Foundations of Physics Letters, VoL 8, No. 5, 1995

THE EVANS-VIGIER FIELD, B(3): DERIVATION OF THE DE BROGLIE MATTER-WAVE EQUATION FROM THE HAMILTON-JACOBI EQUATION

M. W. Evans

Department of Physics University of North Carolina Charlotte, North Carolina 28223

Received August 8, 1994

The emergence of the Evans-Vigier f ie ld S (3} of vacuum electromagnetism has been accompanied by a novel charge quantization condition inferred from 0(3) gauge theory. This finding is used to derive the de Broglie matter-wave equation from the classical Hamilton-Jacobi (HJ) equation of one electron in the electromagnetic f ie ld . The HJ equation is used with the charge quantization condition to show that, in a perfectly elastic photon-electron interaction, complete transfer of angular momentum occurs self-consistently, and the electron acquires the angular momentum ~ of the photon. In this l imi t the electron travels inf in i tesimal ly near the speed of l ight , and i ts concomitant electromagnetic f ields become indistinguishable from those of the uncharged photon. This result independently proves the va l id i ty of the charge quantization condition and demonstrates unequivocally the existence of the vacuum f ie ld B (3)

Key words: de Broglie equation, B (3) f ie ld .

l . INTRODUCTION

I t is well known that an electron translating in f in i - tesimally near the speed of l ight is accompanied by electro- magnetic f ields which are indistinguishable [1,2] from those accompanying the photon. In this Letter, we consider an electron accelerated to re la t i v i s t i c velocit ies by a photon, and in so doing, derive the quantized de Broglie matter-wave

0894-9875/9511000-048|$07.50/9 ¢~ 1995 Plenum Publishing Corporation

482 Evans

equation [3] from the classical but correctly re lat iv is t ic Hamilton-Jacobi equation of the electron in the electromag- netic f ield. The derivation depends on a novel charge quantization condition which is indicated by the novel Evans- Vigier f ield B (~). Recently, i t has been shown [4] that the interaction of one electron with the electromagnetic f ield is described exclusively through an interaction Hamiltonian made up of the product of intrinsic electron spin angular momentum and the newly discovered [5-12] f ield B (3) of vacuum electro- magnetism. Similarly, the induction of orbital angular momentum in the electron is governed entirely by B (3) [6], acting at f i r s t and second order in B (°), the magnetic flux density amplitude of the beam. B (3) therefore is an experi- mental observable because, i f i t were zero in the vacuum, the intrinsic and orbital electron spin could not form an interaction Hamiltonian with the electromagnetic f ield, and fundamental terms in the Dirac and Hamilton-Jacobi equations [6] of e in A~ would be zero incorrectly. In classical, re lat iv is t ic mechanics, i t can be shown [6,11] that the principle of least action leads to a characteristic I~/2 pro- f i le through which B (3) can be isolated unequivocally from the concomitant plane waves B (~) =B (2)" of vacuum electro- magnetism. Here I o is the power density of the beam (in W

mr), and the magnetization •(3) produced in an electron plasma by the magnetic flux density B (3) is proportional to TI/2 from the re lat iv is t ic Hamilton-Jacobi (HJ) equation ~0

[6] in the limit

~< eB ( ° l • (1) / I I o

Here ~ is the beam angular frequency (rad s'), and e/m o is the charge to mass ratio of the electron. The condition (I) can probably be achieved experimentally with a straight- forward modification of the apparatus used by Deschamps et al. [13], a modification designed to increase the peak 3 GHz, microwave pulse power by a factor of at least f i f t y [12]. In the classical re lat iv is t ic theory, however, as in its quantum counterpart, the spinning motion of one electron in the beam is governed entirely, under al l conditions (and not only in the limit ( i ) ) , by s (3), and by no other f ield component.

Therefore B (3} is a new fundamental f ield of vacuum electromagnetism, the f i r s t to be inferred [4-12] since

Evans-Vigier B (3) Field 483

Maxwell. In Sec. 2, the charge quantization condition is used to

derive the de Broglie matter-wave equation [3] from the classical Hamilton-Jacobi {HJ) equation. In Sec. 3, this result is used to write the re lat iv is t ic factor of the HJ equation in terms of a photon-electron collision rather than a classical charge-field interaction; and thereby to show that, in the l imit of a perfectly elastic collision between photon and electron, the latter self-consistently acquires an orb i ta l angular momentum ~ from the former. In th is l im i t the electron is accelerated to the speed of l i gh t and i ts concomitant electromagnetic f i e lds are ind is t inguishable from those of the uncharged photon. Therefore, the charge quant izat ion condi t ion, derived [12] independently using the fundamental theory [14] of gauge geometry, leads to a f u l l y con'sistent resu l t , and provides a l ink between the c lassical HJ equation and quantized de Brogl ie matter-wave equation.

2. DERIVATION OF THE DE BROGLIEMATTER-WAVE EQUATION WITH THE CHARGE QUANTIZATION CONDITION

By considering the classical, re lat iv is t ic , motion of one electron in the electromagnetic f ield, the HJ equation can be used [6,11] to show that the induced orbital elec- tronic angular momentum of the electron is governed entirely by the Evans-Vigier f ield as follows:

(2)

Here ~ is the beam angular frequency and c the speed of light in the vacuum for a hypothetically massless photon. The de Broglie matter-wave equation emerges from Eq. (2) from a consideration of the re lat iv is t ic factor

(a)

For vacuum electromagnetic radiation without mass, the flux density magnitude B (°~ is related to the magnitude of the vector potential, A ~°), by

484

BI°l = ~A(o l =KAsol,. (4) c

so the classical re la t i v i s t i c factor becomes

En 2 = c~y 2 = m~c 4 + c2(e2A(O)~), ( 5 )

where we have mul t ip l ied both sides by the factor c ~. The uni ts of both sides are those of energy squared, because y , from basic HJ theory [6 ] , has the uni ts of l inear momentum.

Equation (5) is now quantized by invoking the usual quantum hypothesis for energy

En = he. (6)

The charge quantizat ion condit ion [6]

eA (o) = ~K (7)

is used for the right-hand side. I t has been shown [6] that Eq. (7) is the result of 0(3) gauge theory [14] applied to vacuum electromagnetism. The 0(3) gauge group is necessary because the conventional 0(2) gauge group is incompatible with the existence of the physical f ie ld B (3} orthogonal to the plane of definit ion of 0(2), the group of rotations in a plane. On the other hand, the three-dimensional group of rotations, 0(3), used in fundamental gauge theory (and non- Abelian electrodynamics [6]), incorporates B (3) self consis- tently, and produces Eq. (7) as a logical consequence.

Using Eqs. (6) and (7) in (5) produces the de Broglie matter-wave equation

~2 m~c4 - + c2K~, ( 8 ) ~2

which is the Klein-Gordon equation, a quantized version of the Einstein equation for energy in special re la t i v i t y .

I f Eq. (8) is to be regarded as a matter-wave equation for the electron of mass m o, then the electron must have acquired a linear momentum ~ from the photon ( i .e . , the quantized electromagnetic f ie ld) in a col l is ion. This inference reveals the physical meaning of the vacuum charge

Evans-Vigier B ()) Field 485

quantization condition, in which the classical momentum eA (°~ introduced by the gauge is identified with quantized photon momentum ~K in the vacuum, and in which e is the scaling constant [14] of 0(3) gauge theory as well as the charge on the electron. I t is now clear that this procedure identifies the re lat iv is t ic factor of the HJ equation as the de Broglie matter-wave equation. Obviously, both e and A (°) are negative [6] quantities. The rest energy of the electron, moc ~, is unaffected by this collision. (Equation (8) can also be one for the photon i f m o is regarded as the photon mass rather than the electron mass. The existence of the Evans-Vigier f ield implies that the photon mass is f in i te [6,15], and this has been shown to be compatible with gauge invariance using spontaneous symmetry breaking of the vacuum.) However, in this context, m o is the electron mass, so Eq. (8) must describe the matter-wave of the electron, a plane wave in the vacuum.

2. CONSERVATION OF ANGULAR MOMENTUM IN A PHOTON-ELECTRON COLLISION

The HJ equation (2) can be rewritten, on using the de Broglie matter-wave equation (8), as

j(3) = ~K ec----~2 • B (~)

(mock+ ~2K2)-~ 0 ̀)2 (9)

in which i t is clear that B (3) is s t i l l an electromagnetic f ield property, and in which ~ is s t i l l the angular frequen- cy of both the f ield and the electron, but in which all other quantities are particulate in nature, properties of the electron rather than that of the field, because we have argued that Eq. (8) must be for the electron's matter-wave. However, since Eq. (8) was derived from Eq. (2), they must describe the same physical process. This can only be so i f the electron has acquired the linear momentum ~K from the photon of the quantized field. On taking the l imit

~K >moC, ( I0)

Eq. (9) reduces to

486 Evans

j(3) _. e c--~2B(3), (ii) (0 2

so the induced orbital angular momentum of the electron is directly proportional to the Evans-Vigier f ield. This produces [6] the characteristic and observable I{/2 depen- dence described in the introduction. The orbital angular momentum acquired by the electron is not there in the absence of an electromagnetic f ield, and so must have been given up to the electron by the field. In the limit (10), this process is experimentally proportional to I~/2. The limit (10) describes that in which ~K, acquired from the photon, is much larger than the electronic rest momentum moC, but the latter is never zero because rest energy never vanishes in special re lat iv i ty.

In Eq. (11), the ratio ~ / c cannot therefore be identi- fied with the wave-vector magnitude K, because moC is not zero. This equation can be checked for self-consistency, however, i f i t is asserted mathematically that moC is identically zero, when and only when K = o / c . On using B ~°) =v~ (°) , Eq. (11) then becomes

j(3) _ eA (o_~) e (-~) . (12) K

The charge quantization condition (7) Finally reduces this equation to

,/(3) = h e ( 3 ) (13)

in which J(~) is the angular momentum of the photon, whose magnitude is ~, the Dirac constant. The maximum orbital angular momentum that the electron can attain is ~, the angular momentum of the photon, but i f and only i f the electron rest momentum moc is ignored. In classical, non- relat iv ist ic , physics, there is no rest momentum, and so the HJ equation reduces to a description in which angular momentum is imparted elastically by the photon to the electron. Angular momentum is conserved. In the same condi- tion, linear momentum is imparted elastically, and ~K of the photon becomes ~K of the electron. I f and only i f moC is zero, the latter becomes the total linear momentum of the

Evans-Vigier B (3) Field 487

electron. This result is achieved through the charge quanti- zation condition (7). This identifies the classical expecta- tion value of ~K of the photon as eA (°) Similarly, in the expression E n = ~ , the classical energy En is the expecta- tion value of the photonic quantum of energy ~ . Because of the irremovable rest-momentummoc of the electron, the photon can never give up all its angular momentum to the electron, and the collision can never be perfectly elastic in the classical, non-relativistic, sense.

3. DISCUSSION

I f the Evans-Vigier f ield B (3) were zero, no orbital angu.lar momentum could be transferred from the photon to the electron, a reduction to absurdity. In experimental terms, there could be no magnetization of an electron plasma by a microwave pulse, in contradiction with experience [13]. Furthermore, the electron's intrinsic (or spin) angular momentum could not interact with the electromagnetic field, and since spin angular momentum is proportional to a magnetic dipole moment [16], this is another reduction to absurdity, because a magnetic f ield always forms an interaction Hamilto- nian with a magnetic dipole moment. This inference is a direct result of the Dirac equation of one electron in the electromagnetic f ield, from which analysis i t can be shown [6] that the interaction Hamiltonian is a direct product of the intrinsic electron spin with the Evans-Vigier f ield B (3) Such a process is also proportional to !~/2 and is observable experimentally, in principle, because i t is a type of Zeeman effect [16].

I t is well known that there is a close relation between the HJ and Dirac equations, the classical HJ Hamiltonian being

Hera,s_ I ]I.]][- eA(°)c, (14) 2rn o

H := p + eA. (14a)

The Dirac equation is, essentially, [14] the quantum mechani- cal equation

488 Evans

Wu=I(°"I~2-eA(°)c]u,2mo (15)

where W= En-mo c~, and where u is a spinor component. In Eq. (15), o is a Pauli matrix, as is well known. The term

(o']~2 = (p+ em) 2 + io'Cp+ eA) x (p + eA) (16)

produces the in t r ins ic spin angular momentum of the electron, which has no classical meaning, because i t is obtained from a commutator whose classical value is zero.

We conclude that the Evans-Vigier f ie ld B TM is the fundamental f ie ld of magneto-optics and governs ent i re ly the induced (angular) and permanent ( i n t r i ns ic ) angular momenta of one electron in the electromagnetic f ie ld . Therefore i t governs magnetization in a l l matter by the electromagnetic f ie ld .

ACKNOWLEDGEMENTS

I t is a pleasure to acknowledge many interesting discussions with several colleagues, among whom are: the late Stanis~aw Kielich, Mikhail A. Novikov, Sisir Roy, and of course, Jean-Pierre Vigier.

REFERENCES

[1] W. K. H. Panofsky and M. Phill ips, Classical Electr icity and Magnetism, 2nd edn. (Addison-Wesley, Reading, Massachusetts, 1962).

[2] J. D. Jackson, Classical Electrodynamics (Wiley Inter- science, New York, 1962).

[3] A. O. Barut, ElectFodynamics and Classical Theory of Fields and Particles (Macmillan, New York, 1964).

[4] M. W. Evans, Found. Phys. Lett. 7, 209 (1994). [5] M. W. Evans, Physica B 182, 227, 237 (1992); 183, 103

(1993); 190, 310 (1993). [6] M. W. Evans and J.-P. Vigier, The Enigmatic Photon,

Volume I: The Field B TM (Kluwer Academic, Dordrecht, 1994); The Enigmatic Photon, Volume 2: Non-Abelian Electrodynamics (Kluwer Academic, Dordrecht, 1995).

Evans-Vigier B (~) Field 489

[7] M. W. Evans and A. A. Hasanein, The Photomagneton in Quantum Field Theory (World Sc ient i f i c , Singapore, 1994).

[8] M. W. Evans, and S. Kiel ich, eds., Modern Nonlinear Optics, Vols. 85(1), 85(2), and 85(3) of Advances in Chemical Physics, I . Prigogine and S. A. Rice, eds. (Wiley Interseience, New York, 1993/1994).

[9] M. W. Evans, The Photon's Magnetic Field (World Scien- t i f i c , Singapore, 1992).

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[12] M. W. Evans, Found. Phys. Lett. 8, 363 (1995). [13] 'J. Deschamps, M. F i ta i re , and M. Lagoutte, Phys. Rev.

Lett. 25, 1330 (1970); Rev. Appl. Phys. 7, 155 (1972). [14] L. H. Ryder, Quantum Field Theory, 2nd edn. (Cambridge

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(Oxford Universi ty Press, Oxford, 1983).