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Transport Theory and Statistical Physics

Relativistic transport-theoretical unification of field equations

Marijan Ribarič, Luka Šušteršič

Published Jan 1, 1995• 4 citations
Publisher

Abstract

Recently, we have proposed the transport-theoretical interpretation of the basic field equations of theoretical physics. This interpretation stemmed from the hypothesis that all physical phenomena are due to streaming and scattering of point-like entities. Therefore, we propose a mathematical method for constructing relativistic transport equations such that the motion of their solutions is described in the strong scattering limit by covariant field equations, in particular, by the Klein-Gordon equation, by the Dirac equation, or by the Proca equation. Such covariant transport equations suggest a mathematical formalism for going beyond the standard model. It seems that their retarded, covariant Green functions allow also for an instantaneous action-at-a-distance; so, their existence weakens the case against those interpretations of quantum mechanics that imply an instantaneous action-at-a-distance.

DOI
10.1080/00411459508205117

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Recently, we have proposed the transport-theoretical interpretation of the basic field equations of theoretical physics. This interpretation stemmed from the hypothesis that all physical phenomena are due to streaming and scattering of point-like entities. Therefore, we propose a mathematical method for constructing relativistic transport equations such that the motion of their solutions is described in the strong scattering limit by covariant field equations, in particular, by the Klein-Gordon equation, by the Dirac equation, or by the Proca equation. Such covariant transport equations suggest a mathematical formalism for going beyond the standard model. It seems that their retarded, covariant Green functions allow also for an instantaneous action-at-a-distance; so, their existence weakens the case against those interpretations of quantum mechanics that imply an instantaneous action-at-a-distance.
We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present, second, part of our investigation, we consider a covariant approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced with (state) sections of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the transport along the identity map of the space\nobreakdash-time) of the state sections in the (total) bundle space. Connections between the (retarded) Green functions of the relativistic wave equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.
We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the bundle description the wavefunctions are replaced with (state) sections (covariant approach) or liftings of paths (equivalently: sections along paths) (time-dependent approach) of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation of the state sections/liftings in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations. Connections between the (retarded) Green functions of these equations and the evolution operators and transports are found. Especially the Dirac and Klein-Gordon equations are considered.
Problems with the extension to relativity. The Klein–Gordon equation The attempt to extend the causal interpretation to cover relativistic quantum mechanics raises several problems of technique and principle. These relate not only to difficulties peculiar to the quantum theory of motion but have to do with still unresolved issues in the underlying quantum mechanical formalism (such as the appearance of infinities in the computation of physical quantities and the lack of a relativistic theory of measurement). Some problems are common to both, such as the difficulty of defining positive densities. The subject has not been extensively investigated and in this chapter we shall discuss only some of the relevant issues. There are two particularly important, overlapping problems: Can the trajectory concept be retained in the relativistic quantum domain? And: how do we interpret relativistic quantum fields? The unresolved issue underlying both questions is the extent to which we should require a quantum theory to be ‘relativistic’. It is known that Lorentz covariant wave equations can have consequences in conflict with relativity. This, as we shall see, is indeed the case with the current relativistic formalism as interpreted, or completed, by the quantum theory of motion. Especially in field theory, the individual processes that build up and explain the covariant statistical predictions of the formalism exhibit features that are in conflict with our entrenched ideas regarding covariance and locality as necessary concomitants of relativity theory. That is, relativity is statistically valid but the individual events do not have an intrinsically relativistic character.
Solitary-particle quantum mechanics' inherent compatibility with special relativity is implicit in Schroedinger's postulated wave-function rule for the operator quantization of the particle's canonical three-momentum, taken together with his famed time-dependent wave-function equation that analogously treats the operator quantization of its Hamiltonian. The resulting formally four-vector equation system assures proper relativistic covariance for any solitary-particle Hamiltonian operator which, together with its canonical three-momentum operator, is a Lorentz-covariant four-vector operator. This, of course, is always the case for the quantization of the Hamiltonian of a properly relativistic classical theory, so the strong correspondence principle definitely remains valid in the relativistic domain. Klein-Gordon theory impairs this four-vector equation by iterating and contracting it, thereby injecting extraneous negative-energy solutions that are not orthogonal to their positive-energy counterparts of the same momentum, thus destroying the basis of the quantum probability interpretation. Klein-Gordon theory, which thus depends on the square of the Hamiltonian operator, is as well thereby cut adrift from Heisenberg's equations of motion. Dirac theory confuses the space-time symmetry of the four-vector equation system with such symmetry for its time component alone, which it fatuously imposes, thereby breaching the strong correspondence principle for the free particle and imposing the starkly unphysical momentum-independence of velocity. Physically sensible alternatives, with external electromagnetic fields, to the Klein-Gordon and Dirac equations are derived, and the simple, elegant symmetry-based approach to antiparticles is pointed out.
Covariant classical particle dynamics is described, and the associated covariant relativistic particle quantum mechanics is derived. The invariant symmetric bracket is defined on the space of quantum amplitudes, and its relation to a generalized Hamiltonian dynamics and to a covariant Schrödinger type equation is shown. Examples for relativistic potential problems are solved. Mathematically and physically acceptable probability densities for the Klein-Gordon equation and for the Dirac equation are derived, and a new continuity equation for each case is given. The quantum distribution for mass is discussed, and unambiguous representations of four-velocity and four-acceleration operators are given.
We underscore the need of relativistic quantum mechanics in dealing with particles. Various notations used in relativity are defined and explained. We derive the Klein–Gordon equation for free particle as well as for particle in electromagnetic field. We obtain a solution and interpret it. We derive the Dirac equation for free particle as well as for particle in electromagnetic field. Plane wave solution to the Dirac equation is obtained. We show how spin of electron, idea of antiparticle , phenomena of pair production and pair annihilation follow from the Dirac equation.