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Rigorous Real-Time Feynman Path Integral for Vector Potentials

Ken Loo

Published Jan 1, 1999• 1 citation
OpenAlex

Abstract

Abstract. In this paper, we will show the existence and uniqueness of a real-time, time-sliced Feynman path integral for quantum systems with vector potential. Our formulation of the path integral will be derived on the L 2 transition probability amplitude via improper Riemann integrals. Our formulation will hold for vector potential Hamiltonian for which its potential and vector potential each carries at most a finite number of singularities and discontinuities. 1. Introduction. In his paper(see [22] footnote 13), Feynman observed that by using wave functions, ill-defined oscillatory integrals can be given rigorous meaning. We will use Feynman’s observation and use wave functions to provide a convergence factor in the derivation of a real time propagator that takes the form of an L 2 transition probability amplitude and the derivation of a real time, time sliced Feynman

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ON THE FEYNMAN PATH INTEGRAL FOR NONRELATIVISTIC QUANTUM ELECTRODYNAMICS

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The Feynman path integral for regularized nonrelativistic quantum electrodynamics is studied rigorously. We begin with the Lagrangian function of the corresponding classical mechanics and construct the Feynman path integral. In the present paper, the electromagnetic potentials are assumed to be periodic with respect to a large box and quantized through their Fourier coefficients with large wave numbers cut off. Firstly, the Feynman path integral with respect to paths on the space of particles and vector potentials is defined rigorously by means of broken line paths under the constraints. Secondly, the Feynman path integral with respect to paths on the space of particles and electromagnetic potentials is also defined rigorously by means of broken line paths and piecewise constant paths without the constraints. This Feynman path integral is stated heuristically in Feynman and Hibbs' book. Thirdly, the vacuum and the state of photons of given momenta and polarizations are expressed concretely as functions of variables consisting of the Fourier coefficients of vector potentials. It is also proved rigorously in terms of distribution theory that the Coulomb potentials between charged particles naturally appear in the above Feynman path integral approach. This shows that the photons give rise to the Coulomb force.

Reviews in Mathematical Physics•Jun 1, 2010•2 citations
Abstract. In this paper, we will show the existence and uniqueness of a real-time, time-sliced Feynman path integral for quantum systems with vector potential. Our formulation of the path integral will be derived on the L 2 transition probability amplitude via improper Riemann integrals. Our formulation will hold for vector potential Hamiltonian for which its potential and vector potential each carries at most a finite number of singularities and discontinuities. 1. Introduction. In his paper(see [22] footnote 13), Feynman observed that by using wave functions, ill-defined oscillatory integrals can be given rigorous meaning. We will use Feynman’s observation and use wave functions to provide a convergence factor in the derivation of a real time propagator that takes the form of an L 2 transition probability amplitude and the derivation of a real time, time sliced Feynman
We will derive a rigorous real-time propagator for the non-relativistic quantum mechanical L 2 transition probability amplitude and for the non-relativistic wavefunction. The propagator will be given explicitly in terms of the time evolution operator. The derivation will be for all self-adjoint non-vector potential Hamiltonians. For systems with potentials that carry at most a finite number of singularity and discontinuities, we will show that our propagator can be written in the form of a rigorous real-time, time-sliced Feynman path integral via improper Riemann integrals. We will also derive the Feynman path integral in a non-standard analysis formulation. Finally, we will compute the propagator for the harmonic oscillator using the non-standard analysis Feynman path-integral formulation; we will compute the propagator without using any knowledge of the classical properties of the harmonic oscillator.
A path , in general, is defined by a determinate trajectory in time, from an initial to a final point. The classical trajectory is only one of the possible trajectories, and in quantum mechanics all the possible paths between the initial and final point come into play. Recall that the probability amplitude is a complex number that is assigned to each determinate path. Indeterminate paths are defined as a collection of determinate paths that are experimentally indistinguishable . In the Dirac–Feynman approach, the inherent indeterminacy of the quantum entity is realized by the degree of freedom – in undergoing time evolution – “taking” indeterminate paths [Baaquie (2013e)]. For a quantum degree of freedom evolving from an observed initial state to the observed final state – and with no other observations made – the Feynman path integral is a mathematical construction that computes the probability amplitudes by summing over all the allowed determinate paths of the degree of freedom – discussed in Feynman and Hibbs (1965), Zinn-Justin (1993), Zinn-Justin (2005) and Baaquie (2013e). Probability amplitude and time evolution Recall that the description of a quantum system, at a particular instant,isgiven by its state vector, namely |ψ>. To avoid confusion with the concept of a state vector, the term probability amplitude is used for describing a quantum entity undergoing transitions in time.
The Feynman path integral for regularized nonrelativistic quantum electrodynamics is studied rigorously. We begin with the Lagrangian function of the corresponding classical mechanics and construct the Feynman path integral. In the present paper, the electromagnetic potentials are assumed to be periodic with respect to a large box and quantized through their Fourier coefficients with large wave numbers cut off. Firstly, the Feynman path integral with respect to paths on the space of particles and vector potentials is defined rigorously by means of broken line paths under the constraints. Secondly, the Feynman path integral with respect to paths on the space of particles and electromagnetic potentials is also defined rigorously by means of broken line paths and piecewise constant paths without the constraints. This Feynman path integral is stated heuristically in Feynman and Hibbs' book. Thirdly, the vacuum and the state of photons of given momenta and polarizations are expressed concretely as functions of variables consisting of the Fourier coefficients of vector potentials. It is also proved rigorously in terms of distribution theory that the Coulomb potentials between charged particles naturally appear in the above Feynman path integral approach. This shows that the photons give rise to the Coulomb force.