natural to associate them with the Landau energy bands. This belief was not shaken when the fractional quantum Hall plateaus were discovered, which have nothing to do with the Landau FS band structure. The quantum Hall theorists went on to invoke a different (CEE) theory of interacting fermions that applies just to the fractional plateaus. However, experiment seems to be telling us a different story. The main facts that we know about the quantum Hall plateaus suggest that these plateaus actually operate in a "boson sea," where the FS limitations no longer apply. These facts can be summarized as follows: (a) The quantum Hall plateaus, with one exception [13], involve odd numbers of flux quanta 4 = Me, and hence have bosonic (CB) current-element states (6) The longitudinal resistance on a quantum Hall plateau decreases by up to 14 orders of magnitude, which indicates superconducting (and hence bosonic) behavior. ( c ) The extreme accuracy of the Hall filling fraction means that each plateau CB state carries this filling fraction, so that the uniform distribution of the external flux quanta 4 is matched by an equally uniform distribution of CB states. This in turn requires that the CBs all have the same "size" (same quantum state), which happens only for bosons. This evidence indicates that a fermion-to-boson statistical phase transition occurs during the formation of a quantum Hall plateau. which has an important consequence: (d) The transformation of the electron wave functions from fermions to bosons decoup les the plateau electrons from their mutual semiconductor "landscape potential" and couples them instead to the magnetic field lines, which logically leads to an increase in their effective mass values m*,as is observed in the CF experiments near v = 112. An increase in m* lowers the energy state of the electron, thus providing an energy gap and stabilizing the fermion-to-boson phase transition. Once we have freed the Hall plateau solutions from the confines of the Fenni sea. we can select, for each Hall plateau, an appropriate (single) Landau bosonic quantum state that accurately tiles the plateau, and thereby obtain a unified approach to both the integer and fractional Hall plateaus, as we describe in the next section.

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11. A close-packed composite boson (CPCB) integer and fractional plateau model The idea of representing a quantum Hall plateau as a boson condensate was first introduced by Girvin and MacDonald [16], and has subsequently been discussed [17]. In this context, we can picture the v = 1 integer Hall plateau as a close-packed hexagonal array of identical single-wavelength CB Landau cyclotron orbitals [18], each of which contains one trapped flux line @ = We.[lO] Since the close-packed CB orbitals have no freedom of movement, their coulomb interactions are unimportant [3]. This CPCB picture [lo] is in agreement with the usual way of viewing the v = 1 IQHE plateau. In order to apply this same close-packed CPCB model to the fractional Hall plateaus v = llm, m = 3,5, , we simply replace the 1h Landau orbitals with suitable mh orbitals [lo]. Experimentally, the fractional Hall plateaus v = 1/3, 1/5, ... are successively reached by increasing the v = 1 magnetic field BI by factors of 3, 5, ... . The Landau cyclotron orbitals are specified by the quantum numbers m and n, where m is the number of orbital de Broglie wavelengths h and n is the radial quantum number (Sec. 16, with m = P). It turns out that the (n,m) = (0,1), (1,3), (2,5), ... Landau orbitals in the magnetic fields BI,3 B,, 5B1,... all have the same area (same value for ) [ 101, and hence each correspond to one conduction electron. Since the rnh orbitals on the v = l/m plateau each trap m# external flux quanta (see 7.1 above), the v = 2/3,4/3,5/3,... fractional filling factors are reproduced in the same close-packed manner as the v = 1 integer filling factor, which is the result that is suggested by experiment.

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12. Multilayer quantum Hall plateaus If we start with the v = 1 IQHE plateau and its BI magnetic field, and then successively lower the field to values of 1/2 BB1, 113 BI, ... , we generate the v = 2, 3, ... IQHE plateaus. As we do this. the l h Landau orbitals grow in size so that they each still contain one flux line $, but they now each contain 2, 3, ... conduction electrons. The question then arises as to the nature of these multi-electron Hall current elements. The solution we offer [lo] is to treat them as 2, 3, ... layers of identical lh-tiled quantum Hall sheets, with each flux line $ penetrating corresponding orbitals on each of the 2, 3, ... sheets. This same situation applies to the FQHE plateaus. Starting with the v = 1 13, plateau, which is a sheet of close-packed 3h orbitals, we generate the v = 2/3,4/3,5/3,... FQHE plateaus as 2, 4, 5, ... layers of 3h orbitals, with a 3$ "flux bundle" penetrating corresponding orbitals on each of the sheets. And so on for the m = 5,7,... FQHE plateaus. It should be noted that this same multi-sheet formalism also occurs in the conventional FS picture of the IQHE. In the FS approach, the v = 1 plateau consists of one fully filled Landau level, with one flux line $ per orbital, The v = 2 plateau consists of two fully filled Landau levels, with one flux line per two orbitals. In order to maintain a microscopic filling fraction, it seems clear that in the FS model, each flux line must be associated with one orbital on each of the two Landau levels, just as for the CPCB model described above. The difference in this FS model is that the orbitals on the second Landau level are larger in size than on the first level. When we consider IQHE plateaus with filling fractions v = 10 or greater. this FS orbital size increase gets very substantial. In the boson models, we can make all of the orbitals on a plateau the same size.

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13. The 2D statistics of composite particles with more than one electron According to the Ehrenfest-Oppenheimer-Bethe rule, a CP composite particle is bosonic (fermionic) if it contains an even (odd) number of elementary fermions. If we picture the magnetic flux lines $ = hle as elementary fermions, then we immediately account for the even-m$ CF and odd-m$ CB composite particles described in Sec. 8. These are single-electron states. But if we now associate more than one electron with a CP, as we do for the v = llm, Hall plateaus that have k > 1, we then have a problem with the EOB rule. On the v = 2, 3, 4 IQHE plateaus, we have 2, 3, 4 electrons bound together with a single flux quantum $. These plateaus are experimentally similar, and it seems clear that they cannot represent alternations between CF and CB Hall current elements. Hence in the 2D Hall geometry, the CP statistics must be determined by exchanges or rotations within the 2 0 geometry of each individual layer on a Hall plateau, so that (e-g.) on the v = 2. plateau, the two electron orbitals (one on each layer) that are combined with a single flux line $represent two associated CBs, and not one CF. 14. Quantum Hall plateau formation The existence of quantum Hall plateaus means that the Hall conductance CH= ve2/h is independent of B over a range of B values. In the FS model, the IQHE plat us are explained by the LD theory, which restricts elemon movement to the central region of each Landau level. However, this LD mechanism does not apply to the FQHE plateaus. In the CPCB model [lo], both the IQHE and FQHE plateaus are attributed to a different mechanism: electron density trapping (EDT) in the plateau CBs. EDT follows from the fact that both the radius and the mean spacing of the plateau CBs vary as B-'" [lo]. Thus the density of CBs on a plateau varies as B. If the electrons are locked into the CB orbitals, then the electron density ne varies as B (instead of remaining constant, as it does off the plateau). Since the flux density ng also varies as B, the Hall filling fraction v = n, 1 n,j is invariant, Hence the Hall plateaus have very accurate zero slopes. 15. External magnetic flux capture, transport, and release

-

In the classical Hall effect. an external magnetic field B = IZ 4 forces an electron 4 current to one side of a 2D Hall bar, where it creates a transverse Hall voltage VH and electric field En, which in turn create a longitudinal Hall current I under the action of the ExB drift [12]. In the drift frame of reference, which moves at the velocity vdrift = WB, the transverse electric field En is canceled out by the moving flux lines $ = hie, which create an opposing electric field via the Faraday effect. In the quantuin Hall effect, as it is envisaged in the CPCB model [lo], the external magnetic field lines @ are captured and carried along by the CBs in the drifting Hall plateau current. These transported field lines create a transverse Faraday-effect voltage as they cross the region where the Hall voltage is measured. There are two different scenarios possible as to how this process takes place: Scenario I. The moving CBs in the drifting Hall current have captured all of the that which external field B, so they move along in a zero-field environment (similar toea

QUANTUM HALL ENIGMAS

345

occurs in the CF region at v = 112. Thus there is no lateral displacement of the CBs. and the Hall voltage is provided solely by the captured and transported flux lines Scenario II. When the captured flux lines @ move along the Hall bar, they create a Hall voltage VH.But when they reach the outlet region of the Hall bar and are released, they "spring back", thus creating an opposing voltage -VH that cancels out the voltage produced by the forward-moving flux lines. The "spring-back field lines @ interact with the Hall-current CBs and force them to one side, thus recreating the voltage VH. When these two scenarios are traced through relativistically, they give the same Hall voltage Vw There is one effect that may distinguish between these two possibilities. When the Hall current reaches a certain critical drift velocity, the Hall plateau abruptly breaks down [2]. This breakdown occurs at the point where the current enters the 2D electron sheet, and it is intrinsically a microscopic phenomenon [19]. It could be a "magnetic viscosity" effect [12] caused by a pileup of flux lines in the inlet region, which would suggest Scenario 11. 16. The enigmatic magnetic-moment term in the Landau energy equation

The Landau Schriidinger wave function Y(r,0) for a 2D electron in a magnetic field B separates into azimuthal and radial components:

The canonical angular momentum operator -iA 3/89 has eigenvalues The radial wave equation in the symmetric gaugeA = 112 Bxr is [6]

$ = 0,f1, -1- 2, ..

..

where oc= Belm* is the cyclotron angular velocity and m* is the effective mass of the electron in the semiconductor. The corresponding eigenvalue equation is

The 112 tho, term on the right-hand side of Eq. (16.2) is the orbital magnetic moment interaction [6]. In order to obtain the standard Landau energy levels in Eq. (16.3), it is necessary to select negative values for C . However, these are not the values that are singled out experimentally. When an electron moves in a Landau cyclotron orbital. the magnetic moment p that is generated is negative. due to the negative charge on the electron. Thus the interaction term W = -p - B [12] is positive, so that C is positive. It is possible to formally obtain negative $ -values by choosing the radial coordinate r as r=x-iy instead of the customary r=x+iy [1,7], but it is not clear that this choice is in agreement with experiment [20]. When the Landau equations are written out using the Landau gauge instead of the symmetric gauge, the energy $-dependence in Eq. (16.3) vanishes. This apparently solves the energy eigenvalue problem, but it raises questions about gauge invariance, as we discuss in the next section.

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MALCOLM H. MAC GREGOR

17. The enigma of gauge invariance in quantum Hal1 theories

In both classical [12] and quantum [6] electrodynamics, a magnetic field B can be formally introduced into the Hamiltonian N = n2/2m by replacing the electron kinetic momentum n with the momenta n = p,+eA, where p,is the canonical momentum, A is the vector potential, and eA is what we can denote as the gauge momentum. The magnetic field B is given by the equation B = curl A. It turns out mathematically that different choices forA can lead to the same value for B, so this procedure has a certain amount of "gauge freedom". This freedom leads to different ways of dividing up the kinetic momentum n(the total particle momentum) into its canonical and gauge components, p, and eA. Since the different choices for A all reflect the same physical system, the correct solutions should be independent of the choice forA; that is, the solutions should be gauge-invariant, both classically and quantum mechanically. A classical example of gauge freedom is provided by 2D non-relativistic electron motion in a quantized cyclotron orbit in a magnetic field Bz.The Newtonian energy for the 1h orbital is E = 1/2 mv2 = 112 ha,. When this problem is calculated in the Hamiltonian formalism, using the symmetric-gauge vector potentialA = (0, 112 Br, 0) in polar coordinates, the electron (instantaneous) kinetic orbital velocity v divides equally into canonical (v,) and gauge (vA)orbital velocity components. When this same problem is calculated using the Landau-gauge vector potential A = (0, Bx, 0) in cartesian coordinates, the kinetic orbital velocity v now divides into a linear x-axis canonical velocity vc with components (-v sin 8,0,0) and a correlated linear y-axis gauge velocity v~w i t h components (0, v cos 0,O). The symmetric-gauge solution gives circular cyclotron motion for both the vc and v~ velocities, whereas the Landau-gauge solution gives harmonic oscillator motion v, along the x-axis and correlated harmonic oscillator motion VA along the y-axis. These two different gauges both add up to the same cyclotron motion and same energy spectrum. The classical solution is gauge invariant. The situation with respect to gauge invariance is different when we examine the quantum mechanical Landau cyclotron equations. The syminetric-gauge formulation of these equations was given in Sec. 16. In this gauge, the kinetic momentum 71 divides equally into canonical p, and gauge eA momenta, just as it did in the classical case. The energy equation has two solutions, depending on whether we select positive or negative values for the angular momentum quantum number C . The observed Landau levels are obtained by choosing negative f? -values, but the magnetic moment interaction indicates positive C -values. When we solve these equations in the Landau gauge A = (0. Bx, O), the wave equation is expressed in terms of plane waves in the y direction and a harmonic oscillator solution in the x direction [q.The electron velocity in the y direction contains both canonical and gauge components, which combine together to give harmonic oscillator motion. The velocity in the x direction contains only a canonical component, which also oscillates harmonically. However, the Landau-gauge eigenenergies contain an enigma. These energies are given by the equation

E = h a , ( n + l I 2 ) , n = 0 , 1 , 2 ,.... (17.1) The energy C -dependence that was shown in Eq. (16.3) has disappeared. Thus the Landau equations as customarily formulated [1,6,7] are not gauge-invariant.

QUANTUM HALL ENIGMAS 18. Final comments about the quantum Hal1 Landau solutions It should be emphasized that the Landau equations for the quantum mechanical motion of an electron in a magnetic field apply only to the possible eigenstates of a single electron. They do not apply directly to a semiconductor ensemble of Fermi sea conduction electrons. The symmetric-gauge Landau equations, for example, are for a set of electron orbitals that all have the same center. Although the energies of the symmetric-gauge Landau electron states can be made degenerate in C by choosing negative C -values in Eq. (16.3), the sizes of these states are not degenerate, and the orbitals increase in size with increasing C -values [lo]. The Landau-gauge electron states are for a series of electron wave functions that are closely-spaced in one direction, and extend the length of the Hall bar in the other direction. These Landau-gauge wave functions are not related to specific angular momentum states C, and each Landau-gauge electron state is in fact a linear combination over all angular momentum values. This makes it difficult to single out the magnetic moment contributions to the Landau-gauge energy eigenvalues. The electron orbitals in a "normal" semiconductor state are quite different from the electron orbitals on a quantum Hall plateau. In the "normal" state, the interactions of the electrons with the substrate are very strong, as manifested by their anomalous effective masses m*. These electrons are trying to establish quantized oc= Belm* cyclotron orbitals, but the orbits are constantly being disrupted by inelastic scattering with the substrate, and stable orbits with stable magnetic moments do not occur. The electrons in a Landau level are synchronized coherently into the Fermi sea by their common averaged interactions with the hills and valleys (the landscape) of the substrate, and they spread out so as to occupy the phase space predicted for them by the Landau equations. Thus they sequentially fill the Landau levels and produce the observed Shubnikov-de Haas oscillations. But they are not neatly occupying the various orbital states that are predicted by the Landau equations for non-perturbed electron motion. They are in what might be termed a condition of "coherent quantum chaos". When a quantum Hall plateau is created, this chaotic condition abruptly changes. The linkage between the electrons and the substrate landscape vanishes, and a new linkage is established with the external magnetic field lines (I = Me. Stable orbitals are now formed, and these orbitals capture flux lines 4 in a ratio that is rigidly defined by the Hall plateau filling fraction v. This requires the resulting composite particles to spread out evenly over the 2D surface of the Hall bar, which in turn requires them to all have the same "size", and hence to be in the same Landau quantum state. This is only possible in a boson condensate, so that the transition from fermions to bosons, when viewed globally, seems to be an inevitable requirement for Hall plateau formation. The coherence between the individual CBs in a Hall plateau boson condensate cannot be attributed to quantum states that are linked together by a single "master equation" (such as occurs for example with the electron states in an atom). Rather, this coherence follows from the close packing of the CBs. in a manner that has been directly observed in experiments with individual quantum dots [21]. Hence the systematics of the quantum Hall plateaus leads notjust to CB condensates, but directly to CPCB condensates [lo].

MALCOLM H. MAC GREGOR

References. 1. Janssen, M., Viehweger, O., Fastenratk U., and Hajdu. J.: Iizcroduction ro the n e o n ?ofthe Irzreger Qmzt~rinHaU Effect, VCH, Weinheim, 1994. 2. Von Klitzing, K.: Nobel Lecture: The quantized Hall effect, Rev. Mod. Phys. 58,519-531 (1986). 3. Stormer, H.L.: Nobel Lecture: The fractional quantum Hall effect, Rev. Mod Phys. 71, 875-889 (1999). Fig. 5 of this paper has the wrong sign for the Lorentz force. 4. Tsui, D.C.: Nobel Lecture: Interplay of disorder and interaction in two-dimensional electron gas in intense nragnetic fields, Rev. Mod. P h y . 7l. 891-895 (1999). 5. Laughlin, R.B.: Nobel Lecture: Fractional quantization, Rev. Mod. Phys. 71, 863-874 (1999). 6. Davies, J.H.: The Physics oflow-Dimerzsional Semicorzd~ctor~s, Cambridge University Press, New York, 1998. 7. Prange, R.E. and Girvin, S.M., eds: The Quantum Hall Ejject, Second Edition, Springer-Verlag, New York, 1990. 8. Chakraborty, T. and Pietilainen, P.: The Quantum Hall Effects; Fractional and Ii~tegral,Second Edition Springer. Berlin. 1995. 9. Das Sarma. S. and Pinczuk A., eds: Perspectives in Quantum Hall Effects. Wiley. New York. 1997. 10. Mac Gregor, M.H.: A Unified Quantum Hall Close-Packed Composite Boson (CPCB) Model, Found. Phys. Lett. 13, 443-460 (2000). 11. Kane, C.L. and Fisher, M.P.A.: Nonequilibrium Noise and Fractional Charge in the Quantum Hall Effect, Phys. Rev. Lett. 72, 724 (1994). 12. Jackson, J.D.: Classical Electrodynamics, SecondEdition, Wiley, New York, 1975. 13. The v = 5R Hall plateau is the only well-established even-denominator Hall plateau [see Ref. 41. 14. Jain, J.K.: The Composite Fermion: A Quantum Particle and Its Quantum Fluids, Physics Today, April 2000, 39-45. 15. Heinenon, 0..ed.: Composite Fenniom: A Unified View ofthe Quantum Hall Regime. World Scientific. Singapore, 1998. 16. Girvin, SM. and MacDonald, A.H.: Off-Diagonal Long-Range Order, Oblique Confinement, and the Fractional Quantum Hall Effect, Phys. Rev. Lett. 58, 1252 (1987). 17. Kivelson, S., Lee, D.-H., and Zhang, S.-C.: Electrons in Flatland, Sci. Amer. 274 (3), 86-91 (1996). The composite boson model illustrated in the bottom right-hand figure on page 89 may be flawed 18. See, for example, Aoki, H.: Quantized Hall Effect, Rep. Prog. Phys. 50, 655-730 (1987), 665, who does not, however, employ a minimum-energy hexagonal (Abrikosov) array. 19. Van Son, P. C., Kruithof, G. H., and Klapwijk, T. M.: Intrinsic sequence in the breakdown of the quantum Hall effect, Surface Science 229. 57-59 (1990). 20. One possible solution to this impasse is to note that the W = -p B term is not the total interaction energy of a magnetic moment in a magnetic field [12], and that if nonadiabatic interactions with a constant-current magnet power supply are taken into account, the sign of this term is reversed [12]. 21. Collier, C. P. et al., Scierzce 277, 1978 (1997); Physics Today, Dec. 1997, 9.

Note This work was performed under the auspices of the U. S. Department of Energy by the Lawrence Livermore National Laboratory under contract number W-7405-ENG-48.

ON THE POSSIBLE EXISTENCE OF TIGHT BOUND STATES IN QUANTUM MECHANICS A. DRAGIC, Z. MARIC Institute of Physics Zemun Pregrevica 118,11080 Belgrade. Yugoslavia J.P. VIGIER Gravitation et Cosmologie Relativistes Tour 22-12,4 erne etage, Boite 142,4 Place Jussieu, 75005 Paris, France

Abstract. The possibility oftight bound states in quantum mechanics has been often emphasized [I]. Two different mechanisms are proposed for their occurrence: strong magnetic interactions at small distances and creation of the "anti-Born-Oppenheimer" states. corresponding to rapid motion of the heavy particles around almost fixed electrons. The observed "excess heat" in cold fusion experiments could be explained by these new tightly bound states [I]. In the present paper both ideas are analyzed.

1. The Barut-VigierTwo-Body Model And Positronium Magnetic Resonances The usual practice in atomic physics treats electromagnetic interactions other than Coulomb (spin-orbit, spin-spin etc.) as perturbations, giving only small corrections to the energy

-

-

levels. Although these terms in Hamiltonian with distance behavior r -3 and r-4 are indeed small compared to atomic scale Coulomb terms, they are comparable or even much higher than later at shorter distances. In principle, a possibility that magnetic interactions at short distances give rise to new phenomena. not explained by perturbative treatment. The first exploration is made by Corben in an unpublished paper. He noticed that motion of a point charge in the field ofmagnetic dipole at rest, is highly relativistic and the orbits are of nucleardimensions. The further investigation has beenundertaken by Schild [2], but the most systematic treatment of this problem is given by Barut (see for example [3]) A two-body system where magnetic interactions play the most significant role is positronium. Both electron and positron have large magnetic moments which contribute to the second potential well in effective potential, at distances much smaller then Bohr radius. Barut and hi coworkers predicted that this second potential well can support resonances [4,5]. A twobody model, suitable for non-perturbative treatment ofmagnetic interactions is presented by Barut[3]and Vigier[l]. 349 R.L. h o r o s o et a1 (eds.), Gravitation ard Cosmology: From the Hubble Radius to the Plunck Scale, 349-356. O 2002 Klulrer A c r ~ l e i cPublishers. Printed in the Netherlar~ds.

A. DRACIC, Z. MARIC AND J-P VIGIER

350

The model essentially represents an extension of the Pauli equation to a twobody system and is defined by Hamiltonian:

where: miis the mass. the momentum, ei the charge, fi the position of the particles (i = 1,2) Ais electromagnetic vector potential and Vdd is dipole-dipole interaction term:

In the center of mass frame and with a normal magnetic moment: ji = s' Hamiltonian (2) becomes:

5

s,

are quantities related to the relative motion of bodies, and where r,p, rn is a reduced mass. The standard Pauli approximation leading to eq.(3) can be improved by keeping the energy term in the Hamiltonian. This correction is essential in the positronium problem, since the reduces mass m is of the order of resonance energies we are interested in. The new Hamiltonian depends upon energy only through the effective mass m* given by:

In terms of total spin and angular momenta Hamiltonian for this self-consistent Barut-Vigier model can be written as:

TIGHT BOUND STATES IN QM where operator Q is:

This most important spin channel for the resonance phenomena is S=l, L=l, J=0, because the attractive spin interactions are strongest and effective potential which appears in the radial Schrijdinger equation:

-

has second potential well at distances of order 1fm. The schematic plot of the effective potential in that channel, obtained using the effective mass corresponding to the energy E = 600 KeV, is shown in Fig. 1. Before attacking the problem of magnetic resonances, the method is first tested in the low energy limit (the usual bound states of positronium) and results are compared with perturbative QED. Energy eigenvalues are determined by an iterative procedure. The approximate energy at each step was obtained by numerical integration of the ScMdinger equation with implementation of the "shooting" method [6]. This energy was put back in the effective mass (eq. (4)) and the procedure was repeated until convergence was achieved. The known bound state energies are reproduced with five digits accuracy. Resonant states are defined as solutions of the radial Schrijdinger equation which satisfy "the outgoing wave only" boundary condition at large distances. As a consequence of a complex boundary condition, the energy is treated as complex quantity: e = E - $I?. The real part E is resonance energy, while imaginary part I' is resonance width. Resonances are investigated by means of conlplex coordinate rotation technique [7]. In contrast to bound state problem. matching of logarithmic derivatives of inward and outward wave functions now amounts to a two-dimensional problem which is treated by Newton-Raphson method. The other difference concerns the iterative procedure, since both the resonance energies and widths must be selfconsistently determined. However, it should be noted that only the real part E of a complex energy enters the effective mass rn*. To start the iterative procedure the lattice of initial guesses of energies and widths is formed, covering energy range of interest and iterative procedure is started at each of this points.

A. DRACIC, Z. M A W AND J-P VIGlER

-

Figure 1: Schematic plot of the effective potential for E = 600 KeV in S = 1, L 1, J = 0 channel. Various features of the potential are drown out of scale in order to be visible.

We have not found any resonances in the energy range 1-2 MeV in the S=l ,L=l ,J=O channel. Stability analysis shows that our iterative procedure is convergent and we can conclude that negative result indeed means that resonant solutions do not exist. In addition, phase shifts are calculated. Two different methods of calculation are used: direct numerical integration of Schrodinger equation (NI) and variable phase method (VPM). Methods are consistent with each other (Table 1) and phase shift does not show any resonant behavior. The details of calculations will be published [8]. 2. Anti-Born-OppenheimerStates The three body problem H z (or D:) has been well understood in the framework of the Born-Oppenheimer approximation. If there is a situation where the motion of the lighter particles (electrons) is suppressed, the energy spectrum would look quite different. A method for estimating energy levels in this "anti-Born-Oppenheimer" approximation (corresponding to rapid motion of the nuclei around almost fixed electron) has been presented by Barut[9].

TIGHT BOUND STATES IN QM

m

M

M Figure 3.

Figure 2. 2.1 MOTIONLESS ELECTRON

The general idea can be most easily seen from configuration with static central particle (electron). For configuration of the three particles shown on fig. 1 and by taking into account only Coulomb interactions, in the c. m. frame Hamiltonian is given by:

~ = 2 - $ + a(? + $ ) ( 7 ) ~ h e Bohr quantization of symmetric circular orbits of the two particles M around the center of mass:

gives

The minimization of the energy

BP

can give us prediction of energy levels.

This method of quantization gives for H- and He ground state energies values very close to the correct ones, In the case of the anti-Born-Oppenheimer situation, the above method gives for HZ :

The ground state energy for H: is

- 28KeV and for D:

-

56KeV.

A. DRACIC, Z. MARIC AND J-P VIGIER

Table 1: Calculated uhase shifts Energy (KeV) 100 150 200 250

300 350 400

450 500 550

600 650

700 750 800 850

Energy levels of ~ z ( s t a t i cconfigmation) Energy levels of D; static configu -56.1:

-14.0: Table 2:

-6.231

-3.5a -2.2Q -1.551 -1.14t -0.876 -0.692

2.2 TRIANGULAR CONFIGURATION

In the more general treatment electron can perform small oscillations between the nuclei(Figure2) As in the Born-Oppenheimer approximation, the motion of the electron is quantized first.

TIGHT BOUND STATES IN QM

The electron Hamiltonian is approximately an oscillator around:

Jc

355

2%) KrLdpr = - 4a(R

with frequency: w = 4 a ( - 2 % ) R- 2s. The quantization o i H by means of f pdg = nh gives

Next, the nuclear Hamiltonian is quantized.

In the radial wave equation for H z (Z=l, z=-I), the effective potential appears:

Kf

The effective potential has two minima. First, for L = 0, RmilL= 4naR2 in the region of the Bohr radius of the electron. The second for L # 0,n = 0 5&c,= 3 M a in the region of the Bohr radius of the proton. Thus, this method predicts the existence of both H z and H:. Following described procedure one arrives at an approximate expression for and D;: energy levels of normal states of

HZ

In the case of H: and 8: energy levels are:

2.3 SPIN INTERACTIONS Spin is encountered for by "minimal substitution":

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A. DRACIC, Z. MARIC AND J-P VIGIER

For static configuration (Figure 2.) with spin Hamiltonian for

is:

For triangular configuration energy levels of H$are calculated by keeping terms linear with respect to the nuclear spin. They are on the same scale as in the static case, Although theory here presented is proposed as a mechanism for occurrence of "excess heat" in the cold fusion experiments, available experimental data can not conclusively be related to it. Also, it is not clear at which experimental situation systems would exhibit behavior predicted by this model. Table 3: Energy levels of H$(triangular configuration)

Table 4: Energy levels of

static configuration with spin interactions)

References [l] Vigier J. P. paper, presented at ICCF4, Hawai 1993. [2] Schild A. Phys. Rev. 131 (1963) 2762. [3] Barut A. 0 . SUN. High Energy Phys. 1 (1980) 113. [4] Barut A.O. and Kraus J. J. Math. Phys. 17 (1976) 506. [S] BarutA.O., BerrondoM. andGarc'ia-CalderonG.,J.Math. Phys. 21 (1980) 1851. Koonin S.E. & Meredith D.C. Computational Physics, Fortran Version. Addison-Wesley, Reading,MA (1990). [7] Connor J.N.L. and Smith A.D. J. Chem. Phys. 78 (1983) 6161. [8] DragiA., BeliA. and MariZ. FactaUniversitatis (in press). [9] Barut A.O. - Inter. Journal of Hydrogen Energy, 15 (1990), 907.

[a

A CHAOTIC-STOCHASTIC MODEL OF AN ATOM

CORNELIU CIUBOTARIU*, VIOREL STANCU Technical University Gh. Asachi of Zasi. Department of Physics. Bv. D. Mangeron No 67, RO-6600 Zasi, Rorrtania CIPRIAN CIUBOTARIU Al. I. Cuza University of Zasi, Faculty of Corrtputer Science, RO-6600 Iasi. Rorrtania, errtail: [email protected]

Abstract. The idea that a 'magnetized' charged particle in interaction with 'resonant' photons operates from an energy level to another higher one by a stochastic acceleration effect suggests that such effects may represent a phenomenological physical mechanism which explains how an electron jumps to higher atomic orbits when it absorbs resonant photons. If we increase the number of iterations of the corresponding nonlinear system of equations, we obtain a Bohr image of an atom. Such (quantum-transition) jumps, their duration and physical mechanism have never been explained by the quantum theory of atoms. We thus offer through such a cascade of chaotic kicked (stochastic acceleration) effects a physical explanation of the quantum model of absorption of energy by an atom. The proposed equations can model a circuit biased with a traveling electromagnetic wave. Such a circuit can also simulate a stochastic acceleration and a chaotic atom.

1. Introduction It is often claimed that the recent (re-)discovery and applications of chaos theory (or ergodic theory if we remind the old name of this subject) constitute the third great

revolution in physics in the twentieth century, the first two being the invention of relativity and of quantum theory. Many systems in nature exhibit classical chaos (i,e. sensitive and exponential dependence on initial conditions) but, for the t i e being, there is no universally accepted definition of quantum chaos (see, for example, [I]-[3]). However, there are different approaches to this concept, such as Quanturn Chaology (QC)' and the Randor~lMatrix Theory (RMT)~.In a way, quantum chaos refers to the * Corresponding author, email: [email protected]

'

Quantum Chaology represents the study of semiclassical behaviour of systems d o s e classical motion

exhibits chaos [4]. It has been shown that the statistics of eigenvalues, and eigenfunctions of classically chaotic systems are similar to infinite symmetric matrices whose elements are random numbers.

357 R.L. Amoroso et a1 (eds.),Gravitation & Cosmology: From the Hubble R~rdiusto the Planck Scale, 357-366 O 2002 Kluwer Academic Publishers. Printed in the Netherlands

C. CIUBOTARIU, V. STANCU & C. CIUBOTAFUU

358

quantum mechanics of cldssically chaotic systems and concerns itself with the quantumclassical correspondence, specifically, deducing quantum properties of a system using only classical quantities (primarily trajectories and periodic orbits). For classically regular (integrable) systems (which form only a subset of measure zero of all Hamiltonian systems?, methods for doing this have been known since the early days of the quantum ideas, before wave mechanics. However, for classically chaotic systems, the methods are still being developed [5]. The purpose of the present paper is essentially threefold: (i) to draw attention that the quantum chaos with all the characteristics of a classical chaos (e.g., the butterfly effect) manifests itself at any quantum level of atomic systems in a specific quantum way (e.g. the quantum butte$y effect); (ii) for a charged particle-electromagnetic field system, the countapart of the quantum resonance is represented, for example, by the classical resonance between the frequency of photons and cyclotron frequency ( w = n a ) ; and (iii) a quantum resonant transition (e.g., in a Bohr's semiclassical theory of an atom) corresponds to a Fenni (stochastic) acceleration mechanism.

2. Poincare Sections and Bogomolny Sections It is well known that the transition from order to classical chaos is studied by the examination of the phase space properties by the way of Poincd-section plots (classical surface of phase-space sections, CSOS). A quantum (or Bogomolny's semiclassical) surface of section (QSOS) is similarly drawn through the (configuration) space of the classical Hamiltonian but now, instead of only marking the points where the different (periodic, quasi-periodic, or chaotic) trajectories cross the surface, one also calculates the semiclassical phase exp(iS/h) which has accumulated since the previous crossing. In other words, one can use the behaviour of classical trajectories from one crossing of a QSOS to the next to compute the energy eigenvalues of the system. We remind that

S, =

4 ~ ~ p i disq the, action accumulated along the trajectory (closed loop) Lk. if a k

Hamiltonian system is classically integrable4 then its trajectories are constrained to t i oben ~applied. In terms of action-angle variables, invariant tori, and ~ ~ ~ - ~ u a n t i z acan Then the EBK the Hamiltonian is a function of the actions Sk only, H = (semiclassical) quantization rules are (k = I , 2,. .. ,1)

Hek).

Generally, in a Hamiltonian system the motion is governed. for example. by Newton-like equations without dissipation. i.e. if it has as many constants of motions in involution (i-e.,their Poisson brackets with each other vanish) as degrees of freedom, where one constant of motion is the total energy E which is equal to the classical Hamiltonian H(p, q) =E. For I-D Hamiltonians (all of which are integrable), the tori arejust the periodic orbits, and the analogous WKB approximation method can be applied. Einstein-Brillouin-Keller quantization

A CHAOTIC-STOCHASTIC MODEL OF AN ATOM

359

where the nk 2 0 are integer quantum numbers, and the integers pk 2 0 are the Maslov indices that count the numbers of caustic%along the trajectories6. The semiclassical (EBK) quantization rules (1) (based on action-angle variables) are not applicable to the chaotic systems since the definition of Skis meaningless for ergodic systems which possess no invariant tori7. The distance between two neighbouring trajectories increases exponentially along a unstable manifold, and decreases exponentially along a stable manifold. The problem is to find a semiclassical quantization approach to the chaotic systems. In other words, the question is to find a classical approach to the quantum-mechanical energy levels of (nonintegrable) ergodic systems. Martin Gutzwiller[6], opened this field by his semiclassical trace formula (for the quantization of chaos)' which sums purely classical information about periodic orbits (i-e., classical orbits which are closed in phase space9) into an expression for the quantum mechanical density of states. The poles of the expression lead to quantum eigenenergies. Thus, semiclassical mechanics is now liberated from the torus and allowed to wander free. In a way, quantum chaos is in fact a semiclassical mechanics off the torus. If we want to explore all phase space, we do not need to use only (Gutzwiller) periodic orbits, but we can use any set of (spray) trajectories with their contributions on a QSOS. We mention that all these methods (Gutzwiller's periodica-bit theory, Bogomolny's QSOS method, dynamical zeta functions etc.) to generate approximations to the quantum energy levels of chaotic systems may be considered as a substitute for the WKB-method for the case of integrable systems and its multidimensional generalization, the EBKquantization conditions.

3. Nonlinear Equations for Unstable Orbits In the framework of quantum chaology, the basic idea is to relate the behaviour of the eigenvalues and eigenfunctions (of a quantum Hamiltonian H)of a quantum system to the physical and formal structure ofthe phase space of the corresponding classical system in the regular and chaotic region. For example, the spectral statistic P(s) of the energy levels is a quantity which can be calculated only from an observed sequence of levels. P(s) is the distribution level (nearest-neighbour) spacings si = Ei+,-Ei of levels El. It has been found that the spectral statistics (or spectralfluctuations) of quantum systems with

6

The motion takes place on a Lagrangian manifold, and pk (the number of conjugate points or the Morse index of a trajectory) is determined by the topology of the Lagrangian manifold in phase space with respect to

configuration space. Individual trajectories in a chaotic dynamical system are ergodic in the sense that they are wandering through

7

phase space in a similar manner to a random (stochastic) variable. 8

The Gutzwiller's trace formula represents the trace of the energy-dependent Green's function which is the

Fourier transform of the time-evolution operator. In the semiclassical limit when A+ 0, the leading contribution to the Feynman path integral [9], [lo] comes from the classical orbits. However, the contributions to the trace of the time-evolution operator come from those classical orbits which are closed in coordinate space.

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corresponding classical (strongly) chaotic behaviour (K-systems) are in agreement with the Gaussian distribution of a Unitary Ensemble (GUE)~'.

whereas quantum analogs of classically integrable systems are described by non-Gaussian (e.g., Poissonian) statistics, P(s) = exp(-s) . (3) In other words, systems which are classically chaotic, and those which are integrable belong to different Universality classes. So far we referred only to the signatures of quantum chaos in spectra. However, the same question can be asserted for the eigenfunctions of H. The distributions of the values of individual eigenfunctions ~ ( qof) H have a Gaussian density in the semiclassical limit when the classical system is chaotic, whereas for classically integrable systems the density is non-Gaussian. Furthermore, a class of chaotic systems was found for which P(s) nearly behaves as it is expected for classically integrable systems. This is the case of the so-called arithmetical quantum chaos and shows us that there do not exist universal signatures of classical chaos that manifest themselves in P(s). It thus seems desirable to introduce a visualization technique that distinguishes quantum systems with chaotic classical limits from those with integrable ones, and which in a more intuitive way displays the random character of P(s) in the former case. It is the aim of this paper to put forward a numerical approach to this subject. We obtain a chaotic generalization of the De Broglie-Bohr quantization of an atom. Instead of quantizing by suspending standing-wave configurations on stable Keplerian orbits, we suspends the standingwave configurations on the infinity of unstable orbits. Such unstable periodic orbits are observed experimentally in some atoms (helium, hydrogen and other systems [I I]). The starting point for our approach is represented by the physical conditions which define the applicability of the WKB semiclassical approximation. We remind that the Schrijd'iger equation for a wave function ry = A exp(iHh) is equivalent to the equation for the function S:

Comparing this equation with the classical Hamilton-Jacobi equation for the action function S,

10

If the chaotic system has a time-reversal symmetry, the distribution corresponds to the Gaussian Orthogonal Ensemble (GOE) of RMT: P(s) =(ns/2)exp(-ns2/4)

A CHAOTIC-STOCHASTIC MODEL OF AN ATOM

36 1

we notice that the last term in equation (4), which is proportional to Planck's constant h, gives minor corrections to the classical equation if the following condition (sen~iclassical approximation) is satisfied:

Considering that p = VS, this condition can be written in the form

In particular, for the I-D case, we get

As p =-J

we can rewrite (8) in the form

Thus, the semiclassical approximation is not applicable for small values of the particle momentum, especially at points where the particle must come to rest ( p + 0, A + 0). Such a situation arises, for example, when a particle in a potential well is reflected at the potential barrier and starts moving in the opposite direction (turningpoint). We consider that, locally, in a semiclassical approximation, an electron of an atom interacting with photons may be modelled by the motion of a charged particle in a field of a transverse monochromatic electromagnetic wave (photons) and within a constant external magnetic field which simulates the curvature of spacetime inside an atom. The nonlinear dynamics of this model is described by the following equations (Maxwell equations, and relativistic equations of motion):

C. CIUBOTARIU, V.STANCU & C. CIUBOTARlU

@Y - -~~,[Pcos (X - T) + l]Px + Hcos (X - T), 4=--@-

(12)

obtained with a non-dimensionalization scheme (see Ref. [12] for notations). 4.

Numerical Results and Bohr Orbits

Since there exist no efficient analytic methods for describing stochastic components of a motion, it is advisable to apply a numerical integration of equations (10)-(12). In this section we propose to analyse numerically the process of the stochastization (chaotization or randomization) of the wave-particle interaction. We initiate the calculus for the following values of parameters: cyclotron frequency = SZB = 0.5, order of resonance = n = 4, initial energy = yo = y4 = 2 , initial momentum = Pxo= PYo= 41.5 The dynamic system of equations (10)-(12) generates a flow in a 3-dimensional (3D) phase space (P,, P, X). Numerical solutions have been obtained by applying a fifth order Runge-Kutta algorithm with an adaptive stepsize control [13]-[15]. The initial

.

valuesfor [T,P , Py,XI are [O-001, fi,0.0011. As may be expected, the particle motion for small dimensionless amplitudes of the external electromagnetic wave (e-g. H= 0.01) remains regular and no acceleration arises. If the amplitude is increased to H = 0.05 the particle motion becomes more complex but retains a regular character. The onset of a stochastization is observed once H exceeds 0.5 but the energy gain remains initially still small. When H = 0.6 there emerges a strong acceleration and a 'gun eflect' is initiated, whereby a sudden expulsion of a particle from the system takes place in a certain direction. The gain in energy becomes significant rising from y4= 2 to y300 = 150 and thus a state of a resonance overlap is obtained.

6,

Figure I : A 3D numerical solution of system (10M12)ibr H=0.9 and T =0..800.

A CHAOTIC-STOCHASTIC MODEL OF AN ATOM Px-T aeribe of a chaotic gun L

A

Figure 2: (P,,I ) time series corresponding to the dynamic system (10)-(12) for the case of the chaotic gun illustrated in Fig. 1.

A major chaotic gun erupts for H = 0.9 (see Fig. 1). In such a case we distinguish three situations: first, a localized chaotic regime (which we name a chaoson) emerges, subsequently, a high frequency oscillation with chaotic modulation of the amplitude appears (see also the time series in Fig. 2) and finally we observe the sharp rectilinear part ofthe trajectory which in fact displays the gun effect. It is easy to show that the electromagnetic gun is indeed chaotic because a small change in the initial value of P,, for instance of lo4, leads to a dramatic change of the phase portrait and the velocity of the particle. At the same time the chaotic motion of a charged particle inside a chaoson will certainly lead to a localised stochastic domain of the electromagnetic field which we may denote as an electrorrlagnetic stochaston. In a stochaston the field has a stochastic or braided structure. A fundamental question arises now: Are these two physical concepts (chaoson and stochaston) related by a reciprocal stochastic induction in the sense that "a chaoson (a localised chaotic motion of a charged particle) induces an electromagnetic stochaston (localised domain of the electromagnetic field) and a stochaston induces a chaoson". If stochastic induction were true we could accelerate particles in chaotic fields. In a way the answer appears affirmative since it is well known that any sufficiently turbulent magnetized plasma may accelerate particles through resonant interactions with chaotic plasma waves. However, the question refers also to the acceleration of particles as from chaotic static fields. We hope to study this problem in a future paper. A very interesting result arises if we increase the number of iterations and the duration of the operation (see Fig 3). We notice that in the process of acceleration the energy is increased by 'quantum jumps' so that a charged particle gains energy within a sequence of a cascade of gun effects. The jump from a 'spiral Larmor orbit' to another one is performed by a gun effect. The radii of the Larmor circles and the corresponding energy of the particle increase. The existence of a multi-gun cascade via different Larmor orbits is illustrated clearly by the time series for P, (Fig. 4), and P, and also through the Poincard sections (Fig. 5). In Fig. 4 the horizontal Lines display the chaotic gun effects between different Larmor orbits represented here by vertical oscillations. From the (P, 7) time series (Fig. 4) we

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deduce that, during a gun regime the 'longitudinal momentum' (P,) is constant and represents the highest value which may be achieved in the current Larmor spiral. This highest value is transferred to the following Larmor spiral in which the particle absorbs again energy from the electromagnetic field until a new gun effect erupts. In fact, on a specific Larmor spiral the particle will go around repeatedly, receiving a kick of energy each time it completes an orbit. We observe a classical Fermi acceleration combined with the gun effect which generates thejumps.

I

I

Figure 3: A 3D numerical solution of the system (10)-(12) for H = 0.9 and T = 0..2150. Phase Larmor spirals displaying orbital Bohr jumps by stochastic acceleration effects (A fifth order Runge-Kutta algorithm with a stepsize control).

In fact a gun effect hides a very localized (in space and time) phenomenon (a chaoson) which leads to a very ordered motion of a charged particle. The long duration of the chaotic gun effect (with respect to the short duration of motion on the Larmor orbits) permits the extraction of an accelerated particle with a certain eneqy. Moreover, the gun timescale, zp, for an escape varies considerably with energy and this variation may not be of the same order as that of other timescales (e.g. the timescale for diffusion, the timescale for advection eta. This situation may have some implications on the steady state and the time evolution of particle distributions in physically realistic situations.

Figure 4: (P, , 7 ) time series corresponding to the dynamic system (10)-(12) for the case of a multi-gun cascade illustrated in Fig. 3.

A CHAOTIC-STOCHASTIC MODEL OF AN ATOM

365

It is also worth noting that the gun effect has a stabilizing effect on the Pycomponent which after each stop of the multi-gun cascade is reduced to the same initial value. Thus, the particle is accelerated and guided preferentially into the direction of the electromagnetic wave vector. In a phase portrait (P, P,) (Fig. 5) the motion seems apparently regular but a sharp sight can distinguish one jump from a Larmor circle to the next one as generated by a chaotic gun effect. All vertical lines display in fact a gun effect. The central vertical segment represents the chaotic gun illustrated in Fig. 1. 5. Conclusion And Comment

The situation described in this paper is similar to that of the semiclassical Bohr orbits in an atom whereby an electron jumps to orbits of larger radii by absorption of electromagnetic energy (photons). Such jumps, their duration and physical mechanism have never been explained by the quantum theory of atoms. We offer through a cascade of chaotic gun effects a phenomenological physical explanation of the semiclassical model of absorption of energy by an atom which can be interpreted as a chaoson ir?lr?lersedin a stochaston. In other words, an atom (or other quantum mechanical systems) represents a semiclassical chaotic (Hamiltonian) system (chuosm) coupled with a stochastic (quantum) field (stochaston). Finally, we notice that, in a very schematic (but general, for a wide range of physical phenomena) way, the nonlinear equations (10)-(12)are equivalent with the equation which describes the motion of a non-linear oscillator perturbed by a plane wave. This situation proves that indeed the proposed stochastic model (a K-system) for an atom appertains to the Gaussian class of distribution".

Figure 5: Projection (phase portrait or orbit) on the (P,, Py)phase space of the particle trajectory shown in Fig. 3. The situation is similar to that of a linear kicked hannonic oscillator. The discontinuous jumps in P, are a result of the kicks caused by photons. 11

We remind that, for example, the coordinate distribution in the ground state of a quantum oscillator is

Gaussian:

I

= -exP[-

Axo

(xI a)].

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C. CIUBOTARIU, V. STANCU & C. CIUBOTARIU

Thus, a quantum system is equivalent with a specialized sensor that works under resonance between frequencies Cl and o in the same way as any circuit based on an analog computer version of the Lorenz or Rossler, or Chua butterfly equations for a classical chaos, or a Josephson junction for a quantum chaos works. In the framework of the present paper, the Pound-Rebka-Snider redshift experiments [16]-[18] as a test of geodesic motion are just an application of the high sensitivity of chaotic quantum systems at the very small variation of specific parameters (frequency of electromagnetic waves).

Acknowledgments The authors are thankful to Professor Richard L. Amoroso (Noetic Advanced Studies Institute, Orinda, CA, USA) for encouragement, and to Professor John Argyris (Institute for Computer Applications, ICA 1, University of Stuttgart) for discussions.

References [I] Steiner. F.: Quantum Chaos. lnvited contribution to the Fetschrift Univer.vitatHanlburg 1994: Schlaglichter der For.vchungz~irtr75. Jahrestag (Ed.R. Ansorge) published on the occasion ofthe 75th anniversary of the University of Hamburg, Dietrich Reimer Verlag, Hamburg 1994, pp. 542-564. [2] Manfredi, V. R. and Salasnich, L.: Different Facets of Chaos in Quantum Mechanics, Int. J. Mod. Phys. B. 13 (18) (1999), 2343-2360. [3] Aurich, R, Bolte, J. and Steiner, F.: Universal Signatures of Quantum Chaos, DESY Report 94-024, February 1994. [4] Berry, M. V.: in Dynnnlical Chaos, The Royal Society, London, 1987. [5] Haggerty, M. R.: Seraiclassical quantization in a srtlooth potential using Bogortzolny'.~quannirtr sutjiace of section, PhD Thesis, Massachusetts Institute of Technology, 1994. [6] Gutzwiller, M C.: Energy Spectrum According to Classical Mechanics, J. Math. Phys. 11 1970, 1791-1806. [7] Gutzwiller, M. C.: Periodic Orbits & Classical Quantization Conditions, J. Math. Phys. 12 (1971), 343-358. [8] Gutzwiller, M. C.: Chaos in Classical and Quanttlrir Mechanics, Springer, New York, 1990. [9] Feynman. R. P.: Space-Time Approach to Non-Relativistic Quantum Mechanics. Rev. Mod. P h y 20 (1948). 367-387. [lo] Feynman, R. P. and Hibbs, A. R. Quanrunl Mechanics and Path Integrals. McGraw-Hill, New York, 1965. [Ill CvitanoviC, P.: Classical and Quantun~Chaos, version 6.0.2, Feb 2,2000, printed April 7,2000 (www.nbi.dk/ChaosBook/). [I21 Argyris. J. Ciubotariu, C.: A new physical effect modelled by an Ikeda map depending on a monotonically time-varying parameter, btt. J. Bif Chaos 9 (1999), 1111-1120. [I31 Butcher, J. C., The Nu~ilericalSolution of Ordinar?. Differential Equations.Wiley, Chichester, 1987. [I41 Koonin, S. E., Cori~putatioitalPhysics. Benjamin Cummings. Menlo Park. 1986, Chap. 4. [IS] Press, W . H., Flannery, B. P., Teukolski, S. A. and Vetterling, W. T. Nurirerical Recipes. Cambridge University Press, Cambridge, 1986. [I61 Pound, R. V. and Rebka, G. A.: Apparent weight of photons, Phys. Rev. Letr. 4 (1960), 337-341. [I71 Pound, R. V. and Snider, J. L.: Effect of gravity on nuclear resonance, Phys. Rev. Lett. 13 (1964), 539-540. [I81 Pound, R. V. and Snider, J. L.: Effect of gravity on gamma radiation, Phys. Rev. B 40 (1965), 783- 803.

SYNCRONIZATION VERSUS SIMULTANEITY RELATIONS, WITH IMPLICATIONS FOR INTERPRETATIONS OF QUANTUM MEASUREMENTS

JOSE G. VARGAS AND DOUGLAS G. TORR Department of Physics University of South Carolina Columbia, SC

Abstract

The incorrect extrapolation into the realm of dynamics of the thesis of conventionality of one-way speeds (i-e. that, unlike two-way speeds, they depend on the choice of synchronization) has been detrimental for developments of physical theories based on an absolute relation of simultaneity. In response to the perceived need for a better understanding of issues such as interpretation of the collapse of the wave function, quantum teleportation, etc., we explore the quantum mechanics (QM) sector of a theory canonically implied by an absolute relation of simultaneity, thus by a preferred frame field (PFF). It constitutes a formidable platform containing a geometric Dirac equation from which to develop a deeper interpretation of the usual, dualistic QM. 1. The Issue of the Relation of Simultaneity of the World.

Participants in this workshop try to harness the potential of teleportation to synchronize clocks at spatially separated points. Looming in the background is the thesis of conventionality of synchronizations. The two camps on the issue remain hopelessly opposed. Is the con-ventionality issue relevant for applications of teleportation to synchronizations? Not in the short term, as one tries to find out whether QM allows phase teleportation using a two level transition. In the longer term, one has to contend with the related issue of conventionality of simultaneity. If simultaneity were a matter of convention, why would quantum teleportation make any sense at all? For the wave function's collapse to teleport from one photon to another, the concept of "at which photon (in an entangled pair) the wave function was made to collapse" must be nonconventional. Workers at the foundations of QM do not share this conventionalist view. They view nature's relation of simultaneity as objective or non-conventional. They believe that it is the one prescribed by standard special relativity (SR) but, unfortunately, they also believe that the experimental evidence c o d m this. 367 R.L. An~lorosoet a1 (eds.).Gravitation a ~ cCosntmnlog~: l Fro111the Hubble Radius to the Plrnck Smle, 367-376. O 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Our resolution of this issue is that whereas synchronizations arefor m o s t p u ~ o s e s about our conventional setting of clocks at a distance, the relation of simultaneity is a dynamical, non-conventional issue that only nature decides. For nlost purposes coordinates are simply markers. We can change from one set of them to another, in particular from the Einstein markers to the (primed) markers for absolute synchronization:

where (v,O,O) is the velocity of these frames with respect to some ab initio PFF. But, if the world abides by an absolute relation of simultaneity, using Einstein's synchronization when finding out at which of two entangled photons did we cause their common wave function to collapse will be a wrong purpose, The right synchronization of clocks can, for instance, be achieved through the use of the candidate PFF, where the slow clock transport and Einstein's synchronizations (implicit in the use of Lorentz transformations) coincide with the absolute one. The objective significance of the relation of simultaneity is embodied in the bundles of frames dual to the coordinate systems, if not in the coordinates themselves. The standard and "absolute" bundles are related by:

(velocity components v,= v,v,v,). The unprimed frames represent the pseudoorthonormal frames of SR and the primed ones correspond to absolute simultaneity. Caban and Rembielinski have shown that one can adapt QM to absolute simultaneity (primed frames), with advantage over the standard QM (unprimed frames)'. In view of this. we shall consider as proven that the relation of simultaneity is not a matter of convention. Assuming then that the actual relation of simultaneity of the world is the absolute one, one can still choosefor most purposes a set of markers (coordinates) which are not dual to the primed frames, but rather correspond to SR, given that the slow clock transport and Einstein synchronizations are the only expedient synchronizations and that both give the same result. The coordinate transformations would then be the Lorentz transformationsfor nlostpurposes even if the actual frames chosen by nature's relation of simultaneity are not the pseudo-orthonormal fi-ames. What experimental evidence determines the actual relation of simultaneity of nature? It must be one which conventionalists would say that it cannot exist, or that they claim to be circular. They rarely have to contend with specific experimental proposals, which, in any case, are not of a patently meaningful and easily recognizable nature. If they were, conventionalists would have conceded that there are meaningful (non circular) experiments that show that nature has a specific relation of simultaneity. Given limitations of space to deal with this issue here, this persistent attitude of conventionalists, intelligent scientists after all, will be viewed as "proof of the statement that the relation of simultaneity of SR has not been demonstrated by experiment in an incontrovertible way.

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2. The Physics Canonically Determined by an Absolute Relation of Simultaneity The post-relativistic postulate of absolute simultaneity implies a PFF, with respect to which the dual coordinate transformations are: x ' = y(X- vT) y' =Y zJ=Z t' = y' i" (3) X,Y,Z,T representing the PFF. The latter canonically determines a rule to compare vectors at different points or affine connection, the one according to which the frames of the PFF are all equal. like the connection on the earth where the rhumb lines are taken to be l i of constant direction. We then have zero affine curvature, to be denoted as teleparallelism (TP). The roundness ofthe earth still shows in the non-zero metric curvature. But, as we now show, the geometry in PFF theories must be Finslerian, even if the metric is Riemannian. The group property is another "non-conventional purpose" for coordinates. Let A denote transformations like (3). taking us from the PFF to "fiducial frames" with velocity (v,v,v,). These frames, and those obtained from them through O(3) rotations, will be called para-Lorentzian (PL). Let RI, and R2 be rotations. A#,-' transforms between fiducial frames of velocities (v,, v,,vJ and (v,,v,, v,,), the transformation Tgiven by

transforms between general PL frames, which can be put in a one to one correspondence with the Lorentz frames. This correspondence speaks of the fact that transformations 4 constitute a non linear representation of the Lorentz group2. The group property extends similarly to large families of flat spacetime structures which differ from SR in not complying with the independent experiments by Michelson-Morley, Kennedy-Thorndike and 1ves-stilwel13. The transformations represented in (4) act on a seven dimensional space, the three additional coordinates being the components ofthe velocity3. This space is the base space of Finsler connections and of Finsler gwmetry. This geometry is about manifolds S(M) which have been constructed from other manifolds M l i e we have constructed our sevendimensional space from spacerime4,5. Remarks: (a) there is affine Finsler geometry not involving metrics, (b) there are Finsler connections on Riemannian metrics and (c) Finsler distances may be irrelevant to physics. As with Riemannian geometry, the metric (nonaffine) origin of Finsler gwmetry is a historical accident. The process followed to obtain the coordinate transformations 4 does not work for their dual frames, as becomes clear by simply trying it. This is a consequence ofhow the frames sit over S(M), Mnow being spacetime. Since this S(M) is seven-dimensional and the set of orthonormal frames is 10-dimensional, there is a three-fold of PL frames related by the O(3) rotations sitting at each point s of S(M), rather than a six-fold of frames related by the Lorentz group. This O(3) group is a remnant of the vanished Lorentz group. It becomes SU(2) in the QM sector of the theory. The Lorentz group survives as a transformation between coordinates in the aforementioned non-linear realization. The Finslerian structure thus is the structure of PFF theories. Furthermore, it can be shown that the Euclidean signature is the canonical signature of Finsler garnet$. This Finsler

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bundle must, however, be reformulated as a 5-d Kaluza-Klein space where it becomes evident that U(1) also is a remnant of the Lorentz boosts on the frames6. It is in this space that the deepest QM must take place.

3. The Unified Theory CanonicaIly Determined by Preferred Frame Fields Unlike caban-~embielinski',we let the canonical mathematics of a PFF, or Finslerian TP, guide us. A superseding, combined Maxwell-Einstein-Dirac theory results7 (in contrast, Riemannian geometry does not take us beyond general relativity (GR)). Let be the affine connection generated by the PFF and the metric. Let a be 1ww P P. forms whose components are the Christoffel symbols. Let p be the contorsion. The P statement that the affine curvature is zero becomes:

The left hand side of (5) is the metric curvature. The parenthesis and the last term yield gravitational and non-gravitational energy-momentum by standard contraction, after identification of the different non-gravitational interactions in the different components of the Finslerian torsion. This one is of the form:

where d is the exterior covariant derivative and dP is the translation form. Ro, 9 contribute to the classical equations of the motion, but R' and Sido not8. These are not, however, terms to be compared with the standard model, because this torsion is a concept from a theory of frames. QM does not deal with these macroscopic objects. Instead of Maxwell's equations, dF=O, SF=j, we now have:

The derivative 6 is the interior derivative, which, applied to tensor valued forms, becomes the interior covariant derivative. It is defined as in the TP version9 of the K2hler calculus10.It replaces the Dirac equation with:

where a and u constitute input and output differential forms (both scalar-valued for electrodynamics). d is the sum of the interior and exterior derivatives. A conserved current is implied by equation (9). for any a. Kahler developed his theory for the LeviCivita connection. One then has that, if dv=O,

SYNCRONIZATION VERSUS SIMULTANEITY

371

This means that u and uv v are solutions for the same a. In particular, the product of solutions for a=O is another solution for a=O. In TP,this is no longer the case, but Eq. 10 remains valid in a weak field approximation9. The reformulation of Finslerian TP over a canonical Kaluza-Klein space yields a "generalized Dirac equation"

where d z is the element of proper time, now viewed as a fifth coordinate, where d g = d P + d q where d P is given by m p and where dv is equivalent to ww,'. Y belongs to a very rich algeb c structure. The unification of micro and macrophysics results. When J is written in terms of Yas prescribed in the Kahler theory, the system of Eqs. 6, 7 and 11 is closed (a given in terms of the metric and P in terms of the torsion). The complexity of the mathematics has impeded the recognition of this program for unification among theoretical and even mathematical physicists in general, except for the Clifford algebra community (see Preface to Proceedings in Ref. 6), where there is a tradition and some success in relating internal symmetries as spacetime symmetries (See examples in Ref. 6). For null J, Equations 7 can be written as dRW4, which thus is a Kihler-Dirac equation for a=O. The expansion of these equations in the PFF and for S=O, i.e. for electromagnetic (EM) vacuum, yields:

We have not tried to find solutions for such sophisticated system. It contains non-linear terms similar to those that cause the beautiful solitons ofthe Muraskin system1' rai

These equations 12 happen to state (not intentionally) that, in TP with Lorentzian metric, d ( p A 4 / @ &@ e v)=0, where the PA are the components of the contorsion. Muraskin obtained different types of numerical solutions of these equations over a 20 year period, solutions which very much depend on the initial values. For a particular set of initial r values, the plot of F',, as a function of x for given values of the other coordinates is shown in the figure, to be used in the next section (no special significance should be attached to this set of indices) We now show how this theory relates to the PFF of absolute simultaneity. In physical TP, the PFF is attached to autoparallel curves. The distribution of matter and charge determines these curves. The PFF at the basis of absolute simultaneity is the one in which

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J. G. VARGAS & D. G. TORR

we neglect "local motion" of matter and consider only the average over very large scales, matter which is neutral. The PFF field of TP in specilk regions ofthe spacetime manifold becomes the PFF of cosmology by disregarding the motion of matter except on the very large scales (comoving matter fame). In view of the unexpected emergence of such formidable theory, we have sidestepped the issue of whether the frames are PL or orthonormal- cum-^^^^. Based on the experience gained, we expect that the correct flat spacetime structure may still come in the wash.

4. Consequences of the Canonical Theory of PFF for Quantum Measurement and for Teleportation.

In this section, we shall overlook that not all the equations have been worked out to the point where precise statements can be made. This need not impede a view of the big picture. We shall proceed in stages of increasing structure. (A) The canonical development of TP leads to the Kahler equation, (9), not the Dirac equation. A major difference is the fact that this equation, based in a calculus of differential forms, has neither gamma matrices nor Hamiltonians. It is geometric. For EM coupling, the geometric character of the Kahler equation

is not apparent because (im+eA)dhc is scalar-valued, and affine geometry proper involves vector-valued forms. It is far fetched to say that the Hamiltonian is built into this equation, given the presence of the factor i in the mass term. There are other differences, l i e the number of components of the wave function, and yet, Eq. 14 solves the hydrogen atom. The energy operator, -(W27ri)#L$ (now with standard 9, is simply the Lie operator relating the solutions of (14) under time translation symmetry. The differences become even more obvious for systems of n particles. In the paradigm. these systems are typically handled with the SchriAinger equation in 3n+l dimensions. The K i l e r equation is an equation in 4 dimensions regardless of the value of n. It is, therefore, of the densityfunctional type, ab initio. In the standard, Hamiltonian-based QM, one feels the need to justify why density functional theory works. Kohn, who does not use Kiihler theory, objects to the concept of multiparticle wave function, as he did in his Nobel acceptance lecturei2, with statements such as "In general the many-electron wave function ~ ( r ..., , , r,) for a systeirl of N electrons is not a legitimate scientific concept, when N W O where NPl@"(ernphasis in original). In Kohn, the concept of multiparticle Hamiltonian goes down the drain with the concept of many-electron wave function. Of come, a QM based on the KWer equation will not be complete without a derivation from it of the multiparticle equation for numbers of particles which are not too large.

SYNCRONIZATION VERSUS SIMULTANEITY

FZgure 1. Muraskin's solitons, with permission from Physics Essays.

(B) The implications of (A) are even richer for spacetimes with torsion. The equation g u v v)=av(u v v ) remains valid, now as a weak-torsion approximation, with v in the role of boson and u as matter, fermions in particular. The wave function for a pair of a measuring instrument (to be called fermion), u, and a boson (say a photon) v, would be given by u v v when apart. When they are together, they constitute a fermion, u v v, for the same a. By the same argument, now forward in time, EQ. 10 admits the interpretation that u v v evolves back into the original fermion and boson. Concepts like confinement, absorption and wave function collapse appear to be related (a) among themselves, (b) to the compliance or not compliance of Eq. 10 and (c) to the fact that the wave function lives in spacetime (complemented by velocity in one way or another), but not in a 3n+l space, as we now argue. Consider the following statements in Diac s classic book on QM'~:(a) "If a system is small, we cannot observe it without producing a serious disturbance..," (p. 4), (P) "...beam of Light passing through a crystal of tourmaline... each photon polarized perpendicular to the axis passes unhindered and unchanged through the crystal, while each photon polarized parallel ..." (p. 5) and (y) "When we make the photon meet a tourmaline crystal. we are subjecting it to an observation... The effect of making this observation is to force the photon entirely into the state of parallel or entirely ..." (p. 7). Suppose we have two parallel tourmaline crystals with their optical axes also in parallel arrangement. Light going through the first crystal will become polarized and. consistently with (P), will pass undisturbed. But according to (y), this is a measurement. So, it is a iiteasurenlent which does not produce disturbance, contrary to (a). The morals of the 7

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story is not to try to invalidate the uncertainty principle, which is implicit here. But one should perhaps be more careful in its statement, or perhaps Dirac could have been more careful. For instance, (y) could be modified with the remark that the observation or the measurement finishes when the intensity of the passing beam is measured by absorption on a screen beyond the second tourmaline crystal. That is where we wanted to be. It is the absorption that counts. The through photons are in a regime where equation (10) works. Photons that are absorbed are in the regime where the torsion is not negligible and u v v does not satisfy equation 9. The wave function now remains a solution of (9) but is no longer u v v. The extreme robustness ofthe photons in modem technology would be a manifestation of the fact that, except perhaps at very high intensities, the regime of Eq. (10) is at work in the form d (v, v vJ=0 if &,=&2=0. Quark-gluon confinement would be the statement that, like in the case d (uv v) a v ( u v v) and unlike in a molecular system (EM interactions), the Kiihler equation does not admit a reformulation in multi-particle format. This is not surprising since the S terms are of a completely different and unusual nature R terms (EM is of the R type). The question is: how can the polarization of photons (relative to the tourmaline's orientation) be a determining factor of whether one or the other regime applies, At this point, we need to go one step further and assume TP. (C) Given the structural sophistication of the canonical PFF theory and the still primitive state of development of its details. we are not in a position to describe even qualitatively what causes a photon's wave function to be absorbed or get through. We may. however, resort to the proxy Muraskin equations to "guesstimate" some form of mechanism at work. In the PFF theory, the EM vacuum equations are d!W=O,i-e. of the type d v=O (when using specific u's and v's, we start to use boldface if applicable). In other words, the vacuum EM field is a spinor -in the sense of solution of the Kahler equation- of the type called harmonic. Except for the fact that we have to use the torsion appropriate for the K-K space14, which limitations of space forbid, the equation dKW=O may be viewed as being of the type dY= d @ v Y with d p vK i.e. d g VOW equal to zero. This is a constraint not unlike requiring that an EM field satisfied AvdA=O. Hence, the boson and fermion equations are unified in TP! Assume for the sake of the argument, that the Muraskin equations were the equations of the EM vacuum, its solitons being the photons (these are not packets or superposition ~a of plane waves of any sort). Since the linearized Muraskii theory yields ~ a , , = ~ the addition of the quadratic terms to the left hand side of d P d = 0 generates the solitons and modulated background of Eqs. (13). The high dependince on initial T s of thetype of solution that one obtains is to be contrasted with the irregular behavior (though bound in magnitude) in the zones between the solitons. Considering that the values of the I"s at any point Q in such zone can be viewed as a set of original conditions for the second soliton, one is led to conclude that the behavior of the solution to the right of Q (i-e. the emergence of the virtually identical second soliton, or other behavior as in the book in Ref. I 1) depends on the existence of conserved algebraic relations between the r s , and thus into classes of solutions. Given the argument of the previous paragraph, it is natural to assume similar behavior for matter. Polarizations of incident photons that are parallel and orthogonal to the axis of the tourmaline are interpreted as being such that the wave

y

SYNCRONIZATION VERSUS SIMULTANEITY

375

function (equi-valently, the set of gammas) of the pair crystal-photon when the two come close evolves into different classes. Different states of polarization would then give rise to wave functions belonging to two different classes of classes, respectively corresponding to whether or not Eq. 10 is satisfied. No collapse is needed, just bifurcation points. The question remains of whether this deterministic picture is affected in any way by Bell's inequalities. Let us start with the observation that Bohm's hidden variable theory satisfies the same inequalities as standard Q M Hence, there are Limits to the applicability of these inequalities, certainly when the hidden variable theory is governed by equations which are quantum mechanical. But, in addition, the "loophole" type limitations are overwhelming in this case. Y depends on the torsion. d(d@), through the connectiondependent operator d and also depends on d g . The dependence on the torsion constitutes a non-local way of depending on the potential, We may then say that the dependence on d@ constitutes a non-local way of depending on the torsion. Furthermore, this equation is not to be viewed as one to be solved for Ywhen 6'and d@ are given, but rather as making part of the system of equations where one has to solve for the (structuredependent) d and d@ at the same t i e as one solves for !E Finally, the system of Eqs. 6, 7 and I 1 is a closed system, but the physical system to which it applies cannot be considered as closed. We send photons into a region of space. They carry "their background" as in the figure, but there is already a background there (This is an issue that has to do with the Cauchy problem of TP7, which is radically different li-om the Cauchy problem of GR because the field equations involve the curvature and not just the Einstein tensor). It should be clear that present-day Bell theorems cannot cope with this sophistication. They would amount to collections of potential loopholes. 5. Conchding Remarks

In canonical Finslerian TP, which is also the canonical theory of PFF, teleportation may not be needed since the information as to the future behavior of a photon when encountering a tourmaline crystal is contained in its wave function (See (c) above). The measurement (absorption of photons) is regulated by deterministic laws, though this determinism is possibly unfathomable in practical circumstances. But even if Copenhagen-type indeterminism is more than a historical accident of twentieth century physics, the great richness of evolutions of solutions of the Muraskin equations speak of the possibility of defining other velocities associated with the propagation of the solitons, in addition to the velocity of the solitons themselves. Thus, an absolute, superlurninal velocity might perhaps be possible, though apparently unnecessary, in this theory. It might be associated in one way or another with the zones among the solitons. But we lack the imagination to go any further in this direction without violating causality. The examples just discussed show that the richness and wealth of information contained in the QM sector of this theory goes way beyond present day QM and the Copenhagen interpretation, not to mention the consequences of replacing gauge geometry (i-e. geometry in secondary bundles) and concomitant internal symmetries with geometry in the principal bundle. Hence there is no need in canonical PFF theory for extreme views

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(at least retrospectively) like "there is no wave function, just probability amplitudes"l5. The arguments put forward cannot be considered anything but qualitative, and even tentative. The reason is that the principle of the theory, Finslerian TP cum Kahler calculus, is extremely removed from any physical concepts. It will take an enormous amount of effort to bring this theory to where its physical consequences can be derived with a great degree of reliability. There are, however, windows of opportunity. One of them is, for instance, the fact that non homogeneous electric fields produce gravitational fields by virtue of source terms that contain the derivatives of this field, rather than by virtue of the energy-momentum contents. An experiment recently performed by T. Datta and M. Yin at this institution has already shown the existence of gravitational fields induced by a non-homogeneous electric field, although we have not yet tried to match it with exact predictions of the theory. Time may soon tell.

Acknowledgments One of us (J.G.V.) deeply acknowledges generous funding from the Office of the Dean of the School of Science and Mathematics of the University of South Carolina at Columbia, and the invitation and funding by JPL to participate in the NASA-DoD on Quantum Information and Clock Synchronization for Space Applications (September. 2000). We thank Professor J. Anandan for reference 15.

References P. Caban and J. Rembielinski, Phys. Rev. 59A, 4187 (1999). J. Rembielinski, Physics Letters 78,33 (1980). 3. J. G. Vargas, Fouridations of Physics 16,1231 (1986). 4. J. G. Vargas and D. G. TON,J. Math. Phys. 34,4898 (1993). 5. J. G. Vargas and D. G. Torr, The Cartan-Clifon Method of the Moving Frame: Firislerian Buridles on Rieamnnian Distances, to be published in Algebras, Grnups artd Geornew (Septemba 2000), as part of the Proceedings of the 1I Romanian Conference on Finsler and Lagrange Geometry. 6. J. G.Vargas and D. G. Torr, Clifford- Valued Clz$5on?~s.A Geortletric Language for Dirac Equations in R. Ablamowicz and B. Fauser (eds.), CliSford Algebras ar~dtheir Applications in Matl~rtlaticalPhysics", Birkhausa, Boston (2000), 135-154. 7. J. G. Vargas and D. G.Torr,Fromthe ComrologicalTerm to the Planclc Comrmir, to be published in a Volume of The Furidartzental Series of Physics, Kluwa Academic Publishers, devoted to the Proceedings of the Third Vigier Symposium. Berkeley. 2000. 8. J. G. Vargas and D. G. Torr. Found Phys. 29. 1543 (1999). 9. J. G. Vargas and D. G. Torr. Fourrl. Phys. 28.931 (1998). 10. E. Kiihler, Rendiconti di Maternatica 21, 425 (1962). 11. M. Muraskin, Physics E.s.sqs 8,99 (1995); Mathertlatical Aesthetic Priririple.~lNoni~~tegrable Sy.stertls" (World Scientific, Singapore, 1995). 12. W. Kohn, Rev. Mod. Phys. 71,1253 (1999) 13- P.A.M. Dirac, Tlie Principles of Quann~rtlMechanics (Oxford U . P., London, 1958) 14. J. G. Vargas and D. G. Torr, Found. Phys. 27,533 (1997). 15. B. Kayser and L. Stodolsky, Phys. Letters B 359,343 (1995). 2.

'

CAN NON-LOCAL INTERFEROMETRY EXPERIMENTS REVEAL A LOCAL MODEL OF MATTER?

*.I.MARTO, **.I.R. CROCA *Departurnento de Fisica da Universidade da Beira Interior 6200 CovilhcZ; Portugal Elnail: jrrzarto @mercury.ubi-pt **Depar?ar~zentode Fisica Faculdade de Cie^ncias,UniversidadP de Lisboa Carnpo Grande, Ed. C1, 1700 Lisboa, Portugal Etrzail: croca @fc.ul-pt

Abstract: In the following we consider the possibility of interpretating non-local interferometric experiments according to the De Broglie causal model. With the help of a simplified mathematical model based on wavelet analysis it is indeed possible to explain it in a causal way. Furthermore we show the distinction between the two formalisms and discuss some experimental conditions that may make these differences evident. 1. Introduction Quantum mechanics has a formalism exhibiting non-separability [I], in that context interferometric experiments with neutrons carried by Rauch [2] [3] qeern to show this peculiar feature. The authors assert that it is possible to get an increase in the visibility of the interference pattern using a proper postselection state process. The interference is observed without superposition, at the mixing region, of the wave packet that represents the neutron. An alternative causal interpretation is possible following the ideas of De Broglie. An early attempt was made [4], in the framework of this theory maintaining however Fourier non-local analysis in the particle description. In it a physical reality was assigned to the plane waves. We feel some uneasiness in understading how is possible to ascribe some character of physical reality to plane harmonic waves that are infinite in time and space. This assumption implies that a wave packet built up by an infinite sum of plane harmonic waves is not separable; therefore it is not possible to individualize real physical systems at a microscopic level. 377 R.L. Amoroso et al (eds.),Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, 377-384 O 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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With those dificulties in mind here we present a causal local particle model and discuss experimental conditions that could settle the problem of non-separability. Whether is it a result of non-local analysis or, due to technical limits in carrying out the experiments that lead to an inability in distinguishing the kind of waves composing the wave packet?

2. Finite pulse superposition In order to built a wave packet we follow a process similar to Fourier analysis, but instead of using non-local infinite plane harmonic waves as the building bricks we use finite waves that represent a quantum particle. Consider, for instance, a point like source emittiing monochromatic finite pulses described analytically, for the sake of simplification, by the Morlet wavelet [5]:

Now suppose also, for simplicity reasons, that the emitted pulses have gaussian wave number distribution. Then the resulting wave packet is obtained superimposing the finite waves:

-w

In non-dispersive medium the relation between the wave number and the frequency of the wave is o = kv, that is what to be expect in a nondispersive medium, so the wave train (2) is after some calculations:

where,

This last relation and its physical meaning, as a more general form for the uncertainty relations, was derived and extensively discussed some time ago by one of us [6], It appears that the possibility of further testing this expression is now possible if some modifications in the experimental apparatus could be made [2][3][7] [8][9].

3. Non-local interferometry Let us consider in figure I a Mach-Zehnder interferometer made assymmetric by a modification in the arm lengths.

NON-LOCAL INTERFEROMETRY

Figure I. Modified Mach-Zehnder interferometer

When the two pulses arrive at the first beamsplitter they are split into two different independent paths. The presence of a phase shifting device in one arm of the interferometer induces a relative shift t) between the two pulses. In such conditions one is allowed to write for the first arm of the interferometer

And for the second arm:

Therefore, at the interferometer output port we will have the sum of the two waves q-,= q ~+ , cpz. To obtain the intensity collected at the detector cp, must be averaged over time,

in such conditions and after some easy calculations we get,

which represents the single pulse at the output port before reaching the optical filter. Recalling the De Broglie postulates [lo], the probability that the particle-singularity cross the interferometer and the filter is given by, r

-.z

1

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3.1. MONOCHROMATORS The monochromator plays a key role in this kind of so-called non-local interferometric experiments. There are monochromators of many types, for photonics we have the optical filters, Fabry-Perot etalons, and crystal analysers for neutrons. In any case the inclusion of this filter at the interferometer exit leads to an increasing of the coherence Length of the output wave train. In the present work our attention is focused on experimental situations where the input sources produces a beam with gaussian wave number distribution. Therefore the practical effect of the mentioned filter is of narrowing the bandwidth of those distributions on the output beam. In some works [2][11], on those matters, we found that the typical bandwidth acceptance on the analysed beam where approximately of the order of Angstrom. The effect of this kind of filter is described by a narrow bandwidth limiting function:

3.2. VISIBILITY OF THE INTERFERENCE PATTERN Assuming a gaussian distribution in the wave number for the pulses and summing over the entire spectrum of the particles emitted by the source, we find the counting intensity at the detector,

The interference pattern visibility for those experiments is by definition,

which by substitution of the intensities leads to the causal visibility:

where

We must point out that this result is formally similar to the one obtained if we had used non-local Fourier analysis [2]. which would give:

The question now is how to establish what is the right formula for describing the experimental visibility obtained. Till present these results have been interpreted in the usual way, that is using non-local Fourier analysis. Now, we have shown that there is an

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alternative way to derive formally the same expression for the visibility. This new process is based in the local wavelet analysis and assumes that the quantum particle is a real physical entity, with a well-defined energy localized in space and time. The obvious way to find out what is the correct model for describing a quantum particle is using a filter which bandwidth goes to zero, akf+ 0. In such case the usual visibility (15) goes to one, for any arbitrary large value in the optical path difference between the two arms of the interferometer. This fact is easily understandable in the light of the non-local Fourier analysis. When the bandwidth of the filter goes to zero this means that in such case the output wave is practically a monochromatic plane wave with an infinite coherence length. Therefore, no matter how long is the optical path difference between the two arms of the interferometer there is always a complete overlapping of the two infinite plane waves and therefore the visibility is always one. Predictions change drastically if instead of considering infinite plane waves one assumes that quantum entities ought to be represented by finite wavelets. In this case when the bandwidth of the fdter goes to zero the causal visibility assumes the form. J-2

which means that the causal visibility depends on the optical path difference. Therefore if the optical path difference is much larger than the size of the wavelet no interference is to be expected and the visibility goes to zero. For shorter path differences the visibility starts growing up till it reaches the value one for no optical delay between the two arms ofthe interferometer. This problem was already analyzed [12], and it was pointed out that what has been interpreted as sound evidence in favor of "postponed compensation", that is time compensation after the actual physical effect has taken place, could as well be explained simply as a classical problem of recovering an interference pattern hidden by noise with the help of an appropriate fdter. 4. Spectral modifications

In order to test the possibility of finite waves description for the quantum particles, we should look for a discrepancy in the experimental visibility. In figure 2 the mentioned discrepancy manifests through the fact that causal visibility no longer tends to 1 for spatial delays larger than the wavelet size o,.

c'

Figure 2. Umal non-local and causal (dashed) visibility as a finction of the limiting bandwidth of the optical filter for three different values of the wavelet sue: a) g = 2t;'; b) op= 5'; c) g = 0,SC.

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From figure 2 we may infer that the causal visibility fall to zero for the shorter wavelet size. What happens now is that in order to reproduce the same result, as with plane waves, the wavelet size o, must be comparable or greater the coherence length 1 / q ' of the plane wave packet. That is for a plane wave packet with a bandwidth ok= okf the wavelet size must satisfy: 1

\2

There is an inconclusive problem with this kind of experiment because we can always say that wavelets explain the visibility obtained if the wavelet size satisfy (17). However there is another way to settle the problem if we consider the study of spectral modifications [7][8][9] on similar non-local interferometry experiments performed with neutrons and photons. In fact it was reported that when the spatial delay in the Mach-Zehnder (figure 1) is greater than the coherence length ofthe wave packet, we could observe a modulation in the spectral distribution. Recalling equation (8) but with lane waves instead of the Morlet wavelet we find for the spectral modulation:

All the experiments mentioned confirm the presence of modulation in the spectral distribution as predicted by (18). but the same experiments reveal an increase of the Consequently the expected minima of spectral modulation depth with the path delay (18) do not go to zero.

c.

Figure 3. Normalized spectral distribution of the input (dashed) and output beams (solid line) in the modified Mach-Zehnder interferometer. A reported experimental result, with 300 pm spatial delay, shows the expected modulation.

The modulation depth was explained [9] by studying the effect of vibrations on the table that sustained the experimental apparatus. We can assume that there is afluctuation v<' in the spatial delay and so equation (18) is modified:

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It is easy to guess that if we consider those fluctuations in the finite wave point of view, we obtain the following result

L

and we will predict that the modulation minima don't go to zero. However the most interesting aspect is that expression (20) predicts also a different behaviour for the modulation. In fact the computation of (19) and (20) can be summarized in the following expression:

where A is the attenuation term, function of the vibration parameter v (both (19) and (20)) and the wavelet size o,(only for (20)). However M(1;') in (21) assumes a very different behaviour for plane waves and finite ones, and if we graphically represent it the difference become evident:

Figure 4. From this figure we can observe that the modulation rate is a linear function of the spatial delay for infmite plane waves, but far finite waves we expect the finction M(t;') to reach a maximum and then go to zero. Shifting the values of the vibration parameter(making a, larger or smaller) change the modulation rate M ( 0 for finite waves, but leave the infinite waves modulation rate unaffected.

It is possible to conceive a conceptual experiment where induced vibrations could open a way to test the idea of finite versus infinite waves in quantum physics.

J. MART0 AND J.R. CROCA 5. Conclusion Non-locality and non-separability are concepts rising from the quantum formalism. and have been the core of a debate old as quantum theory. Some experiments done and mentioned here tried to prove a non-local behaviour for the quantum particle. and seem suitable to test the idea if the undulatory aspect of it is of infinite range. We have presented an examination of those ideas and described an experiment that could get us farther in the elucidation on the nature of non-separability that these interferometric experiments pretend to reveal.

6. References [I] N. Bohr, (1928). in Como Lecture.$, Collected Works, Vol. 6, (North-Holland, Amsterdam, 1985). [2] H. Kaiser, R. Clothier, S. A. Werner, H. Rauch and H-Wiilwitsch,"Coherence and spectral filtering in neutron interferometry," Phys. Rev. A 45, (1992) 31. [3] H. Rauch, "Superposition experiments in neutron interferometry," I1 Nuovo Cimento Bll0, (1995) 557. [4] V. L. Lepore, in Waves aridParticles in Light andMatter, pp. 387-394, (Edited by A. van der Merwe and A. Garrucio) (Plenum Press, New York, 1994). [5] P. Kurnar and E Foufoula-Georgiou. 'Wavelet analysis for geophysical applications," Rev. of Geophysics 35. (1997)385. [q J. R Croca, in FundamentalProblems in Quanhrrn Phpics, pp. 73-82, (Kluwer Academic Publishers, 1995); J. R. Croca, A. J. Rica da Silva, "Dispersion relations for the superposition of monochromatic finite pulses," submit. for publication; J. R. Croca, "The Limits ofHeisenberglsUncertainty Relations," in Causalig atid Localig in Modeni Physics. (Edited by S. Jeffers et al., in print) (Kluwer Academic Publishers, 1997). [7] D. L. Jacobson, S. A. Werner, H. Rauch, "Spectral modulation and squeezing at high-order neutron interferences," Phys. Rev. A 49, (1994) 31%. [8] X Y. Zou. T. P. Grayson and L. Mandel. "Observation ofquantum interference effects in the frequency domain," Phys. Rev. Lett 69, (1992) 3041. [9] D. Narayana Rao and V. Nirmal Kumal, "Experimental demonstration of spectral modification in a MachZehnder interfrometer, '' J. mod Optics 41, (1994) 1757. [lo] L. De Broglie, in The current interpretation ofwave rneclianics (a critical study) Plsevier Publishing Company, 19641 pp. 43-46. [Ill W. Tiller, "Bandwidth measurement of high-resolution Fabry-Perot etalons," Appl. Opt. 23, (1984) 393 1. [I21 J. R. Croca R. N. Moreira. A. J. Rica da Silva,"Recovery of an interference pattern hidden by noise," in Causalig andlocalily in inudenrphysics, (Edited by S. Jeffers et al., in print) (Kluwer Academic Publishers, 1997).

BEYOND HEISENBERG'S UNCERTAINTY LIMITS JOSEE R. CROCA Departanlento de Fisica Faculdade de Ci2ncia.s Universidade de Lisbou Cartlpo Grande Ed CI 1749-016 Lisboa Portugal Enlail: croca @fc.ul-pt

1. Introduction Heisenberg's uncertainty relations state that it is impossible to predict the result of a measurement with a precision lying beyond its limits. In practice this formula defines, in standard quantum mechanics, the limits of the previsional measurement space accessible to any measurement. Beyond it is impossible to perform any concrete measurement. In the last years a strong effort was made to develop causal quantum physics in the spirit of de Broglie and Einstein. These efforts were carried out mostly with the help of a recent mathematical tool, the wavelet analysis, developed precisely to overcome some inconveniencies of the Fourier non-local analysis. As a consequence of this work, a more general expression for the uncertainty relations. containing the usual ones as a particular case, were derived. These more general uncertainty relations define a predictive measurement error space including the usual Heisenberg space. In this new space the prevision of the precision of a result of a measurement depends only of the basic measuring tool used in a concrete experiment. If one wishes to predict the result of a measurement with greater precision it is necessary to change the basic measuring tool. In the language of local wavelet analysis this corresponds to say that one has to use a mother wavelet of a smaller scale. By changing the scale of the basic wavelet it is possible, in principle, to scan all the pre ctive measurement space. For some time those ideas had no experimental support, even if the new approach explained the experimental evidence, which is not a surprise, since the new relations contain the usual ones as a particular case. Now, thanks to the recent development of a new generation of microscopes that have a resolving power that goes far beyond Abbe's limit of half wave length, there are measurements, done everyday, that are outside the Heisenberg's space.

2. Wavelet Analysis And The New Uncertainty Relations The wavelet local analysis [I] was developed in the early eighties by Jean Morlet, a French geophysicist working in oil prospection. He devised this mathematical tool for 385 R.L. Amoroso et a1 (eds.),Gravitation arrcl Cosmology: From the Hubble Radius to the Plarrck Scale, 385-392. O 2002 Kluwer Academic Publishers. Printed in the Netherltrrrds.

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denoising the very sharp seismic signals. These early attempts were developed and formalized by Grossmann, Meyer, and many others till the wavelet local analysis was made into a powerful mathematical tooL competing side by side with the Fourier nonlocal analysis. Today, there is a whole universe of scientific literature dealing with the all-different aspects of wavelets. Thii field is developing so fast that even the very definition of wavelet that at the beginning seemed fairly settled, is today an uncertain ground that has lead some to authors to say that the precise definition of wavelet is a kind of scholastic question. Here, only the basic property of finitude of the wavelet is retained. 2.1. WAVELET LOCAL ANALYSIS VERSUS NON-LOCAL FOURIER ANALYSIS In order to understand the conceptual and practical importance and gasp the deep significance of this new local analysis by wavelets it will useful to make a short comparison with the non-local Fourier analysis. The Fourier analysis must be considered non-local or global because its basic elements, its constituting bricks, are monochromatic harmonic plane waves. infinite both in space and time. The non-local or global character of the Fourier analysis can be easily understood with the following example.

n

At the initial time t,

n

At the time t,

Figure 2.1 - Digitalized images to Fourier analyze

Suppose that it is necessary to record, line by line, the first image shown in Figure 2.1, for instance, in a CD. In order to simplify the problem, let us consider only the line indicated in the picture. The plot of the intensity of the picture along that line is also represented below in the same picture. In order to Fourier represent this variation in intensity it is necessary to find the amplitudes, frequencies and phases of the infinite monochromatic harmonic plane waves that sum up reproducing the initial function. This immensity of plane waves interferes negatively in all points of space except in those two regions where the interference is constructive. Let us now suppose that, for instance, the first ship moves to the right, while the other remains in the same position. This situation is represented in the second image of Figure 2.1. Next, we ask how this motion of the first ship is to be Fourier represented. In this type of analysis, both the first and the second ship are made of the same infinite harmonic plane waves. In these circumstances, a modification in the position of the first ship implies naturally a modiication in the amplitude frequency and phase of all the waves, so that their interference is constructive now only in the new position. This means that, even though the two ships may seem separate entities. they are in fact, treated as a whole entity

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in this process of non-local Fourier analysis and reconstruction. Hence, a modification in the position of one of the ships implies a simultaneous modification of the constituents of the other, no matter how distant they are from one another. This mathematical consequence of dealing with infinite harmonic plane waves, determines the very deep roots of the non-locality of the usual quantum mechanics. The non-local or global character of Fourier analysis has also serious technical drawbacks, namely when, as is the case, one wants to register video images. In general, as is very well known, from video image to video image there are only slight variations. Nevertheless, in this type of nonlocal or global analysis, a minor modification on one region of the image implies the necessity of analyzing and rebuilding the whole image. If, instead of the non-local Fourier analysis, one uses local wavelet analysis, the modification of the position of one ship as nothing to do with the other, if the two objects are sufficiently far one from another. In this case, the group of wavelets that sums up to recreate one object is completely independent from the wavelets corresponding to the reconstruction of the other ship. This is precisely the reason why this type of analysis is called local analysis by finite waves. The difference between the two types of analysis is shown graphically in Rgure 2.2.

Local Wavelets fix)

I 1 I

-

"

X

Non-local Fourier

Figure 2.2-Comparison between local wavelet analysis and Fourier non-local analysis

Another important advantdge of the local wavelet analysis is related to the fact that, in order to represent a given signal, one is f e e to choose different types of basic wavelets. In these circumstances, a wise choice ofthe basic wavelet can further increase the rate of compression and the quality of the reconstruction, or the denoising capability.

2.2. THE NEW UNCERTAINTY RELATIONS Analytically, the usual uncertainty relations are a mathematical consequence of the nonlocal Fourier analysis [2]; if. instead of infinite harmonic plane waves. one uses the local wavelet analysis to represent a quantum particle, the form of uncertainty relations may consequently change [3]. In order to make the derivation of the new uncertainty relations more comprehensible, it is convenient to do it step-by-step. in parallel and with the usual derivation. This process is shown in the following sketch, Figure 2.3

J.R. CROCA

Derivation o f the uncertainty relations Non-local Fourier analysis

Wavelet local analysis

I! Kernel

Sinus and cosinus

Gaussian MorIet wavelet

Representation of the particle

Coeficient function k* --

g(k)= e

ZAP

by substitution and integration

substituting by the quantum values Usual uncertainty relations

New uncertainty relations

Figure 2.3- Derivation of the uncertainty relations.

From the table, Figure 2.3, it is seen that the new uncertainty relations derived with the local wavelet analysis have the form

BEYOND HEISENBERG'S UNCERTAINTY

389

The same analysis could, in principle, be done with other mother wavelets. The Gaussian Morlet wavelet [4] was chosen among the other possibilities because of its interesting properties: - From the mathematical point of view. it has a very simple form. Therefore. the necessary calculations can fully be carried out without approximations. - Another very interesting property of this wavelet comes from the fact that, when its size increases indefinitely, it transforms itself in the kernel of the Fourier transform. In this sense, this local analysis contains the non-local Fourier analysis as a particular case. When the size of the basic wavelet Dxois sufficient large, the new relation transforms itself into the old usual Heisenberg relations. which is a very satisfactory result - It allows a reasonable representation of fair localized particle with a well-defined velocity. - Written in terms of space and time, it is a solution of a Schrijdinger non-linear master equation [5].

The plot of the uncertainty is shown in Figure 3.4. The new uncertainty relations are represented by the solid line for three finite values of the size ofthe basic wavelet. The usual Heisenberg uncertainty relations. which correspond to the case & +a, are shown as a dashed lime,

-

Figure 2.4 Solid line: new uncertainty relation for three different finite values of size the basic wavelet. The

dashed line represents the usual Heisenberg uncertainty relations.

2.3. GENERALIZED MEASUREMENT SPACE The new, more general uncertainty relations were derived in a causal framework, assuming that the physical properties of a quantum system are observer independent, and, even more, that they exist even before the measurement. Naturally due to the unavoidable physical interaction during the measurement process, when the other conjugated observable is to be measured, the quantum system does not remain in the same state. In last the instance, the precision of the measurement, for a non-prepared system, depends

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J.R. CROCA

on the relative size between the measurement basic apparatus and the system upon which the measurement is being done. Until now, the most basic interacting quantum device is the photon. Nevertheless if the photon has an inner structure. as is assumed in the Broglie model. it will imply the ability to perform measurements beyond the photon limit. Since in the derivation of the new uncertainty relations. the quantum systems were assumed to be described by local finite wavelets, the measurement space resulting from those general relations must depend on the size of the basic wavelet used. As the width of the analyzing wavelet changes, the measurement scale also changes. This can be seen in the following plot Figure 2.4.

Figure 2.4 - Wavelet measurement space

From this picture it is seen that, as the width of the basic wavelet Aq, changes, all the measurement space is scanned. This space is only limited by Heisenberg's space. The smaller Axo, the greater is the precision ofthe position measurement. That is: the smaller is the uncertainty Ax, for any value of the error in the momentum. Given that the new relation contains the usual as a particular case, it implies that the measurement space available to the general uncertainty relations is the whole space, as shown in Figure 2.5.

Figure 2.5- Measurement space available to the general uncertainty relation

These results are rather satisfactory, because in this causal pa gm the quantum measurement process depends. in last instance, on a standard used. We are. in principle, free to choose the size of the mother wavelet Axo more suitable for the measurement precision we want to attain.

BEYOND HEISENBERG'S UNCERTAINTY

3. Beyond Heisenberg's Measurement Space The middle of the eighties saw the development of super-resolution microscopes. The first of this new type of microscopes was developed for electrons by Bennig and Roher [6], who received the Nobel Prize for the discovery. The principal characteristic of these microscopes is that their resolution goes far beyond those given by Abbe's rule of half wavelength. Abbe derived his theoretical resolution rule based on the Rayleigh diffraction criteria for the separation oftwo circles in a far field. Super-resolution microscopes were also developed in 1984 by Pohl et al. [7] for the optical domain. Initially, these microscopes had a spatial resolution of 7/20, ten years later [8]they attained resolutions of 1/50 or even better. There are many types of super-resolution optical microscopes, as can be seen in pertinent literature. In most straightforward of these microscopes the light emitted by the sample is simply collected by the sensing probe. In order to see if there are some very special experimental situations were the predicted errors of the two conjugated observables do not lie in Heisenberg's measurement space, it is convenient to consider the well-known Heisenberg microscope in parallel with the super-resolution microscope optical microscope.

Common Fourier microscope Super-resolution microscope Uncertainty in mornentnrn

Uncertainty in position

Product of the two uncertainties

J.R CROCA From the table, one sees that there is a very significative difference of 1/25 in the products of the uncertainties related to the measurement of the two conjugated observables, position and momentum. For the common Fourier microscope, this product lies well within Heisenberg's measurement error space A x A p , 2 h . For the superresolution microscope, we have Ax A p, = hi25 , that lies outside Ax A p, < h Heisenberg's measurement space. and is therefore contained in the more general wavelet precision measurement space.

Acknowledgements

I want to thank Fundaqiio Calouste Gulbenkian. for a grant. allowing me to be present at the Vigier 111 Conference. References 1-A. Grossmann and J. Morlet, SIAM J. Math. Anal. 3 (1989),723; C.K Chui, An Introduction to Wavelets, Academic Press, N.Y. 1992. 2. N. Bohr, (1928) - Como Lectures, Collected Works, Vol. 6, North-Holand, Amsterdam, 1985. 3. J.R. Croca, Apeiron, 4 (1997). 41. 4. P. Kurnar, Rev. Geophpicy, 35 (1997), 385. 5. Ph. Gueret and J.P. Vigier, Letl. Numo Cimento, 38 (1983), 125; J-P. Vigier, Phys Lett. A, 135 (1989), 99. 6. G. Benning and H. Roher, Reviews of Mdel-rr Ph?sics, 59 (1987), 615. 7. D.W. Pohl, W. Denkand M. Lanz,Appl. P h y . Lett. 44 (1984). 651. 8. H. Heiselmann, D.W. Pohl, Appl. Ph?..?.A, 59 (1994). 89.

TOWARDS A CLASSICAL RE-INTERPRETATION OF THE SCHRODINGER EQUATION ACCORDING TO STOCHASTIC ELECTRODYNAMICS K. DECHOUM Departamento de Fisica Moderna Universidad de Cantabria 39005 Santander, Spain Email: [email protected] H.M. FRANCA Instituto de Fisica, Universidade de Sao Paulo CP 66318,053 15-970 Sao Paulo, Brasil Email: hfranca @if.usp.br C.P. MALTA Instituto de Fisica, Universidade de Sao Paulo CP 66318,05315-970 Sao Paulo, Brasil Email: [email protected]

Keywords:

Zero-point fluctuations; Stochastic Electrodynamics.

Abstract We study the statistical evolution of a charged particle moving in phase space under the action of vacuum fluctuations of the zero-point electromagnetic field. Our starting point is the Liouville equation, from which we derive a classical stochastic Schrodingerlike equation for the probability amplitude in configuration space. It should be stressed that we are not deriving the Schrodinger wave equation. An equation formally identical to the Schrodinger equation used in Quantum Mechanics is obtained as a particular case of the classical stochastic Schrodiger like equation. An inconsistency appearing in the standard Schrodinger equation, when vacuum electromagnetic fluctuations and radiation reaction are taken into account, is clearly identified and explained. The classical stochastic Schrodinger like equation, however, is consistently interpreted within the realm of Stochastic Electrodynamics. 393 R.L. Aliroroso et al (eds.), Gravitation and Cosmology: From the Hubble Rcrdius to the Planck Scale, 393-400. O 2002 Kluwer Acuilemic Publishers. Printed in the Netherlarlcls.

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1. Introduction The classical electromagnetic theory has been largely extended by the program of the Stochastic Electrodynamics (SED) [I, 2, 31, due to the inclusion of the effects of the real electromagnetic zero-point radiation. According to the SED picture. there is a clear similarity between the nonrelativistic Heisenberg equations of motion, for a spinless charged particle interacting with the quantized electromagnetic field of Quantum Electrodynamics (QED), and the classical (Langevin type) Abraham-Lorentz equation with real vacuum fluctuation forces. At the same time, it is well known [4] that for a special class of potentials the dynamical evolution of any Schrijdinger wave packet is entirely determined by the laws of Classical Mechanics. Therefore we shall develop a method that is aplicable to any deterministic potential, in order to better clarify the relation between the classical and the quantum theories of the microscopic world. We shall derive a classical Schrijdinger like equation from the Liouville equation, using a procedure similar to that introduced by Wigner [5] for describing Quantum Mechanics in phase space. Our approach introduces a free parameter ti' in the Wigner type transform [6]. We shall show that this procedure enables us to make a clear distinction between the free parameter h' and the Planck's constant h. Only the vacuum electromagnetic fluctuations will depend on the numerical value of the Planck's constant ti.

2. Connecting The Stochastic Liouville Equation To A Schrodinger Like CIassicaI Styochastic Equation The description of classical phenomena by classical statistical mechanics is based on the concept of phase space. The probability density distribution, W(x, p, t ) , evolves in time according to the Liouville equation

where x and pare obtained from the classical Hamilton's equations of motion. Consider an ensemble of systems which consist of a nonrelativistic spinless charged particle interacting only with the electromagnetic field. The Hamiltonian which describes the time evolution of the whole system

RE-INTERPRETATIONOF THE SCHRODINGER EQUATION (particle plus field) is

where e and mare the charge and mass of the particle, respectively, 4(x, t) is the scalar potential, and

is the vector potential. The term A,,, is an external deterministic disturbance. The term A,, is the vector potential associated with the real vacuum fluctuations, and can be written as [3]

(1.4)

where V is the volume containing the particle and the radiation field, k is the wave vector, wk = clkl, X is the polarization index, and t(k, A) are the polarization vectors. The amplitudes a k are ~ taken to be random variables. The random character of the field is contained in these variables which are such that (akA)= 0 and (lau12) = 1/2 (( ) denotes the ensemble average). The term A,, is the vector potential that describes the radiation reaction [l, 31 and H,, is the Hamiltonian of the background radiation field (contains only variables of the field). In the case of zero temperature, HvFcan be written as [3]

i a where E,, = -- -AvF , BvF= V x A,,, so that (H,,) = CkfLwk . at The extension to a non-zero temperature T is obtained by introducing the factor coth(fiwk/2kT). Each particle of the ensemble evolves in time according to the Hamilton's equations dH 1 e jc= ---(p--A) 7 (1-6) ap m C and e2 ~ ~ - e .d 2mc2 (1.7)

I

Substituting the equations (1.6), (1.7) into (1.1) we get the Liouvillian form of the equation governing the time evolution of the ensemble of

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JS. DECHOUM, H.M. FRANCA AND C.P. MALTA

particles for each realization of the stochastic field A,,, namely

Prom the solution of (1.8) the ensemble average over all possible realizations of the field A,, (in order to obtain the average distribution ( W ( x ,p, t ) ) )is performed by considering the average over the random ~ (1.4). Notice that in (1.8) A = A ( x , t ) and Gaussian amplitudes a k in 4 = 4 ( x ,t ) are explicit functions of the variables x and t . Consider the Fourier transform defined by [5, 61

where y is a point in the configuration space and h' is a free parameter having dimension of action. The meaning of the free parameter h' will be discussed further below. Notice that (1.9) corresponds to the well known Wigner transform [5] if h' = h. The use of the conventional Wigner transform within classical SED was discussed in great detail by Franga and Marshall in 1988 [7]. In what follows we shall concentrate our attention in the particular case of very small ti'(ti' << ti). In this case the right hand side of (1.9) is different from zero only if Jy1 is small, and the following approximations will be used

a

Y.zd(x,t)"+(x+~,t)-d(xlt)

,

(1.10)

and

Using (1-9) in (1.8), plus the approximations (1. lo), (1.1 l), we obtain the following Schrginger type equation for $ ( r , t ) 2

t))

$J

+ e)(r,t)@

,

(1.12)

and the corresponding equation for y!P(r,t), with the vector potential A as given in (1.3) and (1.4) (see [8] for details). Therefore, the Schriidinger type equation depends on the Planck's constant only due to its presence in AvF defined in (1.4). In other words, equation (1.12) has terms which are proportional to fi and ti. Moreover, the solutions of (1.12) must be interpreted by considering that the limit ti' + 0 must be taken in the end of the calculations.

RE-INTERPRETATION OF THE SCHRODINGER EQUATION

397

3. Incompatibility Of The Standard Schrodinger Equation With The Zero-Point Field The above derivation shows a clear correspondence between the quantum Schrodinger equation for spinless particles, and the classical stochastic Schrijdinger like equation given by (1.12). The case of neutral spinning particle has been already discussed by Dechoum, Franqa and Malta [9]. The limit h1-+ 0 of the solution of the classical stochastic Schrijdinger like equation corresponds, physically, to classical (non-Heisenberg) states of motion as shown by Dechoum and Franqa [6]. Nevertheless, we shall observe several effects. arising from the vacuum fluctuations. which depend non-trivially on the Planck's constant h. This is better understood by means of very simple examples. One interesting example, discussed in reference [9], is the derivation of the Pauli-Schrodinger equation in the spinorial form, starting from the Liouville eyuation. The experimental results of the Stern-Gerlach experiment, and also the Rabi type molecular beam experiments, were appropriately described and interpreted classically, in the limit ti' -4 0, that is, in the classical limit where the particles have well-defined trajectory, and also continuous orientation of the spin vector. The zero-point field has no1 been included. The best example, however, is the one-dimensional harmonic oscillator discussed in many details in previous works [6, 71. In order to apply equation (1.12) to the charged harmonic oscillator, we shall assume that the scalar potential 4 is the simple static function satisfying e@ = (1/2)&1~, being the natural frequency of the oscillator. We have shown in ref. [6] that by introducing the function @ ( I ,t )

-

Q ( x ,t) we obtain for (1.12) the equivalent

exp

equation

62

d x 2 2m 8x2 + 2 - en: (En,

ihl- = at

+ Ev,)

I

@(x, t ) , (1.13)

where &, and En, depend only on t (dipole approximation). In this ed equation - w ; s is the harmonic force, eE,, = --- ( A R R ) is , the ra-

,.

a ---(A,,), c at e

at

diation reaction force, and eE,, = is the random force. The exact solution of equation (1.13) is a Gaussian coherent state, !PC. exp ( s- I , (t))2mwg/h1]. Here n:, ( t ) is the classical stochastic oscillator trajectory [6] such that = fi/2mW0.Using this exact

-

[-

(6)

K. DECHOUM, H.M. FRANCA AND C.P. MALTA

398

+

( x ) = ( / d x ~ [ (~x -~ xc)2 ~ l ~x g ] ) = 2

-+- : (1.14) fi

fi'

2 w o

2rnw

This gives the correct value at zero temperature, namely { x 2 ) = fi/2pnwo, in the limit h' -+ 0. This is an important result that is very easy to understand within the realm of SED. The inevitable conclusion is that the standard Schrijdinger equation, namely equation (1.13) with fi' = f i , does not give consistent results if the zero-point electromagnetic field EvFis fully considered. 4. Discussion

Dalibard, Dupunt-Roc and Cohen Tannoudji [lo], and Franqa, Franco and Malta [ll] provided an identification of the contribution of the radiation reaction and the vacuum fluctuation forces to the processes of radiation emission and atomic stability. Using the Heisenberg picture and perturbative QED calculations it is possible to show that [lo]

This equation is the quantum generalization of the Larmor formula (2e2r2)/(3c3)for the rate of radiation emission, including the zero-point field effects, of an electron in the quantum state ( a ) (Dirac notation) withenergy ca. We see that the inclusion of the zero-point electromagnetic field simply doubles the rate of the radiation emission. being thus very important for obtaining agreement with experiment [ll, 121. Our last remark is concerned with the necessity of the vacuum electromagnetic field within the Heisenberg picture of the quantum theory. A good discussion of this point is given in the book by P. Milonni [13]. The zero-point electromagnetic field, E,,, is necessary for the formal consistency of the quantum theory, as only its inclusion guarantees the equal-time commutation relation [x(t),p,(t)]= .i f i [13]. This result inti2 a2 dicates that the operator --- already contains some effects of the 2rn ax2 zero-point electromagnetic field when the standard Schrodinger equation (equation (1.13) with fit = f i ) is used. Acknowledgments This work was partially supported by FAPESP and CNPq.

K. DECHOUM, H.M.FRANCA AND C.P. MALTA

References

[I]

Marshall, T.W. (1963): "Random Electrodynamics", k. Royal Soc. (London) A, 276. 475-491.

[21

Santos, E. (1968): "Is there an electromagnetic background radiation underlying the quantum phenomena ?'. Anales de la Real Sociedad Espanola de Fsica y Qumica, Tomo LXIV, 317-320.

[3]

de la Pena, L. and Cetto, A.M. (1996): "The Quantum Dice: An Introduction to Stochastic Electrodynamics" ( Kluwer Academic Publishers).

[4]

Manfredi, G., Mola, S. and Feix, M.R. (1993): "Quantum systems that follow classical dynamics", Eur. J. Phys.14, 101-107.

[5]

Wigner, E.P. (1932): "On the quantum correction for thermodynamic equilibrium", Phys. Rev., 401,749-759. See also O'Comell, R.F. and Wigner, E.P. (1981): "Quantum-mechanicaldistributionfunctions: conditions for uniqueness", Phys. Lett. A 83), 145-158; Lee, H.-W. and Scully, M.O. (1983): "The Wigner phase-space description of collision processes", Found. Phys. 13. 61-72.

[6]

Dechoum, K. and Frana, H.M. (1995): "Non-Heisenberg states ofthe harmonic oscillator", Found. Phys., 25, 1599- 1620. (1996): "Non-Heisenberg states of the harmonic oscillator: Erratum", Found. Phys., 26, 1573-1573.

[71

Franca, H.M. and Marshall, T.W. (1988): "Excited states in stochastic electrodynamics", Phys. Rev. A, 38, 3258--3263. This paper shows that the set of Wigner functions associated with the excited states ofthe harmonic oscillator constitutes a contpleteset of functions. An arbitrarypositiveprobability distribution can be expanded in terms ofthese Wigner functions.

[8]

Dechoum, K-, Franca, H.M. and Malta, C.P. (in press): "The role of the zeropoint fluctuations in the classical Schrodiiger like equation", 11 Nuovo Cimento B.

191

Dechoum,K., Fran\ca, H.M. and Malta, C.P. (1998): 'Classical aspects of the Pauli-Schrodinger equation", Phys. Lett. A, 24,93- 102. This paper discusses various classical aspects of the Pauli-Schrodinger equation. for a neutral spinning particle. including the classical interpretation of the Stern-Gerlach experiments.

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[lo]

Dalibard, J., Dupont-Roc, J. and Cohen-Tannoudji, C. (1982): "Vacuum fluctuations and radiation reaction: identification oftheir respective contributions", J. Physique. 43. 1617-1638. According to Dalibard et al. "If selfreaction was alone, the ground state would colapse and the atomic commutation relation would not hold".

[I 11

Franca, H.M., Franco, H. and Malta, C.P. (1997): "A stochastic electrodynamics interpretation of spontaneous transitions in the hydrogen atom", Eur. J. Phys., 18,343-349.

[12]

Franca, H.M., Marshall, T.W. and Sarnos, E. (1992): "Spontaneous emission in confined space according to stochastic electrodynamics", Phys. Rev. A, 45, 6436-6442.

[13]

P.W. Milonni (1994): "The quantum vacuum: An introduction to quantum electrodynamics, Academic Press Inc., Boston. According to Milonni (see chapter 2. section 2.6). The origin of the Planck's constant in this expression can be traced back to the zero-point radiation with spectral distribution (see also our equations (1.4) and (1.5)

THE PHILOSOPHY OF THE TRAJECTORY REPRESENTATION OF QUANTUM MECHANICS EDWARD R. FLOYD 10 Jamaica Village Rd. Coronado, CA 92118-3208 USA

Abstract. The philosophy of the trajectory representationis contrasted with the Copenhagen and Bohmian philosophies. 1. Introduction The seminal work on the trajectory representation was published in 1982 [I]. The equations of motion for the trajectories are developed from a quantum HamiltonJacobi formulation. These trajectories are deterministic and continuous. Ergo. there is no need for any collapse of the wave function during observation. Recently, Faraggi and Matone [2] have independently generated the same quantum Hamilton-Jacobi formulation from an equivalence principle free from any other axioms. Faraggi and Matone have shown that although all quantum systems can be connected by an equivalence coordinate transformation (trivializing map), all systems in classical mechanics are not so connected. The development of the equivalence principle is beyond the scope of this exposition. We present the philosophical aspects of the trajectory representation of quantum mechanics that distinguish this representation. We exhibit its philosophy, which we contrast to the Copenhagen philosophy and the Bohmian philosophy. Our findings are presented in closed Form mostly, but not exclusively, in one dimension for the time-independent case whenever one dimension suffices. The work in one dimension for the time-independent case is sufficient to refute many of the tenets of the Copenhagen and Bohmian theories. The trajectory representation is not just another interpretation of quantum mechanics because it also predicts results that differ with Copenhagen. Tests have been proposed consistent with Copenhagen epistemology [3,4]. In Section 2, we briefly present the fundamentals of the trajectory representation sufficient to support a philosvphical comparisvn with Copenhagen and Bohm. In Section 3, we present different predictions rendered by trajectories and Copenhagen. We continue in Section 4 to contrast the trajectory representation to the Copenhagen interpretation. In Section 5, we compare the trajectory representation with the Bohmian stochastic representation. A gentler, extended version of this opus with more details is available at the LANL electronic archives [5]. 401 R.L. Amoroso et a1 (eds.), Gravitation and Cosmology: From the Hubble Rtttlius to the Planck Scale, 401-408 O 2002 Kluwer Actrdenzic Publishers. Printed in the Netherltu~ds.

EDWARD R FLOYD

402

2. The Trajectory Representation The trajectory representation is based upon a phenomenological, nonlocal quantum stationary Hamilton-Jacobi equation (QSHJE) in one dimension x. It is a third-order non-linear differential equation given by [6,7]

(w'>2 +l/-Ez 2rn classical HJE

(1) \

/

*

quantum effects; also

-Q of Bohm

where W is the reduced action (also known as the generalized Hamilton's characteristic function), W' is the momentum conjugate to x, V is the potentid, E is energy, m is the mass of the particle, and h = h/(2n)where in turn his Planck's constant. Faraggi and Matone have independently derived the QSHJE from an equivalence principle [2]. We note that W and W' are real even in classically forbidden regions. The general solution for W' is given by [8]

where (a,b, c) is a set of real coefficients such that a, b > 0, and (@,8) is a set of normalized independent solutions of the associated stationary Schriidinger equation (SSE), - h 2 3"/(2m) + (V - E)$ = 0. The independent solutions ($,8) are normalized so that their Wronskian, W ( $ , 8 ) = $8' - 4'8, is scaled to give w2(@, 8 ) = 2m/[fi2(ab- c2/4)]> 0. This ensures that (a$2+ be2 + ~ $ 8>) 0 and that W' is real in the classically forbidden regions (V > E). The motion in phase space is specified by Eq. (2) and is a function of the set of coefficients (a, b, c ) . In general, the conjugate momentum expressed by Eq. (2) is not the mechanical momentum, i.e., W' # mx. Actually, mx = mdE/aW' [1,2]. The solution for the generalized reduced action, W, is given by

w = h arctan

(:!@$$)

+

where K is an integration constant that we may set to zero herein. The reduced action is a generator of motion. The equation of motion in the domain [x,t] is rendered by Jacobi's theorem (often called the Hamilton-Jacobi transformation equation for constant coordinates). The procedure simplifies for coordinates whose conjugate momenta are separation constants. Carroll has shown that for stationarity Jacobi's theorem applies for W' is a Legendre transform [9]. For stationarity, E is a separation constant for time. Thus, the equation of motion for time, t , relative to its constant coordinate T , is given as a function of z by

where the trajectory is a function of a set ofcoefficients (a,b, c) and epoch.

T

specifies the

TRAJECTORY REPRESENTATION OF QM

403

The set ($,8) can only be a set of independent solutions of the SSE. Direct substitution of Eq. (2) for W' into Eq. (I) gives

For the general solution for W', the realcoefficients (a,b, c ) are arbitrary within the limitations that a, b > 0 and from the Wronskian that ab - c2/4 > 0 . Hence, for generality the expressions within each of the three square brackets on the left side of Eq. (5) must vanish identically. The expressions within the first two of these square brackets manifest the SSE. so the expressions within these two square brackets are identically zero if and only if $ and 8 are solutions of the SSE. The expression within third bracket vanishes identically if and only if the normalization of the Wronskian is such that w 2 ( $ ,6 ) = 2m/[h2(ab- c 2 / 4 ) ] . For W ( $ , 8 ) # 0 , 4 and 8 must be independent solutions of the SSE. Hence, and 8 must form a set of independent solutions of the SSE. Equation (5) is independent of any particular choice of ansatz. When comparing trajectories to Copenhagen and Bohm, we have broad selection for choosing a convenient ansatz to generate the equivalent wave picture (nothing herein implies that the trajectories need waves for completeness; only convenience). By Eq. (2), W' is real in the classically forbidden zone. Inside barriers, W' still manifests a trajectory. For bound states of finite V(3;),the trajectories go to turning points at .z. = fmwhere W' + 0 regardless of the ( a ,b, c ) because at least one of ( 4 , 8 ) becomes unbound as x + fm. This is a nodal point singularity. For a given energy eigenvalue. E. of the SSE, there exist infinitely many microstates (trajectories or orbits with turning points at fm ) specified by (a, b, c ) . These microstates are not distinguishable from the SSE for bound states [2,7].

3. Different Predictions between Trajectories and Copenhagen First, we examine impulsive perturbations. Trajectories and Copenhagen render different predictions for the first-order change in energy, El due to a small, spatially symmetric perturbing impulse. XV(x)d(t), acting on the ground state of a infinitely deep, symmetric square well [3]. In the trajectory representation, El is dependent upon the particular microstate ( a ,b, c ) . This has been investigated under a Copenhagen epistemology even for the trajectory theory, where complete knowledge of the initial conditions for the trajectory as well as knowledge of the particular microstate are not necessary to show differences for an ensemble suffi-

404

EDWARD R. FLOYD

ciently large so that all microstates are individually well represented. In the trajectory representation, the first-order change in energy, El,is due to the location of the particle in its trajectory when the impulse occurs. The trajectory representation finds that the perturbing impulse, to first order, is as likely to do work on the particle as the particle is to do work perturbing system. Hence, the trajectory representation evaluates (El),,,,,, = 0. On the other hand, Copenhagen predicts El to be finite as Copenhagen evaluates El by the trace ground-state matrix element XVOoG(O)at the instant of impulse. Due to spatial symmetry of the ground state and V(x), Voo# 0. In an actual test, we do not need perturbing impulses, which were used for mathematical tractability. A rapid perturbation whose duration is much shorter than the period of the unperturbed system would suffice [3]. We also consider a redundant set of constants of the motion. For a square well duct, we have proposed a test where consistent overdetermination of the trajectory by a redundant set of observed constants of the motion would be beyond Copenhagen [4]. The overdetermined set of constants of the motion should have a redundancy that is consistent with the particular trajectory. On the other hand, Copenhagen would predict a complete lack of consistency among these observed constants of the motion as Copenhagen denies the existence of trajectories. Such a test could be designed to be consistent with Copenhagen epistemology [4].

4. Other Differences between Trajectories and Copenhagen As the trajectory exists by precept in the trajectory representation, there is no need for Copenhagen's collapse of the wave function. The trajectory representation can describe an individual particle. On the other hand, Copenhagen describes an ensemble of particles while only rendering probabilities for individual particles. The trajectory representation renders microstates of the Schrodinger wave function for the bound state problem. Each microstate is sufficient by itself to determine the Schrodinger wave function [7]. Thus, the existence of microstates is a counter example refuting the Copenhagen assertion that y!~ be an exhaustive description of nonrelativistic quantum phenomenon. The trajectory representation is deterministic. We can now identify a trajectory and construct, for convenience, its corresponding q!J witb sub-barrier energy that tunnels through the barrier with certainty [lo]. Tunneling with certainty is a counter example refuting Bern's postulate of the Copenhagen interpretation that attributes a probability amplitude to I$. As the trajectory representation is deterministic and does not assign a probability amplitude to q!J, it does not need a wave packet to describe or localize a particle. The equation of motion, Eq. (4), for a particle (monochromatic wave) has been shown to be consistent with the group velocity of the wave packet [I I]. Though probability is not needed for tunneling through a barrier [lo], the trajectory representation for tunneling is still consistent with the Schrodinger representation

TRAJECTORY REPRESENTATION OF QM

405

without the Copenhagen interpretation [lo]. Trajectories differ with Feynman's path integrals in three ways. First, trajectories employ a quantum reduced action while a path integral is based upon a classical reduced action. Second, the quantum reduced action is determined uniquely by the initial values of the QSHJE while path integrals are democratic summing over all possible classical paths to determine Feynman's amplitude. While path integrals need an infinite number of constants of the motion even for a single particle in one dimension, motion in the trajectory representation for a finite number of particles in finite dimensions is always determined by only a finite number of constants of the motion. Third, trajectories are well defined in classically forbidden regions where path integrals are not defined by precept. By the QSHJE, knowledge of a set of initial conditions [W, W W'Jat some point xi is necessary and sufficient to specify E and the quantum motion while for the classical stationary Hamilton-Jacobi equation the set of initial conditions is reduced to [W'J at xi to specify E and classical motion [6]. Thus, the Heisenberg uncertainty principle assumes a subset of initial conditions (x,p) that is insufficient to specify E and quantum motion [the SSE operates in (x,p)-domain; the QSHJE in (x, t)-domain by a canonical transform]. The Heisenberg uncertainty principle is premature since Copenhagen uses an insufficient subset of initial conditions (x,p) to try to describe quantum phenomena [12]. Bohr's complementarity postulates that the wave-particle duality be resolved consistent with the measuring instrument's specific properties. But Faraggi and Matone [2] have derived the QSHJE from an equivalence principle without evoking any axiomatic interpretation of the wave function. Furthermore, Floyd [Iand Faraggi and Matone [2] have shown that the QSHJE renders additional information beyond what can be gleaned from the Schrodinger wave function alone.

5. Trajectories vis-a-vis Bohmian mechanics

The trajectory representation differs with Bohmian representation [13,14] in many ways despite both representations being based on equivalent QSHJEs. We describe the various differences between the two representations in this section. These differences may not necessarily be independent of each other. The two representations have different equations of motion. Jacobi's theorem, Eq. (4), gives the equations of motion for the trajectory representation, Meanwhile, Bohmian mechanics eschews solving the QSHJE for a generator of the motion, but instead assumes that the conjugate momentum be the mechanical momentum, mk, which could be integrated to render the trajectory. But the conjugate momentum, as already shown herein, is not the mechanical momentum [1,2,9,11]. Bohmian mechanics considers to form a field that fundamentally effects the quantum particle. The trajectory representation considers the SSE to be only a phenomenological equation where Il, does not represent a field. To date, no one has ever measured such a $-field. Bohm postulates a quantum potential, Q, in addition to the standard potential,

+

406

EDWARD R. FLOYD

that renders a quantum force proportional to -VQ. [The negative of Bohm's Q in one dimension appears on the right side of the QSHJE, Eq. (I).] But Q , by the QSHJE, is dependent upon E and the microstate (a,b, c) of a given eigenvalue energy E because

Therefore, Q is path dependent and cannot be a conservative potential. Consequently, -VQ does not generally render a force. While Bohmian mechanics postulates pilot waves to guide the particle, the trajectory representation does not need any such waves. Bohmian mechanics uses an ansatz that contains an exponential with imaginary arguments. The Bohmian ansatz in one dimension is I!,J = (w')-Il2exp(iW/h). As Eq. ( 5 ) is valid for any set ( 4 , 8 ) other , ansatze including trigometric forms are acceptable [1,2,5]. Bohmian mechanics asserts that particles can never reach a point where TJ vanishes. On the other hand, trajectories have been shown to pass through nulls of TJ with finite conjugate momentum, W' [I,?]. Bohmian mechanics asserts that bound-state particles have zero velocity because the spatial part of the bound-state wave function can be expressed by a real function. On the other hand, the QSHJE, Eq. (I) is still applicable for bound states in the trajectory representation. For bound states. the trajectories form orbits whose action variables have the Milne quantization J = $ W' dx = nh,n = 1,2,.. . independent of the microstate (a,b, c) [1,7,15]. Bohmian mechanics asserts that a particle will follow a path normal to the surfaces of constant W. Meanwhile, trajectories in higher dimensions are not generally normal to the surfaces of constant W [4,11]. In higher dimensions, trajectories are determined by Jacobi's theorem, Eq. (4) rather than by V W . Bohmian mechanics asserts that the possible Bohmian trajectories for a particular particle do not cross. Rather. Bohmian trajectories are channeled and follow hydrodynamic-like flow lines. On the other hand, the trajectory representation describes trajectories that not only can cross but can also form caustics as shown elsewhere in an analogous, but applicable acoustic environment [16]. The two representations differ epistemologically whether probability is needed. The trajectory representation is deterministic. Bohmian mechanics purports to be stochastic and consistent with Born's probability amplitude [14]. Let us consider three dimensions in this paragraph to examine the familiar stationary auxiliary equation

to the three-dimensional QSHJE. Bohm and Hiley [14J identify R as a probability amplitude and Eq. (6) as the continuity equation conserving probability. Hence, R2VW must be divergenceless. The trajectory representation can now show a non-probabilistic interpretation of R ~ V W Let . us consider a case for which the stationary Bohm's ansatz, = Rexp(iW/h),is applicable. Bohm [13] used R~ = $J

TRAJECTORY REPRESENTATION OF QM

407

U 2 + v2and W = tiarctan(V/U)where U = ?I= ? R, cos(W/ti) ($)and V = S($)= Rsin(W/fi). Hence, by the superposition principle, U and V are solutions to the SSE. Upon substituting U and V into Eq. (6). we get the intermediate step ( R 2 V W ) = UVV - VVU which is a "three-dimensional Wronskian". Whether or not this "three-dimensional Wronskian" renders a constant, it is divergenceless [5]. Therefore, the trajectory representation finds that the auxiliary equation contains a "three-dimensional Wronskian" that satisfies Eq. (6) without any need for evoking a probability amplitude. Bohrn had expressed concerns regarding the initial distributions of particles. Bohm [13] had alleged that in the duration that nonequilibrium probability densities exist in his stochastic representation, the usual formulation of quantum mechanics would have insoluble difficulties. The trajectory representation has shown that the set of initial conditions may be arbitrary and still be consistent with the Schrodinger representation [6]. Stochastic Bohmian mechanics, like the Copenhagen interpretation, uses a wave packet to describe the motion of the of the associated $-field. As previously described herein, the deterministic trajectory does not need wave packets to describe or localize particles. Holland [17] reports that the Bohm's equation for particle motion could be deduced from the SSE but the process could not be reversed. On the other hand, the development of Eq. (5) is reversible. In application, the two representations differ regarding tunneling. Dewdney and Hiley [18] have used Bohmian mechanics to investigate tunneling through a rectangular barrier by Gaussian pulses. While Dewdney and Hiley assert consistency with the Schriidinger representation, they do not present any results in closed form. Rather, they present graphically an ensemble of numerically computed trajectories for eye-ball integration. On the other hand, the trajectory representation exhibits in closed form consistency with the Schrodinger representation. Also, every Bohmian trajectory that successfully tunnels slows down while tunneling even though Steinberg et a1 [19] have shown that the peak of the associated wave packet speeds up whiie tunneling. Our trajectories that successfully tunnel speed up [4.10] consistent with the findings of others [20-231.

Acknowledgement 1 am pleased to thank M. Matone for many discussions. 1 also thank D. M. Appleby, G. Bertoldi, R. Carroll, and A. E. Faraggi.

1. Floyd, E. R.: "Modified Potential and Bohm's Quantum Potential", Phys. Rev. D 26 (1982), 1339-1347. 2. Faraggi, A. E. and Matone. M.: "The equivalence postulate of quantum mechanics", Inf. J. Mod Phys. A 15 (2000), 1869-2017, hep-thB809127.

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EDWARD R. FLOYD

3. Floyd, E. R.: 'Which causality? Differences between tajectory and Copenhagen analyses of an impulsive perturbation", Int. J. Mod Phys. A 14 (1999), 11 11-1 124, quant-phl9708026. 4. Floyd, E. R.: "Reflection time and the Goos-Hiinchen effect for reflection by a semi-infinite rectangular barrier". Found. Phys. Lett. 13 (2000). 235-251. quantphl9708007. 5. Floyd, E. R.: (2000) "Extended Version of 'The Philosophy of the Trajectory Representation of Quantum Mechanics' ", quant-phl0009070. 6. Floyd, E. R.: "Arbitrary initial conditions of hidden variables", Phys. Rev. D 29 (1984), 1842-1844. 7. Floyd, E. R.: "Where and why the generalized Hamilton-Jacobi representation describes microstates of the Schriidinger wave function", F o u t d Phys. Len. 9 (1996), 489497, quant-phl9707051. 8. Floyd, E. R.: 'Closed-form solutions for the modified potential", Phys. Rev. D 34 (1986), 3246-3249. 9. Carroll, R.: "Some remarks on time, uncertainty and spin", J. Can. Phys. 77 (1999), 319-325, q~ant-~h/9903081. 10. Floyd, E. R.: "A trajectory interpretation of tunneling", An. Fond. Louis de Broglie 20 (1995). 263-279. 11- Floyd, E. R.: "A trajectory interpretation of transmission and reflection". Phys. Essays 7. (1994) 135-145. 12 Floyd. E. R.: "Classical limit of the trajectory representation of quantum mechanics, loss of information and residual indeterminacy", Int. J. Mod. Phys. A 15 (2000), 136S1378, quant-ph/9907092. 13. Bohm, D.: "A suggested interpretation of the quantum theory in terms of 'hidden' variables. I", Phys. Rev. 85 (1952), 166-179. 14. Bohm D. and Hiley, B. J.: "An ontological basis for the quantum theory", Phys. Rep. 144 (1987), 323-348. 15. Milne, W. E.: "The numerical determination of characteristic numbers", Phys. Rev. 35 (1930), 863-867. 16. Floyd, E. R.: "The existence of caustics and cusps in a rigorous ray tracing representation", J. Acous. Soc. Ant. 80 (1986), 1741-1747. 17. Holland, P. R.: The Quanmn Theory ofMotion, Cambridge U . Press, Cambridge, UK, 1993, p. 79. 18. Dewdney, C. and Hiley, B. J.: "A quantum potential description of the onedimensional time-dependent scattering from square barriers and square wells". Found. Phys. (1982), 12,2748. 19. Steinberg, A. M., Kwiat, P. G. and Chiao, R. Y.: "Hidden and unhidden information in quantum tunneling", Found. Phys. Lett. 7 (1994), 223-237. 20. Olkhovsky, V. S. and Racami, E.: "Recent developments in the time analysis of tunnelling processes", Phys. Rep. 214 (1992), 339-356. 21. Barton, G.: "Quantum mechanics of the inverted oscillator potential", An. Phys. (New York) 166, (1986), 322-363. 22. Hartmann, T. E.: "Tunneling of a wave packet", J. Appl. Phys. 33 (1962), 34273433. 23. Fletcher, J. R.: '"Time delay in tunnelling through a potential barrier", J. Phys. C 18 (1985), L55-L59.

SOME PHYSICAL AND PHILOSOPHICAL PROBLEMS OF CAUSALITY IN THE INTERPRETATION OF QUANTUM MECHANICS B . LANGE Departamen2 for Logic, Methodology and Philosophy qf Science, University of Gdarisk, ul. Bielahka 580-951, Gdarisk, Poland

1. Introduction The International Conference of Physicists in Warsaw took place from 3dhMay to 31dJune 1938. The Conference was organised by the International Institute of Intellectual Cooperation, and the Polish Commission as its associate body. The talks were chiefly devoted to philosophical issues that emerged along the development of quantum theory and theory of relativity, and other basic issues related to interpretation of new discoveries and theories. The need for organising such meetings resulted from the necessity ofdiscussing and confronting the ideas ofmost prominent scholars in relation to the most basic problems emerging in the period of rapid development of contemporary physics. It was the fmt of the whole pre-planned series of conferences. and it should be pointed out that the very first conference took place in Poland, which is a proof of a hlgh world rank of Polish physics. The conference was initiated and then chaired by an eminent Polish physicist, Professor Czeslaw Bialobneski. The conference was also attended by other eminent scholars. like Niels Bohr. Leon Brillouin, Charles Darwin, Arthur Eddington, George Gamow, Hendrik Anthony Kramers, Paul Langevin, John von Neumann, Eugen Wigner, and others. Poland was represented by Professors Wojciech Rubinowicz. Szaepan Szczeniowski. Jan Weyssenhoff. Ludwik Wertenstein. and F. J. Wisniewski. Complicated political situation of that time, however, was the reason why some invited scholars failed to arrive. During the conference eight papers were delivered, and they were afterwards published together with the abstracted discussion in the volume titled h s Nouvelles Theories d e la Physique [ll. The outbreak ofthe War was the cause why the information and the conference materials did not reach most of the world centres of physics. Therefore the conclusions drawn from the papers and discussion did not become a source offurther debates. Thus the Conference did not have some major impact on further development of the foundations and interpretation of quantum physics. It seems, however, that there is a need to present chief ideas discussed during the conference, particularly since some of them - like the proposition of Bialobrzeski, for instance - were attempts to overcome basic problems in interpreting formalism ofquantum theory. Those issues have not been successfully solved so far, and they still are a point of interest to physicists and philosophers. The following paper aims at presenting the discussion during the Conference, which was joined by Bohr, von Neumann, Bialobrzeski, and Kramers. The discussion developed over the paper of Bohr, whose intervention seemed to have stirred the highest interest among the participants. The proposal of interpretation of the reduction process of the wave function Y presented during the discussion by C. Bialobrzeski [l], seems to be of particular interest here.

409 R.L. Ainuroso et a1 (eds.),Gravitation rrr~dCa~mcdogv:From the Hubble Radius to the Plunck Scale, 409-412. O 2002 Kluwer Actulenlic Publishers. Printed in the Netherlaiuls.

6.LANGE 2. A Dispute Between N. Bohr, J. Von Neumann, Cz. Bialobrzeski, And H. A. Kramers On Causality In Atomic Physics While analysing a problem of the use of the laws of mechanics and thermodynamics in explaining atom's stability Bohr stated that we should make additional assumptions, i.e. that any well defined change in the status ofelectron within the atom depended ofthe shift ofthe atom from one stationary position into another, similar one. According to Bohr, quantum postulates are not only completelyalien to the notions ofclassical mechanics, but they also assume divergence from any casual description of such processes. This particularly refers to possible shift of the atom from a stationary position into another one. Bohr claimed that "the atom faces a choice with no determining circumstances. Situation like that, any forecasts may refer only to the probability of various possible directions of atomic processes, which are susceptible to direct observation" [I]. Further on Bohr claimed that the "utilisation of purely statistical consideration is the only instrument which allows for generalisation of usual description, necessary for explaining individual character expressed by quantum postulates..., (and) this is how we limit classical theory to an extreme case, where the changes ofaction are large as compared to the quantum ofaction. The only reason for formulating such a generalisation was the need for utilisation ofclassical notions in the widest possible scope which could be reconciled with quantum postulates" [I]. Basic assumptions of the Bohr's intervention can be presented as follows: measurement cannot mean anything else but unbiased comparison ofa given property ofthe object under research, with a corresponding property of another system used as a measuring instrument. Properties of the measuring instrument directly result from the definition taken from classical physics. The above mentioned comparison ofthe properties must be convergent with the definition expressed in a common language. Whereas in the field of classical physics such a comparison may be performed without substantial intervention into the object itself, in the quantum theory, however, the interaction between the object and the measuring instrument will basically influence the phenomenon itself. We have to be particularly aware ofthe fact that the interaction cannot be exactly separated from the behaviour ofthe undisturbed object. Thus the necessity of basing our description on classical notions implies disregarding all the quantum effects in the very description, particularly disregarding more precise control over the object's reaction than it might result from the uncertainty principle. Adopting the complementarity point of view we may avoid unnecessary discussion on determinism. There is no need to discuss indeterminism when we state a general scheme ofcausative idea which may aim at the synthesis ofthe phenomena which can be described within the function of the behaviour of the objects irrespective of the mode of their observation. Bialobrzeski, in turn, thought that there were doubts concerning logical coherence of quantum theory. The interpretation of Bohr and Heisenberg presented two, complementary sides of elementary creations of the material world: the corpuscular and the wave sides. It is rightly said that the formalism of quantum mechanics gives synthesis which unifies the two images of the single reality. We must refer to these images to describe natural phenomena within space-timecontinuum. Bialobrzeski thinks, however, that it is possible to find dualism of a different nature within the notions of quantum mechanics. This dualism questions uniformity ofa doctrine, unless it can be explained within the chief postulates of the doctrine. The dualism means - as Bialobrzeski stated - that on the one hand we have a casual seyuence of phenomena governed by the Schrodinger equation, on the other hand, however, when we measure a certain volume of "A", the status represented by function Yis subject to rapid non-casual change. The sudden change infunction Y results from the fact, that we can obtain a result ofa measurement in the form of any eigenstate 5,Y2,Y3... which would correspond to measured value. "We assume," the scholar says, "that we are dealing with a discontinuous system of these states. States YNare

PROBLEMS OF CAUSALITY IN QUANTUM MECHANICS

411

independent from the initial state, decisive only for probability of their realisation expressed by the [...I. If we consider a large number of the systems in the same state of then the formula ( WNj2 measurement ofthe volume 'XA~performed on the set of all these identical systems turns the set into a mixture, which is a proof of non-casual character of the process. The shift from the state Y into one of the states YNmust be either instant or brief, which on the one hand leads to indeterminacy of energy related to time by one of the Heisenberg's uncertainty relations, on the other hand it clearly separates time from the co-ordinates defining position. which is in contradiction to the postulate of relativity" [I]. Bialobrzeski asked a basic question: "What is the mechanism of realisation of the state YNfrom the state Y naturally ifthere exists describable mechanism of the change?" [I]. Bialobrzeski's proposal looks as follows: after accepting the thesis, that von Neumann's analysis defined impossibility(within quantummechanics)of eliminating indeterminismbyintroducing hidden parameters for preserving the description of the measuring process with the use of Schriidinger equation, we musl assume that coupling of the measuring instrument with the object under research is the cause of distortion. The distortion is discontinuous and changeable in such a way, that a probability of realisation of state Y, is strictly defined by the formula (YYN)2. Discontinuity and indeterminacy, which were eliminated from the measurement, reappeared in the distorted operator of energy, which ough~to change in a discontinuous and indeterminable way. As a result, we obtain nothing which could explain the mechanism under discussion. Bialobrzeski concludes his comments with the assumption that the action of taking measurement performed upon the system in which its natural alternation of state is governed by Schrijdinger equation causes another type of alteration of indeterminate character, as it is solely governed by statistical laws. Anyway, it seems to be necessary - as Bialobrzeski believes - to assume the existence ofa basic postulate which refers to the very act oftaking measurement: "quantized physical quantity is basically indeterminate, and we can only recognise the probability of realisation of edch of eigenvalues ofthe quantity. It is just determined by the formula ( w ) ~ . " [I] J. von Neumann noticed that we must have an observer placed somewhere within a given system, therefore it is necessary to draw the borderline between the observer and the observed. It is not necessary, however, that the border overlapped with physical body limits ofthe observing person. He believes that it is possible to "shrink" or "blow up" the observer. We may include, for instance, everything that is happening in the observer's eye and include it all into the part under observation, which is described in a quantum manner. The observer would begin then behind the retina of the eye. We could also include some of the equipment used for physical observation, e.g. a microscope, into the observer. The principle of psycho physical parallelism defines a situation, where the abovementioned border can be shifted as deeply into the body ofthe observer as we wish. As far as the issue discussed by Bialobrzeski was concerned, Bohr thought that the dualism mentioned by him was simply a problem of selecting most adequate description of an experiment. As for von Neumann's comment on differentiation between the phenomenon and the observer, Bohr thought that the differentiation was a natural process with the phenomena we are dealing with within quantum theory, however we try to explain a given phenomenon in a classical way. Kramers in turn briefly referred to Bialobrzeski's comments, and expressed his own opinion on them. He thought that if we adopted calculation scheme utilised, as a matter of fact, in many real physical problems, it would be dubious if we could really talk about independent postulate referring to measurements. Kramers believed that "demanding such a postulate is a consequence of too a classical approach to the problem. We cannot speak about Schrodinger equation and about the ways we build up Hamiltonian unless we think about experiments at the same time. In other words, we cannot reject results of measurements which allowed us to make use ofwave functions. and we cannot neglect the way in which the value of mechanical quantity can be linked with the wave image. To measure is nothing but confirming details in the instrument's state. These details carry the sense

B. LANGE

412

directly derived from the notions of classical physics. According to the notions, the condition of a given system may be either like this or like that. There is neither indefiniteness nor distribution of probability among various possible states. Natural laws, however, say that the result frequently cannot be predicted even if the starting stage of an experiment is defined in the most precise way." [l] Kramers saw in this the role for the quantum ofaction in stabilising Nature. On the other hand, the role of the quantum of action is of the type that does not allow us to talk about some 'mechanism' regulating the selection pertormed by an instrument when we want to measure something. To put forward a question like this means - according to Kramers - a shift into the scheme ofclassical determinism. This determinism, however, had been rejected the moment we accepted wave function. In the further part of discussion Bialob~eskistood by his opinion by saying, that he had not properly understood Kramers's explanation. He noticed that the way of thinking in the field ofquantum theory was the very way we are used to in natural sciences. The situation could always be presented as an interaction between an object and an instrument. The system could be studied - the scholar thought - with the use of additional expression in the hamiltonian formula. [fthe expression exists, the condition changes. The emergence of a spectral line on a photographic plate may be an example here. We have phenomenon which can be described with the expression defining disturbance. Then the whole process of the phenomenon can be defined. Thus we can use this particular postulate, because - as von Neumann said - a division line within a measuring system can be drawn anywhere. In his reply Kramers stated that we could adopt this procedure for more precise analysis of a measuring instrument, we could introduce to hamiltonian words that would define interaction ofthe instrument with variables characteristic for the state ofthe instrument. We could prove by this that we understood the function of measuring instruments well. According to Kramers, however, this did not constitute the 'mechanism' of making choice. In his summary, Bohr said that there were a few ways ofexpressing relations between classical and quantum physics. In the theory of classical electromagnetism, for instance, theoretical description is inseparable from certain experimental methods used to define certain basic notions. In quantum mechanics the situation is different in the sense that interpretation of various measurements which could be performed, whatever their precision be, do not allow for classical definition of the system. According to Bohr it would be very difficult to describe such a situation by saying that it would take additional mathematical postulate to link a result of a measurement with the other principles of the theory.

Bialobrzeski's idea that interpretation difficulties forced us to recognise a choice during measurement of eigenvalue of the measured quantity, as an independent postulate which was mathematically expressed by the formula defining probability of the choice of any of eigenvalues, did not receive recognition. Bialobrzeski did not give up, however, the development of his idea, and made it more comprehensible in his book entitled Cognitive Foundations of the Physics of the Atomic World, [2].

References 1. 2.

Les Nouvelles Theories de Ea Ph.sique, (1939) International Institute of Intellectual Cooperation, Paris. Bialobrzeski C., 11956) Podstaw? poznawczefiqki swiata atomowego, [Cognitive Fouildations of the Physics of the Atomic World,), PWN Warsaw.

THE FORCE c 4 / ~ THE , POWER c5/6 AND THE BASIC EQUATIONS OF QUANTUM MECHANICS

LUDWIK KOSTRO Department for Logic. Methodology and Philosophy of Science, University qf Gdalsk, ul. Bielalska 5, 80-951 Gdalsk, Poland E-mail: fizlk@ univ.gda.pl

1. Introduction

In two recent papers [I-21 the quantity C~/G was interpretated as the greatest possible force in Nature. In a third paper [3] following LR. Kenyon [4] the quantity c5/G was interpretated as the greatest possible power. In the three mentioned papers I have limited myself to classical considerations. I have shown eg. that the classical Newton law and the classical Coulomb law can be rewritten in the following way:

Newton force when m, = m2

when ml#m2

Coulomb force

It was also indicated that the quantities c4/Gand CS/G and their inverses appear in the equations of General Relativity [1,2,3] and Kenyon's interpretation [4] of this fact was presented. Ln my considerations I use the following constants and constant coefficients: c velocity of light in vacuum; G - the gravtational constant; ...- Planck's constant; e - the elementary electrical charge; m - the mass of an elementary particle; K = 1 / 4 n ~..I take into account also the units of lenght, time and mass determined by the following set of constants (c,G,nt) (c,G,e) (c,G, ..) (c, G,ggJ. (c, G, gw). Where in, e, g ~,gw , are the

respective charges of four fundamental interactions. Using the dimensional analyse we obtain the fallowing units: 413 R.L. Anloraw e.? a1 feds.).Gravitation and Cosnlolog?.:From tlw Hzibblt.Rcxius to tltr Planck Scale, 413-418. @ 2002 Kluwer Acude~nicPublislters. Printed in tlte Nethei-luitds.

L. KOSTRO

414

(a) gravitational lenght ZG ,time tG,and mass m~

(b) J.G. Stoney' s lenght Is ,time ts , and mass ms introduced by him in 1874 [5-6]

(c) M. Planck's lenght ZP ,time tp,and mass mp introduced by him in 1899 [7]

ZP

3 112

= (..G/c)

,.

tp= (..G/c)5

I12

,.

ms = (..c/G)

'n

(d) Lenght Is,, time ts, and mass ms,r connected withe the strong interactions

(e) Lenght lw, t i e tw and mass mw connected withe the weak interactions

It is interesting to note that forces F and powers P connected with these units all are equal:

These forces and powers appear especially when the density p of the matter is the greatest possible one. The following formulae present the limiting density of the gravitationally, electrically, strongly and weakly charged matter and the Planck's density.

6 2 ps= (11~2) (c /G )

p,, = (I/. .c)(c6/&)

(Stoney's density)

(Planck's density)

As we can see the respective limiting density depends upon the inverse of the square of the respective charge multiplied by the factor c6/G 2. In other words, it constitutes the product ofthe square ofthe respective charge and the factor c6/G 2

BASIC EQUATIONS OF QUANTUM MECHANICS

2. The Quantities Equivalence

C'/G And C%

And Einstein's Principle Of Mass And Energy

It is interesting to note that the Einstein's Principle of mass and energy equivalence E= me2 can be rewritten in the following way:

This fact shows, may be once again, the dynarnical nature of the matter. [f an elementary particle could deliver its total energy E = me2 acting on the path equal to lG and during the time equal to fc then it could show its greatest force (c4/G) and power (c5/G). If it could happen then the extremal force {c4/e) and the extremal power {c5/G) would be hidden in every particle. Perhaps in the future mankind will find out the circumstances in which it will be possible. At the present day, however, we can ose only to interpretate the two quanties as exremal ones.

3. The Quantities 6'E And C%

And Schriidinger Equation

As is well known Schrodinger equation is the basic equation of the non relativistic Quantum Mechanics. In textbooks it is written in the following way

-

where V can be the Coulomb potential V = ~ 2 / r .In these equations we find the constants .., n ~e, and the coefficient K. Since the constants used in physics and the units determined by them are correlated and interconnected therefore it is not difficult to rewrite the Schrodinger equation in such a way that the quantities c4/G and C'/G and the considered units appear in it. The Coulomb potential can be rewritten as follows

and the Schrodinger equation in the following way

Since lsts = lP t p a (where a = K ~ / . . Cis the fine structure constant) we obtain also

As we can see, in the Schrodinger equation written in such a way, threre appear not only the quantities C'/G and cS/G but also the Planck's lenght and time and Stoney's

L. KOSTRO

416

lenght and time.We see also that Planck's constants ...is related to the quantities c4/Gand 2/G as follows:

When we divide the both sides of the eq. (8) by (I,ip)we obtain

We must be aware, however, that such a division changes the numerical value and the dimensions of the both sides ofthe equation. The threedimensional Hamilton operator

-

H = (..?/2m)[(d 2 r ~ / d x 9 +(aZyl/d$)+ ( d 2 y l / d 2+ V

(1 1)

can be rewritten introducing c4/Gand C'/G as follows

H = - (.l2m)(c4/G)(lPt p )[(&/.x?+

(dzyl/d$)

+(

2

)+ V

(12)

-

When V = - ~ e ' / r= - (c4/G)(1,2/r)= (c5/G)(lStS/r) then we can write

4. The Quantities

And

P/GAnd The Klein-Cordon Equation

Let's consider the Klein-Gorden equatiom written e.g. for the mesons ~t..

- ..z(@*J)

=

- ..~CZ(a2v ~ t ? ~ 2 + ~+d ~ d~y /z

d 2 ) C# + v~ ~ ~

113)

Taking into consideration thequantities C?G and cS/Gthe Klein-Gordon equation can be rewritten:

As we can see the Planck's charge raised to the second power (..c) is related to (c'/G) as follows

Since rn,

2 =(c4/G)IG,=(cS/~)tG, the eq. (15) can be also written as follows

BASIC EQUATIONS OF QUANTUM MECHANICS

417

Since lc, tc, = (lp tP)%x (where %n = Gmn %.c is the coupling constant of gravitational interactions between two particles of the same mass i our case the coupling constant of grawitational interactions between two mesons n) the eq. (16) can be rewritten as follows

Dividing the both sides of eq. (17) by (1,t p )we obtain

We must be aware. however. that such a division changes the numerical value and the dimensions of the both sides ofthe equation. 5. The Quantities &/G And C?/G And The Dirac Equation

The Dirac equation can be written as follows:

where the matrix yP have the following properties

(where

t

means the hermitonian coupling)

Taking into consideration the quantities c4/G and cS/Gthe Klein-Gordon equation can be rewritten:

'

~ gravitational time where t, = (.G/cS) is the Planck's time and tG= ~ m / iscthe

L. KOSTRO 6. Conclusion It was very easy to introduce the quantities (c4/G)and ( c 5 / ~ into ) the basic equations of Quantum Mechanics (and we can even say that such an introduction constitutes a very trivial operation) but it is very dufficult to interpretate their part played in these ) into the classical equations. When we introduce the quantities (c4/G) and ( c 5 / ~e.g. equations of Newton and Coulomb we see immediately their meaning as limiting quantities i.e. the greatest force and the greatest power but when we introduce them into the basic equations of Quantum Mechanics we do not see clearly their physical meaning. They work, however in these equations and therefore we can think that also here they play not only a role of constants but also as Limiting quantities, as the greates force and the greatest power.

References 1.

Kostro L. and Lange. B. (1999) Is c4/Gthe greatest possible force in nature? Physicv Essays. 12. No 1, 182

2-

Kostro L. and Lange B., (1998) The force c4lGand Relativity Theory, in M.C. Duffy (ed), Phj.sical Znterpretatioris of Relativie Tlieo~y.(Proceedings) British Society for Philosophy of Science, Impaial College, London, pp. 183-193

3.

Kostro L. and Lange B., (19%) The Power c5/G and Relativity Theory, in M.C. Duffy (ed), Physical Interpretations of Relativi~Tlleory. (Later Papers) British Society for Philosophy of Science, Imperial College, London, pp. 150-155 Kenyon I.R. (1990) General Relativity, Oxford University Press Barrow J.D. 11983) Q.JlR. Astr. Soc. 24.24 Stoney GJ. (1881). Phil. Mag.. 5,381 Planck, M. (1899) Sit:ungsberichte d. Preus. Akad. Wiss./Mitteilung/S, 440 and Planck, M. (1900) Ann. Phys. 1, 69

4. 5.

6. 7-

PROGRESS IN POST-QUANTUM PHYSICS AND UNIFIED FIELD THEORY JACK SARFATTI Internet Science Education Project & International Space Sciences Organization [email protected]

Abstract

Progress in extending the de Broglie- Bohm-Vigier (AKA dBBV) quantum ontology to the experimental mind-matter problem is reported in Part I. Progress in extending Einstein's classical orthodox holonomic topology-conserving general relativity of 1915 to the unified field theory including topology-changing anholonomic torsion fields from the "hyperspace" of M-theory is reported in Part L1. I also make a conjecture that the empirical duality in the Wesson compared to the Sirag data plots. noted by Gray in 1988, is actually showing the M-theory T-duality R ~u l / 2 ~ n, ff m The Wesson "Regge trajectory" (Kaluza-Klein excitation n -winding number m) mass scale

'

- &M,, +1018~ e -vI, /& - 1 o - cm~ .~ The mass scale - M, /& + 1 0 ~ ' ~ e -v &IP M, =

+ 10" ~

-

e vl p I

-

''dual" Blackett-Sirag magneto-gyro

cm, where a = e2/hc

n: 11137

and

cm. What is clear is that we now

have a new "telescope" directly into the quantum gravity scale showing strong anholonomic unified field effects beyond Einstein's 1915 theory. This is as important as the Hubble flow, the cosmic microwave background, missing mass, gravity waves, and the anomalous acceleration of the universe. A completely new conception of COSMOS is now emerging from the actual data.

Part 1: Progress In Post-Quantum Physics Orthodox quantum theory has many "degenerate2 informal interpretations that appear to have no crucial experimental tests to "lift the degeneracy". The experimental situation is now changing dramatically and quickly with my recognition of the real meaning of data lying around in journals unread for twenty years. All of the contemporary competing

' 11.6 eq. (11.6.2) p. 477 "Intro to Superstrings and M-Theory", M.Kaku, Springer-Verlag, 1999.

* In the sense of atomic spectroscopy with "degenerate energy eigenfunctions" of the Harniltonian operator. 419 R.L. Arnoroso et a1 (eds.),Gravitation and Comolog?.: From the Hubble Rudius to the Planck Scale, 419430. O 2002 Kluiver Academic Publi~her.~. Printed in the nether land^.

420

J. SARFATTI

interpretations3. save dBBV. only have the quantum wave and not the particle in their models of quantum reality. The "particle" or "Bohm point" moving on the landscape formed by the quantum potential and other forces is "e~chewed".~This immediately causes confused thinking, e-g., in the recent articles claiming to divide the charge e ofthe electron into two equal pieces.5 This confusion comes from falsely assuming that the quantum wave "is the essence of an electron" carrying the charge because there is no particle at the micro level. The experimental result of increased current is easily understood intuitively in the dBBV interpretation in which the complete indivisible electron particle, of screened charge e, is completely localized in only one of the two bubbles that the physically real pilot wave divides into. One of the paired bubbles has a real but empty branch. The smaller bubbles move faster through the liquid helium explaining the observed increase of electric current without having to cut the electron in half. Therefore, this experiment seems to lift the degeneracy.6 Bohm's ontology works better for this experiment. In another recent development, Henry stapp7 has proposed a model of ontological collapse8 of the quantum wave with infinite speed in the preferred cosmological rest frame of the Hubble flow9 in the standard cosmological model. This is theory on a classical curved space-time geometry not full blown quantum gravity. Stapp's conjecture is similar to Bohm's and Hiley's in which the quantum potential Q ;its instantaneously in this same preferred frame. There is no reason to suppose that classical Diff(4) local gauge symmetry10 of the 1915 general theory of relativity should be valid when quantum nonlocality is important. However, experiments by is in" et-a1 in Geneva seem to rule out this idea, although Stapp and I are in serious disagreementI2 on how to think about this problem. I am only giving my biased perspective here. Special relativity still works locally. According to the Einstein E.g., Stapp's "'ontological collapse", Penrose's "R" and "'OR", "many worlds" in all of its variations such as David Deutsch's "multiverse", Gell-Mann/Hartle decohering histories, John C ~ m e r ' s"transactional" with weak backward causation consistent with "signal locality" AKA "passion at a distance" (Abner Shimony). "Bohmian Mechanics and Quantum Theory: An Appraisal" Ed. J.T. Gushing, A. Fine, S. Goldstein (Kluwer,

1996) "ew Scientist magazine, 14 October 2000 "This sounds harmless enough, but the implications are staggering. Ifthe bubble split, half of the electron's wave function would be trapped in a c h of the two daughter bubbles .... As the wave function is the essence of an electron, the electron would be split into two. The indivisible would have been divided. ... 'There were more bubbles, and being smaller they were more mobile,' says Maris. Although the total charge in the system remained the same, the smaller bubbles felt less drag in the helium, and thus moved faster. Consequently, the current went up," My solution here is reminiscent of King Solomon's when asked to divide the baby in half. The electron, like the baby, has been thrown out with the bathwater in all the alternative interpretations save dBBV. Discussion by e-mail among Stapp, Stan Klein and myself. "speed ofquantum information" (Gisin et-al, ref 11) In which the cosmic microwave black body background radiation is isotropic to about one part in ld. 10 Integrable holonomic general coordinate transformations that are global 1-1 conserving topology of the 4d s cetime manifold. 1pa@ant-ph/OOolo384 July 2 0 T h e Speed ofQuantum Information and the Preferred Frame: Analysis of Experimental Data" V. Scarani, W. Tittel, H. Zbinden, N. Gisin l2 There is no preferred frame in global special relativity, consequently whether or not nonlocal EPR correlations are observed cannot depend upon the common state of uniform motion of the two detectors relative to any other frame. However, global special relativity breaks down in general relativity where it can only be used locally. Therefore, this issue is an experimental one. Stapp inconsistently tries to maintain global special relativity and the preferred frame ofthe Hubble flow together in my opinion.

'

POST-QUANTUM PHYSICS AND UNIFIED FELD THEORY

421

addition of velocities, an infinite speed in one frame is a finite superluminal speed

v = C'/U

where u is the subluminal speed of the moving "G-frame" relative to the allegedly preferred global rest frame of the Hubble flow. For example. the rotating Earth's motion has a 24 hr periodic projection u. of its CM motion on to the flight line between the G-detectors, of amplitude 300 to 600 kmlse~.'~Therefore, except for 5 seconds every 43,200 seconds, the G-frame qubit speed is less than the minimum 13,

-

-

2

213 x 10' c required dipping down to v = (3 x 1 0 5 ) /(4 x lo2) = 2.5 x 10' km/s

= 103 c

reaching a peak less than 213 x 10' c for most of the data run. In contrast, Gisin cites "a lower bound for the speed of quantum information in this "G-frame" at 2/3 x 10' c " for the parameters of the actual experiment'5 in Geneva of EPR photon pair correlations over a distance 10.6 km. Therefore, G-frame detectors, each with speed u relative to the Hubble flow, separated by a distance 10.6 krn requiring a qubit speed of at least

213 x107 c to travel between them in the 5 picosecond time uncertaintyI6 in the detections, should not show the actually observed EPR correlations most of the time. In fact, the EPR correlations are seen all of the time. This argument assumes that the qubit speed in the Hubble frame is infinite, so that objective collapse with a real qubit speed is a useful way to picture how the nonlocal EPR correlations are maintained. Therefore, Stapp's conjecture here is falsified by the actual experiment. Does this also shoot down the Bohm-Vigier conjecture that the quantum potential Q acts instantly in the preferred Hubble flow frame? Yes, the model that does survive is that of "backward causation" l7 originating in the Wheeler-Feynman 1940 program for classical electrodynamics eliminating independent dynamical degrees of freedom for the electromagnetic field with "Everything is particles."18 However, it appears that Q = (fi/2rn)v2~/~can be redefinedI9 in terms of backward causation. Note that the term "ret" for "retarded" means "from the past", or what Aharonov calls the "history" state vectw. Similarly, the term "adv" for "advanced" means "from the future", or what Aharonov calls the "destiny" state vector. Feynman already used this idea in his original paper on the path integral in nonrelativistic quantum theory.20

Narlikar. p.301 (Cambridge, 1993) Time resolution of 5 picoseconds in footnote [19] of ref 11. l6 Photon pulse temporal width l7 "Feynman zig zag" (0. Costa de Beauregard), "transactional interpretation"(J. 0. "history and destiny state vectors" (Y. Aharonov et-al), Hoyle-Narlikar"advanced response of the universe" in "Cosmology and Action at a Distance Electrodynamics" World Scientific, 19%. 'Time's Arrow and Archimedes Point", Huw Price, Oxford, 1996. Is "Geons, Blackholes & Quantum Foam", Wheeler's autobiography with Ken Ford. l 9 There is no collapse in the Bohm ontology. hence no "qubit speed" for ~mllapse. We have seen that the standard backward causation models without the particle cannot explain the apparent splitting of the charge on the electron as naturally as the Bohm ontology can. 20 Rev. Mod. Phys. 20,267, "Space-Time Approach to Non-Relativistic Quantum Mechanics" (1948) 15

Bohm showed that the pilot wave is a physical field of qubits in the configuration space of the piloted material. The ideas of quantum probability21are not fundamental. God does not play dice with post-quantum r ty. Therefore. unlike the orthodox statistical interpretations relying fundamentally on ensembles of identically prepared simple atomic systems such as particle beams in scattering experiments, Bohm's ontology is ideally suited to explain unique complex highly entangled systems such as the living human brain. There was only one shakespeare2 and to invoke shadow Shakepeares in a multiverse of parallel worlds is "excess metaphysical baggage".'3 This is not to deny the possibility of ''other worlds" close by in material hyperspace, less than a millimeter away, as in M-theory with "3D membranes" folded by anholonomic torsion fields.24One can even imagine traversable wormholes connecting these worlds to each other. Bohm and ai Hiley also emphasize that the quantum pilot field is "nonmechanical" and "organic" with no "preassigned interactions between the parts". In this sense, the pilot field is not at all like a classical machine. The pilot field in configuration space for entangled subsystems is form-dependent and intensity-independent totally opposite to classical fields in z eali and undiminished with increasing separation ordinary space. It is intimate 5. immediate. unlike the classical dynamical force fields of electromag~etism gravity. and torsion confined to ordinary space. These are all desiderata for the "mental field" out of which our thoughts, feelings, and perceptions arise in consciousness. Indeed, the pilot field idea immediately explains how thought can move matter. What Bohm's and Vigier's "causal theory" cannot qu l tatively explain in principle is how matter influences thought to create the inner conscious experience. This is because of an argument Bohm and Hiley ?iveZ6 that the standard statistical predictions of quantum theory for ensembles of identically prepared simple independent unentangled systems" require that there can be no direct reaction or "back-action" of matter on its pilot field. Such a compensating postquantum reaction to quantum pilot action would result in "signal nonlocality" violating the Stapp-Eberhard no-go theorem. The latter forbids the use of quantum nonlocality as a d i i t communication channel for what Einstein called "spooky telepathic action at a distance" in violation of the retarded causality postulate of the classical theory of special relativity. The modem theory of quantum computing, cryptography and teleportation would fall apart if the signal-locality of orthodox quantum theory could be violated. Yet, Feynrnan's phenomenological rules: add complex amplitudes before squaring for indistinguishable alternatives. Square amplitudes before adding for distinguishable alternatives. 22 Whether or not the real Shakespeare was really the brothers Francis and Tony Bacon at the Scriptorium in London with Ben Johnson (Sirag "Shakespeare's doublet reversed in First Folio picture") not the issue. 23 Wheeler refuting "many worlds" he once endorsed. 24 August 9000, Scientific American "The Universe's Unseen Dimensions" p.62, N. Arkani-Hamed et-aL 25 Literally attached to a material configuration like the private mind of an individual living brain. 26 P.30 and Ch. 14 in "The Undivided Universe" (1993) 27 We can call this the "'actuary's limit" of sub-quantal equilibrium of the nonlocal hidden variables, AKA "subquanta1 heat death" in Antony Valentini's Ph.D. dissertation unda Dennis Sciama at Cambridge. 2'

POST-QUANTUM PHYSICS AND UNIFED FIELD THEORY

423

this is precisely what the human mind does when it experiences and knows. The quantum potential Q characterized by action without reaction is "fragile".28 It is this Vragility" that maintains irreducible uncontrollable local quantum randomness29even in nonlocally entangled systems. It is not possible to control a quantum probability at a distance in orthodox quantum theory. Yet, this is precisely what happens in the brain, indeed, in the entire living body beyond the neural transmission of electrical signals and the transport of chemical messenger molecules as important as they are. The pilot Geld of the Bohm-Vigier "causal theory", with deterministic particle trajectories, is an "absolute physical object"3o. Therefore, this pilot wave is just like Newton's absolute space and absolute time before special relativity and just like the absolutely flat spacetime of special relativity before Einstein introduced the direct back-reaction of matter on spacetime geometry to bend it into gravity. The unified field theory goes further twisting the spacetime geometry and changing the topology of the 3D membranes to create and destroy traversable wormholes as one example. Why does quantum theory work so well? 1 propose the following model. Imagine two barriers. One barrier is for the action of the quantum pilot field on the matter it is piloting of height hper qubit31.The other barrier is for the reaction or "backaction" of the matter on its pilot field. When the quantum action is balanced in strength by this new post-quantum reaction, one forms a self-organizing feedback control loop between pilot field and its matter suppressing quantum randomness with "signal nonlocality". Orthodox quantum theory is strongly violated in this situation. The BohmVigier "causal theory" breaks down completely in this new regime. The particle paths are no longer deterministic, rather, they are self-determining . The entangled individual particle paths, inseparable in configuration space, exhibit strangely synchronized motions beyond the local contact forces from classical signal connections when viewed in ordinary space. This is the essence of the "self in biological life. Indeed, the particle paths co-evolve with the changing shape of the landscape formed by the now postquantum potential Q*. Both the particle paths and the shape of the landscape they ride on in configuration space are tweaked by external perturbations from the non-self environment. There is a further mathematical generalization in that the co-evolutionary flow of the nonlocally entangled particle paths on the landscape of their common pilot field is no longer an integrable holonomic path-independent flow characterized by exact differentials. One now has a more complex dynamically changing self-determining topology of closed inexact differential nonintegrable anholonomic flows on the landscape. Indeed one must use the Pfaffian theory of nonintepbility of R. Kiehn32
Bohm and Hiley's term in "The Undivided Universe". The late Heinz Pagels, killed in a way he foresaw in a dream, discusses how quantum randomness prohibits the use of nonlocality as a direct communication channel in 'The Cosmic Code". This book closes with Pagels' strange precognitive dream of his then future death. W "On the Ether" Albert Einstein, 1924 31 A "qubit" is the basic unit of quantum information analogous to the Shannon "c-bil" of classical information associated with negative thermodynamic entropy. The qubit is a relative phase coherent two-state quantum system. The spin of a single electron forms a qubit. A single hydrophobically-caged electron inside the protein dimer molecules tiled around subneuronal microtubule (e.g. Stuart Harneroff's papers) forms the qubit we are specifically interested in. 32 See Kiehn's paper in these proceedings.

J. SARFATTI

424

topological irreversibility33 constituting a self-organizing process of an open system far from thermodynamic equilibrium with memory and learning. I postulate the Ansatz that the height of the reaction barrier of matter back on its pilot qubit information field is

where H is the Hubble cosmological parameter whose dependence on global cosmic time differs for different solutions of the Einstein field equations.34 This Ansatz is interim since the Einstein field equations need to be changed to include the anholonomic unified field from the nonsyrnmetric connection for parallel transport of vectors around closed loops in the deformed spacetime manifold. These closed loops in the manifold project into broken loops with translational gaps in the flat tangent spacetime fiber erected at the common beginning and end of the closed loop in the manifold.35 N is the number of coherently entangled qubits forming a nonlocally connected network of qubits, each of rest mass m and electric charge e, of N nodes. We have post-quantum signal-nonlocality so that the local motion of a single node in the intelligent net is not random, but is synchronized with the simultaneous motion of the other nodes in the approximation of their common rest frame. Galilean relativity works fine here since the dimensionless velocity parameter for each node obeys v/c <<1. The height Abockaction of the backaction barrier is normalized pa. qubit node. The he2 in the denominator is the coherence complexity factor lowering the height of the reaction barrie?6. The fine structure constant a is the strength of the lowest order Feynman diagram connecting two electron nodes with the exchange of a virtual longitudinally polarized near field photon. One needs N(N - 1)/2 virtual photons in the same field oscillator mode to phase coherently connect all N nodes together to lowest order Feynman diagram expansion in the N node many-body problem. I make the second Ansatz that this quantum coherent non-radiating near field is the 40 Hz Crick brain field in the specific application of this model to the the actual living human brain. Since the virtual photons are "off the mass 3 equation w = kc does not apply to them. The Crick near electric field ' "enslaves"39 the N electron qubit nodes into a coherently-phased interferometric array across the whole cortex forming the brain hologram of Bohm and Pribram. Impose the action-reaction principle, in this case the heights of the two barriers are equal forming a resonant two way feedback control loop in contrast to the one way action of orthodox quantum theory

-

Pfaffians of degree 3 and 4 in the Cartan theory of differential forms. N I the backaction barrier is much higher than the action barrier. So quantum theory works. 35 Spin and Torsion in Gravitation" pp. 10, 11 Figs. 1,2,3 V. de Sabbata & C. Sivaram, World Scientific (1994) 36 This coherent lowering of the barrier also applies to Schwinger's theory of cold fusion. 37 For photons, the mass shell Fourier tmsformed to spacetime is the classical light cone. Virtual photons are off the light cone either inside it or outside it. The superluminal virtual longitudinal photons outside the classical light cone dominate the static near field Coulomb force between pairs of charges. Far field radiation of real transverse polarization plays a small role here. The near electric and magnetic induction fields are what are biophysically most significant. 38 'The Astonishing Hypothesis", F. Crick, Scribner's , 1994. 39 "Principles of Brain Functioning", 4.2.2,4.2.3,4.2.4 H. Haken, Springer-Verlag, 1996. 33

34 When '6

-

POST-QUANTUM PHYSICS AND U N F E D FIELD THEORY

425

with uncontrollable randomness from a "fragile" integrable quantum potential field without memory. The nonintegrable post-quantum field, with both memory from the past and "presponse"40 from the future, is robust stabilized by the action-reaction loop.

for the critical nonlocal entanglement complexity needed to obey the action-reaction principle. Stuart Hameroff http://www.consciousness.arizona.eduamoff/ informed me that this is the correct order of magnitude4' for the total number of hydrophobicallycaged electrons in the human brain. Furthermore. from the simple post-quantum model of Bohm and ~ i l thee duration ~ ~ of ~ a single undivided moment of self-organization43is

One must erase the configuration of a billion billion qubits every 0.16 seconds in order to form the Jamesian "stream of consciousness". The required power dissipation to accomplish this is

ergs = 4.5 x 10-~ w m

a 4.5 x lo5

sec

The resting adult human body metabolizes at about 100 Watts, so this is a small power consumption to generate consciousness in this model. What is unique about my model

AKA "presentiment""The Conscious Universe", Dean Radin, Harper Edge, 1997 replicated by Dick Bierman. 4' A billion billion electron qubits. 42 'The Undivided Universe" 14.3. 14.6. Bohm & Hiley op-cit 43 I interpret this as Whitehead's "occasion of experience" i.e. an undivided conscious moment in the intrinsically mental quantum information pilot field.

426

J. SARFATTI

here44 is that human consciousness has a cosmological origin4' in a kind of mental version of Mach's Principle compatible with the Hoyle-Naxlikar "future response of the universe" 4 6 ~ i t hbackward causation based upon a generalization of the WheelerFeynrnan conjecture.

Part 11: Progress In Classical Unified Field Theory This is the bLparticle"part of the dBBV ontological "wave-particle duality". As 1 show below in some detail, Vigier's idea that elementary particles have extended spatial structure in which the center of charge is displaced fmm the center of mass with a finite rest mass of the photon in a superconducting Dirac "aether" of correlated virtual electronpositron pairs is getting increasing experimental confirmation. Bo ~ e h n e r has t ~ ~shown several electromagnetic anomalies such as an effective electric charge density in the classical vacuum needed to explain observed data. 1 suspect that all of Bo Lehnert's observations can be adequately explained by Corum's anholonomic

a:, field4"uation

F', = 5-%+ 2*iVAA axv 'xa Where Ffi, is the Maxwell electromagnetic field tensor and Afi is the electromagnetic 4potential now a local classical observable. The equivalence principle used by Einstein in 1915 only applied to the nonuniform holonomic translational motion with nonrotating local noninertial frames. The use of a nonsymmetric connection comes from extending the equivalence principle to nonuniform anholonomic rotations of local noninertial ~rarnes.'~ Corum's equation is not U(1) gauge invariant. One can make it gauge invariant ' quantization of electric charge from the with a minimal coupling using ~ i r a c ' s ~string magnetic monopole.

44

Compared to Stapp's or Penrose's. connection of the Hubble parameter to the mass of the spatially extended electron has also been noted by R. I. Gray op-cit who derives H"= (4r/3)2(ac/~m),where r = e2/mc2 46 'The Intelligent Universen, Fred Hoyle, Holl Rinehart&Winston (1983). 47 See contribution to this conference. Lehnert is at the Royal Institute of Technology in Stockholm. 48 his third rank tensor field under holonomic Diff(4) gauge symmetry of 1915 general relativity is the antisymmetric part of the connection field for nonintegrable parallel transport of vectors along paths in the manifold. J-Math Physics, 18,4, pp. 770-776, 1977 "Relativistic rotation and the anholonomic object" James FCorm 49 e.g. Gennady Shipov "A Theory of Physical Vacuum" Moscow, 1998 and V l a d i i r Poponin's paper in these proceedings. O'Treometry,Particles and Fields" pp 18,391,491, Bjorn Felsager, Springer-Verlag, 1998. 45 The

"

POST-QUANTUM PHYSICS AND UNIFIED FIELD THEORY

Where Pa is the mechanical 4-momentum and n is a topological winding number. This means that one must violate holonornic Diff(4) gauge symmetry when generating the anholonomic C2:, field in order to preserve the internal UU) gauge symmetry of the electromagnetic force. Kleinert's theory (http://www.physik.fu-berlin.de/-kleiner) of "super-tetrads" for the unified field beyond Einstein's 1915 theory is required. Eq. (7) should also explain the Blackett-Sira~~ f f e c t for ~ ' rotating neutral astronomical objects that show an anomalous magnetic moment (http://stardrive.org/Jack/blackettl.pdf). Sirag's empirical data plot for the magneto-gyroscopic coefficient obeys

The Kerr-Newman solution for a rotating charged black hole has a non-radiating solution with zero Hawking radiation when the Pythagorean theorem is obeyed in control parameter space52.i.e..

51

"Gravitational Magnetism" Nature, Vol. 278,535, April, 1979. 'The gravi-magnetic hypothesis is that a rotating mass," [electrically neutral] -measured in gravitational units (M=G" m) has the same magnetic effect as that of an electrostatic charge, measured in esus moving at the same angular velocity at the same distance." 4-1 "Unified Physics", R. 1. Gray, O.B.E.. in-house report. 1988 from Naval Surface Warfare Center, Dahlgren, VA. This also implies EM radiation from accelerating neutral matter. Also p. 459 eq. (1 1.2.4) ref 1 on Schwinger's classical "dyon". 52 In the sense of Rene Thom's "catastrophe theory" and V.I. Arnold's "singularity theory". Folds and other kinds of controllable catastrophes in the dynamic 3D membrane embedded in 1Od hyperspace are expected.

J. SARFATTI

T is the string tension [energy/length] and of the dilaton field. Changing string tension

T,.

(4)

(4) is the dimensionless vacuum expectation

changes the effective gravity parameter G and the

I conjecture that this right triangle constraint must be generalized to

the general triangle constraint

with torsion in the cross-term for a unified field generalization ofthe orthodox black hole and wormhole solutions beyond Einstein's symmetric 1915 general relativity. Model the bare electron as a non-radiating wormhole mouth with charge &=11.7e=

1 1 . 7 ~ 4 . 8 ~ =5.6xl0-~gesu 16~~ andbaremass

.

. \ l h c / a ~= 11.7 x 2.18 x 10" gm The magneto-gyroscopic coefficient is then

JG @

= = 11.7 . In other Therefore the bare mass that fits the data in (8) is M/ words, the magneto-gyroscopic measurements of astronomical objects act as a ''telescope" directly into the quantum gravity era! This is consistent with Vigier's notion of the spatially extended bare elementary particle in a superconducting virtual dressed electron-positron screening plasma in the quantum vacuum that is not dragged along with the rotating bare core due to zero superfluid viscosity. We have additional empirical astronomical data of P.S. wessonS4

-

not completely bald when we add torsion. Wheeler had J M*in 1955 ''Geon" paper. 54~hys. Rev. D, 23, 8, 1730, April 1981 "Clue to the Unification of Gravitation and Particle Physics". P.S. Wesson. 53 The black hole is

POST-QUANTUM PHYSICS AND UNIFIED FIELD THEORY

429

that fits my 1973~'association of Regge trajectories to tiny non-radiating rotating black holes with strong short-range gravity. The hadronic resonances of the strong quantum chromodynamic SU(3) color force have phadrOn * l ~ e v ". In contrast (12) is -36

-2

- 36

.

Note that the new parameter p. not powers of 10 flatter. i.e. pastraw 10 Gev found in the 1915 Einstein general relativity, has the dimensions of vorticity flux per unit mass for the circulation of the aether flows. Therefore, this shows a strong presence of the anholonomic unified field at all scales of the universe from planets to pulsars to galaxies to clusters and superclusters. The Blackett-Sirag data together with the Wesson data rank in equal importance to the data on Hubble's law for the cosmological redshift and the isotropic cosmic microwave blackbody background radiation. Wesson points out the strong diensionless anholonomic unified field coupling strength parameter and conjectures from his data that

This corresponds to the mass scale

closer to the GUT unification compared to the Sirag-Blackett mass scale of 11.7.~~ The Blackett-Sirag mass scale is roughly dual, in the sense of superstring M-theory, to the Wesson mass scale using the Planck mass as the standard. Therefore, I conclude that the empirical evidence from both Blackett-Sirag and Wesson are effective "telescopes" down to Wheeler's "quantum foam" at the Planck scale contirming M-theory qualitatively. This shows that bare matter is made from non-radiating spatially-extended rotating charged wormholes whose gyrornagnetic properties decouple from the zero viscosity superconducting virtual dressed electron-positron Dirac-Vigier quantum vacuum.57 55

"The Eightfold Way as a Consequence of the General Theory of Relativityn Collective Phenomena, 1, 1974 (edited by H. Frohlich and F.W. Cummings), & "Speculations on Gravitation and Cosmology in Hadron Physics", Collective Phenomena, pp 163-167 (1973); "Quantum Mechanics as a Consequence of General Relativity" IC/74/9 International Centre Theoretical Physics, Trieste, Italy; "Gravitation, Strong Interactions and the Creation of the Universe", Nature-Physical Science (December 4, 1974) "The Primordial Proton", Physics Today letter (May, 1974) 69 also Andrew Salthouse "Is Symmetry Breaking in SU(3) a Consequence of General Relativity", UM HE 73-29 cites my work and fits nuclear data to it. "Space-Time and Beyond" p. 168 (Dutton. 1975) also strong finite range gravity to scale 1 micron p. 129. 137. 56

Not quite superstring dual R + I / R,but the data is not accurate enough yet. Aether in Relativistic Quantum Mechanics", J.P.Vigier, N.C. Petroni, Fdn. Physics, 13, 2, 29 (1983).

57 "Dirac's

J. SARFATTI

Acknowledgements Saul-Paul Sirag, Creon Levit, Vladimir Poponin, Axel Pelster, Bo Lehnert, James Corum, Tony Smith, R. Kiehn, Henry Stapp, Stan Klein, and Hal hthoff have made valuable contributions to this work in private conversations and e-mail. All errors and confusions are solely mine.

POLARIZABLEVACUUMAPPROACH TO GENERAL RELATIVITY H. E. PUTHOFF Institute for Advanced Studies at Austin 4030 W. Braker Lane, Suite 300, Austin, Texas 78759

Abstract Topics in general relativity (GR) are routinely treated in terms of tensor formulations in curved spacetime. An alternative approach is presented here, based on treating the vacuum as a polarizable medium. Beyond simply reproducing the standard weak-field predictions of GR, the polarizable vacuum (PV) approach provides additional insight into what is meant by a curved metric. For the strong field case, a divergence of predictions in the two formalisms (GR vs. PV) provides fertile ground €or both laboratory and astrophysical tests.

1. Introduction The principles of General Relativity (GR) are generally formulated in terms of tensor formulations in curved spacetime. Such an approach captures in a concise and elegant way the interaction between masses, and their consequent motion. "Matter tells space how to curve, and space tells matter how to move [l]." During the course of development of GR over the years, however, alternative approaches have emerged that provide convenient methodologies €or investigating metric changes in other formalisms, and which yield heuristic insight into what is meant by a curved metric. One approach that has intuitive appeal is the polarizable-vacuum (PV) approach [2-31. The PV approach treats metric changes in terms of equivalent changes in the permittivity and permeability constants of the vacuum. E, and p,, essentially along the lines of the so-called "TH8p" methodology used in comparative studies of gravitational theories [46]. In brief, Maxwell's equations in curved space are treated in the isomorphism of a polarizable medium of variable refractive index in flat space PI; the bending of a light ray near a massive body is modeled as due to an induced spatial variation in the refractive index of the vacuum near the body; the reduction in the velocity of light in a gravitational potential is represented by an effective increase in the refractive index of the vacuum, and so forth. As elaborated in Refs. [3-7, PV modeling can be carried out in a self-consistent way so as to reproduce to appropriate order both the equations of GR, and the match to the classical experimental tests of those equations. Under conditions of extreme metric perturbation, however, the PV approach predicts certain results at 43 1 R.L. Anloroso et a1 (eds.,).Gravitation and Cosnrology: From the Hilbble Radiris B I the Planck Scale, 431-446. O 2002 Klu~verAcadernic Publi.sIzers. Printed in the Net/zerl~nds

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variance with the standard GR approach. We discuss these variances in terms of testable implications. both in the laboratory and with regard to astrophysical consequences.

2. The Polarizable Vacuum The electric flux vector

in a linear, homogeneous medium can be written

where E and EO are the permittivities of the medium and of the vacuum respectively. and the polarization corresponds to the induced dipole moment per unit volume in the medium whose polarizability per unit volume is av. The identical form of the last two terms leads naturally to the interpretation of E~ as the polarizability per unit volume of the vacuum treated as a medium in its own right. This interpretation is explicitly corroborated in detail by the quantum picture of the vacuum where it is shown that the vacuum acts as a polarizable medium by virtue of induced dipole moments resulting from the excitation of virtual electron-positron pairs [8]. To represent curved-space conditions, the basic postulate of the PV approach is that the polarizability of the vacuum in the vicinity of a mass (or other mass-energy concentrations) differs from its asymptotic far-field value by virtue of vacuum polarization effects induced by the presence of the mass. That is, we postulate for the vacuum itself

where K is the (altered) dielectric constant of the vacuum (typically a function of position) due to (GR-induced) vacuum polarizability changes under consideration. Throughout the rest of our study the vacuurrz dielectric constant K constitutes the key variable of interest.

2.1. VELOCITY OF LIGHT IN A VACUUM OF V A W L E POLARIZABILITY In this section we examine quantitatively the effects of a polarizable vacuum on the various measurement processes that form the basis of the PV approach to general relativity. We begin by examining a constraint imposed by observation. An appropriate starting point is the expression for the fine structure constant,

a=

e2

, where

4 a ~ ~ h c

c=

&z 1

By the conservation of charge for elementary particles, and the conservation of angular momentum for a circularly polarized photon propagating through the vacuum (even with variable polarizability), e and can be taken as constants. Given that E, can be expected with a variable vacuum polarizability to change to aK) = Kgo, and the vacuum

433

POLARIZABLE-VACUUM permeability may be a function of K, d K ) , c ( K ) = I/,-, constant therefore takes the form

The fine structure

which is potentially a function of K. Studies that consider the possibility of the variability of fundamental constants under varying cosmological conditions, however, require that the fine structure constant remain constant in order to satisfy the constraints of observational data [9-111. Under this constraint we obtain from Eq. (4) p(K) = KcI,; thus the permittivity and permeability constants of the vacuum must change together with vacuum polarizability as1

As a result the velocity of light changes inversely with K in accordance with

Thus, the dielectric constant of the vacuum plays the role of a variable refractive index under conditions in which vacuum polarizability is assumed to change in response to GRtype influences. As will be shown in detail later. the PV treatment of GR effects is based on the use of equations that hold in special relativity, but with the modification that the velocity of light c in the Lorentz transformations and elsewhere is replaced by the velocity of light in a medium of' variable refractive index, c/K. Expressions such as E = mc2 are still valid, but now take into account that c --,c X , and E and m may be functions of K, and so forth.

2.2. ENERGY IN A VACUUM OF VARIABLE POLAREABILITY Dicke has shown by application of a limited principle of equivalence that the energy of a system whose value is Eo in flat space ( K = 1) takes on the value

This transformation, which maintains constant the ratio

@= -/= the impedance of

free space) is just what is required to maintain electric-to-magnetic energy ratios constant during adiabatic movement of atoms from one point to another of differing vacuum polarizability [3]. Detailed analysis shows that it is also a necessary condition in the THsp formalism for an electromagnetic test body to fall in a gravitational field with a composition-independent acceleration WEP, or weak equivalence principle, verified by Eotvos-type experiments) 14-61. Finally, this condition must be satisfied by any metric theory of gravity, which constitutes the class of viable gravity theories.

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in a region where K 1 [3]. This is due to that fact that the self-energy of a system changes in response to changes in the local vacuum polarizability, analogous to the change in the stored energy of a charged air capacitor during transport to a region of differing dielectric constant. The energy relationship given by Eq. (7) also implies, via Eo = moc2,which a corollary change in mass becomes E(K) = rn(~)(c/fC)~,

again a consequence of the change in self-energy. 2.3. ROD AND CLOCK (METRIC) CHANGES IN A VACUUM OF VARIABLE POLARIZABILITY Another consequence of the change in energy as a function of vacuum polarizability is a change in associated frequency processes which, by the quantum condition E = w and Eq. (7), takes the form

This, as we shall see, is responsible for the red shift in light emitted from an atom located in a gravitational potential. From the reciprocal of Eq. (9) we find that time intervals marked by such processes are related by

Therefore, in a gravitational potential (where it will be shown that K > 1) the time interval between clock ticks is increased (that is, the clock runs slower) relative to a reference clock at infinity. With regard to effects on measuring rods, we note that, for example, the radius of the ground-state Bohr orbit of a hydrogen atom

becomes (with c

+c/K, mm,+m, and a constant as discussed earlier)

Other measures of length such as the classical electron radius or the Compton wavelength of a particle lead to the relationship Eq. (12) as well, so this relationship is general. This dependence of fundamental length measures on the variable K indicates that the dimensions of material objects adjust in accordance with local changes in vacuum

POLARIZABLE-VACUUM

435

polarizability - thus there is no such thing as a perfectly rigid rod. From the standpoint of the PV approach this is the genesis of the variable metric that is of such significance in GR studies. We are now in a position to define precisely what is meant by the label "curved space." In the vicinity of, say, a planet or star, where K > 1, if one were to take a ruler and measure along a radius vector R to some circular orbit, and then measure the circumference C of that orbit, one would obtain C < 2.R (as for a concave curved surface). This is a consequence of the ruler being relatively shorter during the radial measuring process (see Eq. (12)) when closer to the body where K is relatively greater, as compared to its length during the circumferential measuring process when further from the body. Such an influence on the measuring process due to induced polarizability changes in the vacuum near the body leads to the GR concept that the presence of the body "influences the metric," and correctly so. Of special interest is the measurement of the velocity of light with "natural" (i-e., physical) rods and clocks in a gravitational potential which have become "distorted" in accordance with Eqns. (10) and (11). It is a simple exercise to show that the measured velocity of light obtained by the use of physical rods and clocks renormalizes from its "true" (PV) value c/K to the value c. The PV formalism therefore maintains the universal constancy of the locally measured velocity of light. 2.4. THE METRIC TENSOR At this point we can make a crossover connection to the standard metric tensor concept that characterizes the conventional GR formulation. In flat space a (4-dimensional) infinitesimal interval is given by the expression

If rods were rigid and clocks non-varying in their performance in regions of differing vacuum polarizability. then the above expression would hold universally. However, a dxo-length measuring rod placed in a region where K > 1, for example. shrinks according to Eq. (12) to dx = d x o / d ~ . Therefore, the infinitesimal length which would measure dxo were the rod to remain rigid is now expressed in terms of the &-length rod as dxo = f i d x . (Such constitutes a transformation between "proper" and "coordinate" values.) With a similar argument based on Eq. (10) holding for clock rate, Eq. (13) can be written

Therefore, the infinitesimal interval takes on the form

where gij in the above expression defines the metric tensor, and

H.E. PUTHOFF

The metric tensor in this form defines an isotropic coordinate systenl, familiar in GR studies.

3. Classical Experimental Tests of General Relativity in the PV Model In the previous sections we have established the concept of the polarizable vacuum and the effects of polarizability changes on metric (rods and clocks) behavior. In particular, we found that metric changes can be specified in terms of a single parameter K, the dielectric constant of the vacuum. This is the basis of the PV approach to GR. In this section we note. with the aid of expressions to be derived in detail in Section 5, how K changes in the presence of mass, and the effects generated thereby. The effects of major interest at this point comprise such classical tests of GR as the gravitational redshift, the bending of light and the advance of the perihelion of Mercury. These examples constitute a good testbed for demonstrating the techniques of the PV alternative to the conventional GR curved-space approach. For the spherically symmetric mass distribution of a star or planet it will be shown later from the basic postulates of the PV approach thal the appropriate PV expression for the vacuum dielectric constant K is given by the exponential form

where G is the gravitational constant, M is the mass, and r is the distance from the origin located at the center of the mass M. For comparison with expressions derived by conventional GR techniques, it is sufficient to restrict consideration to a weak-field approximation expressed by expansion of the exponential to second order as shown. As an example of application of the PV approach to a standard experimental test of GR, we consider the case of gravitational redshift. In a gravitation-free part of space, photon emission from an excited atom takes place at some frequency rue, uninfluenced by vacuum polarizability changes. That same emission process taking place in a gravitational field, however, will according to Eq. (9), have its emission frequency altered (redshifted) to ru = mo/fi . With the first-order correction to K = 1 given by the first two terms in Eq. (17), emission by an atom located on the surface of a body of mass M and radius R will therefore experience a redshift by an amount

where we take GM/R$ K 1. Once emitted, the frequency of the photon remains constant during its propagation to a relatively gravitation-free part of space where its frequency can then be compared against that of local emission, and the spectral shift given by Eq.

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437

(18) observed. Measurement of the redshift of the sodium D, line emitted on the surface of the sun and received on earth has verified Eq. (18) to a precision of 5% [12]. Experiments carried out on the surface of the earth involving the comparison of photon frequencies at different heights have improved the accuracy of verification still further to a precision of 1% [13-141. With the two ends of the experiment separated by a vertical height h, the first-order frequency shift is calculated with the aid of Eqns. (9) and (18) as

where M and R are the mass and radius of the earth. This experiment required a measurement accuracy of ~ ' - 1m0 - l ~for a height h = 22.5 meters. It was accomplished by the use of Mossbauer-effect measurement of the difference between y-ray emission and absorption frequencies at the two ends of the experiment. In similar fashion, we could consider in detail other canonical examples such as the bending of light rays or perihelion advance of planetary orbits near a mass. However, rather than treating these cases individually, we can take a more general approach. In standard textbook treatments of the classical tests of GR one begins with the Schwarzschild metric, which in isotropic coordinates is written [15]

Expanding the metric tensor for small departures from flatness as a Maclaurin series in ( ~ ~ / r we c ~obtain ) ,

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Similarly, in the PV approach one begins with the exponential metric defined by Eqns. (14) - (17),

This, when expanded to the same order as the Schwarzschild metric tensor above for small departures from unity vacuum dielectric constant, yields -2

goo = e

~dvc~ =1-2(5)+2(2) GM -... ,

Comparison of Eqns. (24) - (25) with Eqns. (21) - (22) reveals that, to the order of expansion shown, the two metric tensors are identical. Since the classical tests of GR do not require terms beyond these explicitly displayed, the agreement between theory and experiment is accounted for equally in both the conventional GR and in the alternative PV formalisms. For a charged mass the Schwarzschild metric is replaced by the ReissnerNordstr4m metric in the GR approach, while in the PV approach the exponential metric is replaced by a metric involving hyperbolic functions (see Section 6.2). Again, for the weak-field case. it can be shown that the two approaches are in precise agreement to the order shown in the charge-free case. 4. Coupled Matter-Field Equations

In the preceding section we have seen that the classical tests of GR theory can be accounted for in the PV formalism on the basis of a variable vacuum dielectric constant, K. To carry that out we stated without proof that the appropriate mathematical form for the variation in K induced by the presence of mass is an exponential form. In this section we show how the exponential form is derived from first principles. and. in the process. establish the general approach to the derivation of field equations as well as the equations for particle motion. The approach consists of following standard Lagrangian techniques as outlined, for example, in Ref. 16, but with the proviso that in our case the dielectric constant of the vacuum is treated as a variable function of time and space. 4.1. LAGRANGIAN APPROACH

The Lagrangian for a free particle is given by

POLARIZABLE-VACUUM

which, in the presence of a variable vacuum dielectric constant aid of Eqns. (6) and (8) to read

439

K, is modified with the

This implies a Lagrangian d e n s i for ~ the particle of

Following standard procedure, the particle Lagrangian density can be extended to the case of interaction with electromagnetic fields by inclusion of the potentials

(QA)

The Lagrangian density for the electromagnetic fields themselves, as in the case of the particle Lagrangian, is given by the standard expression (see, e-g., Ref. 16), except that again K is treated as a variable,

We now need a Lagrangian density for the dielectric constant variable K, which, being treated as a scalar variable, must take on the standard Lorentz-invariant form for propagational disturbances of a scalar,

where f(K) is an arbitrary function of K. As indicated by Dicke in the second citation of s we take A= c4/32m~andflK) = 1/if2; Ref. 3. a correct match to experiment r e q u i ~ that thus,

-

H.E. PUTHOFF

-

We can now write down the total Lagrangian density for matter-field interactions in a vacuum of variable dielectric constant,

Ld

- [ ~ ~ ~ + C - q h . v ) n ' ( . -2 ~KPO - - ? [ ~ - K ~ o E ' )

[ K2 (

K

(c

/K)'

(33)

(%)I].

4.2. GENERAL MAITER-FIELD EQUATIONS

I

Variation of the Lagrangian density 6 Lddxdydzdt with regard to the particle variables leads to the equation for particle motion in a variable dielectric vacuum,

We see that accompanying the usual Lorentz force is an additional dielectric force proportional to the gradient of the vacuum &electric constant. This term is equally effective with regard to both charged and neutral particles and accounts for the familiar gravitational potential, whether Newtonian in form or taken to higher order to account for GR effects.' Variation of the Lagrangian density w i t . regard to the K variable leads to an equation for the generation of GR vacuum polarization effects due to the presence of to matter and fields. (In the final expression we use v2K = (1 /~K)(vK)' + 2&v2& obtain a form convenient for the following discussion.)

Of passing interest is the fact that as nao

+0, but v +c/K, the deflection of a zero-mass particle

(e.g., a photon) in a gravitational field is twice that of a slow-moving particle (V important result in GR dynamics.

-+ O),

an

POLARIZABLE-VACUUM

Thus we see that changes in the vacuum dielectric constant K are driven by mass density (first term), EM energy density (second term), and the vacuum polarization energy density itself (third term). The fact that the latter term corresponds to the energy density of the K variable can be seen by the following argument. We start with the Lagrangian density Eq. (32), define the momentum density by = a ,$/a (8~ /t )a , and form the Hamiltonian energy density to obtain

Eqns. (34) and (35), together with Maxwell's equations for propagation in a medium with variable dielectric constant, thus constitute the master equations to be used in discussing matter-field interactions in a vacuum of variable dielectric constant as required in the PV formulation of GR 5. Static Field Solutions

We demonstrate application of field Eq. (35) to two static field cases with spherical symmetry: derivation of the expression introduced earlier for the gravitational field alone, and derivation of the corresponding expression for charged masses. 5.1. STATIC FIELDS (GRAVITATIONAL)

In space surrounding an uncharged spherical mass distribution the static solution (.K/.t = 0)to Eq. (35) is found by solving

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442

where we have u s e d ( v ~ )=~ 4 ~ ( v f i j. The solution that satisfies the Newtonian limit is given by

which can be verified by substitution into the equation for particle motion, Eq. (34). We have thus derived from first principles the exponential form of the variable dielectric constant in the vicinity of a mass as used in earlier sections. As indicated in Section 4, this solution reproduces to appropriate order the standard GR Schwarzschild metric predictions as they apply to the weak-field conditions prevailing in the solar system. 5.2. STATIC FIELDS (GRAVITATIONAL PLUS ELECTRICAL)

For the case of a mass M with charge Q we first write the electric field appropriate to a charged mass imbedded in a variable-dielectric-constant medium,

Substitution into Eq. (35) yields (for spherical symmetry)

where b2 = Q ~ G / ~ . ~ , C ~ . The solution to Eq. (39) as a Function of charge (represented by b) and mass (represented by a = GM/$)is given below. Substitution into Eq. (34) verifies that as r + this expression asymptotically approaches the standard flat-space equations for particle motion about a body of charge Q and mass M.

2

(For b2 > a the solution is trigonometric.) As noted earlier (Section 4), for the weak-field case the above reproduces the familiar Reissner-Nordstr4m metric [17].

POLARIZABLE-VACUUM 6. Strong-Field Tests

As noted in the Abstract, both the conventional and PV approaches to GR problems lead to the same results for small departures from flatness. For increasingly larger departuses from flatness. however. the two approaches. although initially following similar trends, begin to &verge with regard to specific mamtudes of effects. In the PV approach the solution for the static gravitational case yields a metric tensor that is exponential in form, in the conventional GR approach the somewhat more complex Schwarzschild solution. This discrepancy has shown up previously in other general curved-space approaches to GR as we1L3

6.1. ASTROPHYSICAL TESTS A major difference between the Schwarzschild (GR) and exponential (PV) metrics is that ~ prevents radially-directed the former contains an event horizon at R = ~ G M / Cwhich photons from escaping ("black holes"), whereas the latter has no such discontinuity (only increasingly "dark gray holes"). One consequence is that whereas the Schwarzschild solution limits neutron stars (or neutron star mergers) to -2.8 solar masses ( 2 . w because of black hole formation, no such constraint exists for the exponential metric. This raises the possibility that such anomalous observations as the enormous radiative output (-2M@c2, if isotropic) of the gamma ray burster GRB990123 [20] might be interpreted as being associated with collapse of a v e y massive star (hypernova), or the collision of two highdensity neutron stars [21]. The collection of ad tional astrophysical evidence of this and related genres would be useful in the search for discriminants between the standard GR and alternative PV approaches.

6.2. LABORATORY TESTS For small departures from flatness it is useful to express the generalized metric in terns of the PPN (parametrized post-Newtonian) form

where @=GM/rc2and a,P,y and 6 comprise the PPN parameters. For the case of a central mass, both the conventional Schwarzschild and PV-derived exponential solutions

Of special interest is the so-called Einstein-Yilmaz tensor form, in which Einstein's equations are modified by inclusion of the stress-energy tensor of the gravitational field itself on the R.H.S. of the equations, in addition to the usual matterlfield stress-energy [18]. The Yilmaz modification yields exponential solutions in the form derived here by means of the PV approach. The EinsteinYilmaz equations satisfy the standard experimental tests of GR, as well as addressing a number of mathematical issues of concern to general relativists, and are thus under study as a potentially viable modification to the original Einstein form [19].

di

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H.E. PUTHOFF

require by virtue of the classical tests of GR that a = P = y = 1. There exists, however, a predicted discrepancy with regard to the fourth parameter, 8, which is

3 / 4 (GR SchwarzschiId solution) 1 (PV exponential solution) ' As detailed in Ref. [Is], an argument has been put forward that the isotropy-of-mass experiments of Hughes et al. [22] and by Drever 1231, and the neutron phase-shift measurements of Collela et al. [24] yield a value 6 -- 1 f 10" . The data, analysis, and interpretation of such experiments provide yet further opportunities for discriminants between the standard GR and the alternative PV approaches. 7. Discussion

In overview, we have shown that a convenient methodology for investigating general relativistic (GR) effects in a non-abstract formalism is provided by the so-called polarizable-vacuum (PV) representation of GR. The PV approach treats metric perturbation in terms of a vacuum dielectric function K that tracks changes in the effective permittivity and permeability constants of the vacuum, a metric engineering approach, so to speak [25]. The structure of the approach is along the lines of the THE, formalism used in comparative studies of gravitational theories. The PV-derived matter-field Eqns. (34)-(35) are in principle applicable to a wide variety of problems. This short exposition, covering but the Schwarzschild and Reissner-Nordstrflrn metrics and experimental tests of GR, is therefore clearly not exhaustive. Consideration was confined to cases of spherical symmetry and static sources: and important topics such as gravitational radiation and frame-dragging effects were not addressed. Therefore, further exploration and extension of the PV approach to specific problems of interest is encouraged, again with cross-referencing of PV-derived results to those obtained by conventional GR techniques to ensure that the PV approach does not generate incomplete or spurious results. With regard to the epistemology underlying the polarizable-vacuum (PV) approach as compared with the standard GR approach, one rather unconventional viewpoint is that expressed by Atkinson who carried out a study comparing the two [26]. "It is possible, on the one hand, to postulate that the velocity of light is a universal constant, to define 'natural' clocks and measuring rods as the standards by which space and time are to be judged, and then to discover from measurement that space-time, and space itself, are 'really' non-Euclidean; alternatively, one can define space as Euclidean and time as the same everywhere, and discover (from exactly the same measurements) how the velocity of light. and natural clocks. rods. and particle inertias 'really' behave in the neighborhood of large masses. There is just as much (or as little) content for the word 'really' in the one approach as in the other; provided that each is self-consistent, the However, it is known that the PV-related TH~papproachis sufficiently general that results obtained for spherically symmetric gravitational fields can be generalized to hold for nonsyrnmetric conditions as well.

THE INERTIA REACTION FORCE AND ITS VACUUM ORIGIN ALFONSO RUEDA Departtrzent of Electrical Engineering, ECS Building, California State University 1250 Bellflower Blvd. Long Beach, CA 90840, USA. E-mail: [email protected]

BERNARD HAISCH California Institute for Physics and Astrophysics 366 Carrrbridge Ave. Palo Alto, CA 94306. E-rrrai1:[email protected]

Abstract. By means of a covariant approach we show that there must be a contribution to the inertial mass and to the inertial reaction force on an accelerated massive object by the zero-point electromagnetic field. This development does not require any detailed model of the accelerated object other than the knowledge that it interacts electromagnetically. It is shown that inertia can indeed be construed as an opposition of the vacuum fields to any change to the uniform state of motion of an object. Interesting insights originating from this result are discussed. It is argued why the proposed existence of a Higgs field in no way contradicts or is at odds with the above statements. The Higgs field is responsible for assigning mass to elementary particles. It is argued that still the underlying reason for the opposition to acceleration that massive objects present requires an explanation. The explanation proposed here fulfills that requirement. Keywords:Quantum-vacuum,inertia-reaction-force

1. Foreword Among the several proposed explanations for the origin of inertia [I], we review in this article a recent one [2,3,4] that attributes the inertia reaction force to an opposition by the vacuum fields to accelerated motion of any real object that possesses mass. In his recent book, "Concepts of Mass in Contemporary Physics and Philosophy", Max Jammer [I] examines in detail many aspects of the mass concept including the origin of inertia. In particular, in reference to our recent work [2,3,4] he states [5]: 447 R.L. Aii~orosoet a1 (eds.),Gravitation and Cosn~dogyFrmz tlre Hzibble Rudus to tile Planck Scale.447-458. O 2002 Klutver AcudefnicPublisl~ers.Printed in tl~e Netherlairds

A. RUEDA & B. HAISCH "However, debatable as their theory still is, it is from the philosophicalpoint ofview a thought-provoking attenzpt to renounce the traditional priority of the notion ofnzass in the hierarchy of our conceptions of physical reality and to di-~pensewith the concept of r1las.s in favor of the concept [$field. In this respect their theory does to the Newtonian concept of mass what modem physics has done to the notion of absolute space: As Einstein [6] once wrote, 'the victory over the concept ofabsolute space or over that of the inertial systems becari~epossible only because the concept ofthe material object V V ~ P grduallj replaced as the fundamental concept of physics by that of the field. ' "

Here we outline an attempt to show that the inertia reaction force has a contribution from the vacuum electromagnetic field. It strongly suggests that other vacuum fields (weak, strong interactions) do also contribute. A recent proposal by Vigier [7] that there is also a contribution from the Dirac vacuum very much goes along this line. Most of the work outlined here appears in Refs. [2] and [3] (Those two papers we will refer to as RH, and specifically in most cases RH will directly refer to Ref. [2]). Another related paper prior to these is Ref. [4] (This one we denote by HRP). 2. General Comments

We highlight here the main aspects ofthe RH approach [2]. In general the RH analysis is carried out in two complementary but totally independent and comprehensive ways (in addition to the completely different original HRP approach [4] that we shall not discuss in this paper). In the first way (section 3), which is the more intuitive, one calculates the radiation pressure resulting from a non-zero Poynting vector of the electromagnetic zeropoint field (ZPF) as viewed by an accelerated object. This radiation pressure is exactly opposite to the direction ofthe imposed acceleration, and in the subrelativistic case turns out to be directly proportional to the magnitude of the acceleration. The second way (section 4) leads to this same results by showing why and how an accelerating object acquires its four-momentum. This turns out to be directly related to the amount of ZPF energy and momentum contained within the object: it is that fraction of the contained energy that interacts with the fundamental particles comprising the object. From any change in this four-momentum one can straightforwardly calculate the resulting inertia reaction force. Not surprisingly this proves to be exactly the same force as in the first case. In both representations one obtains an (electromagnetic) expression for the corresponding inertial mass that is essentially the amount of ZPF energy, divided by c2, instantaneously contained within an object and which actually interacts with the object. This mass is a factor of 413 too large in the First two (noncovariant) versions (sections 3 and 4) we present below. This requires a correction. The correct form then comes about from a fully covariant derivation that we briefly outline at the end (section 5). There we introduce an important contribution from the ZPF electromagnetic Maxwell stress tensor that was previously neglected.

INERTIA REACTION FORCE

3. The Inertia Reaction Force And The Electromagnetic Vacuum Radiation Pressure On An Accelerated Object We consider a small material object to be undergoing hyperbolic motion, i-e. uniformly accelerated motion, or motion with constant proper acceleration la(= a = constant. . .

Consider the object, that for simplicity we may identify with a particle, at the point (c2/a, 0,O) of a frame S that is rigid and noninertial because it comoves with the particle. At particle proper time z = 0 the particle point (c2/a, 0,O) of S exactly coincides and instantaneously comoves with the corresponding (c2/a, 0,O) point of an inertial Frame that we denote by I* and call the "inertial laboratory Frame." We take the direction of the particle acceleration vector to coincide with the positive x-direction in both S and I,, and in general in all subsequent frames that we will introduce. We consider also an infinitely continuous family of inertial Frames, each denoted by I , and such that each one of them has its (c2/a, 0,O) point instantaneously coinciding and comoving with the particle at the point (c2/a, 0,0) of S at the particle proper time z. Clearly then 1 = I,(z = 0). The acceleration of the particle ar (c2/a, 0,O) of S as seen from I, is a,

=

L;' a,. The frame S is

called the Ridler noninertial frame and as it is rigid, its acceleration is not the same for all its points but we will only be interested in points in the neighborhood of the (c2/a, 0,O) point. The particle undergoes well-known hyperbolic motion, in which the velocity of the particle point with respect to I, is then u, (z) = P,c, or

and then

The position of the particle in I* as function of particle proper time is then given by

and the time in I, when the proper time of the particle is z is

Observe that we select z = 0 at t

= 0.

A. RUEDA & B. HAISCH

450

We refer the reader to RH [2] for a detailed classically exact representation of the stochastic form of the electromagnetic field as well as for the details on the stochastic averaging. In what follows we will also omit a number of subtle points, fully discussed in App. C of RH [2], in which the reader may also find some useful intuitive analogies and a more thorough discussion of essential aspects ofthe arguments sketched here. According to an observer fixed to I, (say at the (c2/a 0,O) point of I,), the object moves through the ZPF as viewed in I, with the hyperbolic motion described above. At proper y

time z the object instantaneously comoves with the corresponding (c2/a, 0,O) point of the inertial frame 1, and thus at that point in time is found at rest in the inertial frame I,. We calculate the 1, frame Poynting vector, but evaluated at the (c2/ay 0,O) point o f,. This allows us to obtain the net ZPF rate of momentum density that is accumulating in the object due to the uniformly accelerated motion. Recall that this is the ZPF of I*. This Poynting vector we denote by

-? , and

where the angular brackets represent stochastic averaging. That only the x-direction is relevant follows from symmetry. There are several subtle points that we must sweep under the rug here, details of which may be found in RH [2], especially App. C. We refer particularly to the so-called k-spheres of integration. Each inertial frame has its own kspheres and even though the ZPF is Lorentz-invariant and it has the same form of energydensity spectrum and is homogeneous and isotropic in every inertial frame, the ZPF of I, does not appear to be that way to an observer of I, and vice versa.

In concise form we present the calculations as follows:

'E

YY*

+B2 +E2 +B2 z,*

z,*

y,*

Observe that in the last equality of eqn. (6) the term proportional to the xprojection of the ordinary ZPF Poynting vector of I, vanishes as it should. The integrals

451

INERTIA REACTION FORCE

are taken with respect to the I, ZPF background (using then the k-sphere of 1, ,of. App. C of RH [2]) as that is the background that the I, observer considers the accelerated object to be sweeping through. The net amount of momentum of the background that the object has swept through after a time t,, as judged again from the I, -frame viewpoint is then

whereg:',

the momentum density, is introduced and V, represents the volume of the object as seen in I,. Clearly then. because of Lorentz contraction. V+= Vo/yC,where Vois the proper volume. In obtaining eqn. (7) from eqn. (6) and for the following step we use the fact that

With the last two equalities of eqn. (8), eqn. (6) becomes

where we used the fact that V+= Vdy, and we introduced the frequency function q(u), where 0 s q(u) I 1 for all w. This represents the fraction of ZPF radiation that actually interacts with the object at a given frequency. Clearly we expect q(u) + 0 as w + co sufficiently rapidly. The force applied by the ZPF to the uniformly accelerated particle or physical object can now be easily calculated:

"'1 =-+[5o

f,"P= dptP --dt* Y , dt*

Am3 V(m)xdm]

a

A

where a = x a is the particle proper acceleration and where use was made of eqns. (1) and (2). We have thus obtained what can be called the ZPF inertia reaction force

with an "inertial mass" ofthe form

A. RUEDA & B. HAISCH

This is an invariant scalar with the dimension of mass. It represents the amount of ZPF radiation enclosed within the object (or particle) of proper volume Vothat actually interacts with it. In eqn. (12) we have omitted a factor of 413 that appears in eqn. (A10). One of the revenues of a fully covariant analysis (of. App. D of RH [2]),that we sketch below in 8 5, is to show that the 413 factor disappears when proper use is made of the electromagnetic stress tensor which has been omitted so far.

4. The Inertia Reaction Force And The Electromagnetic Vacuum Momentum Content Of An Accelerated Object This approach is totally independent from the one above. However, it is strongly complementary. It is "the other side of the coin." One approach requires the other. From Newton's Third Law applied to the accelerated object, when an external agent applies a motive force, f, and thus uniformly accelerates the object, according to the view proposed here the vacuum applies an equal and opposite force.

ffp,in the opposing direction.

i.e.

where the star subscripts just means that we refer here to the laboratory inertial frame I,. Eqn. (13) implies that the correspond'ing impulses, Ap* and - AP?, taken to be zero, say, at time 1, = 0 and z= 0, also obey after a short lapse of time At, and correspondingly a short proper time lapse 4,

Integrating over longer times in l*,from zero to some final time t*, we can write

This allows us to introduce an equation between the corresponding momentum densities where p, = V,g+ and p y = V, g : p , thus

INERTIA REACTION FORCE where we have already confronted gcp in eqn.

453

(3, and g, , as carefully argued in App. B

of RH [2], corresponds to the fraction of the momentum density of the ZPF radiation within the object that interacts with the object. Expressing this momentum density in terms of the corresponding Poynting vector we write

A

where

N, = x N, is the Poynting vector due to the ZPF as meawred in I* at the object's

point, at proper time T of eqns. (3) and (4), in the I, laboratory frame. Because of symmetry again. only the x-component appears. Recall however that we are calculating the ZPF momentum associated with the object. At proper time z the object is instantaneously at rest in the inertial frame I, This means (cf. App. C of RH [2]) that we must perform the integrals over the k-sphere of the I, frame. This becomes more revealing when we Lorentz-transform the field in eqn. (17) from I, to I,:

where again we have a term that vanishes, namely the one proportional to the xcomponent of the ZPF statistically-averaged Poynting vector in I, Recall that integrals are performed with respect to the k-sphere ofIf (App. C of RH [2]). We then have

where as in tj A1 we have used the fact that

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A. RUEDA & B. HMSCH

The prime in o' indicates just that frequencies and fields refer now to those in I, In eqn. (19) we again used the fact that V* =Vdyl and we introduced the factor ~(KJ') again because only the fraction of the ZPF radiation that actually interacts with the particles in the object is relevant. If we differentiate the impulse p, with respect to time we get the corresponding force, f, and with eqn. (13) we obtain the inertia reaction force due to the ZPF as

which reproduces eqn. (10) as should be expected. Now again, the formulae (11) and (12) follow accordingly.

5. A Covariant Approach Here we briefly sketch the argument and calculation that lead to the fully relativistic form ofthe inertia reaction force and that as a byproduct eliminates the bothersome 413 factor. In the previous sections only the contribution due to the momentum density g was included (or equivalently the Poynting vector N = 2g). There is however an additional contribution to the momentum p that was neglected. In order to obtain it, one has to perform a covariant generalization of the previous analysis. We cannot perform here a detailed account of it (see App. D of RH [2] for details). The analysis is a bit more easily grasped by the intuition if the momentum-content approach of 8 4 is used. For this reason we select in this section the momentumcontent approach. The covariant extension of momentum in the radiation field is the four-vector.

Where

We see that p now carries an additional term of the form T.v/c*, where T is the Maxwell stress tensor,

INERTIA REACTION FORCE

455 0

With i, j = x, y, z the angular brackets as usual signil-j stochastic averaging. In P the 0

zero component 9 we find again the energy density U (as in eqn. (8)), but there is now an additional contribution to the energy in the zero component, namely a term ofthe form -v.g. Since we are now considering a relatively small proper volume, Vb written in the I, frame we have

A detailed calculation of p, yields

and then

From this we may obtain the inertia reaction force of eqn. (21)

But now m,is precisely

and the factor 413 does not need to be artificially removed as in eqn. (10) to (12) and (21) where it was de facto removed. Of course in eqn. (30) we again introduced the factor Q(W) for the fraction of the ZPF that interacts with the particles in an object. On the other hand the zerocomponent of the momentum, PO, for sufficiently small proper volume Vocan be written as 0

('I "?)&

P =y, ---

Detailed calculation (App. D of RH[2]) yields then

A. RUEDA & B. HAISCH

Where mi is the mass found in eqn. (30). Putting together eqns. (28), (30) and (32) we recover the conventional form of the four-momentum in relativistic mechanics, namely,

With mi, the "inertial mass" of eqn. (30) that indeed represents the amount of electromagnetic ZPF energy inside the object volume V, that actually interacts with the object. From eqn. (33) we can then obtain the relativistic form of Newton's Second Law

where the star indices have been suppressedfor g e n e r a l i ~ And where p is the relativistic rltorltenturlt

The origin of inertia in this picture becomes remarkably intuitive. Any material object resists acceleration because the acceleration produces a perceived and instantaneous flux of radiation in the opposite direction that scatters within the object and thereby pushes against the accelerating agent. Inertia in the present model appears as a kind of acceleration-dependent electromagnetic vacuum-fields drag force acting upon electromagnetically-interactingparticles.

6. Discussion

In the Standard Model of particle physics it is postulated that there exists a scalar field pervasive throughout the Universe and whose main function is to assign mass by transferring mass to the elementary particles. This is the so-called Higgs field or more specifically, the Higgs boson, and it originated from a proposal by the British physicist Peter Higgs who introduced that kind of field as an idea for assigning masses in the Landau-Ginzburg theory of superconductivity. Recent predictions of the mass that the Higgs boson itself may have, indicate a rather large mass (more than 60 GeV) and this may be one ofthe reasons why, up to the present, the Higgs boson has not been observed. There are alternative theories that give mass to elementary particles without the need to postulate a Higgs field, as e-g., dynamical symmetry breaking where the H i boson is not elementary but composite. But the fact that the Higgs boson has not been detected is by no means an indication that it does not exist. Recall the 26 years which passed

INERTIA REACTION FORCE

457

between the proposal by Pauli in 1930 of the existence of the neutrino and its first detection when the Reines experiment was performed. It should be clearly stated that the existence (or non-existence) of the hypothetical Higgs boson does not affect our proposal for the origin of inertia. In the standard Model attempt to obtain, in John Wheeler's quote, "nrass without ntass" the issue of inertia itself does not appear. As Wilsczek [8] states concerning protons and neutrons: "Most of the ntass of ordinar?,ntatter, for sure, is the pure energy of nroving quarks and gluons. The rowainder, a quantitatively small but qualitatively crucial rmtainder - it includes the ntass of electrons - is all ascribed to the confounding influence of a pervasive nrediu)rt, the Higgs field condensate." An explanation of proton and neutron masses in terms of the energies of quarks motions and gluon fields falls short of offering any insight on inertia itself. One is no closer to an understanding of how this energy somehow acquires the property of resistance to acceleration known as inertia. Put another way, a quantitative equivalence between energy and mass does not address the origin of inertial reaction forces. And the manner in which, say the rest mass of the neutrino, is taken from the Higgs field, does not at all explain the inertia reaction force on accelerated neutrinos. Many physicists apparently believe that ow conjecture of inertia originating in the vacuum fields is at odds with the Higgs hypothesis for the origin of mass. This happens because of the pervasive assumption that inertia can only be intrinsic to mass and thus, if the Higgs mechanism creates mass, one automatically has an explanation for inertia. If inertia is intrinsic to mass as postulated by Newton, then inertia could indeed be considered to be a direct result of the Higgs field because presumably the Higgs field is the entity that generates the corresponding mass. and inertia simply comes along with mass automatically. However. if one accepts that there is indeed an extrinsic origin for the inertia reaction force, be it the gravity field of the surrounding matter of the universe (Mach's Principle in senso stricto) or be it the electromagnetic quantum vacuum(or more generally the quantum vacua) that we propose, then the question of how mass originates - possibly by a Higgs mechanism - is a separate issue from the property of inertia. This is a point that is often not properly understood. The modem Standard Model explanation of mass is satisfied if it can balance the calculated energies with the measured masses (as in the proton) but obviously this does not explain the origin of the inertia reaction force. It is the inertia reaction force associated with acceleration that is measurable and fundamental, not mass itself. We are proposing a specific mechanism for generation of the inertia reaction force resulting from distortions of the quantum vacua as perceived by accelerating elementary particles. We do not enter into the problems associated with attempts to explain inertia via Mach's Principle, since we have discussed this at length in a recent paper [9]: A detailed discussion on intrinsic vs. extrinsic inertia and on the inability of the geometrodynamics of general relativity to generate inertia reaction forces may be found therein. It had already been shown by Rindler [lo] and others that Mach's Principle is inconsistent with general relativity. and Dobyns et al [9] further elaborate on a crucial point in general relativity that is not widely understood: Geometrodynamics merely defines the geodesic that a freely moving object will follow. But if an object is constrained to follow some different path, Gwmetrodynamics has no mechanism for creating a reaction force. Geometrodynamics has nothing more to say about inertia than does classical Newtonian

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A. RUEDA & B. HAISCH

physics. Geometrodynamics leaves it to whatever processes generates inertia, to generate such a force upon deviation from a geodesic path, but this becomes an obvious tautology if an explanation of inertia is sought in Geometrodynamics. We would like to point out that Mach's Principle in senso strict0 is, as described above, the hypothesis that inertia here is due to the overall matter in the distant Universe that produces a net gravitational effect so that the inertia reaction force is generated on an accelerated object by the gravitational field of all the Universe. This Mach's Principle in a strict sense is not compatible with our proposal that inertia is generated by the vacuum fields [9]. However, a more broad interpretation of some of Mach's ideas is the view that inertia is notjust inherent to mass but due to an external agent that acts on the accelerated massive object. Such agent is different from the accelerated massive object itself and should reside in the external Universe. This view is then perfectly compatible with the view that we propose, namely that the vacuum fields are the entities responsible for producing, on the accelerated massive object, the inertia reaction force. We finally, acknowledge that Newton's proposal that inertia is intrinsic to mass looks, superficially at least, more economical (Occam's razor) but it is also oversimplistic as one may always continue asking for a deeper reason for the operation of physical processes or for more fundamental bases for physical laws. The question of why the mass associated with either matter or energy should display a resistance to acceleration is a valid question that needs to be addressed even if the Higgs boson is experimentally found and confirmed as the origin of mass.

Acknowledgement We acknowledge NASA contract NASW 5050 for support of this research. AR acknowledges additional support from the California Institute for Physics and Astrophysics (CIPA).

References 1. M. Jammer, "Concepts of Mass in Cor~ter)lporar?. Physics arid Philosophj" , Princeton University Press (2000)2. A. Rueda and B. Haisch, Foundations of Physics 2& 1057 (1998). 3. A. Rueda and B. Haisch. Physics Lett. A 240, 115 (1998). 4. B. Haisch, A. Rueda and H. E. Puthoff, Phys. Rev. A 49,678 (1994). 5. See pg. 166 of Ref. [I]. For a whole discussion see pp. 163-167 of Ref. [I]. 6. A. Einstein, Foreword in M. Jammer --Conceptsof Space" (Harvard Univ. Press, 1954 or Dover, New York, 1993) p xvii. 7. J.-P. Vigier, Found. Phys. 25, 1461 (1995). 8. F. Wilsczek, Physics Today, Nnv. 1999 p 11 and Jan. 2000 p. 13. 9. Y. Dobyns, A. Ruedaand B. Haisch, Found. Phys. (2000), in press. 10. W. Rindler, Phys. Lett. A 187,236 (1994) and Phys. Lett. A 233,25 (1997).

ENGINEERING THE VACUUM

TREVOR W . MARSHALL Dept. qf Mathematics, Manchester University, Manchester M13 9PL

Abstract. As a continuation of the programme reported by me to the first Vigier conference, 1 report that a theory of nonlinear optical phenomena, based on a real zeropoint or "vacuum" electromagnetic field, has successfully explained a wide range of allegedly nonlocal experimental data. The same theory is seen to be capable of making quite accurate predictions of new phenomena not foreseen by the prevailing photon theory of Quantum Optics, as well as having some important engineering applications.

L. The vacuum is full of light waves "Vacuum" is the name commonly given to a region of space from which all atoms and all "photons" have been extracted, the latter by cooling to zero Kelvin. It is a bad name, because we now know that this "vacuum" actually contains a lot of radiation, known as the Zero Point Field (ZPF). The ZPF [I, 21 was a hypothesis put forward by Max Planck in 1911. and developed, by him and Walther Nernst, between 1911 and 1916, because they did not believe the phenomena of light emission and absorption could be adequately explained by Einstein's hypothesis of "light quanta", which subsequently became photons. In 1947 and 1948 the effect of the ZPF was directly demonstrated in the Lamb shift and the Casirnir effect. After more than 50 years these very clear demonstrations of the ZPF go unacknowledged. Meanwhile the very objections, made by Planck to the photon hypothesis, have been transformed, by its enthusiasts, into what they consider to be great achievements. They claim to have proved that our world is "nonlocal", when all they have actually done is demonstrate that the photon description is nonlocal. For a discussion of the experiments purporting to have proved nonlocality, notably the atomic-cascade experi459 R.L. Amoroso et a1 (eds.), Gravitation and Co.osmology:From the Hubble Radius to the Plrurck Scale. 459-468 O 2002 Kluwer Acudenlic Publishers. Printed in the Netherlands.

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TREVOR W. MARSHALL

ments of Clauser and Freedman and of Aspect, see my article for the first Vigier congress [3] and also my web page[4]. Thanks to the techniques of nonlinear optics, the last 50 years have witnessed great technical achievements which rely on the ZPF, but, as with heat engines two hundred years ago, our advance in engineering has ous tripped our scientific understanding. The first, and most spectacular achievement is the laser. Then, with the intervention of nonlinear crystals, we have seen that certain modes of the ZPF may be squeezed below their normal (that is Uncertainty-Principle) amplitude. Also, certain pairs of ZPF modes may be amplified in a correlated manner, when they are nonlinearly coupled to a pumping laser. This phenomenon is known as Parametric Down Conversion (PDC). The description of PDC correlations offered by photon theorists hardly qualifies as an explanation, and, I shall argue, has deservedly given the physics of the latter half of the last century a reputation for mysticism. On the other hand, the ZPF description is now sufficiently advanced. not only to explain in detail the experimental data of PDC, but also to predict a new phenomenon, called Parametric Depletion of the Vacuum (PDV). Willis Lamb[5] said, in 1996, In 1947 I proved the vacuum does not exist.

He was thereby associating himself with the point of view I have just expressed. The ZPF is real and fills the "vacuum", which we should now really call a plenum. Schematically his experiment of 1947 established that

H atom + ZPF

+ Lamb

shift.

According to the ZPF description each mode of the vacuum electromagnetic field has an average energy of Tiw/2. At h s t glance this is not too different from the description of Quantum Electrodynamics (QED), but there is a crucial difference. In the ZPF description this mean energy is subject to fluctuations, so that it has a standard deviation, also of fiw/2. Thus this "ground state" of the radiation field is not to be interpreted as an energy eigenstate. More radically we must recognize that such a continuum of energies is incompatible with the concept of a photon, since there is no way in which a quantum of Tiw, superposed on such a ZPF background can give anything other than a continuum of energies in all modes of the field. The simplistic, that is corpuscular, notion of the photon underlies all of the magical "nonlocal" interpretations of optical phenomena, From the Clauser-Freedman experiment mentioned above to such exotic recent arrivals as teleportation[6]. The incompatibility 1 speak of has been long recognized by exponents of the ZPF. For example Planck said, in a letter to Einstein in 1907[7]

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I am not seeking the meaning of the quantum of action (light quantum) in the vacuum but rather in places where emission and absorption occur, and 1 assume that what happens in the vacuum is rigorously described by Maxwell' s equations. More recently Willis Lamb[8], in an article entitled Antiphoton, made an essentially similar point. 2. Parametric down conversion (PDC)

PDC is a process, depicted in Fig.1, in which a (very small) percentage of

wo )Iwl +hw2

W 1 + ~ 2 ,

=

L o ,

Plwl sin B1 = hwz sin 82 ,

Solving these equations gives a rainbow B1(wl). Figure I . PDC - the photon version. A laser photon down converts into a conjugate pair of PDC photons with conservation of energy and momentum. These latter conditions determine the direction 81 at which a given frequency (31 emerges.

photons (wo)from a laser are converted, by a nonlinear crystal (NLC), into photon pairs (wl , w z ). Different photons emerge in different directions and the relationship between w l and B1 is determined by the requirements of energy and momentum conservation, after taking account of the different refractive indices of the various photons within the crystal. Where any photon comes from is already a mystery. We need only consult[9] Feynman's discussion with his own father - Father Atoms emit photons. Right? - Son Yes. - Father So do atoms contain photons?

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TREVOR W. MARSHALL

- Son No. Not exactly. - Father So where do photons come from? - Son We don't know.

One could pose the same questions about the photons that come out of a nonlinear crystal. The difference is that in this case, we are speaking of a macroscopic object, whose diameter is about lcm, instead of an atom. The macroscopic nature is further emphasized by our ability, literally, to cook the PDC rainbow by changing the temperature of the crystal, thereby changing no, nl and nz. That in turn changes O1(wl).Every "photon" is produced by the entire crystal. Furthermore, I emphasize that the process of photon cookery, as well as being macroscopic, is very much a disordering one. As such it is not at all like what occurs in a superconductor. 3. The mind boggling experiment

In 1991 Wang, Zou and Mandel (WZM) performed an experiment[]01 which they dubbed Induced coherence without induced emission. My colleagues and I 111, 12, 13, 14, 15, 16, 1 7 have published a series of articles showing that all of the mystifying results claimed by photon theorists have a ready explanation once one recognizes the reality of the ZPF. I believe the WZM experiment deserves singling out for two reasons; it has been so singled out by the photon theorists themselves, and also the apparently paradoxical result, once it has been demystified, is the basis of a new prediction which. once verified, will give a convincing new demonstration of the ZPF. Greenberger, Home and Zeilinger[18] call the WZM experiment "mind boggling", which may be translated as "mind frightening". Reading their article one cannot help wondering whether they believe it is the actual experimental data that should frighten us, or whether perhaps, after the manner of Uri Geller, they think we should be frightened by the magical powers possessed by WZM. The title of the article is Two photon interference, and in view of Dirac's[l9] famous statement about photons never interfering with each other, it indicates something of the boggling situation in which they find themselves. The WZM experiment is depicted in Fig.2. Photons from a laser v are separated so that they go either into channel vl or into v2. Some of those going into vl undergo PDC (see Fig.1) at the nonlinear crystal NLCl and down convert into (sz,il), while some of those going into v2 down convert at NLC2 into (sz, i2).WZM observe interference fringes between sl and sz by varying the two optical paths from the crystals to the beam splitter BS2, where these two signals are combined. This may be done, for example, by varying the orientation of the phase shifter PS thereby producing a

ENGINEEFUNG THE VACUUM

Fagum 2. The mind boggling experiment.

variation in the counting rate at the detector D. When WZM replaced PS by an absorber, the fringes disappeared. The "explanation" of these results offered by WZM is that sl and sz interfere because they are indistinguishable. We cannot tell which path a given photon detected at D has travelled in order to arrive there. Indeed, it is an important condition of the experimental arrangement that the unobserved (so called "idler") photons, il and i2, are made indistinguishable by very careful alignment of NLC2 with NLCI. But, so the "explanation" continues, replacement of PS by an absorber renders il and i2 distinguishable. We could, if we wished, put a second detector in channel i2; if it clicked we would know that PDC had occurred in NLC2, and if it did not we would know it had occurred in NLC1. This is an example of a kind of observer subjectivism inevitable in any purely quantum mechanical (that is ignoring the presence of fields) analysis. Not only does measurement cause a "collapse" of an information wave function; the possibili~of making such a measurement also causes collapse. Incidentally, another feature of this type of argumentation is a kind of schizophrenia which finds difficulty in distinguishing real from thought experiments. Only in the purest fantasy do detectors of the above eitherlor type actually "exist". 4, Parametric Amplification of the Vacuum

To begin the process of demystification we have to return to Fig.1, and recognize the active role played in the phenomenon known as PDC by the zero point field. I claim that the failure to understand PDC is so deep, among the school of photon theorists, that the phenomenon should really be

TREVOR W. MARSHALL

464

given a new name. I call it Parametric Amplification of the Vacuum (PAV) in recognition of the fact that, not only would there be no phenomenon at all if the "vacuum" were truly vacuous, but also it is not correct to say that a laser photon "down converts"; rather we should say that a laser mode (since there are no photons) interacts with a ZPF mode to give a signal mode at the difference frequency. Of course, it is precisely at this point that the nonlinearity of the crystal plays a crucial role. The PAV process is depicted in Fig.3. It is really two independent

Figure 3. Parametric Amplification of the Vacuum. The phenomenon known as PDC really consists of these two statistically independent processes.

processes. In one of them a mode 22, of frequency w2, together with the laser mode of frequency wg, polarizes the crystal so that it vibrates at frequency wo - w2. The polarization current radiates the signal sl at this latter frequency, and the inducing mode 22 has its amplitude modified to i 2 . In the second process the roles of signal (s) and idler (i) are reversed. The amplitudes of the outgoing modes may be calculated by the techniques of classical nonlinear optics[20]. The modes which are substantially modified are those for which the partial waves from the various parts of the crystal interfere constructively. This gives rise to Phase Matching Conditions, which, apart from a cancelling factor of TI, are identical with the energy and momentum conservation conditions given with Fig. 1. We have shown that sl il and 32 ip have intensities greater than xl and x2. It may be assumed, as explained in Ref.[3], that only intensities above the zeropoint level give rise to detection events, which is why we see strongly correlated events in the two outgoing channels. We have here a causal de-

+

+

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465

scription of the process known as PDC; the incoming laser field, together with the ZPF, polarizes the crystal, and the resulting polarization current radiates into a mode whose frequency is the difference of the frequencies of the exciting modes. 5.

The mind unboggled

We now apply the realist analysis of the previous section to the Mind Boggling Experiment of WZM. The vital missing link required to progress from the nonlocal "'explanation" offered previously is the component ZQ of the ZPF added in Fig.4. The field v is split into mutually coherent

Figure 4. The mind unboggled.

components ~ 1 m, , which go into both channels. These interact with the same component q of the ZPF at the two crystals NLCl and NLC2 to give coherent outputs sl,s2. When PS is replaced by an absorber, then NLCl and NLC2 still have coherent vl and vz incident on them, but the ZPF components, zo and zl, are independent. So sl and s2 are then mutually incoherent. Again we must change the name given by photon theory to a phenomenon; instead of the name given to the Mind Boggler by its discoverers we should call it Induced emission with induced coherence. The alignment of NLCl and NLC2, may, as they said, ensure indistinguishability of the photons il and i2, but that has nothing to do with the explanation of the phenomenon. Physics is a science, and it requires causality, not teleology. The real achievement of ZWM's alignment of the crystals was in ensuring that the same component of the ZPF interacted with both vl and v2, inducing thereby both the emission and the coherence.

TREVOR W. MARSHALL

6. New Science

In the sequence of articles I cited above[3, 11, 12, 13, 14, 15, 16, 17J we have shown that the whole family of allegedly nonlocal optical phenomena, from atomic-cascade coincidences to teleportation, has a natural explanation in terms of a wave theory of light which is simply the one bequeathed to us by Fresnel, Faraday and Maxwell, the only novel feature being the recognition of Planck's zeropoint field. I claim that we have thereby achieved the only convincing explanation of all these phenomena, and it is only because, in the twentieth century, physicists abandoned their duty, as scientists, to find explanations that our results have received so little recognition. It was the received wisdom of that century (thankfully we are now in a new one!) that science should concern itself only with prediction, and that explanation belonged to a bygone immature phase. Fortunately biologists took no notice of this message from the "senior" science. Nevertheless, prediction has its place, so I will now describe a fairly straightforward application of our theory leading to a striking new prediction, which I think could not have been made from the current photonbased version of Quantum Optics. We just saw that the process, within the crystal, which produces the PDC signal sl is the polarization current at the difference Erequency wo - wz. Now another polarization current at the frequency wo + w2 also exists within the crystal. Indeed, in the case that the a mode has an intensity comparable with that of the pump wo, this sum mode produces an "up-conversion" signal which has already been observed[20]. Now the calculation of the expected intensity, when this second mode is reduced to its zeropoint level, is a straightforward extension. and the question of whether a detectable signal emerges is settled by calculating whether the mode intensity is or is not amplified above the zeropoint level. I have done this calculation, and reported its results[21]. As in the case of PAV, described above, we have to study two processes together. namely wo and u p

+

~1

wo and w l

+

w2 .

(ul> wo) ,

We find that the ZPF mode with the higher frequency, that is w l , actually has its intensity reduced below the vacuum level. I propose to call this Parametric Depletion of the Vacuum (PDV); it is somewhat different from squeezing, since in this case both quadrature modes are reduced in intensity, while in squeezing one of them is reduced at the expense of the other. Such a phenomenon will not easily be demonstrated experimentally, but the other mode, that is w , is amplified. So we need to know where to look for wz, and what intensity to expect. I have published sufficient details of this on

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467

the Los Alamos archive, as cited above, so all you have to do is look at the predicted angles and you will see the phenomenon. 7. New technology?

I think that, once the ZPF has been fully accepted as a red physical object, it will have an engineering, as well as a scientific impact. A prime candidate for this is the field of laser design. To emphasize the relevance of the ZPF in a Laser, I point out that a laser is a device which selectively amplifies certain modes of the ZPF. Indeed the acronym "lasef' stands for light ampl[fication by stimulated emission of radiation, and it is precisely the ZPF which does the stimulating! It is no accident of history that effectively the first working laser (called, at that time a maser) was the apparatus used by W. E. Lamb and colleagues in 1947 to measure the Lamb shift. As we have seen, Willis Lamb is outstanding, among the pioneers of Quantum Electrodynamics, in his recognition both of the importance of the ZPF and of the inadequacy of the Photon concept. Lasers are extremely inefficient heat engines. They convert incoherent (that is thermal) Light energy from the pump into a coherent output ("work?'), but typically, in a narrow-band continuous-wave laser, about 10 kilowatts of pump power is required to produce 500 milliwatts of coherent output. Good coupling of the pumped atoms to the ZPF, both outside and inside the laser, is achieved by good cavity design, and I think the standard Fabry-Pemt cavity can be immensely improved, once we have understood how the ZPF may be engineered. There is a historical parallel here. Heat engines played a central role in the Industrial Revolution between 1770 and 1850, but scientists at that time accepted a deeply incorrect theory of heat; it was thought to be a material substance called "caloric". Then Thermodynamics was correctly formulated and heat engines became even better, culminating in the automobile. We have made a lot of progress with lasers since 1947, even though our theory of Optics has become dominated by mythical objects called "photons". The time has come, when we should recognize, in the zeropoint field of Max Planck, a new branch of Thermodynamics, that of zero degrees Kelvin. 8.

Theses on Magic

- No competent magician believes in magic. - A competent magician is really a scientist pretending to be a magi-

cian. - A competent magician is one who knows how the rabbit got into the

hat ...

468

TREVOR W. MARSHALL

- . . . and to distinguish between a scientist and a competent magician is a problem in ethics rather than science. References 1. L. de la Pefia and A. M. Cetto, The Quantum Dice, (Kluwer, Dordrecht, 19%) 2. P. W. Milonni, The Quantum Vacuum, (Academic, San Diego, 1993) 3. T. W. Marshall and E. Santos. The myth o f the photon in The Present Status o f the Quantum T h e o ~ yof Light, eds. S. ~iffersletal'(Kluwer, Dordrecht, 1997) pages 67-77. T. W. Marshall, www.demon.co.uk and homepages.tesco.net/-trevor.marshal1 W. E. Lamb, interview in La Nueva EspaAa, Oviedo, Spain (July, 23, 19%) D. Bouwmeester and A. Zeilinger. Nature 388, 827-828 (1997) G. Weihs. T. Jenneswein, C. Simon, H. Weinfurter and A. Zeilinger, Phys. Rev. Lett, 81, 5039 (1998) A. Pais, Subtle is the Lord, page 384 (Clarendon, Oxford, 1982) W. E. Lamb, Appl. Phys. B, 60, 77-82 (1995) R. P. Feynman. You must be joking Mr. Feynman L. J. Wang, X. Y. Zou and L. Mandel, Phys. Rev. A, 44, 4614 (1991) A.Casado, T.W.Marshal1, and E.Santos, J. Opt. Soc. Am. B, 14, 494-502 (1997). A.Casado, A.FernBndez Rueda, T.W.Marshal1, R-Risco Delgado, and E-Santos, Phys.Rev-A, 55, 3879-3890 (1997). A.Casado. A.Ferndndez Rueda, T.W.Marshal1. R.Risco Delgado. and E-Santos. Phys.Rev-A, 56, R2477-2480 (1997) A. Casado, T. W. Marshall and E. Santos, J. Opt. Soc Am. B, 15, 1572-1577 (1998) A.Casado, A.Fern&ndez Rueda, T.W.Marshal1, J. Martinez, R.Risco Delgado, and E-Santos, Eur. Phys. J., D11,465 (2000) K.Dechoum T. W. Marshall and E. Santos, J. Mod. Optics. 47, 1273 (2000) K. Dechoum, L. de la Peiia and E. Santos, Found. Phys. Lett., 13, 253 (2000) D. M. Greenberger, M. A. Home and A. Zeilinger, Phys. Today, 46 60.8,22 (1993) P. A. M. Dirac, Principles of Quantum Mechanics, page 9 (Clarendon, Oxford, 1958) B.E.A.Saleh and M.C.Teich, Fundamentals of Photonics, (John Wiley, New York, 1991) T.W. Marshall, http://xxx.lanl.gov/abs/quant-ph/9803054.

THE PHOTON AS A CHARGE-NEUTRAL AND MASS-NEUTRAL COMOSITE PARTICLE Part I. The Qualitative Model

HECTOR A. MUNERA Department of Physics Universidad National de Columbia A.A. 84893, Bogota, Columbia

Keywords: Photon, Aether, Composite photon model, Maxwell equations, Advanced fields, Charge neutral particles, Mass neutral particles.

Abstract. In the context of a 4D aether model, where rest mass is associated with a flow of primordial mass (preens), the photon was described as an electron-positron pair. Such a composite particle is then a charge-neutral and mass-neutral entity; thus accounting for photon standard properties: zero charge and null restmass. The electromagnetic field of such photons contain both advanced and retarded components, without any causality breach. The model obeys conventional Maxwell equations.

1. Introduction

The idea that the photon may be a composite particle is not new. Long ago De Broglie (1932) suggested that the photon was a composite state of a neutrino-antineutrino pair; such pair, however, did not obey Bose statistics. To avoid this difficulty Jordan (1935) introduced neutrinos with different momenta. Over the years, additional adjustments were made by Kronig (1936), Pryce (1938), Barbour et al. (1963), Ferreti (1964), Perkins (1 9651, Bandyopadhyay and Chaudhuri (1971). Since the photon rest-mass is zero, or very small (Vigier, 1997), neutrinos are chosen as its components. However, 469 R.L Amoroso et a1 (eds.),Gravitation aircl Co.snlology: From the Hubble Radius to the Planck Scale, 469-476 O 2002 Kluwer Acaclemic Publishers. Printed in the Netherlm~cls.

470

HECTOR A. MUNERA

one would nziively expect that if the photon is a composite particle it may under some conditions decay or be separated into its components. Indeed, photon pair production leads to an electron-positron pair, but not to a neutrino-antineutrino one. However, from the viewpoint of total rest mass, an electron plus a positron can not be the components of a low energy photon. There is a clear difficulty for composite models along this line of thought. On the other hand, the idea that the modern "vacuum" (= aether in this paper) is of hydrodynamic nature is a recurrent one. Recent examples: a superfluid of particle-antiparticle pairs (Sinha et al., 1976), a fluid of "stuff' particles (Di Marzio. 1977), and a variety of fluids (Shekhawat. 1976: Widom and Srivastava, 1990; Winterberg, 1997; Ribaric and Sustersik, 1997). From such fluids, some authors derive electrodynamic and particle models: Thomson (1931), Hofer (1998), Marmanis (1998), Dmitriyev (1999). This author recently proposed a four dimensional (4D) hydrodynamic model that allows for a variable component of the 4-velocity along the time axis (Munera, 1999), which leads to a LCD-force as the gradient of the 4pressure; the 3D-electromagnetic force is a particular case (Mdnera, 2000). Also, we have argued elsewhere (Chubykalo. Mirnera and Smirnov-Rueda. 1998) that, in the context of Maxwell's equations, the concept of zero charge in vacuum may be interpreted as neutrality of charge almost everywhere, rather than as complete absence of charge. In this note we take one step further. The photon is modelled as a source-sink pair (into and out of our 3D world), having a zero net mass flow into our 3D world. thus accounting for the photon zero rest mass. Next section 2 summarizes the 4D aether model, and section 3 sketches the photon modeL A final section 4 closes the paper.

2. A Four Dimensional Fluid

Let us assume the existence of a fourdimensional (4D) flat Euclidean where the time dimension w=vwtbehaves exactly the space space Z=(w,xSySz), same as the 3 spatial dimensions (Mdnera, 1999; 2000). Further, let Z be filled with a fluid of preons (= tiny particles of mass m and and Planck length dimensions). These particles are in continual motion with speed ~ ( V ~ , V ~ , V ~ VN~O ~a V priori , , , ,limits V ) . are set on the speed vw of preons along the w-axis. (Notation: 4D-concepts and vectors are represented either by calligraphic or by Greek uppercase letters, while 3D-vectors are in the usual bold face). Note that the Limitations of the special theory of relativity (STR), if applicable, refer to v = (v;+v,?+v, 2) 112 (= the speed of particles in 3D) not to vw= the projection of the 4D-velocity '1/ on the w-axis. Here, we extend the

PHOTON AS COMPOSITE PARTICLE

notion of absolute space to 4D (= (ct,x,y,z), i.e.

471

R~),whereas the spacetime of STR is

R'.~.

Motion of individual preons in T: is governed by a 4D-equation of motion, given by the matrix expression (Miinera, 1999):

where p = nm is the preonic fluid mass density, n is the number of preons per unit 3D-volume, the column vector V = [v,,vx,vy,vJ is the 4D-velocity of individual preons, Va = [cDvX,vy,v,l refers to the time-arrow, the vector is a 4D gradient. the 4x4 matrix T~~ is the 4Doperator dp[&,V],&dl& stress tensor, and P =P (w,x,y,z)is the pressure generated by the preonic fluid; the Greek index p = (w,x,y,z).Finally, the energy-momentum tensor pWa(a 4x4 matrix) results from the dyadic product Wa. Consider now a 3D-hypersurface formed by a projection of the 4Duniverse onto the w-axis, say w = wo= v,oto (Fig. 1 ) . The plane w-r may be interpreted in two complementary ways: Interpretation 1 (Fig. la). At a fixed time to (say the present), the Line w=wo divides the plane into three classes of particles: preons moving with vw> V , O (upper region), preons moving with vw< v , o(lower region), preons moving with vw=v , o (on the horizontal line). Interpretation 2 (Fig. lb). For the class of preons moving with vw= vw,o, the line w = wo divides the plane into three periods of time: the future for t > to(upper half-plane), the past for t < to(lower half-plane), the present t = to (on the line). The conventional worldlines of STR and the space underlying Feynman diagrams belong to interpretation 2 with v ,unspecified. ~ If we postulate that we live in a 3D-hypersurface where vw,o= c, then all preons in our world move with constant speed c along the time axis. This brings in a novel intrpretation for constant c: the speed at which our hypersurface slides from the past to the future (Interpretation 2 above). The meaning of the w-r plane under Interpretation 1 can now be rephrased as: at a given to (say the present) our 3D-world separates superluminal from subluminal preons. Furthermore, as seen below, there is a continuous exchange of p r a m between our hypersurface and the two half-spaces above and below. For events inside our hypersurface, eq. (1) reduces to

472

HECTOR A. MUNERA

where the elements of T~~~associated with the 3D spatial dimensions are given by the conventional ~3~~ viscosity matrix. The elements associated with the w-dimension are

where S=(S,S,SJ is a (displacement) energy flux along axes x,y.z (dimensions: energy per unit time per unit area), and the source/sink ~ ,isfa concentrated energy flow along the w-axis (dimensions: energy per unit time), and 8 represents the position of the energy source/sink (positive/negative respectively), and a(.) is a 3D-Dirac's delta function (dimensions: (l~m~ht)-~). Eq. (3) may be interpreted as a transfer of energy by displacement from the waxis into the spatial axes (or the other way around), whereas eq. (4) is a transfer of energy along the w-axis. Therefore, the 4D-source S = (s,~,s) simply represents a "convective" tramfer of preons from one region of the 4Dfluid into another, i.e. there is conservation of energy in the whole 4Duniverse. Note that other fluid theories contain expressions similar to our eq. (2) (for instance, eq. 3 in Ribaric and Sustersic, 1998). However, our approach is fundamentally different because we allow for interaction between our world and other regions of ):with v, $ c (described by the more general eq. 1). This interaction gives rise to the 4D-source 5 described by eqs. (3) and (4). By analogy with the standard 3D-case, the 4Dpreonic fluid exerts force, and performs work along the four dimensions (w,x,y,z), via its hydrodynamic pressure P (in this sense, P is interpreted as potential energy per unit volume). This immediately leads to a component of force along the w-dimension, which is responsible for the appeareance of sources and sinks in our hypersurface, via the following mechanisms: Sources Sf are produced by the fourth component of force, which acts upon preons outside our hypersurface, via two mechanisms (Fig. 2a): preons moving with v, > c are decelerated to enter our world at t=to with v, = c, preons moving with v, < c are accelerated to enter our world at t=to with v, = c.

PHOTON AS COMPOSITE PARTICLE

I subluminal

r = (x,y ,z) a) condanttirne 1 = t

= [X,Y,Z~ b) constant speed v=, vOw

0

Fig. 1

Figure I. Four-dimensional representation of universe as an r-w diagram. The projection on the w-axis is a 3D hypersurface. This horizontal line partitions the universe into two half-spaces. Part a) For a given time to (say, the present) the upper (lower) space corresponds to universes with higher (lower) speeds on the w-axis. Part b) For a given v,,, = c, the upper (lower) space corresponds to the future (past). See the text.

Fig. 2

Figure 2. The four mechanisms for producing sources and sinks (see the text).

474

HECTOR A. MUNERA

Sinks S-are produced by the fourth component of force, which acts upon preons in our hypersurface that move with vw =c for t < to. Two mechanisms (Fig. 2b): @accelerationofpreons which leave our world at t = to with vw> c, and edeceleration of preons which leave our world at t = to with vw< c. For additional details see Mlinera (1999; 2000). 3.

The Photon As A Source-Sink Dipole

It is widely known that Maxwell equations (and also the homogeneous wave equation) have two sets of solutions: retarded (= outgoing) and advanced (= incoming) ones. The latter are typically neglected on the grounds of causality violations (for instance, Panofsky and Phillips, 1962, p. 244). To account for the advanced solutions in the presence of charges, Wheeler and Feynman (1945, 1949) located an absoher ofradiation at some distance from a charge (see also Panofsky and Phillips, 1962, ch. 21). In the model for the photon described next, we also allow for advanced and retarded solutions without causality violations. In the 4D aether described in previous section, a particle (antiparticle) is a S 1 ), Energy is source (sink), whose rest mass is proportional to Sw+( 1 [ transported into (or out of) our 3D world by a flow of preons. Such representation immediately explains away the difficulties associated with infinities in potential energy (gravitational. electrical. or otherwise) Let the photon be a pair electron-positron, both of them inside a small region of diameter ro. Since the rest masses of the constituent particles are identical, then Sw+-+ Sw-= 0,which implies that the effective rest mass of the campsite particle is zero. Evidently, the net charge is also null, except inside a neighbourhood ro. A simple 3D-analogy is a water filled vessel, with a source Q+ and a sink Q- allowing equal flows of water in and out of the vessel. Consider a differential volume located at an arbitrary point P connected to Q+by a line of flow. Water particles emitted from Q+at t, reach P at a later time according to the speed of propagation of the particle, thus giving rise to a retarded field of pressure. Likewise, consider a particle that is absorbed by the sink Q- at a time t,along a line of flow coming from P. Of course, such a particle was at P at an earlier time, and gives rise to an advanced pressure field. However, there is no causality breach. Simply put, some of the particles located at a given time inside the small volume at P may come from the source, some may reach the sink at a later time, and many other particles may have quite a different fate.

PHOTON AS COMPOSITE PARTICLE

475

The photon is then a rotating dipole in the normal 3D Euclidean space. Let the timedependent retarded (and advanced) electric field associated with the electron (positron) be N (P). It can be shown that fields N and P obey the symmetric Maxwell's equations that we described some time ago (Mdnera, 1997). Such symmetric system is tautologically equivalent to the conventional Maxwell equations, provided that the standard electric and magnetic fields be defined as:

P- N E = -----,B = P + N

.

2

2

4. Concluding Remarks

In the context of a 4D aether model, where rest m a s is associated with a flow of primordial mass ( p n s ) , the photon was described as an electronpositron pair. Such composite particle is then a charge-neutral and massneutral entity, thus accounting for the photon standard properties: zero charge and null rest mass. The electromagnetic field of such photon contains both advanced and retarded components, without any causality breach. The model leads to a symmetric system of Maxwell's equations (Mdnera. 1997). containing two sources (electron and positron). Such system immediately leads to the conventional Maxwell equations (the details will be published elsewhere). Falaco solitons were recently reported (Kiehn, 2000) as pairs of solitons that exist on the surface of a fluid (water), and are interconnected through the third spatial dimension. Our model for the photon is a pair of 3D solitons interconnected through the fourth dimension. An open question to be pursued at a later stage is the connection between the equation of motion describing the 4D aether proposed here and the BohrnVigier (1958) relativistic hydrodynamics.

References Bandyopadhyay, P. and P. R. Chaudhuri, "The photon as a composite state of a neutrinoantineutrino pair'', Phys. Rev. D 3, No. 6 (1971) 1378-1381. Barbour, I. M., A. Bietti, and B.F. Toushek, Nuovo Ciirrento28 (1963) 453. Bohrn, D. and J.-P. Vigier, "Relativistic hydrodynamcis of rotating fluid masses", Phys. Rev. 109, NO. 6(1958) 1882-1891.

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HECTOR A. MUNERA

Chubykalo, A. E., H, A. Miinera, and R. Smirnov-Rueda, "Is the free electromagnetic field a consequence of Maxwell's equations or a postulate?", Found. Physics Lett. 11, No. 6 (1998) 573-584. De Broglie, L, Compt. Rend. 195 (1932) 862; Compz. Rend. 199 (1934) 813. Di Marzio, E. A., "A unified theory of matter. I. The fundamental idea", Fotmd Physics 7 (1977) 511-528. "11. Derivation of the fundamental physical law7', ",Found. Physics 7 (1977) 885-905. Dmitriyev. V. P.. "Turbulent advection of a fluid discontinuity and Schriidinger mechanics". Galilean Electrodynatnics 10, No. 5 (1999) 95-99. Ferreti, B., Nuovo Ci~nento28 (1964) 265. Hofer, W. A., "Internal structures of electrons and photons: the concept of extended particles revisited", PhysicaA 256 (1998) 178-1%. Kiehn, R. M., paper at Vigier 2000 Symposium, University of California, Berkeley, USA (August 2000). Kronig, P. Physica 3 (1936) 1120. Marmanis, H., "Analogy between the Navier-Stokes equations and Maxwell's equations: Application to turbulence", Phys.Fluids 10. No. 6 (1998) 1428-37. "Erratum". Phys.Fluids 10, No. 11 (1998) 3031. Miinera, H. A. "A symmeric formulation of Maxwell's equations", Mod Phw. Len. A 12, NO. 28 (1997)2089-2101. Miinera, H. A. "A realistic four-dimensionalhydrodynamic aether interpreted as a unified field equation", presented at the International Workshop Lorentz Group, CPT and Neutrinos, Universidad Aut6noma de Zacatecas, Zacatecas, Mexico (June 1999). Published in the Proceedings edited by A. Chubykalo, V. Dvoeglazov, D. Emst, V. Kadyshevsky and Y.S. Kim. Miinera, H. A. "An electromagneticforce containing two new terms: derivation from a 4D aether7',Apeiron 7, No. 1-2 (2000) 67-75. Panofsky, W. K. H., and M. Phillips, Classical Electricity andMagnetisrn , 2nd edition, Addison-Wesley Publishing Co. (1962) 494 pp. Perkins, W. A., Phys. Rev. 137 (1965) B1291. Pryce, M.H.L., Prm. Roy. Sm. (London)165 (1938) 247. Ribaric, M. and L. Sustersik, "Transport theoretic extensions of quantum field theories", Eprint archive: hep-tW9710220 (Oct. 97) 36 pp: "Framework for a theory that underlies the standard model". LANL electronicfile hep-th/9810138 (Oct. 1998). Shekhawat, V., "Some preliminary formulations toward a new theory of matter", Found. Physics 6 (1976) 221-235. Sinha, K. P., C. Sivaram, and E. C. G. Sudarshan, Found. Physics 6, No. 1 (1976) 65-70. Thomson, J. J., "On the analogy between electromagnetic field and a fluid containing a large number of vortex fdaments", Phil. Mag.. Ser. 7 12 (1931) 1057-1063. Vigier, J.-P., "Relativistic interpretation (with non-zero photon mass) of the small ether drift velocity detected by Michelson. Morley and Miller", Apeiron 4. No. 2-3 (1997) 71-76. Widom, A. and Y. N. Srivastava. "Quantum fluid mechanics and quantum electrodynamic^^^, Mod Phvs. Lett. B 4 (1990) 1-8. Wheeler, J. A. and R. P. Feynman, Revs. Modern P h y 17 (1945) 157, and Revs. Modem Phys. 21 (1949) 425. Winterberg, F. "Planck aether", Zeistch fur Naturforsch. 52a (1997) 185.

PREGEOMETRY VIA UNIFORM SPACES W.M. STUCKEY and WYETH RAWS Departr~lentof Physics & Engineering Elizabethtown College Elizabethrown, PA 17022

Abstract. We begin by motivating a pregeometric approach to quantum gravity. A pregeometry is then introduced over denumberable sets which employs the discrete uniform space Dx and a uniformity base UB induced by a topological group. The entourages of Dxprovide a non-metric notion of locality consistent with the open balls of a differentiable manifold, thereby supplying a pregeometric basis for macroscopic spacetime neighborhoods. Per the underlying group structure, entourages of U, provide a pregeometric model of quantum non-localitylnon-separability, thereby supplying a pregeometric basis for microscopic spacetime neighborhoods. In this context, a robust pregeometric correspondance between microscopic and macroscopic spacetime structures is enumerated. Finally, we show how the pseudometric generated by a uniform space may be used to provide the M4 geodetic structure. This affine structure may produce a metric structure by requiring the covariant derivative annihilate the spacetime metric per standard Riemannian geometry. An example on s2 is provided.

1. Introduction There is as yet no definitive course of action in the search for a theory ofquantum gravity (QG). Since QG must satisfy the correspondence principle with general relativity (GR) and quantum mechanics (QM). it must provide a basis whence the locality of GR and the non-locality of QM. Demaret, Heller. and Lambert write [I]. In this Section we analyse fundamental concepts of quantum mechanics. We show that they lead to some problems which force us to modify the usual notion of spacetime. ... The second problem is related to the famous E.PR. paradox which introduces the idea of non-locality or more precisely of non-separability with respect to space. In fact, in quantum mechanics space cannot be viewed as a set of isolated points. These problems lead to a deep modification of our representation of "quantum" space-time. Stuckey has argued [2] that a non-local, reductive model of reality precludes the fundamental use of trans-temporal objects. Since trans-temporal objects are fundamental to kinematics and kinematics is fundamental to dynamics, concepts such as mass, momentum, and energy are excluded from the foundation of a rational reductionist theory of QG. It is difficult to imagine where to begin modeling reality without reference to the 477 R.L. Amoroso et al (eds.),Gravitation and Cosmdogx: From the Hubble Radius to the Plunck Scale, 477-482. O 2002 Khwer Acadeiriic Publishers. Printed in the Netherlands

478

W.M. STUCKEY AND WEYTH RAWS

concepts of dynamics. Weinberg writes [3]. "How can we get the ideas we need to formulate a truly hndamental theory. when this theory is to describe a realm where all intuitions derived from life in space-time become inapplicable?' Accordingly, QG may require an approach a la Wheeler's pregeometry. That is [4], "the features of the conventional space-time, such as its continuity, dimensionality, and even causality and topology, should not be present from the beginning, but should emerge naturally in the transition process from pregeometry to the usual space-time dynamics of our conventional physical theories." Further, it may behove us to base the program in discrete mathematics. Butterfield and Isham write [5], "For these reasons, a good case can be made that a complete theory of quantum gravity may require a revision of quantum theory itself in a way that removes the a priori use of continuum numbers in its mathematical formalism." Au writes 161, "One can see how a discrete theory could reduce to a continuum one in the large scale limit, but to shed light on a discrete theory while working from the perspective of a continuum one seems difficult to achieve." And Sorkin writes 171, "The dynamical principles learned from quantum mechanics just seem to be incompatible with the idea that gravity is described by a metric field on a continuous manifold." Thus, we are motivated to search for QG using pregeometry based in denumerable sets.

2. The Model We are guaranteed that a uniform space U may be constructed over any denumerable set X by introducing a group structure and the discrete topology over X. And. the discrete uniform space Dx induces the discrete topology over X while its entourages provide a conventional, but non-metric, definition of a ball centered on x E X. Thus. Dx induces the topology required for U while providing a pregeometric definition of macroscopic spacetime neighborhoods. Given that the introduction of a group structure over X underlying Dx provides a uniformity base U, for U. we have the means to define microscopic spacetime neighborhoods independently of. but consistently with. macroscopic spacetime neighborhoods. Specifically 181, for x and y elements of X. a symmetric entourage V is a subset ofX x X such that for each (x, y) E V, (y, x) is also an element of V. Dx is the collection of all symmetric entourages. For (x, y) E V the distance between x andy is said to be

less than V . The ball with center x and radius V is {y E X

1 (x, y)

E

V) and is denoted

B(x. V). A neighborhood of x in the topology induced by Dx is Int B(x.V). so all possible balls about each x E X are established. This is precisely in accord with the conventional notion of locality, i.e., open balls about elements of the spacetime manifold. Therefore, B(x. V) is a perfect pregeometric definition of a macroscopic spacetime neighborhood of x for denumerable X (cf. Sorkin's finitary topological spaces [9]). In section 4, we will show that this definition of macroscopic spacetime neighborhoods accommodates the topological priority of causal chains over metric balls per Finkelstein [lo]. To show that the introduction of a group structure G over X underlying 4( allows for the construct of U, we construct its uniformity base Us via neighborhoods of the identity e ofG in the following fashion [l 11. The entourage A, of U is ((x, y) E X x

PREGEOMETRY VIA UNIFORM SPACES

X

I

479

xym' E 2) where 2 is a neighborhood of e in the topology over X. When X is

denumerable of order N, ((w, y) E X x X entourages A, (x

E

X such that x

1

w # y) is partitioned equally into the

# e) for the N - 1, order-two neighborhoods of e, i-e.,

I

Ax is generated by {e. x). The entourages A, and C / {(x, x) x E X) constitute a base U, for U. Entourages generated by larger neighborhoods of e are given by members of U,,i.e., {e, x, y) generates A, U A,, etc. While for some group structures all members of U, are elements of Dx, e.g. the Klein 4-group [12], this is not true in general. In fact, A, E Dx # x V X such that x = x-I. This, since for (y, z) E A, such that y z, yz-' = x and therefore, zy-' = x" = x j (z, y) E A,. For the base members A, and A,, such that x = y-l, we have A;' = Ay where

A-'

=

((w, z)q

(z, w) E A). This, since for (w, z) E A, such that w + z, wz-'

=

x and

therefore, zw-'= x-l = y =3(z, w) E Ay We may now construct the largest element of Dx via multiplication of the members of Us. With A a subset of any entourage (uniquely and axiomatically), we have in general for entourages A and B that A C A B and B C AB where AB / {(x, z)

1

(x, y) E

A and (y, z) E B). Next. consider {(& y), (y, z)l (x, y) E % and (y, z) E A, with x Z y and y # z). In addition to c these account exhaustively for the elements of A, and A,. For any such pair (x, y) and (y, z), (x, z) E A:A, by definition and (x, z) E As,, since sw = ( ~ ~ - ' ) ( ~ z= " )x i ' . The N pairs (x.z) with C account exhaustively for the elements of kW and. excepting the impact of C on AsAy, the N pairs (x. z) account exhaustively for the elements of &A,. Again, the impact of C on Ask, is to render A, c A,A, and A, c A,Aw. Therefore, A,Aw = A, U A, V A., So, ifG is cyclic with generatorx, A '-: = Ax U Ay U ... U A, where y = 2 and z = xN-'. Ax U Ay U ... U A, is of course the largest element,V of Dx. [This is of particular interest, since the cyclic group structure Z,.. exists for all N E N and is the unique group structure for N prime.] IfG is not cyclic, one may produce V,via -1 A, u U A, U AwA, U ... U A,A, for x = x-', ..., y = 6', w = z , ...,s = v-I, since &A, = A, v 4 when w = z-'. We are also guaranteed to produce V, via some variation of A,A,...A,where {x, y, ..., z ) = X, according to G. It should also be noted that, as implied supra, the entourage A, U A, V ... U A, of U is generated by the entire set X

...

3. Consequences Should we define microscopic spacetime neighborhoods with the members of U, analogously to macroscopic spacetime neighborhoods per the symmetric entourages of D,, we note the following interesting consequences.

W.M. STUCKEY AND WEYTH RAWS

1. A, = A;' for s = w-',so when s f w the distance between elements of A, is non-separable from that ofA,,,, lest we compromise the symmetry of our pregeometric notion of distance. For s = w, i.e., S = s-l, A, E DX and our microscopic spacetime structure accommodates locality. Thus, the degree to which our spacetime is to accommodate quantum non-separability is determined by the choice ofG. 2. The choice of G over X underlying Dx is all that is needed to produce the microscopic spacetime structure embedded in the macroscopic spacetime structure. 3. The members of the base Us ofthe microscopic spacetime structure U may be combined via entourage multiplication to yield the largest element V,, of the macroscopic spacetime structure Dx. Complementing this, , V is equivalent to the entourage of U generated by the entire set X. Thus, a robust pregeometric correspondance between the microscopic spacetime structure and the macroscopic spacetime structure is provided. 4.

Nexus to Physics

First, we show how a pseudometric induced by 4( yields the geodetic structure of M4. We borrow from a proof ofthe following theorem [13]:

...

For every sequence Vo, V1, of members of a uniformity on a set X, where Vo = X x X and (vi+,13 c Vi for i = 1.2, ..., there exists a pseudometric > on the set X such that for every i 2 1

( ( X ~ Y )p~ ( x , ~ ) < ( l ~ )c~ vi I c((x,Y)

I

p &y)~(lf2)'1-

To find >(x, y), consider all sequences of elements ofX beginning with x and ending with y. For each ad~acentpair (h, xMl) in any given sequence, find the smallest member of (Vi) containing that pair. [The smallest Vi will have the largest i, since (vi+d3 C Vim] Suppose V, is that smallest member and let the distance between x, and x,,+~be (1/2)m. Summing for all adjacent pairs in a given sequence yields a distance between x and y for that particular sequence. According to the theorem, >(x,y) is the smallest distance obtained via the sequences. While this pseudometric is Euclidean rather than Minkowskian, it may be used to define geodesics €or either space, since their affine structures are equivalent. Thus, we define a geodesic between x and y to be that sequence yielding >(x, y). Since some sequences might contain 'distant' adjacent pairs, our definition is suitable only for M4.[TO consider curved spacetimes, we would have to restrict our attention to sequences harboring only 'local' adjacent pairs.] The finest resolution would result when considering sequences Vo, V1, of maximal length. The construct ofmaximal sequences is possible with X denumerable and finite. Of course, this immediately suggests a pregeometric foundation for the path integral formulation of QM. And, should one consider various sequences of entourages satisfying (Vi+1)3 C Vb SO that the combinatorics of both element and entourage sequences are considered, then a pregeometric foundation for quantum field theory is also

...

PREGEOMETRY VIA UNIFORM SPACES

481

intimated. Since each sequence of entourages produces a pseudometric, a combinatoric formalism over entourage sequences is analogous to the path integral formulation of quantum cosmology per Hartle and Hawking [14]. The transition from pregeometry to classical spacetime dynamics might be initiated by constructing an affine definition of 4-momentum Let a worldline be

5.

defined by a sequence of elements of X. Then, 4-velocity i? is defined by adjacent pairs in the sequence with direction specified via ascending order ofthe sequence. Thus, for a particle of mass m we have j j l m? . To accommodate curved spacetimes the definition of >(x,y) would have to be restricted to sequences constructed of 'local' adjacent pairings. Speculatively, an equivalence relation might be used to partition X into cells providing this restriction. These local affinely M4 frames would then be pieced together so that the spacetime metric is consistent with the 4-momentum distribution a la Einstein's equations. Since this demands a relationship between affine and metric structures, a reasonable axiom is that of Riemannian geometry as noted by Bergliaffa et al. [15], i.e., the covariant derivative annihilates the spacetime metric. We have

V i (&0'@ O k ) = & * i O~J f yields gsw,i - g .

r :,- gsk r , k

j ) ~ - & k :lr 0'

Zj

Zl

=

d and

ae

0' @ m S = 0 w h i c h

= 0-

As an example, consider coordinates. Let

@ a -kg j k r :

s2with the standard polar (8)and azimuthal (p)

d z2 = . The affine structure yields V

3~

1

Zl

=

r {I

= 0 and the other Christoffel symbols can be computed by using parallel transport and

rk(viz,)= r:

,where r 1 = d 6 a n d r 2 = d p . w e k n o w

I E2,sog12=g2,

= 0. Combined with the annihilation condition, we have gll,]= 0 and glI2= 0.Thus. gll is a constant. The annihilation condition also gives gll(cose sine) = g22cot@,or gz2= gl Isin2@, thus completing our example.

Acknowledgement This work was funded in part by a grant from AmerGen Energy CompanyIThree Mile Island Unit 1.

References J. Demaret, M. Heller, and D. Lambert, Foutd Sci. 2, 137 (1997). W.M. Stuckey, "Pregeometry and the Trans-Temporal Object," to appear in Snrciies on the structure of tittle: Frollt physics ro psycho(patho)logy, R Buccheri, V . Di Ge& and M. Saniga, eds., (Kluwer, Dordrecht, 2001). 3. S. Weinberg, Sci Amner. 281.72 (1999). 4. Ibid I. 5. J. Butterfield and C.J. Isharn, "Spacetimeand the Philosophical Challenge of

482

W.M. STUCKEY AND WEYTH RAWS

Quantum Gravity," grqcf9903072 (1999). 6. G.K. Au. 'The Quest for Quantum Gravity," gryc/9506001 (1995). 7. R Sorkin in i7w Creation of Ideas in Physics, J. Leplin, ed, (Kluwer, Dordrecht, 1995) 167. 8. R. Engelking, General Topology (Heldermann Verlag, Berlin, 1989). 9. R. Sorkin, Inr. J. Theor. Phys. 30,923 (1991). 10. D. Finkelstein. Phys. Rev. 184. 1261 (1969). 1I. R. Geroch, Mathematical Physics (Univ of Chicago Press, Chicago, 1985). 12. W.M. Stuckey, Phys. Essays 12,414 (1999). 13. lbid & 14. S.W. Hawking in 300 Years of Gravitation, S.W. Hawking and W. Israel, eds.. (Cambridge University Press. Cambridge. 1987) 631. 15. S.E.P. Bergliaffa, G.E. Romero, and H. Vucetich, Int. J. Theor. Phys. 37, 2281 (1998).

A ZPF-MEDIATED COSMOLOGICAL ORIGIN OF ELECTRON INERTIA M. IBISON Institute for Advanced Studies at Austin 4030 Braker Lane West, Suife300 Austin, TX78759, USA

Abstract Support is found for a fundamental role for the electromagnetic zero-point-field (ZPF) in the origin of inertia. Simply by requiring that that a universal noise field be selfconsistent in the presence of the lightest charge, it is shown that this field must be the ZPF, and that the mass of that charge must be close to 10"' kg. The ZPF functions as homeostatic regulator, with the electron mass decided by cosmological quantities. The calculation validates Dirac's second Large Number hypothesis.

1.

Introduction

Several speakers at this conference have been pioneers championing the cause of an electromagnetic zero-point-field (EM ZPF) origin for inertia. Notable amongst these have been Haisch, Rueda and Puthoff [l-111. Currently the implementations are classical, with a 'classicized' ZPF as conceived within the program of Stochastic Electrodynamics (SED), (see Kalitsin [12], Braffort [13] and Marshall [I41 for the original works, and Boyer [15] for a review of this field). Epistemologically, a common theme of their work is that the ZPF is the cause of resistance to acceleration. In mathematical practice though, the end result is an inertial mass-energy that attributable to the ZPF. Brwddly, the ZPF is seen as an external, energizing influence for a local degree of freedom, which, classically, is the co-ordinate of the particle whose mass we wish to explain. Thus the program has - in part - some of the flavor of Mach, because the ZPF provides a 'background' against which the acceleration can be measured. The particle, once energized, is conceived as having attributable energy, and therefore inertia. Although Haisch et a1 [I 11 in particular have made a distinction between the inertial and energetic aspects of matter, this distinction appears to be largely epistemological; since any 'localized' packet of energy is found to resist acceleration, it is sufficient to explain within the context of this program how EM ZPF energy can become localized. Even so, within this program. there are two quite different possible implementations distinguished by different degrees of non-locality for the origin of mass. To date. 483 R.L. Amoroso et a1 (eds.),Gravitation and Cosmology: From the Hubble Rudius to the Platzck Scale, 483-490. O 2002 Kluwer Acndemic Publishers. Printed in the Netherlurzds.

484

M. IBISON

despite the active role of the ZPF, current implementations result in a value for the inertial mass that is intrinsic to the particle in question. In contrast, the purpose ofthis paper is to argue for an alternative implementation, wherein the value of the inertial mass is determined entirely by external - cosmological - factors. For this reason the approach described here is much closer to the original conception of Mach (ca 1883) than previous ZPF-as-background approach. In the following is given contrasting descriptions of the intrinsic and extrinsic approaches to ZPF-originated inertia. These are followed by a calculation supporting the latter, wherein one of Dirac's large number hypotheses [16,17] is derived and interpreted as evidence of a cosmological ZPF-origin of the inertial mass of the electron.

2.

Role Of The ZPF I n A Model Of Locally Determined Inertial Mass

In a locally determined, ZPF-originated, model of inertia, there exists a local dynamical degree of freedom, such as an oscillator amplitude [I], or a resonator excitation level. This co-ordinate is conceived initially as quiescent, and having no intrinsic energy. Subsequently, if the ZPF is switched on, then the oscillator or resonator is energized, and the ZPF-originated energy that is now associated with the oscillator or resonator can be regarded as the 'rest' mass. In some work, the local properties ultimately deciding the mass of the charged particle enter as a Fourier form factor governing the spectral response to the ZPF [3,4,5]. This has the advantage of leaving open whether the response is due to the energetic resonance of an oscillator or geometric structure. But in either case it is taken to be a local property. The end result is the same in that not only is there resistance to acceleration, but there is also a localized energy density that can be associated with the particle in question. Clearly, in this approach, the object has zero true rest mass, whilst the ZPF-energized mass may nonetheless be statistically at rest due to the homogeneity and isotropy of the ZPF. Also, thanks to the peculiar k-space distribution of the ZPF - the ZPF retains the same homogeneous energy density in every inertial frame - it follows that with suitably chosen dynamics it should be possible to make the rest mass a fully invariant scalar. These original ideas have stimulated new thinking about the origin of inertia and brought forth some encouraging responses [18,19,20]. However, it is premature to claim that the origin of inertia has been found in the ZPF, because there are some unresolved and unsatisfactory aspects of the current approach: 1. The particle oscillator or resonator must contain electrical charges capable of interaction with the ZPF, so the model cannot describe a neutral elementary particle such as a massive neutrino, for example.

2. The electron also poses a problem unless it is admitted that it has some - as yet unobserved - structure.

3. The values of the intrinsic mass (of the electron, muon, and tau say), are not predicted, but must be inserted by hand. The final mass - the energy stored in the oscillator or resonator - is decided by intrinsic qualities i-e.: locally, wherein combinations of charge-field coupling, geometric form factor, and spectral form factor, must be chosen to give the desired final mass. (The existence of these energy-storing

ORIGIN OF ELECTRON INERTIA

485

'degrees of freedom' rests upon the presumed existence of a structure for the particle in question - points 1 and 2 - and neither this structure nor these coupling and form factors are explained.) 4. There is no clear path of development for the theory by which it can unite the inertial and gravitational aspects of mass. There ir the hope that it also has something to do with the EM ZPF, as first suggested by Sakharov [21,22]. But to date there have been no successful implementations of a ZPF origin for gravity. Perhaps the most attractive feature of the current thinking along these lines is that the proposed energetic source by which means both gravity and inertia may perhaps be united - the ZPF - is a 'ready-made', omnipresent, influence. In the next section is investigated a different implementation which retains this foundational feature, but which overcomes some of the above enumerated difficulties.

3.

A Model For Non-Locally Determined Inertial Mass

By non-local model is here meant that inertia is conceived not as an intrinsic, unitary property, but as arising out of a non-local rnutual interaction. Just as the mutual interaction energy of charges, current elements, and (gravitating) masses cannot be assigned to either partner in the interaction, so - it is suggested - inertial mass-energy cannot be ascribed to a single particular particle, but results from the multiple mutual (pair-wise) interactions with distant partners. Like the foregoing examples, the Casimir and van-der-Waals energies are also mutual, yet these are different in that they exist only by virtue of the ZPF. Though an apparently intrinsic Casimir energy does exist for a conducting curved surface embedded in the ZPF [23], one may regard this energy as arising out of the mutual interaction of local elements of the curved structure, just as in a Casimir cavity. To date, no one has identified a mutual yet distant interaction energy of electromagnetic origin that can explain inertia. And this is the reason why the ZPFinertia advocates have concentrated on local, 'unitary' qualities that might cooperate to localize ZPF energy. A detailed description of the distant interaction believed to be responsible will be given in a future document. Briefly, a consequence of that work is that, like the local models, the positional / motional particle degree of freedom may be regarded as 'energized' by the ZPF. But unlike local models, the energy of interaction turns out to be mutual, involving all distant particles. A good metaphor is provided by van-der-Waals binding energy, except that the rate of radial fall-off precludes it from candidature. For now, the following calculation is presented as evidence to support the claim that inertial mass is a non-local energy - with the ZPF as its rrzeans.

4.

Derivation Of The Electron Mass

In the following calculation it will be assumed that associated in some way with a charged particle is a resistance to acceleration equal to a final renormalized (not bare) inertial mass me.It will be assumed that underlying this mass is a particle in micromotion (commonly, but not very accurately termed zitterbewegung). We also need to

486

M. LBISON

assume that an EM noise field and the micro motion of the charge are consonant in that the 'in' fields impacting the source and the 'out' fields leaving the source have the same statistics. In other words, it will be assumed that the EM noise field has attained a selfconsistent state in the presence of the particle micro-motion. (A causal flow is not implied here: the particle's motion does not cause the field nor does the field cause the particle's motion. Rather, the field and motion are to be viewed as mutually consistent) So far this sounds rather like the SED program that results in a ZPF-energized, but nonetheless intrinsic-valued, inertia. However, the particle employed here is deemed to have no intrinsic structure, and therefore cannot have an intrinsic-valued inertia - with or without the ZPF. Very broadly, this can be concluded simply from the absence of any length scale that could conceivably be associated with a mass. More specifically, it can be shown that a massless classical point charge dropped into the ZPF leaves the electromagnetic spectrum and energy density unchanged from that of the charge and ZPF considered apart from each other - unless the charge is permitted to interact with other charges. It must be admitted at the outset that the particle that will be singled out by this calculation is the electron. This is because it is the lightest charged particle. As a consequence it gives the largest acceleration per unit field, and therefore the largest out field per unit in field. It follows that, provided the in and out fields are universally selfconsistent, the fields must be maintained predominantly by the electron. Therefore, in the following calculation, it will be assumed that N loR0electrons, sprinkled approximately uniformly throughout the visible universe, cooperate in the maintenance of an EM noise field. Further, it can reasonably be assumed that the micro-motion will have a coherence length somewhere between the classical electron radius and the Compton wavelength. From this it follows that, although widely varying, the local environment of electrons is to a good approximation of no consequence to the presumed micro-motion, since the coherence length of the latter is relatively so shoa In this paper the self-consistency calculation will be simplified by assuming that the micro-motion is non-relativistic. For this to be true for all electrons from the perspective of our own earthly reference frame, we must necessarily consider only a static universe - i.e. without expansion. This is because the electrons near the Hubble radius will turn out to dominate the self-consistent field calculation. (A more complicated calculation admitting expansion gives a very similar result, as discussed below.) With this restriction the electromagnetic noise-induced acceleration is approximately (c=4m0=1)

-

a ='in

(elme)

(1)

for which the outgoing radiation, in the far-field, has electric field

The corresponding 3D orientation averaged energy density from both the magnetic and electric fields, as viewed in the frame in which the particle's expected position is always at the origin, is

OFUGLN OF ELECTRON INERTIA

487

In the presumed static cosmology there are N sources approximately uniformly distributed throughout the universe of static radius R. i-e.. p(r)d3r= ( 3 ~ / 4 r r R ' ) & r . The expected energy density from all the sources is therefore (&NP)

= Jd3rp(r)(&, (r, f)) =

3N e2a2

R

4r

ldrr2 o

IVe2a2

=.

2nR2

(4)

By contrast, the energy density of the in field at the particle in question can be expressed in terms of the acceleration using Eq. (I):

For self-consistency the energy density of the in field must, at all locations, equal the energy density due to all the out fields:

Consequently, one obtains the Dirac large number hypothesis [I61

me= & ? e 2 / ~ .

(7)

With N =loE0, and R set to the Hubble radius of 1 0 ~ ~ c m this , computes to kg, i.e. 40% of the observed value of the electron mass - well within the 0.36 x tolerance set by the uncertainty in (which expressed as a factor is between about 0.5 and 2).

5.

Discussion

The above calculation establishes a linear relations p between the in and out fields. It follows directly that it does not matter how strong or weak is the noise field: the electron mass given by Eq. (7) would have the same value whatever. Another consequence of the linearity is that the computed electron mass is also insensitive to the energy spectrum of the ZPF. (A consequence of the fact that the charge-field scattering is elastic.) A relativistically correct version of this calculation performed in a flat expanding universe turns out to give, apart from a numerical coefficient of order unity, the same Dirac relation, and therefore the same electron mass as a function of cosmological constants, with the former remaining independent of time. Specifically, taking into account the special role played by the ZPF in maintaining homeostasis, the second Dirac hypothesis [I61- that N1'*e2!Ris constant - can be validated. Dirac's suggestions seem to have been rejected mostly on the basis of his first relation, which predicts a

time-dependent gravitational constant, and is considered to be incompatible with observation; see [24] and [25] for reviews. Although not directly impacting the validity of the second relation under investigation here, it is nonetheless interesting that similar ZPF-mediated arguments have led Puthoff [25,27] to claim a time-independence for the first relation. It is hoped to reproduce elsewhere the detailed calculations and qualifying cosmologies discovered to maintain constancy of the second relation. In those calculations it turns out that appeal must be made to velocity-invariant statistics of the EM noise field. That is, the self-consistent field must be, at least at the level of expectations of quadratic field operators, the electromagnetic zero-point field as furnished by second-quantized Maxwell (henceforth the ZPF). The reader may be alert to the fact that a proper relativistic treatment necessitates the use of the Lorentz-Dirac equation with non-linear radiation reaction terms, or - in the quantum domain - the corresponding Heisenberg equation of motion [26].This, and related issues concerning the bandwidth of the self-consistent field and the origin of time asymmetry, require a much more detailed treatment, and will also be addressed elsewhere. A concern expressed by some is that the cosmological distribution of matter is such that any alleged derivation of particle constants from cosmology will suffer from an unacceptable level of frame dependent, or perhaps time dependent, variability. A rough estimate of the variability in the predicted value of mass is to entertain fluctuations in N, which are likely to be of order f 1 2 . This gives rise to corresponding fluctuations in N'I2 of order unity, i.e. one part in lo4'. Therefore, at least by this mechanism, cosmological variability does not lead to a detectable variability in the mass. It must be emphasized that nowhere in the above was inertial mass 'explained'. Rather, this calculation tells us only that if the fields are to be self-consistent, the electron mass could not have any value other than the one it is observed to have, given the cosmological numbers. The calculation does not explain the mechanism of the mass. Despite these caveats, the success of the calculation provides support for the novel emphasis placed on the ZPF by Haisch and others in their work on inertial mass. This work continues that effort, but with a different role for the ZPF. Here, the ZPF is the means by which homeostasis is maintained; it is the means by which the electrons throughout the universe come into electromagnetic equilibrium with each other, whereby the electron mass attains universal consistency.

6.

Cosmological Origin Of Length-Scale

In natural units where e = 1, mass has units of (length)-'; the classical length corresponding to the electron mass (the classical electron radius) is, in S.I. units, e?(4n~~rn~~), which is about 3 x 10'15 m. In the previous calculation based upon , cosmological self-consistency this length is found from Eq. (7) to be R / ( ~ N ) ' ' ~where both R and N are cosmological constants. Despite all the talk of electromagnetism and ZPF-induced micro-motion, the final result constructs a very small length out of cosmological constants. If the cosmic somehow determines this length, as implied in this work, then there should be a direct cosmological interpretation for this very small length, without any reference to electrons. This is the focus ofthe following discussion.

489

ORIGIN OF ELECTRON INERTIA

Notice that the integral in Eq. (4) may be regarded as computing the expectation of

r-2over the Hubble volume. For a homogeneous distribution,