[advances in imaging and electron physics] volume 101 || the dual de broglie wave

ADVANCES IN LMAGING AND ELECTRON PHYSICS . VOL . 101

The Dual de Broglie Wave

MARCIN MOLSKI

Deparfment of Theoretical Chemistry. Faculfy of Chemistry. Adam Mickiewicz Universify Grunwaldzka 6. PL 60-780 Poutah. Poland

I . Introduction . . . . . . . . . . . . . . . . . 11 . Wave-ParticleModelsofMassiveParticles . . . . . .

A . De Broglie-Bohm Wave Theory . . . . . . . . . B . Mackinnon Soliton . . . . . . . . . . . . . C . Jennison-Drinkwater Electromagnetic Theory . . . . D . Corben Tachyonic Theory . . . . . . . . . . . E . Horodecki-Kostro Model and the -0-Wave Hypothesis F . Das Model and Pseudovelocity . . . . . . . . .

G . ElbazModelandLorentzTransformations . . . . . H . Generalized Barut Approach . . . . . . . . . . I . Conclusions . . . . . . . . . . . . . . . .

111 . Wave-Particle Model of Photons . . . . . . . . . . A . Photon as a Bradyon-Tachyon Compound . . . . . B . Conversion of Light into B- and D-Waves . . . . . C . lko-WaveModelofLongitudinalPhotons . . . . .

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. . . . . . . . . 165 D . Massless Photons and Ponderable Matter . . . . . . . . . E . Extended Proca Theory . . . . . . . . . . . . . . . . F . Conclusions . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . A . Three-Dimensional Rectangular Space Cavity . . . . . . . . B . Three-Dimensional Spherical Space Cavity . . . . . . . . . C . One-Dimensional Linear Time Cavity . . . . . . . . . . . D . Three-Dimensional Spherical Time Cavity . . . . . . . . . E . Two-Dimensional Square Space-Time Cavity . . . . . . . . F . Conclusions . . . . . . . . . . . . . . . . . . . .

V . Extended Special Relativity and Quantum Mechanics in a Local L-Space A . Special Relativity in L-Space . . . . . . . . . . . . . . B . Quantum Mechanics in L-Space . . . . . . . . . . . . . C . Conclusions . . . . . . . . . . . . . . . . . . . .

. . . A . Kaluza-Klein Field Theory . . . . . . . . . . . . . . B . Charged Particle as a Five-Dimensional Tachyonic Bootstrap . . C . Conclusions . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . A . Tachyo-kinematic Effect . . . . . . . . . . . . . . . B . Particle on a Line . . . . . . . . . . . . . . . . . . C . Particle in a Scalar Field . . . . . . . . . . . . . . . D . Uncertainty Principle . . . . . . . . . . . . . . . . E . Many-Body Problem . . . . . . . . . . . . . . . . .

IV . Electromagnetic Model of Extended Particles

VI . Two-Wave Model of Charged Particles in Kaluza-Klein Space

VII . Extended de Broglie-Bohm Theory

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143 Copyright @ 1998 Academic Press . Inc . All rights of repduction in any form reserved .

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F. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 230 VIII. Infons? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

IX. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 232 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . 234 References . . . . . . . . . . . . . . . . . . . . . . . . . . 234

swiatlu zalamanemu w krysztale czasu To the light refracting in the crystal of time W. Sedlak (1985)

I. INTRODUCTION

The idea that not only an ordinary de Broglie wave (B-wave) but also a matter wave of the second kind is associated with a moving particle has a long and fascinating history. The first suggestion of this matter emanated from Louis de Broglie, who between 1924 (the date of publication of his doctoral thesis) and 1927 developed a theory of double solution known also as a causal theory, or, in its simplified form, as a pilot-wave theory (de Broglie, 1960, p. 97).

According to this concept, we associate with a uniformly moving particle waves of two types (de Broglie, 1960, p. 99): a probabilistic B-wave of continuous amplitude and subjective character, and a singularity wave of finite amplitude and objective character, centered about a point (or a pointlike region) at which the associated particle is localized. The B-wave, commonly employed in conventional wave mechanics, represents no physical reality and has only statistical meaning, whereas the singularity wave is a true physical representation of a physical entity, a purricle. This particle is localized in space as in the classical picture, but is also incorporated in an extended wave phenomenon. According to the phase connection principle (de Broglie, 1960, p. 99) the B-wave has the same phase as the singularity wave; one may then state that a particle during its motion is guided in space-time by the B-wave playing a role of a pilot wuve.

The theory of a double solution was extended by Bohm (1952a,b, 1953), Bohm and Vigier (1954), Bohm and Bub (1966a,b), Bohm and Hiley (1973, and Bohm et al. (1987) to the form known as the theory of hidden variubles, which with de Broglie’s causal theory laid the basis for the Paris school of interpretation of quantum mechanics, competitive with the Copenhagen school (Bohr, Born, Heinsenberg, and others). According to the former authors, the fundamental constituents of matter (e.g., electrons, protons, neutrons, etc.) are both waves and particles following continuous and causally determined trajectories. Thus they are composed of a particle satisfying certain equations of motion and a wave satisfying a SchrWnger equation. Both particle and wave (represented by the wave function) are taken to be objectively real whether or not they are observed. In particular, Bohm and Vigier (1954) suggested that a particle can execute a random movement (resembling Brownian motion) as a result of interactions with a subquantum level characterized by unknown (hidden) parameters. In

THE DUAL. DE BROGLIE WAVE 145

this connection the probability density for the particle to be at a certain position is a steady-stare distribution ultimately resulting from random movements of the particle.

This has raised the possibility that quantum mechanics can be understood essentially in terms of ontological assumptions concerning the nature of individual systems so that the epistemological and statistical content would then take on a secondary role as in Newtonian mechanics. ( B o b et al., 1987)

Contrary to the Paris school, proponents of the Copenhagen interpretation assume that there is no way to describe or to understand any physical microprocess or microentity in intuitive and imaginative terms as we must use classical language and classical concepts (as a result of our macroexperience) to do so. Hence quantum theory yields only statistical predictions of results of measurements of quantum systems (Bohr, 1934, 1958). In particular the square of the modulus of the wave function associated with a particle is interpreted as the density of the probability of finding the particle at a point in space-time. Despite critical opinions by de Broglie, quoted below, this approach has been intensively developed since 1926, and is dominant in the scientific community.

As the work of other scientists led to further progress in wave mechanics, it became daily more evident that the @ wave (B-wave) with its continuous amplitude could be used only in statistical predictions. And so, little by little, there was an increasing trend towards the purely probabilistic interpretation, of which Born, Bohr and Heisenberg were the chief advocates. I was surprised at this development, which did not seem to me to fulfill the explanatory aim of the theoretical physics .... (de Broglie, 1960, p. V) The @ wave (B-wave) usually employed in wave mechanics cannot be a physical reality; its normalization is arbitrary; its propagation, in the general case, is supposed to take place in an obviously fictitious configuration space, and the success of its probabilistic interpretation shows clearly that it is merely a representation of probabilities dependent upon the state of our knowledge and suddenly modified by every new piece of information. (de Broglie, 1960, p. 91)

Further investigation in the field, leading directly to the concept of a dual de Broglie wave, was undertaken by Kostro (1978, 1985a,b, 1988), who proposed a three-wuve particle model in the framework of the Paris interpretation of quantum mechanics and Einstein’s theory of relativity, in which the existence of a covariant erher is assumed. According to Kostro’s model, one assumes that three waves are associated with a moving particle: one internal nondispersive standing Compton wave (C-wave) and two external ones, one ordinary B-wave having a superluminal phase velocity, and a wave of the second kind endowed with a subluminal phase velocity. The C-wave is responsible for inertial properties of ponderable objects, whereas a matter wave of the second kind is assumed to result from interaction between Einstein’s hidden subquantum ether (excited by the B-wave) and a particle associated with C-wave. The extra wave is supposed to be a carrier of the

146 MARCIN MOLSKI

particle inertia (connected with the C-wave) through space-time in which the way is prepared for it by the pilot B-wave (Kostro, 1985a.b).

Kostro’s model was modified and extended significantly by Horodecki ( 198 1, l982,1983a,b, 1984, 1988a,b,c, 1991). who introduced to the wave theory of matter the notion of a dual de Broglie D-Wave (Horodecki, 1981). According to this concept, a matter wave of the second kind (D-wave) appearing in Kostro’s approach is assumed to have a subluminal phase velocity equal to the B-wave group velocity identified with the classical velocity of an associated particle; the group velocity of the D-wave is assumed to be superluminal and equal to the phase velocity of the B-wave, a cross correspondence. Consequently, according to the wave picture a moving massive particle is represented by a nonlinear nondispersive wave packet (called a C-wave) that involves an internal characteristic spectrum of matter waves, including B- and D-waves. Because the C-wave may be considered a nonlinear superposition of B- and D-waves, according to Horodecki’s model the number of independent waves associated with a moving particle decreases to two, providing a conceptual background for a two-wave particle model, a nonlinear wave hypothesis, and an extended space-time description of matter (Horodecki, 1988a, 1991). According to the latter concept, a massive particle in motion is described approximately in terms of a timelike component, characterized by a timelike four- momentum associated with a B-wave and a spacelike component characterized by a spacelike four-momentum associated with a D-wave (Horodecki, 1988a, 1991). Because a group velocity of the D-wave according to Horodecki’s model is assumed to be superluminal, the spacelike particle constituent becomes endowed with typical tachyonic properties (Horodecki, 1988a).

A two-wave particle model including B- and D-waves was investigated by Das (1984.1986, 1988, 1992) and Elbaz (1985, 1986,1987, 1988); they started from two points representing the corpuscular and wave aspects of the problem.

Exploiting the fundamental concept that “the particle picture and the wave picture are merely two aspects of one and the same physical reality” (Jammer, 1974), Das (1986,1988) pointed out that the correspondences

energy t) frequency momentum t) wavelength

velocity t) group velocity ? t) phasevelocity,

familiar for particle-wave duality, are incomplete, as they fail to include parameters of like number in the particle and wave pictures. To restore parametric symmetry, Das (1986, 1988) assumed that a moving massive particle might be endowed not only with an ordinary velocity but also with a nonobservable superluminal pseudovelocity equal to the phase velocity of the associated B-wave. Because a particle having a pseudovelocity must also have a nonobservablepseudomomentum and pseudoenergy, and because these must be associated with a pseudowave via

THE DUAL DE BROGLIE WAVE 147

extended wave-particle corresponences, Das deduced a matter wave of a new type called the transformed Compton wave. Detailed analysis indicated that it was endowed with the same wave characteristic as the D-wave of Horodecki’s model.

A similar incompleteness, but in conventional wave theory, was found by Elbaz (1983, who considered transformational properties of a B-wave phase under a Lorentz boost. He realized that according to de Broglie’s theory, a B-wave function of a moving particle is derived by means of a time Lorentz transformation to a timelike periodic element associated with a particle at rest (de Broglie, 1960, p. 4). Proceeding analogously, one may derive a matter wave of the second kind, applying a space Lorentz transformation to the spacelike periodic element associated with the same particle. The amplitude wave (Elbaz, 1985) obtained in this way has wave characteristics identical to those of the transformed Compton wave of Das’s theory and the D-wave of Horodecki’s model.

Because attribution ad hoc of a spacelike periodic element to a particle at rest is artificial and groundless form a physical point of view, Elbaz (1985) assumed that a photonlike standing wave, being a superposition of purely timelike and spacelike waves, is associated with a particle at rest . This system of waves under time and space Lorentz transformations produces B- and D-waves associated with a uniformly moving particle.

All these concepts gained strong support in independent investigations by Mack- innon (1978, 198 la,b, 1988), Jennison and Drinkwater (1977), Jennison (1978, 1983, 1988), Jennison er al. (1986), Corben (1977, 1978a,b) and Recami (1986,

On the basis of de Broglie’s phase connection principle and a particular superposition of de Broglie waves, Mackinnon (1978,198 la,b) constructed anondispersive wave packet that fails to spread with time and constitutes a particlelike solitary wave endowed with characteristics identical to those of a C-wave according to Horodecki-Kostro theory. Jennison (1983) showed that Mackinnon’s soliton represented a massive particle having a sharp and finite boundary and was entirely consistent with a model of an electromagnetic phase-locked cavity (Jennison and Drinkwater, 1977; Jennison, 1978, 1983, 1988; Jennison er al., 1986). According to this concept, the inertia of all finitely bounded material particles originates in an echo effect of a feedback process occurring for c-velocity waves, which is intrinsic to phase-locked particles (Jennison, 1983). The electromagnetic model of a massive particle proposed by Jennison is compatible with a fundamental assumption (Elbaz, 1985) that there is a photonlike standing wave associated with a particle at rest.

Controversial ideas of a superluminal pseudovelocity proposed by Das, an extended space-time description of matter developed by Horodecki, and a dual de Broglie wave appearing in all models considered are consistent with a tachyonic theory of elementary particle structure developed by Corben (1977, 1978a,b). Ac- cording to this concept, a moving particle is considered to be a composite object comprising both a bradyonic component (timelike, slower-than-light particle) and

p. 111).

148 MARCIN MOLSKI

a tachyonic component (spacelike, faster-than-light particle’) associated with a system of time- and spacelike waves having similar, and under some conditions (Molski, 1991, 1993a) the same, wave characteristics as those of B- and D-waves according to the Horodecki-Kostro, Das, and Elbaz models.

A complete synthesis of all these approaches is presented here. The two-wave particle model, an extended space-time description of matter, a nonlinear wave hypothesis, a tachyonic theory of elementary particle structure, and an electromagnetic concept of matter become unified to yield a consistent wave-corpuscular model of both massless photons and massive particles.

We are concerned with the formulation of extended special relativity and quantum mechanics in a local L-space, which seems the simplest and most natural way to introduce tachyon mechanics and a D-wave concept. Moreover, we consider an internal structure of charged particles and reinvestigate standard problems of quantum theory in the framework of extended de Broglie-Bohm theory, including a D-wave concept and a tachyonic theory of matter.

11. WAVE-PARTICLE MODELS OF MASSIVE PARTICLES

To increase our insight into wave-particle models of matter presented in the preceding section, we consider their brief mathematical formulation, confined to fundamental information in the field.

A. De Broglie-Bohm Wave Theory

The starting point of all models considered is a wave-particle dualism of matter represented by the correspondence

in which

pw = ( E / c , p) = mod’, (2)

up = ( y c , yv), y = (1 - p ) - ” 2 , p = v/c, (3)

k p = (w/c , k), (4)

w = 2nv, k = 2n/A. ( 5 )

In the abundant literature on tachyons, as an introduction we recommend papers by Bilaniuk er al. (1962), Bilaniuk and Sudarshan ( I 969). and Kreisler ( I 973); experimental aspects of the problem were considered by Feinberg (1970), Meszaros (1986). and Clay (1988). Resources for this subject are presented by Feldman (1974). Perepelitsa (1986), and Recami (1986).


In these equations p, denotes the timelike four-momentum of a particle endowed with rest mass mo, moving in four-dimensional space-time x p = (ct, x , y, z) of signature + - -- at velocity v ; k@ is the wave four-vector of an associated B-wave of frequency u and wavelength A.

These four-vectors fulfill the invariant relations

(6) 2 2 P,P@ = moC ,

V,V@ = c2, k,k, = (%/c)’, (7)

in which WO/C = moc/h = ~ I T / A o ; A0 is the Compton wavelength, and the B-wave is additionally characterized by a group (phase) velocity given by

d w 0 vg = - < c,

d k (8)

To derive a wave equation governing propagation of a B-wave, one applies a = k ’ c.

vgvp = c2,

quantization procedure,

pp + p P = ihar, (9)

to Eq. (6) to produce a timelike Klein-Gordon wave equation

[a,W + ( m ~ c / h ) ~ ] + = 0,

a , a p = a; - a,‘ = a; - A = m,

+(xc”) = A exp [ ik ,xp ] .

(10)

(1 1) with a timelike B-wave solution

(12) A wave equation for the field of a charged particle is obtained on substituting into

(13)

Eq. (10)

a, --+ D, = a, + i e / (hc)A, ,

yielding

[D, Dp + ( m ~ c / h ) ~ ] + = 0, (14)

In a nonrelativistic regime ( v << c) Eq. (14) reduces to a time-dependent in which Ah = ( V / c , A) is an electromagnetic four-potential.

SchrWnger equation

[ih: - eV - (2mo)-’ fi - -A + = 0, ( : ,’I which, after generalization to a 3 ( N + 1)-dimensional equation [see Eq. (512)], forms a fundamental equation of conventional quantum mechanics employing statistical interpretation of a B-wave +.

150 MARCIN MOLSKI

According to de Broglie-Bohm theory, alternative to that above, one assumes that a wave function of a massive particle endowed with rest mass MO is given in a factorized form (de Broglie, 1960; Bohm, 1952a,b),

q ( x @ ) = R ( x @ ) exp[(i/h)S(x’)], (16)

and satisfies a timelike Klein-Gordon equation,

[o + ( M O C / W 2 ] @ = 0, (17)

in which R(x’) and S(x’) are real functions to describe the amplitude and phase of an associated matter wave.

Introducing Eq. (16) into Eq. (17) and then separating the latter into real and imaginary parts, one obtains two equations (de Broglie, 1960, p. 112):

[80S(x@)I2 - [VS(x’)I2 = M;c2 + h 2 n R ( ~ ’ ) / R ( ~ ’ ) , (18)

&R(x’)8oS(x’) - VR(x’)VS(x’) = -1 /2R(x”)uS(~@), (19)

of which the first is Jacobi’s relativistic equation for a particle endowed with a variable rest mass (de Broglie, 1960, p. 116),

This variable rest mass has interesting properties (de Broglie, 1960, p. 135); for example, the energy of a particle endowed with a mass mo,

moc2 = JW’

remains finite for u = c and real for u > c; consequently, the kinematics of such an object are subject to no limitation involving the velocity of light as an upper bound. In the nonrelativistic regime, the following approximations hold (de Broglie, 1960, p. 121):

mo 2: MO + Q(x’ ) /c2 , E 2: Moc2 + Q(P) + 1/2M0v2, (22)

h2 U R ( x ” ) Q(x’) = --,

2Mo R(x’)

in which the quantity Q(x’) is generally interpreted as a quantumpotential related to a quantumJield existing even in the absence of a field of classical type (e.g., gravitational or electromagnetic).

THE DUAL DE BROGLIE WAVE 15 1

In the case of a charged particle moving in an electromagnetic field A’, Eqs. (1 8) and (19) take the form

2 2

[%S(x”) - - V ] e - b S ( x ’ ) + ‘A] = Mic2 + h 2 ~ R ( x ’ ) / R ( x c ” ) , (24) C C

According to de Broglie-Bohm theory, the guidance formula (de Broglie, 1960, pp. 107, 113)

plays an important role in determining the relativistic velocity of a particle interacting with an external electromagnetic field.

For a nonrelativistic many-body problem, Eqs. (16), (24), and (25) become generalized to the form ( B o b and Hiley, 1975)

* ( x , t ) = R ( x ) exp[(i/h)S(x, r)l, (27)

in which x = (a,), a = x , y, z, denotes the set of 3N position coordinates. Equation (28) is a Hamilton-Jacobi equation for a system of N particles, on

which not only a classical potential V ( x ) but also a quantum potential Q ( x ) act. The latter introduces an additional action that is a major discriminant between classical and quantal theories (Bohm er al., 1978). Equation (29) describes the conservation of probability density p with currents

determined in configuration space M(1,3N). Hence velocities of particles in the system are determined by a nonrelativistic counterpart of a guidance formula

MARCIN MOLSKI

From the context of these considerations, the physical meaning of amplitudal and exponential parts of a wave function (27) emerges: R ( x ) determines both the probability density and the quantum potential, whereas S(x, r) determines the mean momenta of the particles and their velocities.

B. Mackinnon Soliton

The explicit form of R ( x ” ) and S(x”) functions for a freely moving particle was obtained by Mackinnon (1978, 1981a,b). The point of departure for this concept was relativistic invariance of a phase S(x” ) of matter wave under a Lorentz transformation. According to Mackinnon (1978):

The motion of a particle in space-time does not depend on the motion relative to it of any observer or of any frame of reference. Thus, if the particle has an internal vibration of the type hypothesized by de Broglie, the phase of that vibration at any point in space-time must appear to be the same to all observers, i.e. the same in all frames of reference. Each observer or reference frame will have its own de Broglie wave for the particle. The phase of the particle’s vibration must, by definition, be the same as that of all possible de Broglie waves at the point where the particle is. By superimposing all these possible de Broglie waves, a wave packet is formed centred in space on the particle.

The superposition of B-waves detected by a set of observers moving relative to the laboratory frame at velocity v E (-c, +c) in the x’-direction produces a wave packet

r , = d(x1- ~ x o ~ y 2 + xz + x:, (34)

in which m = moc/h, which has no spread with time and propagates at a group velocity equal to the velocity of an associated particle. Mackinnon’s construction constitutes a particlelike solitary wave and fulfills all requirements of de Broglie- Bohm theory provided that they make the identifications

THE DUAL DE BROGLLE WAVE 153

One readily verifies that in Minkowski M (1,3) space Mackinnon’s soliton satisfies a photonlike d’Alembert equation (Mackinnon, 1981b).

O@(X’”) = 0, (36) whereas in M( 1, 1) space it is a solution of a nonlinear field equation,

considered first by Gueret and Vigier (1982). Mackinnon’s model was reinvestigated by Czachor (1989), who showed that

one can construct a wave packet competitive to Eq. (33), which for a particle at rest takes the form

sin(mx’) sin(mx2) sin(mx3) @ ( x q = exp[imxO].

mx1 mx2 mx3 Equation (38) reveals that Mackinnon’s and Czachor’s constructions become identical only in M( 1,l) space, whereas in quadridimensional M( 1,3) space they differ significantly.

C. Jennison-Drinkwater Electromagnetic Theory

An interesting wave-particle model of matter is Jennison-Drinkwater electromagnetic theory (Jennison andDrinkwater, 1977; Jennison 1978,1983,1988; Jennison et al., 1986). according to which a massive particle is considered a relativistic phase-locked cavity with an internal standing electromagnetic wave. They showed that such trapped radiation has inertial properties of ponderable matter, and that all particles having inertial mass may be considered to consist of trapped radiation.

According to Jennison and Drinkwater (1977):

Movement of the perfectly reflecting wall of the cavity into the radiation falling upon it from the internal waves will create a small excess force from the radiation, for it will appear Doppler shifted to the blue and the rate of energy flow is increased relative to the equilibrium value when the wall was at rest. Thus one of Newton’s laws appears naturally at this stage.. . .

In Jennison’s (1988) mathematical formulation the excess force S F and kinetic energy T of a node moving at velocity u = Bc are given by

SF = C ( g - -

T = - CnA [ (1 - + B ) ” 2 + (1 - - ”‘”1 2 1 - / ? 1 + B

,

(39)

I54 MARCIN MOLSKI

in which C is a proportionality factor, and nA, n = 1,2, . . . , denote a length of the trapped wave. According to Eqs. (39) and (40). one obtains

2T 2v c2(1 - p)'/2z'

SF =

in which 6r = nA/c is a feedback echo period. As T is half the total internal energy of the system of trapped waves, whereas the second term is a Galilean acceleration A, Eq. (41) may be rewritten in the form

Equation (42) is merely a relativistic Newton's law for a particle endowed with rest mass mo moving at velocity u; a system of trapped electromagnetic waves is thus endowed with a rest mass. This result is fully consistent with de Broglie's fundamental hypothesis: that wave properties of an elementary particle are manifested in both moving and rest frames.

D. Corben Tachyonic Theory

A trend is developed to assume that tachyons play a role in the structure of elementary particles (Hamamoto, 1974; Corben, l977,1978a,b; Rosen and Szamosi, 1980; Horodecki, 1988a; Recami, 1986, p. 111 and references cited therein). Corben (1977,1978a.b) showed that a free bradyon with rest mass mo and a free tachyon with rest mass2 mb can trap each other in a relativistically invariant way. If mo > mb, the compound particle is invariably a bradyon with rest mass

M~ = drn; - m f ,

described by a wave function (43)

* = JIJI'. (44)

satisfying a wave equation

[awa" + ( M ~ C / A ) ~ ] 9 = 0, (45 ) with respect to an invariant interaction condition

a,+ar*' = 0.

Such a bradyon-tachyon compound is considered two coupled particles associated with a timelike JI wave and a spacelike JI' wave being solutions of Klein-Gordon

Bilaniuk and Sudarshan (1969) stated that the notion resr mass is somewhat a misnomer for tachyons, as a superluminal particle has no rest frame; they suggested the term proper mass.


and Feinberg (1967) wave equations

The + and +’ fields interact and lock to form a plane wave Q, which is timelike whenmo > mb; internal motion of associated bradyonic and tachyonic constituents can be either of oscillatory type or of Kepler type (Recami, 1986, p. 116). A single timelike state can become locked with at most three spacelike states, leading to a timelike state of mass (Corben, 1978a)

The number 3 reflects the condition expressed in Eiq. (46) and the number of spatial dimensions of the M( 1,3)-space in which a bradyon-tachyon compound is formed. One cannot combine two time(space)like states in this way, because application of condition (46) to such states leads to imaginary momenta and exponentially increasing (not normalizable) wave functions. ’ b o bradyons or two tachyons cannot thus trap each other to yield a bound system of particles.

This tachyonic theory holds under substitution (13) to produce a wave equation for a bradyon-tachyon compound interacting with an electromagnetic field A,,

[D, D , + ( M ~ c / ~ ) ~ ] Q = 0, (50)

D,+D,+’ = 0. (51)

The appearance of tachyonic components and an associated superluminal velocity is no inconsistency of Corben’s theory:

Some wave equations based on perfectly valid representations of the Lorentz group lead to space-like solutions, thereby forcing us to deal with tachyons, if only to get rid of them. If relativity theory allows the existence of tachyons, and if they exist one can study them; if they do not exist we learn that something is wrong with our understanding of relativity theory. (Corben, 1978a)

According to well-known models of Salam-Weinberg type, gauge symmetry is spontaneously broken, filling the vacuum with tachyons. In this case such objects are Eggs-field particles that can be considered formally tachyons converted into bradyons (Recami, 1986, p. 117). A created tachyonic field may be interpreted as a component of a virtual massive photon (or Higgs-like particle), of which the four-momentum is orthogonal to the four-momentum of the bradyonic component (Horodecki, 1988a). Such authors as Barut and Nagel (1977) showed that spacelike particle states may play an important role in second-order processes such as the Compton effect, electromagnetic polarizabilities, and Delbriick scattering.

156 MARCIN MOLSKI

E. Horodecki-Kostro Model and the lho- Wave Hypothesis

The Horodecki-Kostro model was developed in the framework of de Broglie’s postulate of wave-particle duality, the many-wave hypothesis (Kostro, 1978, 1985a,b; Horodecki, 1981, 1982, 1983), and Einstein’s theory of relativity, in which the existence of a unitary information ether (Horodecki, 1991) is assumed. This approach provides a conceptual background for an extended space-time description of matter, a two-wave particle model, and a nonlinear wave hypothesis.

According to an extended space-time description of matter, a massive particle is described in terms of a timelike component characterized by an ordinary timelike four-momentum pp = ( E / c , p) associated with a B-wave, and a spacelike component described by a spacelike four-momentum p’p = (E’ / c , p’) related to a D-wave. It implies a double, two-wave structure of a massive particle, characterized by the generalized equation (1) (Horodecki, 1988a, 1991)

p @ = hkp, p‘@ = hk’p, (52)

kp = (w/c, k), k’@ = (w’/c, k’) (53)

in which

are wave four-vectors of B- and D-waves, respectively, and k’ = 2n/I’, w’ = 2n u’, in which I’ and u’ denote the wavelength and frequency of a D-wave.

The time and spacelike four-momenta satisfy the following conditions:

p @ = r n o u p , p‘@ = rnou’p, (54)

u p = ( Y C , yv), U’P = ( y u , y c ) , ( 5 5 )

PpP‘fi = 0, (57)

in which c/c is a unit vector along the direction of the particle three-velocity.

time and spacelike wave equations: On applying quantization law (9) to Eqs. (56)-(57), one obtains the following

[spa" + ( r n o ~ / h ) ~ ] = 0, [apap - ( r n o ~ / h ) ~ ] e’ = o (58)

and a field interaction condition

in which

+ ( x p ) = A exp [ i k , x @ ] , + ’ (xp) = A‘exp [ik&xp] (60)

are B- and D-wave functions, respectively.


Substitution of wave solutions y? and +‘ into wave equations (58) produces dispersion relations for B- and D-waves,

k,k” = (o/c)~ - k2 = (oo/c)~, klk” = ( w ’ / c ) ~ - k’2 = -(w/c)~, (61)

and a wave velocity interrelation

Cross correspondences (63) and (64) imply particle relations (Horodecki, 199 1)

p = hk = ~ U ’ C / C = E’c/c, p’ = hk’ = ~ U C / C = Ec/c , (65)

and the frequency relations

(66)

These equations form a conceptual background for a nonlinear wave hypothesis (Horodecki, 1988a, 1991) according to which a massive particle in motion represents a nonlinear nondispersive wave packet (C-wave) that involves an internal (characteristic) spectrum of matter waves. A de Broglie oscillation of frequency w excites in the vacuum medium an oscillation of frequency w’, and Doppler-shifted frequencies w- and w+ arise (Horodecki, 1988a, 1991). The quantities w and w’ are interpreted as harmonics of the internal spectrum of a de Broglie extended particle, being a nonlinear C-wave excitation of the vacuum field identified by Horodecki with a unitary information ether (Horodecki, 1991).

2 wo = w- * w+, w- = w - w’, w+ = w + w‘.

E Das Model and Pseudovelocity

According to Das’s model (Das, 1986, 1988, 1992), one assumes that a massive particle in motion is simultaneously associated with velocity of two types: the actual velocity v = c drldxo, and an unobservable velocity v‘ = c dxo/dr (called a pseudovelocity), interrelated by

(67) d = c .

According to Das, the wave-particle dualism of matter represented in E = h u, p = h/A, v, = v , and up = c2/u > c clearly demonstrates that the idea of pseudovelocity v’ is not trivial and is indeed necessary to complete the set of correspondences:

2

158 MARCIN MOLSKI

u + E , )c + p , ug e u, and up * u’ among parameters involved in wave (u , A , ug, u p ) and particle (E, p , u, u’) characteristics.

Employing Eq. (67), we find that the spacelike four-velocity and four-momentum appearing in the Horodecki-Kostro model can be identified with the four-pseudovelocity and four-pseudomomentum given by

U’P = (y ’c , y’v’) = ( y u , y c )

y’ = (8’2 - l)-I/z, p’ = V ’ / C ,

p’@ = (mocy’, mov’y’) = (moyu, rnoyc). (70)

Taking advantage of the extended wave-particle dual correspondences given in Eq. (52), one introduces a matter wave of the second kind (the transformed Comp- ton wave) characterized by a wave four-vector

identical to that obtained for a D-wave in the Horodecki-Kostro model. Hence one may identify the transformed Compton wave introduced by Das with a D-wave in Horodecki-Kostro theory. Consequently, Eqs. (63) and (64) become written in the forms

B = v D - - u < c , ugD=upB=v’>c.

G. Elbaz Model and Lorentz Transformations

The starting point for an extended wave model of a particle developed by Elbaz (1985, 1986, 1987, 1988) was an observation that the fundamental de Broglie hypothesis of wave-particle duality is incomplete by itself. On introducing the concept of a B-wave, de Broglie (1960, p. 4) assumed that in a rest frame one attributes to a particle endowed with energy E = mgc2 a periodic phenomenon of frequency 00 = E / h and phase S = o0xo/c . When a particle moves uniformly in the XI-direction at velocity u = c d x l / d x o relative to a laboratory frame, the phase of an associated wave undergoes a relativistic time Lorentz transformation

(73)

(74)

X0 + (XO - p x ’ ) y ,

S(X’) = WOX’ /C 4 S(X’, x’) = WO(X’ - / ~ X ’ ) Y / C = koxo - k l x ’ ,

yielding an ordinary B-wave characterized by


Although Lorentz transformations act on space and time coordinates, in these calculations only a time transformation is used; the space transformation

(76)

Following this conclusion, Elbaz assumed that a standing wave of photonlike

(77)

OW(X0, x ' ) = 0, (78)

x 1 - (XI - /3xO)y

is ignored (Elbaz, 1985).

type is associated with a particle at rest (Elbaz, 1985):

q ( x 0 , X I ) = + ( x o ) + ' ( x ' ) = exp [ iooxo/c] exp [ i w o x ' / c ] ,

in which the spatial part of Eq. (77) is called by Elbaz an amplitude wave.

according to Eq. (76): When a particle is moving, the phase of an associated amplitude wave transforms

S ' ( X ' ) = O O X ' / C - S'(x0, X I ) = % ( X I - /3x0)/c = k ' , ~ ' - k&xo, (79)

(80) kl, = W ' / C = w y / 3 = OOY', k' = W O Y / C = ~ ~ / I ' Y ' / C ,

providing the wave function

+/'(xo, X I ) = exp [i ( k i x ' - khxo) ] , (81)

whereas the time-dependent component +(xo) of the standing W ( x o , X I ) wave transforms into an ordinary B-wave component,

(82)

The type of superposition of +' and + waves associated with a moving particle is photonlike; the d' Alembert wave equation (78) remains satisfied.

Equations (65), (71), and (80) reveal that a D-wave, a transformed Compton wave, and an amplitude wave introduced by Elbaz are endowed with the same wave characteristics.

+(xo, X I ) = exp [ i (koxo - k l x ' ) ] .

H. Generalized Barut Approach

Mackinnon's soliton theory was recently generalized by Barut (1990). who constructed exact localized oscillating energy solutions,

of the massless d'Alembert equation

160 MARCIN MOLSKI

that fail to spread and that behave like a single massive particle. The factor f (w0) appearing in Eq. (83) ensures that the solution has a finite energy (Barut, 1990).

The time-independent function F ( r ) appearing in Eq. (83) satisfies the purely spacelike equation

(85 ) [A + (00/c)2] F0-1 = 0

and hence has the form

F ( r ) = C~mr-'J2JI+~/2(oor/c)P;f(cosO) exp[im4]. (86) Im

In the moving frame the solution (83) for 1 = 0 becomes

in which

(88)

w = w y , k = m y p , ~ = v / c . (89)

rv = [r2 + 8' y2x,2 - 2y2x0p + r + y2(p r12] 112 ,

For 1 = 0 and one-dimensional motion along the x'-axis, solution (87) reduces to the form

which differs from Mackinnon's soliton by the factor f ( o / y ) .

1. Conclusions

This summary of the most important concepts in the field clearly shows that all are consistent in many points and lead to the following final conclusions.

1. A massive particle in motion may be considered a system composed of a bradyonic component associated with a timelike B-wave and a tachyonic component connected with a spacelike D-wave. These particle constituents can trap each other in a relativistically invariant way to produce a bradyon-tachyon compound described with a superposition of the associated matter waves.

2. Models of a massive particle by Horodecki and Kostro, by Das, and by Elbaz are viewed as special cases of a more general tachyonic Corben theory that reduces to the former when masses of bradyonic and tachyonic constituents are the same.

3. Superposition of B- and D-waves forms a nonlinear nondispersive wave packet (C-wave), identified by Horodecki and Kostro and by Das with Mackinnon's soliton, of which propagation is governed by nonlinear propagation law (37).


4. A particle at rest is associated with a system of photonlike standing waves. This assumption is compatible with both Elbaz’s model and Jennison-Drinkwater theory, in which a particle is considered a system of standing electromagnetic waves trapped in a phase-locked cavity. Radiation thus trapped has inertial properties of ponderable matter; and, vice versa, all massive objects are considered to consist of trapped radiation.

5. The B- and D-waves are identified with exponential and amplitudal waves appearing in de Broglie-Bohm theory; Eq. (17) for MO = 0 reduces to d’ Alembert’s equation describing propagation of a C-wave identified with Mackinnon’s soliton. De Broglie-Bohm theory and a two-wave concept might thus be unified into one generalized theory. This problem is considered in detail in Section VII.

III. WAVE-PARTICLE MODEL OF PHOTONS

Wave-particle models of a massive particle considered in the preceding section may be extended to include massless objects of luminal type (photons, neutrinos, and gravitons) called luons. Photons (spins = 1) and gravitons (s = 2) belong to particles of integer spin in the class bosons, in contradiction to neutrinos (s = i), which are fermions. Consequently, their formulation is based on second-order (Klein-Gordon) and first-order (Dirac) equations, respectively. Considerations presented in this section are limited to photons.

Characteristics of a photon and its conjugated wave are given approximately by a four-momentum and a wave four-vector.

related through waveparticle correspondences

p ; = A f f i , vg = v, = c, (92)

in which w = 2nv, and f = 2n/A,; v, and A, are the frequency and wavelength of the electromagnetic wave, and E, and p , denote the energy and momentum of an associated photon.

Electric- and magnetic-field components of an electromagnetic wave obey Maxwell’s equations,

V E = 4np, V x H - 4nc-’j - aoE = 0 (93)

V . H = O , VxE+aoH=O, (94)

162 MARCIN MOLSKI

in which j and p denote current and density of electric charge. A direct consequence of Eqs. (93) and (94) are the wave equations

a,a% + 4n (.-'ad + vp) = 0,

i3,a.H + 4nc-'V x j = 0,

When p = j = 0, valid for a space free of charge and current densities, Eqs. (95) and (96) become

t l p a p W ( x f l ) = 0, Q ( x p ) = Aoexp [if#], (98)

in which A0 = EO or HO denotes the amplitude of an electromagnetic wave.

AP = ( V / c , A), whose components satisfy (Jackson, 1975) In a description of the electromagnetic field one may use the four-potential

(99)

a,WA"(xfi) = 0, A"@,) = AI; exp [if,x"], a = 0, 1,2,3. (100)

For an electromagnetic wave propagating in the x3-direction, Eq. (100) and its solutions reduce to a two-component equation (a = 1,2),

(101)

H = V x A, E = -V* V -&A,

a,aWA"(xo, x3) = 0, A"(xo, x3) = A: exp [ i ( foxo - f3x3)>1,

A. Photon as a Bradyon-Tachyon Compound

Detailed analysis of Corben's tachyonic theory reveals that photons may be treated as a special case of bradyon-tachyon compounds, of which the particle constituents are endowed with the same rest mass (Molski, 1991,1993a. l994,1995a,b). When the bradyonic constituent is at rest and the tachyonic component attains a limiting infinite velocity, the associated field + is periodic in time and independent of position, in contrast to a field +' that is static in time and periodic in space. Hence Eqs. (45)-(48) reduce to the simple forms

[a: + (moc/h)*] +(xo) = 0, +(xo) = exp[irnocxo/h], (102)

a,+(xo)#'$'(r) = 0, (104)

[A + (moc/h)2] $(r)' = 0, +'(r) = exp[fimocr/h], (103)

a , a p W ( x p ) = 0, Q(x") = +(xo)+'(r) = exp [ i f P x " ] , (105)

p~ = (moc, moc), f p = (rnoch-', rnoch-I), (106)

(107) f, f, = (w/c)2 - f 2 = 0,

in which c/c denotes the unit vector along a direction of wave propagation.


From Eq. (107) one obtains

The results indicate that a superposition Q of time- and spacelike waves and @' propagates at a phase (group) velocity equal to the velocity of light, whereas the associated compound of a bradyon at rest and a tachyon at infinite speed, termed transcendent (Recami, 1986, p. 23), is endowed with zero rest mass. Such a bradyon-tachyon compound may thus be treated formally as an object of luminal type; and, vice versa, photons may be considered objects composed of both bradyonic and tachyonic components endowed with the same rest mass. This result, c o n k e d in independent investigations by Dutheil(1984,1989,1990a,b, 1993), plays an important role in our considerations, presented below.

B. Conversion of Light into B- and D-Waves

Taking into account Eqs. (54) and ( 5 3 , one derives the relativistic mass formulas (Molski, 1991)

mu = moy, m: = moy' = mo/?y, ( 109)

(1 10)

that relate masses of time and spacelike components of a particle. Equation (1 10) shows clearly that mu and m: may be termed an internal mass spectrum of a particle, so that a massive particle in motion can be regarded as a composite object comprised of both bradyonic and tachyonic components.

m: = (mu - rn:)(rn, + m:), m: = /?mu,

Equation (1 10) leads to a conclusion that at a luminal velocity,

B=1, m : = m , , m o = O ; (1 11)

hence lurons appear to be objects without mass. This case features a great re- semblance to conversion of particle-antiparticle pairs (of identical masses) into photons, accompanied by disappearance of the rest mass. According to a two-wave model, in the reverse phenomenon (e.g., Delbriick scattering), in addition to an ordinary bradyonic component associated with a B-wave, a tachyonic component connected with a D-wave is created.

To prove this hypothesis, we consider a one-dimensional phase-locked cavity filled with standing electromagnetic waves, such as according to a Jennison- Drinkwater model (Jennison and Drinkwater, 1977; Jennison 1978). When the motive reflector element on the wall of the cavity is moved forward at a velocity V I relative to the laboratory, the internal frequency wg undergoes a Doppler shift

164 MARCIN MOLSKI

according to the formula 1 /2 w1=w0(- ) 1 + UI/C .

1 - U'/C The internal frequency 00 is received at the motive reflecting element at a frequency w1, whereas in the laboratory system it is reflected at the frequency

1 / 2 ~ = w l ( - ) " ~ = % ( - ) 1 + Ul/C 1 + UI/C =m(-) 1 + u / c , (113) 1 - UI/C 1 - UI/C 1 - u / c

in which

u = 2Ul/(l+ u:/c2>

is the velocity of the following mirror element (placed on the opposite wall of the cavity) moving at velocity v = c/? = dx3 /dxo; the frequency w is restored to the original value 00 (Jennison and Drinkwater, 1977; Jennison, 1978). If we restrict our considerations to one-dimensional motion along the x3-axis, the electromagnetic wave characterized with a four-potential A, = (0, AX, A)', 0) propagates inside the cavity with a Doppler-shifted frequency w, in accordance with a luminal Maxwell equation

a,aw(xo, x 3 ) = 0, w ( x 0 , x 3 ) = A; exp [ ~ o , c - ' ( x ~ - x O ) ] , (115)

Exponential part of Eq. (1 15) is written in the alternative form

= exp [iooc- 'y(l+ p ) ( x 3 - x o ) ]

= exp [iw0c-'y(/?x3 - x o ) ] exp[iwc-'y(x3 - /?xO) ] , (117)

which with use of the Jennison-Drinkwater (1978) result rno = wo h/c2 becomes

= exp [ i ( rnoc/h)y( /?x3 - xo) ] exp [i(rnoc/h)y(x3 - /?xo)]

= exp [ih-' (p3x3 - pox')] exp [ih-' ( p i x 3 - pbxo)] = + $ I . (1 18)

Substitution of Eq. (1 18) into wave equation (1 15) yields

a,a,++' = +lapap+ + +a,ap+l + 2a,+a~+' = 0, (1 19)

which on differentiating @ and +' with respect to time and space coordinates provides the set of equations


These are merely field equations for a timelike B-wave and a spacelike D-wave appearing in a two-wave particle model (Horodecki, 1988a. 1991; Das, 1984, 1986, 1988, 1992; Elbaz, 1985, 1986, 1987, 1988) and a tachyonic theory of matter (Corben, 1977,1978a.b). Hence conclude that radiation trapped in a phase- locked cavity under motion of the motive reflector wall is Doppler-shifted and transforms into a system composed of B- and D-waves that lock to the form of the luminal wave. According to a corpuscular interpretation, photons in a phase- locked cavity undergo conversion into bradyon-tachyon components coupled to each other in a relativistically invariant way. The compound particle has photonlike characteristics: It moves at the velocity of light and has zero rest mass. The same holds for a cavity at rest. In such a case trapped radiation may be considered a system of purely timelike and purely spacelike waves having the same wave characteristics as those of the amplitudal wave considered by Elbaz (1985). A massive particle at rest may consequently be associated with an intrinsic system of standing luminal-type waves characterized by the wave four-vector

identical to that given in Eq. (77). This conclusion is consistent with models of Elbaz, of Jennison and Drinkwater, and of Corben, and with a Horodecki's two- wave hypothesis of the structure of matter.

C. Two- Wave Model of Longitudinal Photons

The preceding considerations are extensible to include radiation trapped in waveguides that are two-dimensional phase-locked cavities with the third dimension open. Thus one can investigate not only the staircase dynamics of trapped radiation (Jennison and Drinkwater, 1977) but also one-dimensional kinematic phenomena of energy-momentum transfer mediated by longitudinal photons.

Free photons are relativistic particles par excellence moving in vacuum at the velocity of light, characterized by zero rest mass and spin J = 1. The electromagnetic vector fields associated with such photons (known as transverse photons) are perpendicular to the wave three-vector determining the direction of wave propagation. The transverse photons associated with a transverse electromagnetic (TEM) wave are exactly massless; the extraterrestrial limit on the photon mass obtained by Goldhaber and Nieto (1971) is mo I 4 x g; an experiment by Williams et al. (1971) yielded a similar result, mo 5 1.6 x g. In contrast, photons that have longitudinal components of electromagnetic fields cannot be massless (de Broglie, 1951; Perkins, 1982, p. 96). The latter case may be realized, for example, inside waveguides during transmission of transverse electric (TE) and transverse magnetic (TM) waves.

166 MARCIN MOLSKl

We consider a rectangular vacuum waveguide of transverse dimensions a' and a2 with perfectly conducting walls and an interior free of charge. If a TE wave is excited in the waveguide and propagates in the x'-direction. Maxwell's and Helmholtz's equations for the longitudinal magnetic H, component take the form (Jackson, 1975)

(122)

(123)

(124)

8pi3fl@,!,(xo) exp [i(knx3 - k0xo)] = 0, (Y = 1,2,

( A + K ~ ~ ) + ; J X ~ ) = 0, no = 0,1,2, . . . ,

K ; ~ = ki - k:, in which ko = o/c = 2nv/c and k' = 2n/A; v and A denote the frequency and wavelength of the TE wave.

An explicit form of solution of Helmholtz's equation, satisfying boundary con-

With a critical frequency (cutoff frequency)

Wn, = CKna 9

the group (phase) velocity of TE wave is defined as (Jackson, 1975) 2 112

2 -112

~g = cdko/dk = ~ [ l - (Wn, /o ) ] 5 C,

2 c, up = cko/k = c[l - ( w n a / o ) ] whereas Maxwell's equation (122) takes the form

( a p V + K ~ ~ ) @ ( X ' , x') = 0, @(xo, x3) = exp [i(k3x3 - koxO)]

These equations yield thus pertinent conclusions. Phase and group velocities of transferred radiation satisfy a relation upug = c2 identical to that for a B-wave associated with a particle moving at velocity vg . The wave equation (131) is identical to the timelike Klein-Gordon equation for a particle endowed with rest mass rn;, = h ~ , , /c moving in the x'-direction. As the x3-direction coincides with


the direction of propagation of the longitudinal H, component of the TE wave, we identify m:a with a mass of the associated longitudinal photon.

Such a photon has bradyon-type characteristics (Molski, 1991).

E = poc = tto = mjlacz[l - (ug/c)2]-’ /2 , ( 132)

and the same relativistic properties as those of ordinary ponderable matter; for instance, it cannot move at the velocity of light because this case holds for w = 00;

for w = unW, the longitudinal photon is at rest ug = 0, up = 00, and in the case of the luminal TEM wave, A+’(x”) = 0, K = 0, ug = up = c; the associated (transverse) photon is exactly massless.

These astonishing conclusions become clarified in the context of the Jennison- Drinkwater theory considered in Section: Solutions of Helmholtz’s equation (K # 0) describe an electromagnetic wave standing between the xz and yz planes of a waveguide. An application of the boundary conditions given in Eq. (125) leads to the appearance of standing-wave modes, directly related to the quantization of an associated mass,

Such trapped radiation has inertial properties of ordinary matter, with the possibility of excitation of various inertial states, in full accordance with Jennison- Drinkwater theory. This concept is consistent with de Broglie’s (1960, p. 102) theory of constrained particle states. According to this concept, the constrained states

present an illuminating analogy to the circumstances one encounters for photons enclosed in a waveguide, whose motions correspond to rest masses which vary according to the form of the waveguide and the type of waves propagated-rest masses much greater than the normal rest mass of the photon, which is zero or undetectably small. (de Broglie, 1960 p. 102)

Equations (122), (123), and (131) reveal that they are compatible with Corben’s tachyonic theory and a two-wave particle model. The solution +Au of Helmholtz’s equation, being static in time and periodic in space, may be interpreted (Molski, 1993a.b) in terms of a D-wave associated with two transcendent tachyons of infinite speed, endowed with rest masses nttnl /a~ and nAnz/az, moving back and forth in oscillatory motion between the xz and yz planes of a waveguide. The exponential timelike solution + is interpreted as a B-wave connected with a bradyon of rest

168 MARCIN MOLSKI

mass m& moving at velocity ug along the x3-dimension of the waveguide. Such time- and spacelike fields interact to form a Q,,= wave (interpreted according to the nonlinear wave hypothesis as a C-wave),

2

q n m ( x f i ) = HO n cos (5) exp [i (k3x3 - k o x 0 ) ] , (136) f f = l

which satisfies the luminal Maxwell equation (122) and is associated with a bradyon-tachyon compound of zero rest mass:

2

Mo = \I (m;a)z - "c ($) =o. f f= l

This conclusion becomes more comprehensible if we consider the simplest case of a TE1,o wave (9 , , ,= l .n2=0 wave) propagating at group velocity ug = 0. Then according to Eq. (133), k3 = 0 and ko = K ~ , , whereas longitudinal and transverse components of the TE1,o wave become (Jackson, 1975)

H, = zf0 cos - exp [ i K l s o X o ] , (T') ik3a3

H x - lr HO sin ($) exp[i~l,~x'] = 0,

If a waveguide with a TEl .0 wave excited inside moves relative to the laboratory frame in the x'-direction at velocity u = c/l = dx'/dxO, the associated field components undergo relativistic transformations:

XO - x ' p


producing

Amplitudal and exponential functions appearing there are solutions of space- and timelike wave equations

to be identified with a D-wave +La associated with a tachyonic photon component of rest mass nA/(a'c), and a B-wave + connected with a bradyonic photon constituent of rest mass ~ 1 . 0 h / c . The interaction condition

ap+;vaf i+ = o (147)

The case of a TM wave endowed with a longitudinal E, component of the is fulfilled in both rest frames.

electromagnetic field (Jackson, 1975)

satisfying boundary conditions \u(x" = a") = 0, provides the same conclusions. This analysis is extensible to include TE and TM waves transferred inside a

cylindrical waveguide of internal radius ro. The longitudinal E, component of a TM wave is a solution of the wave equation (Jackson, 1975)

( 149) (a; - a,2 - , - - l a , , - r -2 a+ 2 - a ~ ) \ u m n ( x o , r , # , x 3 ) = o , satisfying the boundary condition @ (ro) = 0 and given in terms of Bessel functions

Qmn(Xo, r, $J, x 3 ) = EoJ,,, (xm, , r ; l r ) exp[firn#] exp [ i (k3x3 - koxo) ] , (150)

in which Xm,, rn = 0, 1,2,3,. . . , n = 1,2,3,. . . denote the roots of Bessel

Jm ( r ) ,

function J m ( X m n ) = 0.

170 MARCIN MOLSKI

Proceeding analogously to the case of a rectangular waveguide, one decomposes the luminal superposition given in Eq. (150) into space- and timelike solutions

(8: + r-'% + r-2ai + xi,ri2)+k,(r, 4) = 0, (151)

(152)

(153)

+(xo, x 3 ) = exp [i (koxo - k 3 x 3 ) ] , ( 154)

in which components of the B-wave four-vector k@ = (kO, 0, 0, k 3 ) obey the dispersion formula

(155) 2 2 - 2 - 2 -2 ko - k3 - K,, - x,,ro .

+k,n(r* 4, x3 ) = Jm ( x m n r i ' r ) exp [ z t imbl ,

(a: - a; + K ~ , ) + ( X O , x 3 ) = 0,

The solution +k,, of Helmholtz's equation (151) is interpreted as a transcendent D-wave associated with a tachyon of infinite speed and rest mass hx,,/cro moving in combined radial and circular motions about the bradyonic constituent of rest mass ~ K , , , ~ / C associated with a B-wave + propagating in the x3-direction. These time- and spacelike fields interact to form a Q,,, wave (C-wave) propagating as an excitation of luminal type associated with a bradyon-tachyon compound of zero rest mass,

The results indicate that this two-wave model of longitudinal photons is consistent not only with a two-wave hypothesis of matter, a nonlinear wave hypothesis, and a tachyonic theory of matter but also with de Broglie-Bohm theory. Comparison of Eqs. (136) and (146) with Eq. (16) allows us to identify the amplitudinal part +' of TE and TM waves with R ( x @ ) function, whereas

(157)

When the guidance formula given in Eq. (26) is applied to Eq. (157) one finds that

S(x'L) = pox0 - p 3 2 .

This result shows that the amplitude wave +' and the exponential wave + are also interpretable in terms of singular and probabilistic waves appearing in de Broglie's theory of double solution. Consequently, a two-dimensional region of space ( x , y ) or (r, 4) confined by walls of a waveguide represents an extended two-dimensional massive photon moving in the open x3-direction of a waveguide at velocity u = ug .

The most important conclusion is that in the process of conversion of massless transverse photons into ponderable longitudinal photons, not only is an ordinary


timelike component created but also a spacelike component associated with a tachyonic field called a matter wave of the second kind @-wave).

D. Massless Photons and Ponderable Matter

A process of conversion of massless photons into ponderable matter occurring in a phase-locked cavity may be explained in a purely relativistic framework (Molski, 1991). Free photons and an associated coherent beam of electromagnetic rays are exactly massless, but for photons moving in various directions, the situation differs.

We consider N identical photons in a system characterized by the total four- momentum

N

Nhwlc, ni hwlc P," = i=l

(159)

in which ni is a unit vector along the photon motion. The photon system as a whole has a nonzero rest mass

N

p o = ~ w c - ~ NZ - C cos(eij), cos(eij) = ni nj, (160) 4 i. j = 1

where

eij = cos-'(ni, nj) # 0, i + j . (161)

Such a cloud of electromagnetic radiation composed of massless photons is endowed with a nonzero rest mass, generates its own gravitational field, and is af- fected by external gravitational fields. For a two-photon system, Eq. (160) reduces to

(162)

which indicates that a nonzero rest mass 2 hw/c2 may be attributed to an associated standing electromagnetic wave (01.2 = n). Hence we understand the result of Jennison and Drinkwater, who proved that standing electromagnetic waves trapped in a phase-locked cavity have inertial properties of ponderable matter.

po = h ~ c - ~ [ 2 ( 1 - C O S ( ~ ~ , ~ ) ) I ~ / ~ ,

E. Extended Pmca Theory

The preceding results indicate that the conventional Proca theory3 (Jackson, 1975) of massive timelike longitudinal photons is extensible to include their spacelike

m a ' s works are listed by Goldhaber and Nieto (1971).

172 MARCIN MOLSKl

constituents. Maxwell’s equations for timelike massive photons, which play a fundamental role in Proca’s theory, read

V . E = 4np - K’V, V x H = 41rc-’j - K’A, (163)

V * H = O , V x E = - a o H , (164)

in which K = rnoc/h and mo denotes the mass of a timelike photon constituent, whereas A,, is a four-potential, defined in Eq. (99).

Proca’s equations for a massive vector fields have the form (Jackson, 1975)

4n . (a,,ag + K’)A, = --J,,,

c

4n . apF,,, + K’A, = - J , , c

a,, jj’ = 0, Fpv = apAv - &A,. (167)

For a free massive electromagnetic field there exists a conserved energy-momentum density

E = [E2 + H2 + K’(A’ + A:)], 81r

1 P = -[EX H+K’AoA],

4RC

so that

aoE + v . P = 0. (170)

To extend this Proca theory to include spacelike objects is difficult. A fundamental requirement of such a generalized theory is invariance of the electromagnetic tensor FF’’ and the four-potential A” under superluminal Lorentz transformations. Such operations change a timelike tangent vector into a spacelike vector, and vice versa (they invert the quadratic-form sign), and form a new extended group G including subluminal (ortho- and antichronous) Lorentz transformations (Recami, 1986, p. 54). However, extended relativistic theories including superlu- mind inertial frames present interpretive difficulties unless they are formulated in pseudo-Euclidean M ( n , n) space having space and time dimensions of the same number (Recami, 1986, pp. 39, 118). Hence only a two-dimensional representation of the G-group, acting in M( 1, 1) space, has a clear physical interpretation (Maccarone and Recami, 1982).

If one assumes Ffi” and A,, to be G-invariant.


the ordinary Maxwell equations retain their form for spacelike objects (Recami and Mignani, 1974, p. 277).

According to a two-wave model of matter, one can write basic equations for the extended Maxwell and Proca theory, including spacelike components of massive photons (Molski, 1991),

V . E = ~ ~ ~ + K ~ V , V x H = 4 n c - ’ j + ~ ~ A (172)

V * H = 0, V x E = -&H, (173)

4n . (apap - K ~ ) A ~ = - j p ,

C

being the spacelike counterpart of timelike equations (163)-(167). Proceeding as in the timelike case, one writes the energy-momentum density of

free electromagnetic fields associated with the spacelike component of a massive photon

- [ E 2 + H 2 - ~ ’ ( A 2 + & ) ] , 1 877

1 4nc

’ P I = - [E x H - K~AoA],

(177)

which imply the continuity equation

a,,€’ + v . PI = 0. (179)

For spacelike objects, Eq. (177) and (178) have a physical meaning only for

E2 + €I2 2 K’(A~ + A:). E x H # K~AoA; (180)

the case

E’ + H~ = K ’ ( A ~ + A:) (181)

corresponds to an infinite group velocity allowed for tachyons (Molski, 1991). A simple extension of Proca’s original theory developed here is confirmed by

Horodecki and Horodecki (1995), who showed that the timelike Proca equation and its spacelike counterpart are derivable from Dirac electrodynamics with spontaneous gauge symmetry breaking, and that Proca’s extended theory is fully consistent with Corben’s tachyonic theory and a two-wave particle model.

174 MARCIN MOLSKI

E Conclusions

A photon is considered to be a composite object having both bradyonic and tachyonic components endowed with equal rest masses, which trap each other in a relativistically invariant way to yield a particle endowed with zero rest mass. Ac- cording to the wave picture, a photon is associated with a superposition of time- and spacelike waves that lock to form a luminal-type wave propagating at a phase (group) velocity equal to the velocity of light in vacuo.

A longitudinal Doppler effect for an electromagnetic wave emitted from a moving source or reflected from a moving mirror is explained as a result of transformational properties of time and spacelike fields incorporated in a light wave.

A massless transverse photon imprisoned in a phase-locked cavity or waveguide undergoes conversion into a bradyon-tachyon pair of which the timelike constituent is interpreted as a massive longitudinal photon. According to a wave picture, trapped electromagnetic fields are considered a nonlinear system of B- and D-waves that lock to form a C-wave propagating as an excitation of luminal type.

The timelike component of a longitudinal photon is endowed with the same properties as those of ordinary ponderable matter-it has rest mass and moves at a subluminal velocity that cannot exceed the velocity of light. Consequently, such objects are considered relativistic models of massive particles (Molski, 1993b).

Relativistic equations ( 132)-( 139) are derived using neither a Lorentz transformation nor reference to geometrical properties of space-time, whereas the Klein- Gordon equation (131) is obtained without applying a standard quantization formalism. One can hence derive fundamental equations of special relativity and quantum mechanics in a purely electromagnetic framework (Molski, 1993b).

Interpretation of wave-particle phenomena appearing in a phase-locked cavity and a waveguide interior is consistent with a two-wave particle model, a nonlinear wave hypothesis, Corben’s tachyonic theory, and models Jennison and Drinkwater and of Elbaz for a massive particle.

The appearance of a spacelike constituent in the spectrum of mass associated with a longitudinal photon indicates that Proca’s conventional theory is extensible to include spacelike photon states. Such a generalization is made in the framework of a two-wave hypothesis of matter or Dirac electrodynamics with spontaneous breaking of gauge symmetry.

Iv. ELECTROMAGNETIC MODEL OF EXTENDED PARTICLES

Although according to standard concepts of physics one attributes a pointlike character to material objects, this approach generates unphysical effects of infinite


self-energy and self-field; to overcome these difficulties, Dirac distributions or a renormalization procedure is used. To avoid these complicated and abstract operations, several models are proposed (Jehle, 1971, 1972, 1975; Post, 1982, 1986; Jennison, 1983; Jennison et al., 1986; Elbaz, 1987, 1988, 1995; Molski, 1991, 1993a, 1994, 1995a,b) in which an elementary particle is treated as an extended structure, not as a pointlike object.

Because radiation trapped in a waveguide is endowed with kinematical properties of ordinary matter and associated longitudinal photons with two-dimensional spatial extension, such a system accurately reproduces fundamental properties of extended particles, although neglecting their extension in a third space dimension. Therefore, the approach described above should be generalized to include electromagnetic fields trapped in three-dimensional resonator cavities, which seems the best geometrical representation of these objects (Molski, 1993a, 1995a.b).

A. Three-Dimensional Rectangular Space Cavity

We consider arectangular electromagnetic space cavity @-cavity) of size a, (a = 1,2,3) with perfectly conducting walls and an interior free of charge. Under these assumptions the imprisoned electromagnetic fields satisfy the luminal Maxwell equation (Jackson, 1975)

in which mn,c is a resonator frequency and equation

are solutions of Helmholtz’s

(A + m ~ a ) $ A o ( ~ u ) = 0, IZ, = 0, 1,2,. . . . ( 1 W

If the TE mode, endowed with the longitudinal H, component of a magnetic field, is excited in a S3-cavity, solutions of Eq. (184) satisfying suitable boundary conditions take the form (Jackson, 1975)

Equation (1 85) in this form ensures that boundary conditions are satisfied not only for the longitudinal component of the TE wave, but also for the transversal ones, E l and HL. to be obtained from Eq. (185) and Maxwell’s equations (93) and (94).

176 MARCIN MOLSKI

With the results obtained in Section 111, a nonzero rest mass

becomes attributed to electromagnetic fields imprisoned in an SN-cavity. This associated mass has inertial properties of ordinary ponderable matter, with the possibility of quantization of various inertial states (Jennison and Drinkwater, 1977; Jennison, 1978). Consequently, all intrinsically stable massive objects are considered modes of imprisoned radiation.

If we trap an electromagnetic field within a rectangular S3-cavity, spatial field transmission in the external cavity domain vanishes; Eq. (182) reduces to a time- dependent formula

(a: + miu)+(xo) = 0, +(xo) = exp [im,,xo] (187) describing propagation of outer or radiating fields (Elbaz, 1988) associated with a massive electromagnetic S3-cavity along the xo-axis. Because in four-dimensional space-time the direction of field propagation coincides with the worldline along which the associated object moves, the xO-axis appears incidentally as a worldline of the S”-cavity. Consequently, the timelike Klein-Gordon equation for outer fields associated with a freely moving S3-cavity in the x3-direction at velocity u = CB = c dx3/dx0 is obtained on application of Lorentz’s time transformation to Eq. (187), yielding

(%ap +mi,,) exp [ik,xp] = 0,

Simultaneously under a uniform motion, the trapped internal S3-cavity fields, called inner or selfjelds (Elbaz, 1988), transform into time and spacelike fields (Molski, 199 1-1 995b).

of which superposition yields a TE wave (C-wave) q n , , = $Aa+ propagating in accordance with wave equation (182) as an excitation of luminal type.

Functions (190) and (191) satisfy space and timelike Klein-Gordon equations


consequently, from Eq. (182) one obtains the invariant interaction condition

a p + ; J x q a q b ( x p ) = 0. (193)

Functions (190) and (191) are also given in covariant forms:

in which p p is the bradyonic four-momentum and p'p, p i , and p i denote the tachyonic four-momentum and momenta of transcendent tachyons, respectively.

Equations (191) and (192) are fully compatible with a two-wave description of matter and a tachyonic theory of the structure of an elementary particle. In particular, +Aa is interpreted as a superposition of three independent spacelike states (D-waves),

associated with tachyonic particle constituents related to three mass states m:. = nttc-'n,a;'.

When a cavity is at rest, these spacelike objects move back and forth between cavity walls at infinite speed along the x,-axes, whereas associated fields are described with functions periodic in space and independent of time. In contrast, @(xo) is interpreted as a B-wave periodic in time and independent of position and is associated with a bradyonic constituent at rest of mass m:--an orthorhombic lattice with spacings h / p l , h / p , , and h / p , (Corben, 1978a). Interaction between such fields is analogous to scattering of a wave at a diffraction grating (Corben, 1978a). Three values of luffice spacings in three directions of space correspond to masses of three spacelike states that combine with one timelike state (Corben, 1978b; Recami, 1986, p. 113). In particle terms we interpret it as conversion into a bradyon-tachyon bound system of particles that trap each other in a relativistically invariant way, yielding compound particles endowed with zero rest mass:

178 MARCIN MOLSKI

A TE mode excited inside an $-cavity is then treated as a system of B- and D-wave that lock to form a photonlike C-wave *,,a propagating as excitation of luminal type. This tachyonic interpretation of an SN-cavity interior agrees with that of Corben (1977, 1978a.b) and with other theoretical results in the field (Recami, 1986. p. 1 11 and references cited therein). Thereby a free bradyon can trap at most three free tachyons (Corben 1978b). or in wave terms, that at most three spacelike states are superimposable on one timelike state to produce another particle [see Eq. (48), cf. also Pagels. 1976; Hoh, 19761.

This approach is applied to a TM mode for which the Lorentz transformed E, component of electric field, satisfying suitable boundary conditions, takes the form

Thus these considerations hold true for a TM mode excited inside a rectangular S'-cavity.

B. Three-Dimensional Spherical Space Cavity

Extended particles in a rectangular geometry, treated as electromagnetic S'-cavities, contradict intuitive expectations and a standard concept attributing to all particles a spherical geometry rather than a rectangular one. Therefore, we treat a construction, in an electromagnetic framework, of an improved model of extended particles, which takes into account not only their wave-corpuscular character but also a spherical geometry.

To proceed we consider a spherical cavity of internal radius a , which has perfectly conducting walls and an interior free of charge. If a TE wave is excited inside the cavity, its field component Eg propagates according to Maxwell's equation (Jackson, 1975).

{a: - [a,? + r-2( sinV'a6 sinBa6 - ~inB-~)]}rE+ = 0, (203)

(204)

in which m,,lc, n, 1 = 0,1 ,2 , . . . is the resonator frequency of the TE,/ wave and oY(f3) denotes associated Legendre polynomials with m = f l ; J,,/(m,/r) are spherical Bessel functions of order f satisfying

Eg = Eo(m,rr)-'Jfl/(mfl/r)0(8): exp [im,/xo],

[a: + m:/ - Z(1+ l)r-2] J,l(m,/r) = 0, J,/(O) = 0. (205)


The remaining field components H, and He are calculated on making use of Maxwell's equations (93) and (94), providing

(206)

(207)

Imposing on a electric Eg component of TE wave the boundary condition E4 (a) = 0, one obtains the resonator frequency

H, = i Eol( l+ l)(mnlr)-2Jnl(m,,lr)@(0)l exp [imnrxo], He = -iEo(m,r)-2r-'a,J,l(m,,r)0(0): exp [im,lxo].

in which x , ~ denotes root n of a spherical Bessel function Jnr(xnl) = 0. In the simplest case of the TE,o wave, the amplitudal part of Eq. (204) becomes

J,o(mnor) = sin(m,or), (209)

and the resonator frequency and field components of the TE wave take the forms nn a

m,o = -,

H, = 0.

Equation (2 12) reveals that He, having a normal orientation relative to the cavity surface, satisfies the boundary condition a,r He (a) = 0, providing the same values of the resonator frequency as those of a field component E4.

Proceeding as in the case of a rectangular cavity, we attribute a nonzero rest mass

hnlr m:o = -

ca

to radiation trapped inside a spherical cavity, which depends on the mode characteristic n and the cavity dimension a. The mass mlo associated with imprisoned fields is quantized, fully in accordance with the results of Jennison and Drinkwater (1977) and Jennison (1978). Hence the electromagnetic spherical cavity may be employed as a wave-corpuscular model of extended massive particles endowed with spherical geometry.

Exploiting this idea, we identify the fields trapped inside the cavity (r E < 0, a) ) with inner (self) fields propagating as excitation of a luminal type, whereas outside the cavity (r E (a, 00)) spatial propagation of fields vanishes and Maxwell's equation (203) reduces to a purely timelike equation,

(a: + mio) exp [imnoxo] = 0, (214)

180 MARCIN MOLSKl

describing the propagation of outer fields in the external cavity domain along the xo-axis being, incidentally, the cavity worldline. A wave equation governing propagation of outer fields, associated with a cavity moving in the x’-direction at velocity u = c d x l / d x o , is obtained on application of Lorentz’s time transformation, producing

(215)

(216)

Equation (2 15) is merely a Klein-Gordon wave equation describing propagation of a timelike B-wave associated with a freely moving particle of rest mass rno = rn,o h/c and spin zero.

To obtain the Lorentz transformed inner fields associated with a moving cavity interior, we note that not only is @‘no a solution of the wave equation (203) but it also satisfies

(a,W + rn;,) exp [ik,xfl] = 0, kp = {rn,o(l - /?2)-’/2, rn,op(l - p2)-1/2, 0, O } .

r = Jm, (218) with a d’Alembertian in terms of Cartesian coordinates

0 = a; - a; - a; - a;. (219)

Simultaneous application of Lorentz time and space transformations to Eq. (21 1 ) provides inner fields in the form

rv = d ( x 1 - bx0)2/(1- 8 2 ) + x; + x z . (22 1 )

Equation (220) shows that inner fields associated with a moving cavity satisfy


identical with those in Corben’s (l977,1978a,b) tachyonic theory of matter, a two- wave particle model and Mackinnon’s soliton theory. +Ao and + fields are then identified with D- and B-waves, respectively; their superposition yields a C-wave Qn0 propagating as an excitation of luminal type. According to a particle picture, it is interpretable as conversion of a photon into a bradyon-tachyon system of which the particle constituents trap each other in a relativistically invariant way, yielding a photon-type particle endowed with rest mass zero, as

For a spherical cavity a tachyonic component created has colaritudal(O), azimuthal (#), and radial ( r ) degrees of freedom; it can move on the surface of a sphere centered at the bradyonic constituent, but the radial degree of freedom also enables the tachyon to travel between infinitesimally close spherical surfaces. Hence such a bradyon-tachyon system created in a spherical cavity structurally resembles a hydrogen-type (H-type) compound.

According to Mackinnon’s soliton model, a Qn0 field is interpretable as a nonlinear nondispersive wave packet that has no spread with time and that constitutes a particlelike solitary wave. Gueret and Vigier (1982) showed that such a soliton wave follows a geodetic in an external gravitational field; so it behaves as a singularity of the gravitational field (test particle). Because Mackinnon’s soliton is endowed with a three-dimensional spatial extension and an inertia property, it is proposed as a wave-corpuscular model of extended massive particles (Mackinnon, 1978, 1981a,b, 1988). Because in M(1, 1) space propagation of a Qn0 wave is governed by a nonlinear propagation law (Gueret and Vigier, 1982), it may be interpreted in the framework of a nonlinear wave hypothesis as a C-wave involving B- and D-waves as internal spectrum waves associated with the cavity interior.

Consistent with all models mentioned in Section II, a wave picture of the interior of a spherical cavity emerges. Radiation trapped inside the cavity undergoes conversion into a timelike B-wave + and a spacelike D-wave +,,o that lock to form a solitary photonlike wave Qn0. Such imprisoned inner fields form a nondispersive wave packet that has no spread with time and that travels at a group velocity equal to that of the moving cavity.

We focus our attention on correlation between geometrical characteristics of spherical cavities and of extended particles. For this purpose, on the basis of the rest-mass formula given by Eq. (213). we derive the relationship

in which d is a cavity diameter and A0 = h/rnE0c denotes a Compton wavelength characterizing the massive cavity at rest.

Equation (229) indicates that for the fundamental T E l o mode the diameter of a spherical cavity at rest is equal to the Compton wavelength of an associated mass

182 MARCIN MOLSKI

and corresponds to the width of a wave packet measured at the first zero points of a function W10(xo, a ) . Hence standard parameters characterizing extended particles are strongly correlated with the geometry of spherical cavities.

The geometry of the cavity and its associated mass vary in the moving frame. To find suitable relations, we consider for interpretative simplicity of inner fields Q,," projected onto M( 1, 1) space, which is governed equation

propagation by the wave

(230)

Equation (230) reveals that the radius of a moving cavity undergoes a relativistic deformation and compresses in the direction of motion,

a - a J v , (23 1)

in strict conformity with a transformation of the Compton wavelength,

Simultaneously, the mass associated with a cavity varies according to the equation

which indicates that radiation trapped in a spherical cavity behaves as ordinary ponderable matter, and the relativistic mass problem may be considered in a purely geometrical framework (Molski, 1991, 1993a).

For a de Broglie wavelength associated with a moving cavity 1 = h / p l , PI being a component of the cavity three-momentum, one obtains

for which p' = p - I ; the de Broglie wavelength measured in diffraction experiments provides important information about the geometry of particles considered as spherical cavities.

Because the width between the first zeros of the amplitude of the bradyonic soliton (230) is equal to

h -- - h AX' = 2aJ- =

m:&?'y' A p ; '

the uncertainty principle for a spacelike momentum is

(235)


We investigate geometrical characteristics of a bradyon-tachyon H-type compound created inside a cavity. Because the spacelike field associated with a tachyonic component is delocalized in space (Recami, 1986, p. 56), one obtains useful information about the internal structure of a particle system by calculat- ing the radial distribution of a tachyonic field in space surrounding a bradyonic constituent. Taking advantage of the formula

dp(r) = 4 ~ r ~ J , o ( r ) ~ d r , (237)

and introducing into Eq. (237) the radial function given by Eq. (209) for u = 0, we obtain

which provides the probability of finding a tachyonic constituent within a spherical shell of radius r and thickness dr , or within volume 4nr2dr at radius r . The function given by Eq. (238) becomes zero both at the bradyonic constituent and at the cavity wall and has n maxima at points

2k + 1 r k = ( y ) a , k = 0 , 1 , 2 ,..., n - 1 . (239)

For the fundamental mode (n = 1) the radial distribution attains a maximum for ro = a / 2 ; the greatest probability to finding a tachyon moving in space surrounding a bradyon occurs at half the cavity radius.

These considerations are extensible to a TM wave endowed with field components ( H # , Eo , E r ) . In this case the boundary conditions (Jackson, 1975)

&(a) = 0, a,rH@(a) = 0, (240)

imposed onto a TM,o wave yield a resonator frequency c(n + 1 / 2 ) n .

a mnlc =

then the geometrical correlation given in Eq. (229) is absent.

C. One-Dimensional Linear Time Cavity

Derivation of a timelike Klein-Gordon equation for outer fields associated with a massive electromagnetic $-cavity is divided into two stages. On the basis of Maxwell’s and Helmholtz’s equations, one obtains a time-dependent wave equation (214) describing a massive cavity at rest; application of Lorentz’s time transformation yields a timelike Klein-Gordon equation (2 15) for outer fields associated with a moving cavity.

Because bradyon-tachyon symmetry is strictly related to time-space symmetry (Recami, 1986, pp. 34,35), one derives (Molski, 1993a) within an electromagnetic

184 MARCIN MOLSKI

framework a spacelike Klein-Gordon equation employing mirror operations of time imprisonment of electromagnetic fields and Lorentz’s space transformation.

Although a time-trapped field seems speculative, this possibility deserves attention, as it plays an important role in quantum theory of spacelike states. For example, Horodecki (1988b) considered in a nonrelativistic regime time quantization of spacelike fields trapped in an impulselike rectangular well, and Vyiin (1977a) investigated quantization of spacelike states on a closed timeline.

As the time- and spacelike representations of a Poincark group are SO3 and S02,1, respectively, tachyons are not localizable in our ordinary space (Duffey, 1975,1980; Vygin, 1977a.b) and appear more to resemble fields than do particles of fmite spatial extension (Recami, 1986, p. 59). Tachyons invariably admit reference frames (called criticulframes) in which they appear (at a speed u = 00) as points in time extended simultaneously in space along a line (Recami, 1986, p. 56). Hence the concept of a tachyon as a time cavity with trapped electromagnetic fields seems reasonable from a physical point of view. However, we refrain from comment on how such time imprisonment may be implemented practically.

To realize this concept, we consider an electromagnetic wave characterized with the four-potential Ap = (0, AX, AY, 0), which propagates in the fx3-direction. Nonzero components of A” satisfy a luminal Maxwell equation

(242) in which f p = {w /c , 0, 0, f 3 } denotes the wave four-vector of an electromagnetic field with f3 = fo = w / c . After such a wave becomes trapped in a one- dimensional T’ cavity of dimension ao, placed on the xo-axis, the imprisoned fields obey the equation

a,apAu = 0, A ~ ( x ~ ) = A; exp [i(f3x3 r foxo)],

apavQ(xp) = 0, Q(xp) = A;$,,.,,(x0) exp [if3x3], (243)

(a: +m~o)$no(xo) = 0. (244)

in which $,,.,,(x’), no = 0, 1,2, . . . are solutions of the time-dependent Helmholtz equation

Taking into account a boundary condition for time (Horodecki, 1988b),

$,,JXO = aO) = 0, (245) similar to the spatial one, we find solutions of Helmholtz’s equation (244) in the form

If a nonzero rest mass


is associated with radiation trapped in time, Maxwell’s equation (243) for outer fields associated with a T ‘-cavity exterior reduces to

(8; + mio) exp [ikax3] = 0, k3 = m n o c K 0 1 . In the next step, application of Lorentz’s space transformation to Eq. (243) yields a spacelike Klein-Gordon equation (Feinberg, 1967),

(0 - m,2,,)+’<x@>, +’(x”) = exp [ikLx”] = 0, (249)

describing propagation of spacelike outer fields associated with a freely moving TI -cavity at superluminal velocity u’. Simultaneously, inner fields associated with a T ’ -cavity transform to yield

in which an amplitude function +,,, satisfies a timelike Klein-Gordon equation

Equation (25 1) reveals that an electromagnetic field trapped inside a T ‘-cavity can be considered to be a system of D- and B-waves, but by comparison with space- trapped radiation the amplitude wave has timelike characteristics, whereas the exponential one is a wave of spacelike type. Such trapped electromagnetic fields reproduce fundamental properties of tachyons (ie., ponderability, localization in time, and superluminal kinematic). This approach fails to reproduce the three- dimensional extension of tachyons, which seems to be a genuine property of all particles, independent of their time- or spacelike characteristics.

D. Three-Dimensional Spherical Eme Cavity

To construct an electromagnetic model of three-dimensional tachyons, we generalize the approach proposed by Horodecki (1983a), who constructed a one- dimensional tachyonic soliton dual to the soliton constructed by Mackinnon. In the superluminal rest frame the tachyonic soliton has the form (Horodecki, 1983a)

sin(mocxO/h) mocxO/ h

W(x ,x ) = exp[imocx’ / h] , (253)

186 MARCIN MOLSKI

whereas in a Lorentz frame moving at a velocity u = cj? = c dx ' / d x o , it becomes

in which y = (1 - j?')-'/'. Such a nondispersive solitary wave, viewed as a dual C-wave (Horodecki, 1983a), does not spread during its superluminal motion; it may be viewed as an extended model of one-dimensional tachyons.

To generalize Horodecki's approach to three dimensions, one finds ordinary M(1,3) space-time inadequate, as it assumes the existence of a single time dimension. Several authors (Cole, 1977, 1979, 1980a.b; Demers, 1975; Recami, 1986, p. 130 and references cited therein; VySin, 1995) support the idea that a theory of relativity involves three time dimensions. Cole (1980a) showed that six- dimensional Lorentz transformations are derivable in which two extra variables are interpreted as time coordinates. According to Cole (1980b), the metric of such pseudo-Euclidean M(3,3) space has the form

6

ds2 = dyj dyJ = c2 (dr: + dr i + dr:) - dx: - dx; - dx; , (255)

whereas a link with a standarddescription in M( 1,3) space is achieved by assuming the particle three-velocity to be

(256) v = C--, dr = ldtl = d m . in which d t / d t determines the direction of a time displacement in six-dimensional space. These relations indicate that a transition from vector time of Cole's theory to scalar time appearing in our physical M( 1,3) space can be realized in the same manner as passing from Cartesian coordinates to spherical ones:

j =1

d r dr

( x ' , x2, x 3 ) -, (r, o,4), r = dx: + x i + x f . (257)

Hence a subspace M(3,O) has spherical symmetry, resulting in the space-time metric form

(258) ds2 = dp2 + p2(dp2 + sin2 19) - dx: - dx; - dx:,

p = ct = c d t ; + t; + r f , (259)

crl = p sin(I9) cos(p), cr2 = p sin(I9) sin(p), cr3 = p cos(I9). (260)

We consider a spherical electromagnetic wave trapped in a three-dimensional spherical time T'-cavity of dimension po, which satisfies a six-dimensional Maxwell equation

[p-2app2ap + P - ~ ( sin 8-'a,~ sin I9ao + sin I9-'8: - a:)] Q ( y i ) = 0. (261)


For the simplest case of purely radial solutions and one-dimensional motion along the XI-axis, one obtains

sin(m,!p) q,o, X I ) = exp [ i m , ~ ~ ' ] , n' = 1 ,2 ,3 , . . . , mn'P

in which the amplitudal function is a solution of a time-dependent Helmholtz equation

(p-'aPp2ap + m;f)+,,r(p) = 0, (263)

n'n

PO m,,l = m:, A/c = -,

and we attribute rest mass m!, to radiation trapped in time. Bearing in mind Eq. (259), we write the function (262) alternatively as

which is a three-dimensional version of a tachyonic soliton (253) constructed by Horodecki (1983a). In M( 1, 1) space, Eq. (265) reduces to Eq. (253).

According to Eqs. (253) and (265), an extended tachyon may be viewed as a lump of energy localized in a closed region of time. Total imprisonment of energy carried by an electromagnetic wave produces a transcendent tachyon endowed with zero energy and nonzero momentum m:,c. As the width between the first zeros of the time-dependent amplitude of the tachyonic soliton (254) is equal to

hc -

h Axo = 2poJ1-gz = - - -

mjj,cy A E '

it implies the uncertainty principle

A E A t = h

of conventional quantum mechanics.

E. Two-Dimensional Square Space-Time Cavity

Because wave-particle duality appears as a genuine property of matter, including objects both massive and of zero rest mass, we construct in an electromagnetic framework an extended wave-corpuscular model of photons. The starting point is the fact that photons may be viewed as a special case of bradyon-tachyon compounds of which the constituents endowed with the same rest mass couple with each other in a relativistically invariant way (see Section 1II.A). Such a compound particle has photonlike characteristics; it moves at the velocity of light, has zero rest

188 MARCIN MOLSKI

mass, and is associated with a system of waves propagating as an excitation of luminal type. With time- and spacelike objects as electromagnetic SN- and T1-cavities, respectively, one combines these two approaches to construct a wave-corpuscular model of a photon treated as an extended object and not as a pointlike structure.

We consider a simple model in which bradyons and tachyons are considered to be S’ and T’ phase-locked cavities of dimensions a3 and ao, respectively, with internally trapped electromagnetic fields. In the case of photons, the notion cavity wall has only an auxiliary value, as massive objects (walls) cannot travel at the velocity of light.

S’ -cavity: Proceeding as in Section IV.C, we generate the following equations:

a p a ” A z # n , ( X 3 ) exp [i foxo] = 0,

(A + m i , ) # n , , ( ~ 3 > = 0, n 3 = 0, 1 , 2 , . . . ,

According to results of preceding sections, nonzero rest masses m:, and mXo are attributed to space- and time-trapped electromagnetic fields, whereas associated inner fields are given in the general form

w p ’ ) = 4 ( X 3 ) * ( X O ) , W(XP) = # ( X O ) @ ( X 3 ) . (276)

With this solution one may build a nonlinear superposition of time- and spacelike fields

(277) 5 2 ( x f i ) = Q ( X @ ) Q ’ ( X f l ) = # ( x ’ ” ) x ( x p ) ,

#W) = 4 ( X 0 M ( X 3 ) , X ( X 9 = * (x0 ) * ’ (x3 ) , (278)

52 ( x p ) = A: sin (0) m o x o sin (7) m 3 x 3 exp [i ( foxo q= f 3 x 3 ) ] (279)


associated with a bradyon-tachyon system of particles treated as one-dimensional phase-locked cavities. This function and its field components 4 and x satisfy the lurninal wave equations

a,aw(xq = o (280)

a,afix(x”) = 0, a,ah$(x,) = 0, (281) provided that associated bradyonic and tachyonic constituents are endowed with the same rest mass,

and that an interaction condition

a , ~ ( x ~ ) a ~ x ( x c ” ) = 0 (283) is fulfilled. A requirement for equality of rest masses of the interacting bradyon and tachyon given by Q. (282) implies correspondences

0 3 = ao, n3 = 110, (284) so that dimensions of the space cavity must equal those of the time cavity and so that trapped radiations must have the same mode characteristics.

Introducing in explicit form the wave functions x and 4 into Eq. (283), and employing Eqs. (271) and (275), we obtain

or in an equivalent form,

Equation (286) is satisfied for

(287) This condition determines the initial position of a wave x = exp[i(koxo 7 k3x3)] propagating in the 7x3-direction at a constant phase velocity equal to the velocity of light (Coulson and Jeffrey, 1977). Hence the x wave propagates in an external space-time cavity domain: It can be identified with outer or radiating fields, in contrast to a 4 wave representing inner fields associated with a space-time cavity interior. Consequently, extended photons appear in our formalism as two- dimensional space-time (S’ T ’) square cavities of equal space and time dimensions. This condition is a consequence of identical mass of bradyonic and tachyonic photon constituents. As such imprisoned inner fields are endowed with an effective

3 0 x f x = x 3 f c t =a3=ao.

190 MARCIN MOLSKI

zero rest mass, outer fields obey the luminal Maxwell equation, in agreement with our knowledge of the field. Hence, an electromagnetic square S' TI -cavity is considered a wave-corpuscular model of a photon as it reproduces its fundamental properties: zero rest mass, nonpoint extended structure, and luminal kinematics.

If we take into consideration the de Broglie relation pp = hf p that is valid for massive and massless particles, one obtains

E = p c = hnna- c, (288) 0 I 3 p' = hnna-I,

or in an equivalent form,

(289) 0 2 E = hw = m,c = nhwo, w0 = n a - l c ,

These formulas for a E (0; 00) reproduce the full spectrum of waves associated with photons, indicating a rigorous connection between wave and geometric characteristics of associated luminal particles viewed as an S 1 TI-cavity: values of parameters such as frequency or wavelength are related to SIT1-cavity dimensions. Moreover, these formulas not only provide a value of the elementary energy-momentum quantum (for n = 1) but also predict the possibility of the existence of its integral multiple (multiphoton) for n = 2,3, . . . . Multiphotons appear, for example, in Planck's emission, in photoelectric effect, and in Compton scattering, and can spontaneously decay to the ground state (n = I), producing n spatially separated but coherent single photons. Chin and Lambropoulos (1984) showed that when an isolated atom in an excited state decays spontaneously to its ground state, the emitted light must be a single photon having a well-defined energy and frequency and that the process must occur within a characteristic dura- tion comparable with the period of oscillations of the emitted photon. Hence the spatial dimensions of the photon must be on the order of its wavelength.

Equation (275) reveals that a tachyonic component of a photon, endowed with infinite momentum p 3 = h f = m!,]c and zero energy, is responsible for momentum carried by a luminal wave. In contrast, Eq. (27 1) indicates that energy of a c-velocity wave is rigorously connected to a bradyonic component endowed with zero momentum and energy E = m!,c2. The equation pp = hf p was first considered for photons by Einstein and extended by de Broglie to include massive particles, providing a conceptual background for a fundamental hypothesis of wave-particle duality and a formulation of quantum mechanics. These results indicate that wave-particle duality for photons may be accounted for by de Broglie relations (1) plus an energy-mass relation giving w = h-'m:lc2, f = h-'rn:,,c. Hence a tachyonic theory can explain that zero rest mass M0 = d- = 0 is attributed to photons, but we associate with photons a nonzero mass mo = hu/c2 via de Broglie's and Einstein's relations.


We focus attention on transformational properties of fields associated with an extended photon under a Lorentz transformation. To this effect, we assume that a photon treated as an S' TI-cavity is observed in a reference frame moving in the fx'-direction at velocity u = cp = c dx' ldx' . The Lorentz-transformed fields and the corresponding wave equation take the form

m fv = f o p

z/l'rs' (293)

in which fu denotes the Doppler-shifted frequency of a luminal wave associated with a photon. Equation (291) reveals that in the moving frame both time and space dimensions of an S1 T1 -cavity undergo relativistic compression

the masses of associated bradyonic and tachyonic components both increase but remain equal to each other. For this reason a photon and its associated luminal waves observed in a moving frame invariably travel at the velocity of light. In the limiting case ( p + 1) photons become pointlike objects of infinite energy and momentum.

Thus the concept of a photon as an object of invariant properties in all inertial reference frames is invalid; the zero rest mass and the velocity of light attributed to a photon remain invariant in moving and rest frames, but their geometric characteristics become modified according to Eq. (294).

Summarizing these considerations, we indicate the possibility of verifying this model of a photon. To accomplish this objective, we consider photons (treated as S1 TI-cavities) in a beam incident on a periodic crystal-like structure and reflected from its planes. In such circumstances, interaction between a nonpoint luminal particle and a periodic obstacle may be considered an elastic collision; the angle of reflection equals the angle of incidence. Cavity dimensions are invariant under reflection, as an elastic collision excludes dissipative processes. Values of the angle 00 leading to a large reflected intensity of a photon beam are determined on making use of Bragg's law, 21 sin 00 = nk, (Kittel, 1966), in which 1 is a distance between crystal planes and 00 is the angle measured from crystal planes to incident and reflected beams.

When we use Eq. (271) in the form

(295) -1 f' = f' = 2n/k, = nn3a3 ,

192 MARCIN MOLSKI

in which A, is the wavelength of a luminal wave, after simple mathematical operations we find that

2423 = n31,, (296)

a3 = 1 sin&, (297)

and Bragg’s formula (n = n3) takes the form

relating only geometrical parameters of a periodic structure and interacting photons, Equation (297) indicates that Bragg’s reflection may appear for only photons of internal space dimensions a3 5 1. This geometric version corresponds to a wave condition (Kittel, 1966)

&I2 5 1 (298) obtained for n3 = 1, which clearly shows that Bragg’s reflection may be interpreted in a purely geometrical framework in which a photon is considered to be extended a nonpoint object of internal space size a3 corresponding to the S ’ T ’ - cavity dimension.

Equation (297) is valid for only photons emitted from a source at rest relative to the laboratory frame in which a crystal is placed. Because the geometry of photons emitted from a moving source varies according to Eq. (294). for this case Eq. (297) becomes replaced with

a3 Jg = I sine,, (299) in which 8, is the angle measured from crystal planes to incident and reflected beams of Doppler-shifted photons.

Dividing EQ. (297) by Eq. (299), one obtains sin 00 1 sine,, - d m = y ; --

hence the angle B,,. measured in a dynamical Bragg experiment, varies according to

(301) The formula conjugate to Eq. (300) provides an excellent way to determine the velocity of a moving source given by

e,, = sin-] [(I - u2/c2) s in~o] .

sin2 eo sin2 e,,

These formulas are obtained employing only geometric properties of photons treated as extended relativistic objects.

So far our considerations are confined to photons interacting with ordinary timelike periodic obstacles; Eqs. (275) and (295) reveal that similar interactions


occur for a hypothetical periodic structure composed of spacelike objects. From Eq. (275) in the form

(303)

2ao = no& (304)

a. = 1'sin8;, (305)

f 3 = 2n/& = f O = nnoa,' ,

one obtains

and the spacelike counterpart of Eq. (297),

in which 1' is the spacing within a spacelike periodic obstacle. The results indicate that a photon is a universal object that may interact with

particles of both bradyonic and tachyonic types. In particular, photons interact with a hypothetical periodiclike structure composed of spacelike objects. If such an interaction is viewed as an elastic collision between an extensive photon treated as an S' T -cavity and a periodic obstacle, one may evaluate the reflection angle leading to a large reflected intensity governed by the law

21' sin 0; = nAp, (306)

which is a spacelike counterpart of Bragg's law in ordinary timelike form. This electromagnetic model of photons is also applied to a geometric interpreta-

tion of the cosmological red shift observed for spectral lines of distant galaxies and nebulae. For this purpose we consider an isotropic and homogeneous expanding space-time characterized by the Robertson-Walker metric

ds2 = d x i - R ( x ' ) ~ 1 + E - [dr2 + r2(d02 + sin2 8 d ~ $ ~ ) ] , (307) ( :)i in which R(xo) is an expansion parameter; E = +1,0, -1 for spherical, flat, and hyperbolic geometries of the universe.

Maxwell's equation for electromagnetic waves propagating in space-time conformally coupled to (307) reads (Parker, 1972)

(spa" - 2r)A, = 0, (308)

in which x o is replaced by a conformal time variable xoR(xo)-'. Solutions of wave equation (308) take the forms (Parker, 1972)

(309)

(3 10)

(311)

A, = exp [fifoxoR(xo)-'] F(r, 8, qj),

fo = (f2 - 2€)'/2,

= n 2 - 1 , ~ = + 1 n = l , 2 , 3 ,..., f 2 = -AF(r , 8, C$)/F(r, 8 , # )

E = -1

194 MARCIN MOLSKI

Introducing the frequency formula (271) into Eq. (309), we find that

A, = exp fi-x F(r , 8, q5), [ a3;:9 O] (312)

indicating that frequencies of electromagnetic modes in an expanding universe decrease as the universe expands. This red shift is familiar for spectral lines of distant galaxies and nebulae in the expansion process. Equation (312) reveals that decreasing frequency modes result from stretching of the space dimension of an S' TI-cavity (representing a photon) due to an expansion

a3 + unR(x"), R(0) = 1. (313)

This interpretation may be applied not only to an electromagnetic field but also to all massless fields of arbitrary nonzero spin s; the reason is that wave equations governing propagation of such fields are all conformally invariant (Parker, 1972). For example, one may consider for s = the two-component neutrino equation, and for s = 2 the vacuum solutions of Einstein's field equations. This geometric interpretation of the cosmological red shift hence applies to photons, neutrinos, and gravitons.

These considerations are readily generalized to include an extended model of a photon considered as a superposition of bradyonic and tachyonic solutions given by Eqs. (217) and (265) projected onto a M( 1, 1) space. Proceeding as in the case of a square S' TI-cavity, one may construct a nonlinear superposition of waves of luminal type, which in the moving frame takes the form

(3 14) (315)

in which m! and m!, denote masses of associated time- and spacelike S1 T '-cavity constituents.

Amplitudal and exponential functions appearing in Eq. (3 14), and their superposition Q,,,,l, satisfy the luminal wave equation provided that

(3 16)

0 0 m,, = m,c/h = nlr/ro, m,,! = m,,c/h = n'Tr/po.

3 0 r0 = po, n = n', x - f x = r0 = po.

These restrictions are identical to those obtained for the square S' TI-cavity.

E Conclusions

The electromagnetic approach enables one to construct wave-corpuscular models of extensive massive and massless particles, and improves our understanding


of their internal structure. In this context, timelike objects appear as three- dimensional electromagnetic cavities with space-trapped electromagnetic fields. Such imprisoned radiation undergoes conversion into a system of B- and D-waves that lock to form photonlike inner fields occupying the cavity (particle) interior. According to the particle picture, the photon becomes converted into a bradyon- tachyon system, of which constituents are self-trapped in a relativistically invariant way, yielding a compound particle of photon type. The three-dimensional space cavity has a bradyon-type characteristic, i.e., it behaves like a massive subluminal extensive object associated with time-like outer fields interpreted as matter waves of the first kind (B-waves).

Exploiting a mirror method in a similar manner, one may construct an electromagnetic model of a spacelike object considered to be a cavity with time-trapped electromagnetic fields. Such imprisoned radiation is endowed with a rest mass depending on cavity dimensions, whereas the outer spacelike fields associated with the cavity exterior can be interpreted as matter waves of the second kind (D-waves).

In both cases space and time cavities with trapped radiation reproduce well the fundamental properties of ponderable matter, i.e., inertia, mass quantization, three-dimensional spatial extension, and time(space)-like kinematic characteristics. Hence they may be considered wave-corpuscular models of massive particles.

This approach is consistent in many points with well-known theoretical results, and predicts, for example, the following:

1. Conversion of massless photons into ponderable matter in accordance with Jennison-Drinkwater (1977) theory.

2. A factorized form of the particle wave function identical with that in the de Broglie-Bohm model.

3. The presence of B- and D-waves in the spectrum of states associated with the cavity (particle) interior, which is compatible with a two-wave particle model.

trapped inside a moving cavity, in full accordance with a nonlinear wave hypothesis and Mackinnon’s (1978, 1981a,b) theory.

spectrum associated with imprisoned radiation, which is consistent with an extended space-time description of matter developed by Horodecki (1988a, 1991) and Corben’s (1977, 1978a.b) tachyonic theory.

In particular, the electromagnetic model provides an explanation of why a particlelike solitary wave obtained by Mackinnon satisfies the luminal d’ Alembert’s equation but fails to satisfy the timelike Klein-Gordon equation (Gueret and Vigier, 1982). Mackinnon’s equation governs the propagation of inner fields associated with the cavity (particle) interior. Because the spectrum of inner fields includes

4. Formation of a nonlinear nondispersive wave packet (C-wave) by fields

5 . The appearance of bradyonic and tachyonic constituents in the mass

196 MARCIN MOLSKl

both time- and spacelike waves in the form of a nonlinear superposition, its propagation is governed by the luminal d’ Alembert’s equation. As outer fields associated with the cavity exterior are endowed with a timelike characteristic, they propagate according to the Klein-Gordon equation for massive subluminal particles. Both equations are correct, but in distinct cavity (particle) domains.

This approach also permits construction of a relativistic wave-corpuscular model of a photon that may be treated as square two-dimensional electromagnetic space- time cavities. Such imprisoned inner fields have characteristics of luminal type: zero rest mass is attributed to them, and the associated outer fields propagate as c-velocity excitations. According to a particle interpretation, a photon is considered an object composed of both bradyonic and tachyonic components, of the same rest mass, trapping each other in a relativistically invariant way. Such a compound particle is endowed with a characteristic of luminal type; it behaves as a massless object moving at the velocity of light. This model is consistent in many points with the tachyonic model of photons proposed by Dutheil (1984, 1989, 1990a,b, 1993). Therefore, photons are treated as objects of time and spacelike faces, which live on the border between two kinematically inpenetrable subluminal and superluminal worlds. This view is consistent with the well-known fact that photon worldlines separate the Minkowski cone into domains of time and spacelike kinematic characteristics. Hence photons seem to be universal objects interacting with both bradyons and tachyons; for example, photons should interact with a hypothetical periodic structure constructed from spacelike objects. If such an interaction is viewed as an elastic collision between a nonpoint photon and a periodic obstacle, one can evaluate the reflection angle leading to a large reflected intensity using a spacelike Bragg’s law so far unknown in the domain of theoretical physics.

The proposed model of extended photons also predicts the following properties:

1. Quantization of photon energy and momentum experimentally confirmed in

2. The possibility of existence of multiphotons appearing in Planck’s the blackbody radiation law and Compton scattering.

emission, the photoelectric effect, Compton scattering, and multiphoton ionization of atoms.

3. Photons with finite dimensions, which may be viewed as localized packets of electromagnetic energy of a size comparable to the photon wavelength, this property being consistent with the photoelectric effect, Compton scattering, and the Hunter-Wadlinger (1988) model of photons.

of extended photons.

spacelike Bragg’s law and a cosmological red shift predicted for neutrinos, gravitational waves and light.

4. A strict correlation among wave, corpuscular, and geometric characteristics

5 . The possibility of a purely geometrical interpretation of a time and


The first three points are consistent with best established characteristic of a quantum of light formulated by Einstein and quoted by Diner et al. (1984):

The energy of a light ray spreading out from a point source is not continuously distributed in space, but consists of a finite number of energy quanta which are localized at points in space, which move without dividing, and which can only be produced and absorbed as complete units.

The results obtained in this section have important methodological consequences worth discussing. The timelike Klein-Gordon equation and its spacelike counterpart for a particle considered to be an electromagnetic cavity are derived in a framework of Maxwellian electromagnetism and the special theory of relativity. Hence equations fundamental in quantum mechanics have an electromagnetic nature; consequently, ‘‘a de Broglie wave may prove to be as real as a classical electromagnetic wave, and [d’ Alembert’s] equation may prove to be of more importance to quantum mechanics than has hitherto been supposed” (Mackinnon, 1981a).

Conversely, this electromagnetic approach permits one to consider the relativistic mass problem in a purely geometric framework and explains why the mass of an extensive timelike particle increases with velocity. An internal photonlike structure of an extended, massive particle is consistent with Weinberg’s (1975), Winterberg’s (1978), and Sedlak’s (1986) concepts treating a photon as a fundamental particle of nature. The following onto logical gradient of fundamental aspects of the existence of matter:

wave + geometry + particle

becomes a consequence of the fact that corpuscular properties of matter can be derived from a wave theory of light with suitable boundary conditions (geometry).

The electromagnetic concept of matter permits also one to construct a quantum theory without Planck’s constant h. According to this idea (Barut, 1992), it is possible to formulate quantum mechanics without fundamental constants such as h, mo, ore, as a purely wave theory, provided that a quantum system is characterized by a fundamental characteristic frequency ~ 0 . Barut (1992) showed that all well- known equations in conventional quantum theory, such as Schrodinger’s or Dirac’s equation and their eigenvalues can be expressed in terms of wo instead of a mass h q / c 2 . A nonrelativistic Schriidinger equation for a freely moving particle and for a particle moving in a Coulombic field take the form

in which w is the Rydberg frequency

e2 ch

w=001112/2, (Y = - (3 19)

198 MARCIN MOLSKI

and a! is the fine-structure constant. The eigenvalues of the wave equation (318) are given as a frequency spectrum

The solutions for space-trapped electromagnetic fields become viewed as a special case of solutions (83) obtained by Barut (1990). whereas associated masses become expressed in terms of the resonator frequency WO. Consequently, the electromagnetic approach enables one to formulate a relativistic quantum mechanics of extended spin-0 particles without the Planck constant h.

An important general conclusion is that a pointlike characteristic attributed to material objects seems to be only a rough mathematical approximation of real properties of particles appearing in nature. The wave-particle duality of matter, including both massive particles and radiation, becomes enriched with the geometrical aspect of their internal structure, because the wave, corpuscular, and geometric pictures are merely three aspects of the same physical reality.

v. EXTENDED SPECIAL RELATIVITY AND QUANTUM MECHANICS M A LOCAL L-SPACE

Our research on the D-wave so far is confined to classical electromagnetism, which seems to be the best framework in which to construct an extended relativity em- bracing superluminal motion and an extended quantum theory, including spacelike particle states. The fundamental problem arises as to whether there exists a physically well-established theory other than Maxwellian electromagnetism that can verify an extended wave-particle model of matter and answer the main question: Can a D-wave exist in nature?

A solution of this problem was obtained in the framework of special relativity plus the de Broglie postulate on wave-particle duality (Horodecki, 1988a) or by extension of de Broglie’s theory (Elbaz, 1985; Das, 1992). Detailed analysis indicates that they do not reclassify the basic problem from the category hypothesis to a physically well-established theory.

Hence we seek to reformulate a conventional theory of special relativity and quantum mechanics in the framework of local space (L-space) (Molski, 1995c), which appears to be the best framework to introduce concepts of the D-wave and spacelike particle states.

A. Special Relativity in L-Space

Our point of departure is a notion of absolute space introduced into physics by Newton to explain inertia and then generalized to a four-dimensional absolute


space-time by Einstein within special and general theories of relativity. The notion absolute means that "the space-time is physically real but also independent in its physical properties, having a physical effect, but itself not influenced by physical conditions" (Einstein, 1955).

The absoluteness of space-time seems well established within general relativity; de Sitter's and Godel's solutions of Einstein's field equations indicate that a space- time continuum can exist without mass and that all masses in the universe rotate within an absolute space-time (Einstein, 1955). For special relativity the situation differs; its formulation does not refer explicitly to the notion of absoluteness that appears in this theory as a hidden property of space-time. Because a transition from general to special relativity can be made easily (Einstein, 1955), it becomes obvious that in such an operation the basic equations of the latter are completely deprived of information about absoluteness.

To consider this problem mathematically, we assume that a test particle of rest mass rn0 follows a geodesic in four-dimensional space-time endowed with a metric G,,, and signature (+ - --). The metric G,,, evaluated by solving Einstein's equation (Einstein, 1955),

rigorously determines the particle kinematics in curved space; its four-velocity v, is calculated on integrating the equations of motion,

in which s is a four-dimensional affine parameter along a classical trajectory and vi", denotes an initial value of the four-velocity.

When we consider a particle moving with a four-velocity u: in the local flat space-time called by Einstein a Galilean region, relation (322) reduces to

(323) G,, + v: = const.

Because Gi,, may in general contain both diagonal and off-diagonal nonzero elements, a transition to the fundamental metric of Einstein's relativity (E-metric),

(324)

is realized with the operations (1) diagonalization procedure GE, + g:, = SiS,,, and (2) scaling g:, --+ S,,, in which

0

3

ds2 = S,,dx'dx" = dxi - dx: - dxi - dxf , ,v=0

2 0 B, = g,,, 800 = 1, 611 = 822 = 833 = -1, S,, = Oforp + v. (325)

These relations indicate that a correlation between the kinematic characteristics of a particle in uniform translational motion and a geometry of a local flat space

200 MARCIN MOLSKI

(L-space) may be given by elements of the metric GE,, or after diagonalization, by the diagonal elements gi,, and scaling leading to the E-metric S,, completely removes the possibility of finding geometric and kinematic relationships.

To find such a geometry-kinematic connection, we note that the two-dimensional L-interval obtained after diagonalization of the metric G i v

(326) ds2 = (Bo)2dxi - ( /11)2d~:

is invariant4 under generalized Lorentz transformations (Molski, 199%)

0 B O Z O - BBIZ.' x =

pod- '

g , d m ' 1 f i 1 .F ' - BBoZo

x =

in which ' = Bc is the velocity of a reference frame. Dividing Eq. (3 Eq. (327), one obtains

XI b(Z'/Zo) - B BI - Sl p- = p = - - , p = - . xo 1 - BB(K1/Ko)" P O B O

(327)

This equation indicates the conclusion that if we assume Bxl appearing in Eq. (329) for p < 1 and /I > 1 to be treated as a contracted or expanded light distance XI

equal to the light time xo = ct , then XI/XO = Zl/Zo = 1, and Eq. (329) reduces to

These equations are only transformational relations for /? measured in two inertial frames, Their form is identical with that of the velocity addition theorem, provided that we identify (Molski, 1995~)

B = B I / B O - V I I C , s =B1/Bo - i jI /C. (331)

In the case of four-dimensional space-time the L-metric (326) takes the form 3

ds2 = Bidxi - C pidxi , (332) k= I

which is given equivalently as

(333) k = l

In the case of &,-space, the term invariance implies that we seek a transformation relating suitable quantities in two inertial frames and preserving (xo, r, jJo, P I ) H (xh, r', jJh,

the quadratic form of Eq. (326).


in which we use the relation dxk = dxo = ds # 0 valid in L-space. Employing Eqs. (333) and (331) generalized to the form

B = (v I /c , v2/c7 % I C ) vk/c = Bk/BO, (334)

we express PO and B k in terms of V k :

allowing us to correlate geometric characteristics of L-space g:,, with kinematic characteristics of a particle in uniform translational motion. By comparison of Eq. (335) with Eq. (3), we see that parameters Bp become expressed in terms of components of the particle four-velocity

Bp = v,/c. (336)

In the framework of L-space we derive not only conventional relativity for timelike particles but also its form extended to include spacelike objects. Thus we introduce the time(space)like Lagrangian

in which q = + 1 for timelike and q = - 1 for spacelike particles, respectively (Recami, 1986, p. 50). With the four-momentum formula

for q = f l we obtain a time(space)like four-momentum appearing in the Horodecki-Kostro and Das models [see Eqs. (52). ( 5 3 , and (70)]. Hence the L-space concept is compatible not only with a postulated extended space-time description of matter (Horodecki, 1989a, 1991) but also with subluminal and superluminal theories of relativity (Recami, 1986, p. 21).

The L-space theory also works for limiting cases Vk = 0 and v k = 00. Then the interval (326) reduces to

Uk’O lim ds2 = dx i - (0 . dx1)2, U k + b o lim ds2 = ( 0 . dxd2 - dx:, (339)

whereas generalized Lorentz transformations (327) and (328) take the forms

202 MARCIN MOLSKI

Under transformations (340) and (34 I ) , the purely timelike and purely spacelike intervals (339) transform to

(342)

(343)

Bearing in mind Eq. (336), we write the operation (342) alternatively in the form

ds2 = dxi - (0 . dX1)2 -+ dS2 = SidXi - BfdX:,

ds2 = (0 - dxo)2 - dx: + dd2 =&di i -$di : .

( , - B 2 d i : ) , (344) ds’ = dx, 2 + dS2 = (1 - B2) -1 ’2 d i 2

discussed in detail in Section VII. The concept of L-space allows one to explain the phenomenon of inertia. As the

geometry of L-space rigorously determines kinematic characteristics of a particle in uniform translational motion, such a space is explicitly absolute, and inertness of massive objects appears as a consequence of the relation

which may be viewed as a relativistic formulation of the first principle of newtonian mechanics.

B. Quantum Mechanics in L-Space

We consider matter waves associated with a particle uniformly moving at velocity v in an isotropic (PI = 8 2 = 8 3 ) L-space characterized by the line element

ds2 = Bidxi - /3:dr2, (346)

in which

BIlBO = B = V I C . (347)

Taking into account de Broglie’s (1924) idea that “nature treats all particles in the same way with regard to particle-wave duality, whether their rest mass is zero or not,” one assumes that d’ Alembert’s equation constructed in L-space,

(348)

not only describes the propagation of electromagnetic waves (B = 1) but also governs the propagation of a B-wave (B < 1) and a D-wave (j3 > 1) .

According to this hypothesis, by differentiating the wave function appearing in Eq. (348), we obtain the dispersion formula

g;v8p8,, exp [ik,xc’] = 0,

g;”k,k,, = 0, (349)

equivalent to

p2k2 - B?k2 = 0. 0 0 (350)


Taking advantage of Eq. (347) and the relativistic equation

v/c2 = p/E, (35 1 )

valid for subluminal and superluminal particles, from Eq. (350) we obtain

This result expresses the wave-particle correspondences for timelike ( u < c), photonlike ( u = c), and spacelike ( u > c) objects, derived here with an accuracy proportionality to the factor A.

In the next step we consider the wave equation (348) in the space dS2 = ds2&*, conformally coupled to (346)

introducing into it the useful relation

derived from Eq. (338), which is valid for bradyons ( q = +l ) , photons ( q = O) , and tachyons ( q = - 1). We thus produce a wave equation that after differentiation of the wave function twice with respect to space coordinates yields

Equation (355) for q = + 1, 0, - 1 gives the Klein-Gordon, Maxwell, and Feinberg ( 1967) wave equations, respectively, describing propagation of matter waves associated with the subluminal, luminal, and superluminal particles moving at a velocity cp. This general wave equation includes wave equations governing propagation of B- and D-waves. With this approach they are obtained with no standard quantization formalism and without referring to quantum-mechanical operators.

The derived wave equation (355) is generalized to include a charged particle interacting with an electromagnetic field A,. To proceed we consider the guidance formula (26) and a relation (de Broglie, 1965, p. 112)

2 2

[aoS(xU) - -V] e - [ V S ( x @ ) + !A] = qrn;c2 C C

that is rewritten in the equivalent form

(357)

204 MARCIN MOLSKI

Substituting Eqs. (358) and (13) into Eq. (348), we obtain

g["D,D, exp[ih-'S(xP)] = 0. (359)

From this compact wave equation one may derive, by differentiation of the wave function twice, its expanded form

[ D ~ D P + q ( m o ~ / h ) ~ ] exp[ih- '~(xP)l= 0.

(S2Di - D: - 022 - Df)@P = 0

(360)

To consider the Dirac equation for a charged particles of spin we assume that

(36 1 )

Eq. (359) in the equivalent form

is satisfied by @P components of Dirac's bispinor function

JI = (@o, @ I , $2, $3) = ( U O , U I , u2, ~3)exp[ih- 'S(x~)l , (362)

in which the uP denote components of bispinor amplitude. We find the first-order differential operator

a!$ Do + a!' D' + a2 D2 + a 3 D3,

uODO - IY~ D' - CY~D' - u3D3

(363)

which when multiplied on the left (right) side by an operator

(364)

produces the operator part of Eq. (361). Then we arrive at the matrix relations

U,'CY, + CX,U$ = 28B2, u ~ ( Y , + CX,,(Y,, = 28, p, v # 0, (365)

(366) a0 +(-)a, - a! P O a!+(-) = 0, p # 0, in which aPr 8, and 0 are 4 x 4 matrices; the second is a unit matrix.

Because

B2 = B-B+ = B+B-, (367)

in which e

DoS(X') = &S(X') - -V, C

(370)

we achieve the matrix representation

THE DUAL DE BROGLIE WAVE

or equivalently,

205

in which a = fa*. The matrices appearing in the equations above are defined as _ _ follows~

0 0 0

0 -1 O ;), 0 0 0 - 1

0 0 0 - i

0 -1

- 1 0 0 0

a-=.[ ; -; ; !), (373)

a1 = (. 0 1 0 0 ' O 0)

a3= [. ; ; -i). (375)

0 0 0

0 0 0 1

(374)

1 0 0 0

0 0 1

0 -1 0

Introducing the explicit form (371) of the matrix a: into Eq. (363),

and then differentiating $ once with respect to the time xO-coordinate, we obtain Dirac's equation,

(Do + a1D' + a2D2 + a3D3 + i f i m o c a / h ) + = 0, (377)

valid for timelike ( q = +1) and spacelike ( q = - 1) particles (Molski, 1992). Similarly, from Eq. (364) we derive

( D 0 - a l D ' - a 2 D 2 - a j D 3 - i , / i lmoca/h)$ = O ; (378)

hence the product of operator parts of Eqs. (378) and (377) yields the second-order operator appearing in Eq. (360).

Matrices a,, are conventionally denoted (p , a,, aYaz),

206 MARCIN MOLSKI

Finally, we write compactly Dirac’s equation (363) in the highly symmetric form

d‘ D,$(x’*) = 0, (379) in which no mass term i,/ilmoccr/h appears explicitly.

C. Conclusions

These results indicate that the L-space concept is the best way to formulate extended special relativity and wave theory, including spacelike particle states. The tachyon kinematic and D-wave mechanics emerge in a simple and natural manner without introductory assumptions. The L-space concept provides a theoretical and conceptual basis for an extended space-time description of massive particles and a two-wave hypothesis (Horodecki, 1989a, 1991; Das, 1989, 1992).

If we require space-time to be explicitly absolute, we obtain via generalized Lorentz transformations the velocity-dependent L-metric, and then by generalization of Maxwell’s equation, wave equation (353). which includes the Maxwell, Klein-Gordon, and Feinberg equations as special cases. Additionally, de Broglie wave-particle correspondences are obtained with an accuracy proportionality to the factor A .

In a conventional formulation of a wave theory of matter, these equations (for mo # 0) are obtained on postulating the following:

1. De Broglie wave-particle correspondences p , = hkp 2. The form of quantum-mechanical operators i), = i ha, and f, = x, 3. Jordan’s‘ rules p, + i),, x , + f, 4. A general form ofthe wave equation P(i), , f,)$ = o On comparison of these approaches we conclude that formulation of wave theory

in L-space lessens the number of its basic postulates and allows a consistent wave- particle description of micro-objects, whether or not they possess rest mass. For instance, putting u = c into Eq. (353). we find Maxwell’s equation, whereas for u # c, Eq. (353) gives the Klein-Gordon or Feinberg equation. Instead of three wave equations for time-, photon-, and spacelike objects, we need only one, d’ Alembert’s equation (353). to describe matter waves of all types that may appear in nature. This achievment confirms de Broglie’s fundamental insight in assuming a similarity between massless photons and particles with nonzero rest mass, and indicates that propagation of de Broglie waves in L-space seems to have much in common with propagation of ordinary electromagnetic waves.

For a product of two noncomutating operators, Jordan’s rule must be supplemented by a sym- metrization rule to produce a Hermitian operator; it fails for products of three operators, leading to the Temple paradox (Julg, 1993).


The proposed approach generates methodological consequences. It is well known that from the chronological and methodological points of view, wave theory and classical mechanics, including both general and special relativity, developed independent of each other. Whereas classical mechanics is based on geometric space-time properties, the basis of wave theory has no connection with space-time features. Consequently, its fundamental equations, such as the wave equation, the correspondence rule, or the form of quantum-mechanical operators, are not derived or deduced from geometric space-time properties. Then L-space theory allows one to construct a wave theory in a purely geometric framework with an initial supposition that physical space is an explicitly absolute four-dimensional space-time continuum.

The Dirac equation derived for spacelike fermions raises a question about tachyon spin. Although this problem seems speculative, it is widely discussed (Recami, 1986, p. 115 and references cited therein) and deserves attention for the reason that spacelike objects play a role in internal structure of particles endowed with spin (Costa de Beauregard, 1972; Corben, 1995). For example, in quantum electrodynamics based on Joos-Weiberg higher-spin wave equations, some solutions for integer-spin particles are of the tachyonic type, whereas the Bargmann-Wigner equation holds not only for time- but also for spacelike particles (Recami, 1986, p. 115). Tachyons are generally considered to be spinless (scalar) particles, or objects with infinite number of polarization states (Recami, 1986, p. 115); contrary opinions are also published (Corben, 1977).

VI. NO-WAVE MODEL OF CHARGED PARTICLES IN KALUZA-KLEIN SPACE

The classical, semiclassical, and quanta1 models of a charged particle are well investigated but they still animate discussions. The fundamental problem to be solved is to explain how charge and mass are structurally related inside a particle, and the nature of the force binding charge and mass into charged matter to give a stable system of finite self-energy and self-field. To answer this question, several models are proposed that, according to the particular method, are developed in the framework of Maxwellian electromagnetism and Newtonian gravity (Visser, 1989; Robinson, 1995), theories exploiting the Casimir effect (Boyer, 1968), electromagnetic approach (Jehle, 1971, 1972, 1975; Post, 1982, 1986), and unified field theories of the Kaluza-Klein type (Kaluza, 1921; Klein, 1926).

Kaluza-Klein theory treats gauge fields and gravitation on a five-dimensional M4 x S' manifold (five-space), of which M4 denotes an ordinary four-dimensional space (four-space) and S' is a compact space with the topology of an extra dimension as a circle of radius R.

208 MARCIN MOLSKl

In the five-space the position of a particle is described with coordinates in a set

(380)

that for any integer n denotes the same point; the geometry of five-space is characterized with a metric (Toms, 1984)

(381)

x i = ( x ~ , x 4 + 2scnR), i = 0, 1 , 2 , 3 , 4

d t 2 = gijdx'dxj = g,,,dXfldX" - (KA,,dxp + d ~ ~ ) ~ ,

K = C - 2 d m , (382)

in which A,, denotes the electromagnetic four-potential and G the gravitational constant; g,,, describes Einstein's gravitation.

A. Kaluza-Klein Field Theory

According to this theory, an electric charge acquires a purely kinematic interpretation (Rayski, 1965); namely, it can be related to a momentum p4 canonically conjugate to a compact extra x4 variable. To investigate this concept in detail, we consider a complex scalar field * ( x i ) associated with a charged particle of rest mass Mo, moving in five-space (381).

The transition to a field description is made via a Lagrangian formalism starting with the action (Toms, 1984)

in which g5 = det gi, . Restricting consideration to a q ( x i ) field in the absence of gravitation, from Eq. (383) under a variational principle ASS = 0, we find a wave equation in five-space,

(384) [(a,, - ~ ~ , , a ~ ) ( a p - K m q ) - a; + ( M ~ C / A ) ~ ] Q ( X ~ ) = 0,

which for A" = 0 reduces to a simple form

[05 + (Moc/h)2] * ( x i ) = 0, (385)

(386)

As a consequence of assuming of a closed extra space, according to Kaluza-Klein theory every field must be a periodic function of the fifth coordinate. To derive the field equation in four-space we make a Fourier analysis of the \Ir field,

5 0 = aiai = a; - a; - a; - a; - a:.

. +m


hence W is expanded into a complete set of harmonics with four-dimensional fields taken as expansion coefficients. Introducing Q. (387) into Eq. (383) and then integrating the extra variable x4 with orthonormality properties of the harmonics, we obtain the action in four-space (Toms, 1984),

nz-m J ,=O

in which

g4 = detg,,, D, = a, + in KR-’A,. (390)

This procedure is called dimensional reduction. Employing a variational principle, AS4 = 0, and assuming absence of gravitation, from Eq. (388) we derive the field equation in four-space,

[(a, + inKR-lA,)(alL +inKR-’A’*) + ( m ~ c / h ) ~ ] @ ( x ’ ) ~ = 0, (391)

which in the absence of an electromagnetic field reduces to

[a$, + (moc/h)*] +(x% = 0. (392)

The derived equation (391) has a form identical to that of a wave equation well known in electrodynamics,

(393)

to describe a charged particle of 0-spin interacting with an A , field, provided that charge q associated with the particle is quantized in discrete units of elementary charge e ,

[(ac, + i ( q / h c ) A , ) ( P + i(q/hc)A’) + ( m ~ c / h ) ~ ] @ ( x ’ ) , , = 0,

ChK R

q = ne = n-. (394)

This result enables us to express the mass term mb appearing in Eq. (389) and the radius of the compact manifold in terms of the fine-structure constant a, the Planck mass m p [ , and the Planck length 1,1, as follows:

mb = e(16~rG)-’/~ = a’ /2m, l , (395)

(396)

2.2 x lOW5g.

(397)

R = h(l6~rG)’/~/ec = a-’/21pl,

a = e2/hc = 1/137.03604,

1,1 = J 1 6 ~ r G h c - ~ x 1.6 x 10-33cm, mpl = d m

210 MARCIN MOLSKI

Detailed analysis of the transition from Eq. (384) to Eq. (390) indicates that one may formally associate with charge an operator of infinitesimal displacement of the closed extra x4-coordinate

q + a 4 = -ih&, (398)

having eigenfunctions \I, ( x i ) and eigenvalues

q ~ ( 1 6 n G ) - ' / ~ = - i h & Q / q , (399)

nmbc = -iha4Q/Q. (400)

given in an alternative form as

These equations allows akinematic interpretation of electric charge as amomentum p4 o( q conjugate to the compact extra x4 variable. As from Eq. (389) one can derive the relation

l4 = mbcR = nh, (40 1 )

electric charge is also interpretable as an angular momentum I , conjugate to an extra angular coordinate 8 of period 2n (Rayski, 1965).

B. Charged Particle as a Five-Dimensional Tachyonic Bootstrap

As all charged particles appearing in nature (excluding quarks) are endowed with charge q = ne, n = + 1 , 0, - 1 , expansion (387) becomes reduced to

@ ( x i ) = (2nR)-'/*{ $ ( x p ) + exp[+ix4/R] + +(x@)- exp[-ix4/R]}, (402)

which contains the functions

including the mass term mb = crl'zmp,. Under detailed analysis, Eqs. (385) and (392) and the field component given

by Eqs. (403) and (404) indicate that these equations can be viewed as a five- chmensional version of Corben's tachyonic theory (Molski, 1997). In particular, @ ( x p ) may be interpreted as a B-wave associated with a bradyon of mass m0 moving in a four-space M4 and described by the wave equation

(405)

whereas +'(x4) is a purely spacelike D-wave associated with a transcendent tachyon of mass mb and infinite speed moving in a compact space S' about a

[O + ( m o c ~ ) ~ ] @ ( x p ) = 0,


bradyonic constituent and satisfying the wave equation

[a$ - ( m b c / ~ ) ~ ] +(x4) = 0. (406)

These two free objects can trap each other in a relativistically invariant way, yielding in the five-space M4 x S’ a bradyon-tachyon compound of mass [see Eq. (389)]

M O = d m : - m t , mb = crI/2mpl, (407)

described by the wave function

* ( x i ) = +(x9$7x4), (408)

satisfying the five-dimensional wave equation

[ 0 5 + (Moc/h)2] * ( x i ) = 0. (409)

with respect to the invariant interaction condition

a;$(x9a;$’(X4) = 0. (410)

According to a particle picture, the quantity called charge (in conventional theory) is a purely spacelike object endowed with mass mb = a’ /2mpl and momentum p4 = &m&. which moves in an S1 space of internal radius R = c r - ’ / 2 ! p j . Such ultraheavy tachyons might be created (Molski, 1993c) from hypothetical particles called maximons (Markov, 1982) in the process of expansion of the universe (Minn, 1990). The quantity crA’2m,,1 with a negative sign appears in a classical model of an electron, coupling ordinary electromagnetism with a self-interacting version of newtonian gravity, and is called the bare mass of an electron (Visser, 1989). Although the fine-structure constant and Planck mass are both quanta1 concepts, their combination crA/2m,j is purely classical (independent of h).

If bradyonic and tachyonic constituents of a charged particle interact through a field similar to a gravitational field (Recami, 1986, p. 112 and references cited therein), the motion of a charge at infinite speed corresponds to a fundamental state of a system of particles in which the trapping force holding charge on a circular orbit tends to zero. Hence a charged particle in Kaluza-Klein space may be considered to be a relativistically invariant tachyonic bootstrap (Corben, 1978a; Chew, 1968,1970) of two free particles in the foim of a Kepler-type (Recami, 1986 p. 116) system.

According to a field interpretation, a free complex scalar field \i, is viewed as a superposition of and $‘ fields that interact and lock to form a $$’ field whose characteristics depend on masses mo and mb of interacting particles

timelike, mo =- cr’f2mpl

spacelike, mo < a ‘ f 2 m p ,

photonlike, mo = a’j2mpl .

212 MARCIN MOLSKI

For timelike particles appearing in our physical four-space, the condition

mo << a’ /2mpl (412) invariably holds; consequently, in M4 x S’-space a bradyon-tachyon system is endowed with an imaginary rest mass

and @ ( x i ) is spacelike.

a tachyonic interpretation of a large mass problem: This peculiar property of a @ field according to Kaluza-Klein theory is merely

It is impossible to describe a test particle with a charge e and effective mass less than the Planck mass, unless the test particle follows a space-like geodesic in five dimensions (Gegenberg and Kunstater, 1984). Despite much effort, this problem cannot be solved satisfactorily (Gegenberg and Kunstater, 1984).

C. Conclusions

According to the standard paradigm of physics, charge and mass are attributed to the same material object but without explaining how charge and mass are structurally related inside the particle. The tachyonic model permits penetration into the particle interior to answer this question. Moreover, we explain the nature of the force binding charge and mass into a stable system. From the context of our analysis the following conclusions emerge.

Kinematics of charge and mass are treated separately in our approach, and charge kinematics are subject to no limitation involving the velocity of light as an upper bound. Elementary charge in a particle moves at a superluminal velocity. Motion of a charge inside a charged particle generates a ring current circulating azimuthally at a superluminal angular velocity. An elementary manifestation of electric charge is simultaneously of a stationary and dynamic nature, as a transcendent tachyon representing it transports not energy E = 0 but momentum p4 = mbc.

Those conclusion agree with results obtained by Post (1982.1986), who considered a charged particle to be a ring current involving elementary charge circulating azimuthally as a tabular sheath of distributed charge. Application of the Post model enabled retrieval of the Dirac moment of an electron up to the first-order QED anomaly a/2n. and produced conclusions identical to those above. The results are also compatible with those reported by Davidson and Owen (1986), who attempted to clarify a geometrical relationship between mass and charge


considering charged particles in the Kaluza-Klein space to be tachyonic modes of higher dimensions.

With a proposed tachyonic model of charged objects we assume that a test particle follows a spacelike five-dimensional geodesic in Kaluza-Klein space, but without conflict with known laws or experimental facts; superluminal motion along a five-dimensional spacelike geodesic appears to be a genuine property of the system under consideration (Recami, 1986, p. 91 and references cited therein).

VII. EXTENDED DE BROGLIE-BOHM THEORY

Detailed analysis of the de Broglie-Bohm, Horodecki-Kostro, and Corben concepts presented in Section I1 indicates a possibility of their unification into one generalized theory. The point of departure to realize this objective is an important property stipulated by Corben (1978a)-that two time (space)like waves cannot be locked one to the other, or, according to a particle picture, that two bradyons (tachyons) cannot trap each other in a relativistically invariant way to yield particles in a bound system. If they did, the momenta of such interacting particles would become imaginary, leading to nonnormalizable solutions (Corben, 1978a). Considering the de Broglie-Bohm factorized function (16) to be a superposition of amplitudal and exponential waves, we recognize that if one of them is spacelike, the other must be timelike. We assume that for a freely moving particle, these functions satisfy time- and spacelike wave equations

[IJ - ( r n b ~ / h ) ~ ] R ( x p ) = 0, (414)

whereas their superposition is a solution of Eq. (17). In such circumstances one obtains from Eq. (20)

[IJ + ( r n ~ c / h ) ~ ] exp[(i/h)S(xP)l = 0,

in full agreement with Corben’s theory. For MO = 0, Eqs. (414) and (415) reproduce the fundamental equations of Mackinnon’s soliton theory.

Exploiting this analogy we link the quantum potential (23) with a spacelike solution (414) to generate an equation

valid in a nonrelativistic approximation.

214 MARCIN MOLSKI

In the case of a tirne-independent spacelike field R ( x a ) , the wave equation (414)

(4 18)

reduce to

[A + ( r n b ~ / h ) ~ ] R(x") = 0,

whereas the quantum potential (23) takes the form

This result indicates that carriers of a field R(xP) and a source of quantum potential Q(xP) can be spacelike particles. This result plays a vital role in an extended de Broglie-Bohm theory developed in succeeding parts of this article.

A. Tachyo-kinemaric Effect

We reinvestigate Corben's theory of particles composed of both bradyonic and tachyonic components (see Section 1I.D). If a bradyon at rest endowed with rest mass mo and described by

(420)

absorbs a tachyon (antitachyon) of a rest mass mb and infinite speed, characterized

[a; + ( r n o ~ / h ) ~ ] +(xo) = 0, +(xo) = exp [irnocxo/h]

by

[A + ( r n ; , ~ / h ) ~ ] +(r)' = 0, +'(r) = exp[fimbc. r/h], (42 1 )

the bound system of particles has a mass

MO = d m i - rnt and is described by the wave function

~ ( x p ) = +(xo)+'(r) = exp [i/h(rnocxO f mbc r)], (423)

satisfying the wave equation

[apap + (Mo~/h)~] W(X') = 0, (424)

under an invariant interaction condition

aP+(xo)aP+'(r) = 0.

According to the generalized Stiickelberg-Feynman switching principle (Recami, 1986, pp. 32,33), the signs f in the wave function (421) correspond to the tachyon and antitachyon, respectively, moving in opposite space directions.


Wave equation (424) and its solution (423) have the equivalent form

215

which when compared to a standard form of the Klein-Gordon equation

provides the identification

first mentioned by Corben (1978b). According to a corpuscular picture, relation (428) is a result of a four-momentum conservation principle

implying that a free bradyon absorbing a free tachyon (antitachyon) not only alters its mass according to Eq. (422) but also attains a velocity the value of which depends on masses of interacting particles. This phenomenon, henceforth called the tachyo-kinematic efect, allows us to reinterpret the kinematic and wave- mechanical properties of ordinary timelike particles in terms of Corben's tachyonic theory.

Timelike Theorem. Each slower-than-light particle of rest mass MO is considered a bound system of a bradyon at rest endowed with mass mo and a transcendent tachyon (antitachyon) of mass m; < mo. These two free objects trap each other to yield a timelike particle of rest mass MO = J- moving at a velocity v = cmb/mo determined by masses of interacting particles.

As the phase of a B-wave associated with a moving microobject is expressed in Eq. (426) only with masses of interacting time- and spacelike particles we derive Lorentz's time transformation making use of mass-velocity relation (428). To prove this thesis, we consider a definition of a mass center of a bound system of N particles endowed with masses mi and described by coordinates in the set ff;, a! = x , y , z :

216 MARCIN MOLSKI

in which Mo denotes the total mass of the system. This equation for a bradyon- tachyon compound formed in two-dimensional pseudo-Euclidean M ( l,l)-space reduces to the form

given equivalently as

x o f (mb/mo)x'

M. XCM =

Taking advantage of Eq. (428). we arrive at Lorentz's time transformation,

which describes the position of the mass center of a bradyon-tachyon compound moving in M ( 1 , 1)-space.

To obtain Lorentz's space transformation, we consider similarly the process necessary to form a superluminal bradyon-tachyon compound. For this purpose we assume that a tachyon at infinite speed and of rest mass mb absorbs a bradyon (antibradyon) of rest mass mo. If mo < mb, the compound particle moves at a superluminal velocity, whereas Eqs. (420)-(424) take the form

[a; + ( m ~ c / h ) ~ ] $ ( x o ) = 0, $ ( x o ) = exp [ f i m o c x o / h ] , (434)

(435) [A + ( m b ~ / h ) ~ ] $09' = 0, $'(r) = exp[im;c . r / h ] ,

q(xP) = $(xo)$'(r) = exp [ i / h ( m b c . r f mocx')],

[aPaL( - ( M h ~ / h ) * ] W'(xP) = 0.

(437)

(438)

According to a switchingprincipk (Recami, 1986, pp. 32,33), the signs f that appear in the timelike wave function (434) correspond to a bradyon and antibradyon, respectively, moving in opposite directions of time.

Equation (438), given equivalently as


takes a form identical to that of a spacelike Feinberg (1967) equation [see Eq. (58)]

for a D-wave associated with a superluminal particle moving at a velocity cp'. Then we identify

These results enable us to formulate the spacelike theorem.

Spacelike Theorem. Each faster-than-light particle of rest mass Mt, is considered a bound system of a bradyon (antibradyon) at rest endowed with mass mo and a transcendent tachyon of mass mb > mo. These two free objects trap each other to yield a spacelike particle of rest mass Mh = d-2 moving at a velocity v' = cmb/mo determined by masses of interacting particles.

Following the same procedure as for Lorentz's time transformation, in the first step we determine the mass center of a bradyon-tachyon system moving in M(1, 1)-space,

mbx' f moxo ECM = JR'

which is given alternatively as

x ' f (rno/mb)x" .fCM = d w *

x ' f pxo

Taking Eq. (441) into account, we arrive at Lorentz's space transformation,

WCM = J1-s"

(443)

in which /J = j?-', to describe the position of the mass center of a bradyon-tachyon compound moving at superluminal velocity cp' = c/J-' in M( 1, 1)-space-time. Hence

pp' = 1 V U ! = c2, (445)

so d becomes identified with the Das pseudovelocity [see Eq. (67)]. According to our approach, it is related to a tachyonic component of a particle, not with the particle itself.

218 MARCIN MOLSKI

As a bradyon can trap at most three tachyons [see Eq. (49)], with the foregoing approach we derive Lorentz's time transformation acting in M(1,3)-space. To accomplish this we consider a system of bradyon at rest and three transcendent tachyons (antitachyons), the mass center of which is described with the equation

X C M = (446) mOxo f (mix' + m;x2 + mix')

Jrn; - mi2 - m$ - mf

m i , m i , mi < mo.

in which

(447)

The four-dimensional version of Lorentz's time transformation is obtained from Eq. (446),

in which we introduce

B = W , / m o , m;/mo, rn;/mo) = (v l /c , v2/c, vnlc). (449)

(450) reveals that such an operation changes a mass term mo + Mo, and this condition contradicts the fact that the rest mass of a micro-object is relativistically invariant. To avoid this inconsistency, we reconsider the basic wave equation (424) in the form

[n + (m; - m$)c2/h2] exp [i/h(mocxo f mbc . r ) ] = 0. (451)

Detailed analysis of a transition from wave equation (420) to (424), [a: + ( m o c / h ~ ~ ] + (xo) -+ [ a , ~ + ( ~ o c / h ) ~ ] q ( x f i ) = 0,

If we make the conformal transformation

x p + P = x p d 1 - m t / m i , mb < mo, (452)

(453)

of space-time coordinates x" and wave equation (451), we obtain

ds2 = dx,dxfi -+ dS2 = d i ,d .V ,

[n + (mt - mL2)c2/h2] exp [i/h(m,cZo f m:.c. F ) ] = 0, (454) in which m, and m: are conformal masses of bradyonic and tachyonic constituents of a particle

Hence m, 2 -mF = m i

(456)


and Eq. (454) takes the well-known form

Comparing Eq. (456) with the four-momentum relation (6) in the form

( E / c 2 ) 2 - = m i , (458)

m, = E / c 2 , mi = p / c , (459)

we make the important identifications

to correlate suitable quantities appearing in ordinary and extended special relativity. Equation (457) shows that formulation of the problem in conformally coupled

space-time (453) preserves invariance of the bradyonic rest mass and kinematic mass relation (428). The latter is a consequence of invariance of the three-velocity under the conformal transformation (452)

dr di: m:. mb dxo dIo m, mo'

- - -

With Eq. (428), conformally coupled coordinates 2' in the form

attain a purely relativistic interpretation: They are Lorentz-contracted coordinates xp of a laboratory frame.

This procedure also holds for a spacelike bradyon-tachyon compound described by wave equation (439), but in this case transformation (452) takes the form

xfi --+ ifi = x p J 1 - mi/rn$, mb > mo. (462)

Consequently, time (space) Lorentz transformation of the phase of a B(D)-wave

is equivalent to two operations: tachyon (bradyon) absorption

s(x0) = mOcxo + S ( X P ) = mOcxo f mbcxl,

~ ' ( x ' ) = m&xl + ~ ' ( x p ) = mbcx' f mOcxo (464)

220 MARCIN MOLSKI

and transformation (452) (or (462) for a spacelike bradyon-tachyon compound)

yielding the same final result as application of the time (space) Lorentz transformation in Eq. (463).

A transition to conformal masses and space-time coordinates does not affect the tachyo-kinematic interpretation of Lorentz transformations. Then Eqs. (43 1) and (442) take the form

m , i o f mLZ1 i0 f (m:/rnc)Z-' (466) - XCM = -

J- J 1 - (m:/m,)2'

m: . i l f m , ~ ' Z ' f ( r n , / r n : ) ~ O

J- = J I - (m,/mk)2' . f c M = (467)

so by virtue of Eq. (460) from Eqs. (466) and (467). we obtain time and space Lorentz transformations.

A transition to conformally coupled space-time becomes more comprehensible if we consider it in a framework of the L-space concept. If a bradyon at rest absorbs a tachyon of infinite speed, the compound system attains a velocity according to Eq. (428). Consequently, associated with a bradyon the L-space undergoes transformation (344) to the form

ds2 = d x i + dS2 = [1 - ( rnb /m~)~] -" ' (dZ i - (mb/mo)2dZf). (468)

The right side here reveals that an L-metric associated with a moving bradyon- tachyon compound is expressed in terms of conformal coordinates (452). From Eqs. (468) and (452) we recover the original form of the L-interval before conformal transformation:

The necessity to formulate the basic problem in conformally coupled space- time (453) has important interpretive consequences. During formation of a bradyon-tachyon compound, three effects appear: alteration of the rest mass of a bradyon on absorbing a tachyon, alteration of the motionless state of a bradyon that attains a velocity on absorption of a tachyon, and Lorentz contraction of spatial and temporal dimensions. These effects together produce an entity that in conventional relativistic mechanics is called a moving particle in the space time.


So far we treated relativistic wave mechanics of freely moving time- and spacelike objects. To derive a nonrelativistic counterpart of wave equation (457), we note that for u << c , or equivalently, mb << mo. the following approximations hold:

m , = mo + m:/(2mo) + 3m:/ (8mi) +. . . % mo + m” 0 /(2mo),

x mb. (47 1 )

(470)

m: = mb + m!/(2mi) + 3m$/ (8mi ) + . - a

Consequently, in the nonrelativistic regime, Eq. (454) takes the form7

exp { -i /h [ ( T + moc2)1 F p - r] } = 0, (472)

in which

(473)

are interpreted, according to a conventional approach, as kinetic energy T = (rn011)’/2rno and momentum p = mov of a particle.

Equations (459), (472), and (473) form a basis to reinterpret conventional quantum mechanics in terms of a tachyonic theory of matter, presented as follows.

B. Particle on a Line

We consider the simplest quantum-mechanical problem, a particle trapped in a one- dimensional infinitely deep potential well of length a. Then the time-dependent Schrodinger equation (472) has solutions

wk(r, x) = (2/a)’ / ’ sin(knx/a) exp [-ih-’ (mot' + T k ) f ] , (474)

(mbc)’ (khn)’ T k = - - - - k = l , 2 , 3 , ...

2mo 2moa2’ (475)

that satisfy the boundary condition $(x = 0, a) = 0. The amplitude function appearing in Eq. (474) is a solution of the time-independent Schrodinger equation

(476) (A + ~ ~ ) ( 2 / a > ’ / ~ sin(knx/a) = 0,

in which

’ To keep the notation simple we r e t m to the original variables x P instead of .V

222 MARCIN MOLSKI

Taking Eq. (475) into account, we rewrite Eq. (476) as

[A + (m;c/h)'] (2/a)'/' sin(knx/a) = 0, (478)

which represents a purely spacelike equation for a tachyon at infinite speed and endowed with a quantized nonrelativistic mass

khrr ca

mi = - , k = l , 2 , 3 ,...

and momentum kAn

pk = fm;c = &-. a

(479)

Accordingly, we show that a particle on a line can be considered to be a compound of a bradyon at rest endowed with mass m, = mo + Tk/c2 and a tachyon of mass m: = mi moving at infinite speed back and forth between the potential walls. This controversial conclusion becomes more comprehensible when we consider the transformational properties of the field associated with a particle trapped on a line. Thus we assume that a line with a particle trapped thereon moves in the x ' -direction at a velocity V relative to the laboratory frame. Space (time)like fields associated with a particle undergo a space (time) Lorentz transformation, yielding

in which B = V / c ; satisfy wave equations

and $ transform as typical space- and timelike fields that

and

for subluminal and superluminal particles endowed with masses rn: and m,, respectively.

In conventional quantum mechanics, wave function (474) is identified with an ordinary B-wave of only statistical meaning. According to the Copenhagen interpretation it determines the probability of finding a particle between the points x and x + d x , which is equal to

d P = **9 dx = (2/a) sin'(nkx/a) d x . (485)


Equation (485) shows that fork 2 2 the wave function has nodes. In particular, for k = 2 the node appears at the middle of the line; hence the probability of finding a particle between x = a / 2 and x = a/2 + dx is zero. If a particle really moves between the ends of the line, a question arises as to how it moves from one side of the node to the other.

Several solutions to this problem are proposed (Nelson, 1990), of which pro- posals of Bohm (1952a,b) and White (1934) merit detailed analysis. In the de Broglie-Bohm theory, the wave function (474) is given in a factorized form (16) in which

If we consider nonrelativistic guidance formula (32). by virtue of Eq. (486) one finds that u = 0; hence a particle trapped on the line is at rest,' so it cannot traverse the nodes.

According to White's (1934) interpretation, the infinitesimal interval in which the particle is between x and x + dx in a stationary state is equal to QQ* dx = p d x . Hence its average velocity during crossing left to right, and vice versa, is given by u = dx/d t o< p - ' ; at the nodal point at which p = 0, u = 00 (i.e., the particle traverses the node infinitely quickly). This result remains contradictory to a classical picture assuming a subluminal velocity for a particle moving between the ends of a line.

Detailed analysis of these concepts indicates that they describe a basic problem from two points of view, related to the time- and spacelike constituents of a particle. If a moving particle is a system of a bradyon at rest and a transcendent tachyon, the following consequences apply. An amplitudal function R ( x ) = (2/a)'I2 sin(knx/a) is a purely spacelike wave associated with the tachyonic Component of a particle, endowed with mass mi = knh/ac and momentum P k = fm;c = f k n h / a . An exponential function @ ( t ) = exp[-ih-'(moc2 + Tk)t] describes a bradyonic component of a particle, remaining at rest and having rest mass m, = mo + Tk/c2. A superposition of these fields Q ( r , x ) = R(x)*(r ) satisfies the nonrelativistic time-dependent Schrodinger equation for a particle of mass rno moving at velocity u << c on a line.

These conclusions explain the results of de Broglie and Bohm and of White. The guidance formula imposes zero velocity for a bradyonic constituent of the particle; the node is traversed by its tachyonic component but not by the particle itself. The problem of transit of a particle from one side of the node to the other consequently disappears.

* This result was criticized by Einstein (de Broglie, 1960, p. 136), but Bohm (1952a.b) asserted that it holds only for a particle moving in the box with perfectly rigid walls. It does not apply to a particle in a real box, since the thermal motion of the walls prevents the particle from being at rest.

224 MARCIN MOLSKI

The constant velocity attributed to a particle trapped on a line is explained on comparison of Eq. (475) with Eq. (417) to yield the important relation

Hence the source of quantum potential on a line is a tachyonic constituent of the particle, and a quantum force F generated on the line by the quantum potential Q equals zero: Because neither external nor internal forces act on the particle, it moves at constant velocity.

C. Particle in a Scalar Field

The proposed interpretation of a particle on a line also holds for a particle interacting with an external scalar field endowed with a potential U(r). Then the relativistic four-momentum formula

( E - - p2/c2 = m i (488)

(m, - u/c 2 2 ) - rn: = mi,

takes the equivalent form

(489)

in which m, and m i are defined in Eq. (459). In the nonrelativistic regime Eq. (489) becomes

m: a m ; = A C JG. (49 1 )

Hence the associated tachyonic component is endowed with variable rest mass rnb. depending on the particle’s position, in contrast with the bradyonic component that for stationary systems (E = const) is endowed with constant rest mass m,. Consequently, the time dependence of particle velocity in conventional mechanics corresponds to space dependence of mass of a tachyonic particle constituent,

rnov(r) = mb(r)c. (492)

Because by differentiating Eq. (492) with respect to time, we obtain

and taking into account the relation valid for conserved fields,

(493)

dv dt

mo- = -VU = F, (494)


we find an important identity

225

as a direct consequence of Q. (490). These results indicate that two representations of the particle dynamic are pos-

sible: a conventional one for interactions with a field of potential U and another assuming the existence of a tachyonic field T interacting with a particle. Accord- ing to a classical picture, both descriptions are equivalent and yield an equation of motion (493) of the same form as Eq. (494).

In a quantal description, the Schrodinger equation for a particle interacting with an external scalar field takes the form

h2A> 2m0 ( i h k - moc 2 - U ( r ) + - R ( r ) exp [ - i l k ( & + moc2)t] = 0, (496)

in which the amplitudal term is a solution of the spacelike wave equation

[ A + (m;c/h)*] R ( r ) = 0 (497)

for a tachyonic component endowed with variable rest mass

Hence E k denotes a quantized energy of the particle system, resulting in mass quantization of a tachyonic component. Taking Eq. (498) into account, one may introduce a quantal counterpart of a classical potential T = - ( m b ~ ) ~ / 2 m o ,

(499)

which differs from quantum potential (29) in the de Broglie-Bohm theory by sign. Consequently, forces generated by Tk(r) and Q ( r ) are equal but act in opposite directions.

To summarize these considerations we propose a dynamical theorem.

Dynamical Theorem. Quanta1 interaction of a particle with a field endowed with potential U ( r ) is interpreted as a result ofparticle interaction with a quantum potential Tk ( r ) associated with a transcendent tachyon of quantized variable mass rn; = ( h / c ) , / - A R ( r ) / R ( r ) .

To show how the dynamical theorem works, we consider a particle moving in oscillatory motion along the x-axis in a field endowed with harmonic potential U ( x ) = 1 /2moo2x2, in which o denotes the frequency of vibrations. The solution

226 MARCIN MOLSKI

of wave equation (496) takes the form

mow 2 q(r ,x)k = Akexp [ - - i / f t (Ek +moc2)rl~k(x)exp [ - x x ] = 0, (500)

in which H k ( x ) , k = 0, 1,2, 3, . . . are Hermite polynomials, whereas

are the normalization factor and vibrational energy, respectively. The amplitudal part of Eq. (500) is a solution of a spacelike wave equation (497),

including quantized variable rest mass

(503) 2rno[hw(k + 1/21 - 1/2mow2x2]

C2

Wave equation (502) and its solutions transform as typical spacelike expressions, turning in a frame moving at velocity V = c B along the fx-axis, to

1 [O - (mi~/h)~]AkHk[(x f Bxo)G]exp

2mo(hw(n + 1/2) - 1/2mow2[(x f Bx0)Gl2} C2

m i =

in which G = (1 - B 2 ) - ' / 2 . Comparison of Eqs. (502) and (505) shows that tachyonic variable rest mass is not relativistically invariant but varies in the moving frame according to formula (505).

Superposition (500) of a spacelike field (502) and a timelike field described by the exponential part of wave function (500) produces a field of timelike type, associated with a bradyon-tachyon system moving at a velocity

Equation (506) shows that at the turning points

h(2k + 1) mow

x = f I (507)

the tachyonic mass and the velocity of a system of bradyon and tachyon reduce to zero, whereas in the region beyond the turning points, which is unattainable in


classical mechanics, the tachyon acquires an imaginary mass so that the compound particle moves in the forbidden region at an imaginary velocity

v = ic {mow2x2 -my(2k + I) (508)

As a particle endowed with such a velocity travels areal distance d x in an imaginary interval i d t , we conclude that a tachyonic interpretation of the tunnel effect is equivalent to that employing extrapolation to imaginary time (Bjorken and Drell, 1964; Freed, 1972; Jackiw and Reebi, 1976).

The proposed tachyonic interpretation of particle interaction holds true for all scalar fields endowed with a potential U ( r ) . Thus the proposed model is applicable to describe electromagnetic, strong, and gravitational interactions, provided that we can solve a suitable wave equation to determine a spacelike field R ( r ) .

D. Uncertainty Principle

We examine Heisenberg's uncertainty relations

A E A t Z h , A p , A a ? h , C Y = X , Y , Z (509)

which with Eq. (459) become

o h h Am,Ax 1 -, Am'zAa ? -.

C C (5 10)

Relations (509) can be interpreted to be a result of delocalization in space-time of conformal masses of bradyonic and tachyonic constituents of a particle. This conclusion becomes clear when we recall that free bradyons invariably admit subluminal reference frames (rest frames) of a particular class from which they appear as points in space extended in time along a line. Free tachyons invariably admit subluminal reference frames from which they appear as points in time extended in space along a line (Recami, 1986, p. 56). The reason is that localization groups (little groups) of time- and spacelike representations of the Poincark group are SO3 and S02.1; hence tachyons (bradyons) are not localizable in space (time) (Recami, 1986, p. 59).

If a region of possible motion of a particle is restricted to size a, the spatial uncertainty relation enables us to estimate qualitatively its tachyonic mass

h Am; -,

ac

in accordance with mass formula (479) obtained for a particle trapped on a line.

228 MARCIN MOLSKI

E. Many-Body Problem

A fundamental question emerging from this discussion is whether our approach can be generalized to include the most general case of the many-body problem. To answer this question, we consider a stationary system of N interacting particles of mass ms described with a generalized Schriidinger equation (15)

in which

is the total mass of the system, whereas the wave function tL.es the form

Q ( x , t ) = R ( x ) exp [-i/h(Ek + moc2) t ] . (5 14)

In Eq. (514) E k denotes the overall quantized energy of a system depending on quantum numbers k = (kl, k2, . . .), whereas the amplitudal function is a solution of a spacelike wave equation

Because Eq. (512) is formulated in configuration space M ( 1,3N), we can interpret it within a generalized Corben’s (1977) approach, assuming that several spacelike states with mass m&, can combine with a timelike state mo to produce a subluminal particle of mass

I N

s=l cr=x.y.z

Accordingly, proceeding along lines of the one-body problem, we define the mass of tachyonic component of a system,

that allows calculation of local velocity ua, of particle s:

(517)


directly from amplitude function R(x), and not from the phase S ( x , t ) via guidance formula (32). We also derive a quantum potential Tk(x),

and with Eq. (517), the classical potential N

- U ( x ) - E . T ( x ) = -x- (m:d2 - s=l 2mS

According to de Broglie-Bohm theory, the amplitudal function R(x) determines the quantum potential Q ( x ) and probability density for particles to be at certain po- sitions in space-time (Bohm etal., 1987). Equation (518) clearly shows that in our approach R ( x ) additionally determines the velocities of particles, which depend on the state of the entire quantum system described by R(x). Although particles follow continuous and causally determined trajectories, obtained by integration of Eq. (5 18). they execute random movements, resulting in a statistical interpretation of R(x). For points at which R(x) = 0, the quantum potential Tk(x) = 00 and the particle velocity attains an infinite value according to Eq. (5 18). The probability of finding a particle there equals zero; conversely, if R(x) at some point takes a large value, the particle velocity tends to zero and the probability of finding it there increases. This interpretation is fully compatible with our intuitive (classical) understanding of the notion probubilify.

If wave function (514) is factorizable, N N

s=l s= I

and the basic wave equation (512) decomposes into N independent equations

[ i h k - msc2 - Us(rs) + -As Rs(rs) exp [-i/h(Ek,, + mSc2)t] = 0, 2mS h2 1

interpretable within a framework of the one-body problem considered in Sec- tion VI1.C. In such circumstances the quantum potential Tk(x) reduces to the sum of terms

Consequently, we consider an N-particle system as N bradyons of mass E k , / c2 + m,, interacting independently with N tachyons endowed with mass ml .

230 MARCIN MOLSKI

For a nonfactorizable wave function (5 14) this interpretation is inapplicable. In such circumstances the system of interacting particles forms a holistic structure [unbroken wholeness (Bohm and Hiley, 1975)], which cannot be considered in terms of separate and independent parts. However, having solved Schrodinger's equation (512) in an approximation scheme, one may determine a spacelike field R ( x ) and an associated quantum potential Tk(x), and then calculate velocities and trajectories of particles in the system according to formula (5 18).

E Conclusions

Our results indicate that we may construct in the framework of a two-wave model of a particle and tachyonic theory of matter a generalized de Broglie-Bohm theory in which an amplitudal wave R ( x ) is identified with a spacelike D-wave, and an exponential wave + ( x , t ) = exp[ih-'S(x, r ) ] is identified with a timelike B-wave. Thesetwo wavesinteractandlocktoformatimelikewave q ( x , r ) = R ( x ) + ( x , r ) , conventionally employed in the causal interpretation of the quantum mechanics.

Carriers of the spacelike field R ( x ) are tachyons of infinite speed and generally of variable rest mass, which are also a source of a quantum potential Tk(x) and its classical counterpart T ( x ) . The former has properties identical to those of the quantum potential Q ( x ) in de Broglie-Bohm theory; however, Tk ( x ) generates quantum forces acting in directions opposite that of forces generated by Q ( x ) . This fact has important ontological and methodological consequences.

First, if Tk(x) # 0, a quantum system cannot appear in a motionless state, as de Broglie-Bohm theory predicts for a particle trapped in the box (Bohm, 1952a.b) or an electron in state ns of a hydrogen atom (de Broglie, 1960, p. 126). Con- sequently, even though a guidance formula attributes zero velocity to a quantum system, its components can move at a velocity determined with an amplitudal wave R ( x ) and not with phase S ( x , t) of an exponential wave associated with the system.

Second, in the classical limit (h + 0) the de Broglie-Bohm quantum potential Q ( x ) may be neglected; then a particle is influenced only by the classical potential V ( x ) (Bohm and Hiley, 1975). In an extended approach one may distinguish a quantum potential Tk(x) and its classical counterpart T ( x ) , both well defined but in separate ontological domains.

As the potentials Tk(x) and Q ( x ) differ only in sign, the former includes all well-known properties of the latter. For example, the quantum potential produces no vanishing interaction between components of quantal systems; even distant objects may still have strong and direct interconnections. As the quantum potential depends on R ( x ) , being a generally nonfactorizable function of position coordinates of all components of a system, it also depends on the quantum state of the entire system. In such circumstances a quantum many-body system cannot properly be analyzed into independently existing parts, with fixed and determinate dynamical relationships between them (Bohm and Hiley, 1975).


As with the spacelike field R ( x ) are associated transcendent tachyons playing the role of carriers of particle interactions, the quantum potential Tk(x) introduces instantaneous superluminal interactions at a distance interconnecting all particles in the quantum system. For this reason the behavior of each constituent may depend nonlocally on all others, no matter how distant they may be. According to Bohm et al. (1987):

If one reflects deeply and seriously on this subject one can see nothing basically irrational about such an idea. Rather it seems to be most reasonable to keep an open mind on the subject and therefore to allow oneself to explore this possibility. If the price of avoiding non-locality is to make an intuitive explanation impossible, one has to ask whether the cost is not too great.

The proposed tachyonic interpretation of particle interactions can be applied not only to microparticles but also to large-scale systems, including the universe as a whole. Then all macro- and micro-objects are seen to be in direct connection, depending, in an irreducible way, on the state of the entire system. According to this picture, transcendent tachyons form a universal background for phenomena of energy-momentum transfer between parts of a system, and the universe appears as a multidimensional self-trapping tachyonic bootstrap (Chew, 1968, 1970; Corben, 1978a). This conclusion is fully consistent with the results of experiments (Freed- man and Clauser, 1972; Aspect et al., 1982) that test Bell's theorem (Bell, 1964). Bell-Kochen-Specker theorem (Bell, 1966; Kochen and Specker, 1967) and other theoretical results in the field (Bell, 1982,1987; Stapp, 1971,1972,1975,1977). It is also compatible with the concept of synchronicity first proposed by Jung (1953, which indicates a possibility of extension of the interpretative area even to living systems.

VIII. INFONS?

The results obtained in Sections 1I.A and VI1.A show that one can propose that all particles appearing in nature may be considered to be systems composed of a bradyon at rest and a tachyon at infinite speed. Then if we express time- and spacelike four-momenta of particle constituents in the form

pP = hot, I), I = ( O , O , O ) , (524)

the associated B- and D-waves are given as

232 MARCIN MOLSKI

Consequently, the bradyon-tachyon system, representing a particle, becomes described with a superposition of time- and spacelike fields

Q(XP) = exp [ih-’ (mocx’ f mbc. r)] I ( X P ) , (528)

I * - I* = 0. (530)

This result indicates that the Q(xP) wave is invariably coupled with an additional photonlike I ( x P ) wave associated with an object having zero energy and zero momentum. This additional wave is characterized with an infinite wavelength and zero frequency.

Characteristics identical to these were attributed to hypothetical particles called infons (Stonier, 1990). According to this concept, infons possess neither mass nor energy and consist only of information, and may be envisaged to be photons that appear to have stopped oscillating, so that their wavelengths become infinitely stretched.

Equation (528) reveals that such a hypothetical particle may be attached to objects of all types, whether or not their mass is zero and whether they move at subluminal, luminal, or superluminal velocity. Consequently, our tachyonic theory of matter provides a conceptual background for the hypothesis of infons assumed to be a particular form of information manifesting itself, for example, in variation of organization (Stonier, 1990).

IX. CONCLUDING REMARKS

The concepts of a D-wave and an associated spacelike particle constituent, even if speculative, deserve attention, as they may play an important role in the internal structure of microparticles; their incorporation into quantum mechanics and special relativity enables us to understand many problems of contemporary physics. These include extended wave-corpuscular models of massive and massless particles, the internal structure of micro-objects, the nature of mass and charge and the force binding them in a particle, a formulation of extended special relativity, and quantum mechanics, including spacelike particle states and superluminal particle interactions.

The former two problems were considered in the framework of Maxwellian ,electromagnetism and Einstein’s special relativity. They are sufficient to construct simple extended wave-corpuscular models of massive and massless particles from light.


The idea that mass has an electromagnetic origin was developed long ago (Jackson, 1975) but still animates discussions (Weinberg, 1975; Sedlak, 1986; Winterberg, 1987). In cosmological models of the universe, for example in Klein- Alfven and Charon’s models, and in the standard model, one assumes that all particles originated from photons of high energy at the beginning of the universe. Although the proposed approach seems controversial, it is fully consistent with classical electromagnetism and special relativity, whereas interpretation of the obtained results employs well-known concepts in the field: a two-wave particle model, an extended space-time description of matter, a nonlinear wave hypothesis, a tachyonic theory of elementary particle structure, and an electromagnetic theory of matter.

The third problem, concerning the internal structure of charged particles, is considered in a framework of five-dimensional Kaluza-Klein theory, which seems to be the best framework to unify gauge fields and gravitation. The proposed model of a charged particle as a relativistic self-trapping tachyonic bootstrap is consistent with Post’s model formulated in ordinary four-dimensional space, but it advances in trying to explain the nature of electric charge and the force binding charge and mass to produce a stable charged particle.

The fourth problem is realized in a framework of the L-space concept and extended de Broglie-Bohm theory. The L-space concept enables one to formulate extended special relativity and quantum mechanics in the simplest and most natural manner, whereas the extended de Broglie-Bohm theory generates a bootstrap model of physical reality, including nonlocal superluminal interactions. The proposed tachyonic interpretation of wave mechanics and the results presented should be treated as an outline reporting only the most important ideas in the field worth of further investigation. However, the preliminary results, which seem only the tip of an iceberg, are sufficient to suppose that interpretation of wave mechanics in this manner might be useful to explain, for instance, the Einstein- Podolslq-Rosen (1935) paradox or interactions at a distance (Kocis, 1994), appearing, for example, in Pfleegor-Mandel(1967), Freedman-Clauser (1972), and Aspect-Dalibard-Roger (1982) photon experiments. According to the proposed connection, the tachyonic components of quantum systems are responsible for the existence of non-message-bearing quantum correlations over a spacelike interval involving holistic features of nature (Horodecki, 1994). This part of the article is a realization of Recami’s (1986, p. 160) program:

For future research, it looks, however, even more interesting to exploit the possibility of reproducing quantum mechanics at the classical level by means of tachyons. In this respect even the appearance of imaginary quantities in the theories of tachyons can be a relevant fact, to be further studied.

The results presented here and our proposed interpretations indicate that a D- wave concept may be key to open a door to an unknown physical reality that seems to be faster than hitherto supposed.

234 MARCIN MOLSKI

ACKNOWLEDGMENTS

This work is dedicated to the memory of Professors W. Sedlak and R. Dutheil. Thanks are due to Professors E. Recami, H. C. Corben, and V. VySin and Dr. L. Kostro for making it possible to become acquainted with research work indis- pensable for this article. I am indebted also to Dr. M. Wnuk who brought the infon concept to my attention, and to Professor J. Konarski and Dr. R. Horodecki for constructive discussions. Special thanks are also due to Profes- sor Ch. Marcinkowski for stimulating correspondence, and to my family Z. Z. Skupniewicz and Skupniewicz-Molska for support.

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