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World Scientific series in 20th century physics

Covariant Fock equations

Paul Taunton Matthews, Abdus Salam

Published May 1, 1994• 0 citations
Publisher

Abstract

A new type of amplitude-Feynman amplitude-is defined for any state. These amplitudes are shown to propagate in the Feynman manner. It is shown further that they satisfy an infinite set of four-dimensional, covariant, coupled integral equations, analogous to the three-dimensional Fock equations between the ordinary probability amplitudes.
DOI
10.1142/9789812795915_0007

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The Schrodinger equation is not covariant. Nevertheless, quantum field theory is often formulated using the Schrodinger equation to describe the time evolution of the system, which is equivalent to using Feynman path integrals. It is well known that scattering theory gives covariant results provided that the interaction vanishes in the asymptotic limit of t→±∞t \rightarrow \pm \inftyt→±∞, but there are other situations of interest that do not satisfy those conditions. As an example, the Aharonov-Bohm effect is derived here using second-quantized field theory to describe all the electrons as well as the electromagnetic field, which allows the effects of retarded vector potentials to be calculated in a straightforward way using the Feynman propagator. The results are in agreement with the usual expression for the Aharonov-Bohm effect when the retardation of the electromagnetic field is negligible, but they predict a fractional phase shift, a lack of covariance, and violations of causality when retardation effects are significant. These results suggest that quantum field theory based on the Schrodinger equation or Feynman path integrals is incomplete in the sense that it cannot give a correct description of all observable phenomena.
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.
Abstract The formulation of quantum field theory in terms of the Feynman principle is discussed. It is shown that the operators defined in terms of this principle satisfy the equations of motion. A definition of canonically conjugate momenta is given in terms of the principle and is shown to be equivalent to the usual definition. The canonical commutation relations are then deduced and the equivalence of this formulation and the canonical formalism is thereby established. The equations for Feynman amplitudes are also obtained. In conclusion some difficulties of the theory and some possible extensions are discussed.