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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

Covariant Fock equations

Paul Taunton Matthews, Abdus Salam

Published Jan 7, 1954• 42 citations
Publisher

Abstract

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
10.1098/rspa.1954.0010

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Apr 6, 2017•2 citations
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.
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.
In a number of previous papers authors have introduced quasiparticle of radio- and optical systems. We have called this quasiparticle by ‘radion’. The basis for this is the representation of Green’s function of equation of quasioptics by Feynman integral. It means that radion has quantum mechanical properties. In particular in approximation of quasioptics one can interprete amplitude of electromagnetic field as amplitude of probability of radion. In the paper presented we describe new features of radion which we have established recently namely we have found some exact solutions of evolutionary equation for field of quantum mechanical averages of momenta of ensemble of radions possessing by average of momentum in initial state depending on average of coordinate in it. We demonstrate three the simplest solutions from this set of exact solutions. These solutions in the form of power series on dimensionless time are applicable up to the moment of gradient catastrophe. For representation of solutions of parabolic equation in approximation of quasioptics Remizov method of quasi-Feynman formulae has been used. All these results due to suggested by us physical analogy are valid for free one-dimensional nonrelativistic quantum mechanical particle too. Also in the framework of Remizov method a number of new formulae for statistical radiophysics and optics has been obtained. ‘Natural’ fractality of radio- and optical systems has been discussed.