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Reviews in Mathematical Physics

ON THE FEYNMAN PATH INTEGRAL FOR NONRELATIVISTIC QUANTUM ELECTRODYNAMICS

Wataru Ichinose

Published Jun 1, 2010• 2 citations
Publisher

Abstract

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.

DOI
10.1142/s0129055x1000403x

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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.
The definitions of the Feynman path integral for the Pauli equation and more general equations in configuration space and in phase space are proposed, probably for the first time. Then it is proved rigorously that the Feynman path integrals are well-defined and are the solutions to the corresponding equations. These Feynman path integrals are defined by the time-slicing method through broken line paths, which is familiar in physics. Our definitions of these Feynman path integrals and our results give the extension of ones for the Schrödinger equation.
We consider Feynman's path integral approach to quantum mechanics with a noncommutativity in position and momentum sectors of the phase space. We show that a quantum-mechanical system with this kind of noncommutativity is equivalent to the another one with usual commutative coordinates and momenta. We found connection between quadratic classical Hamiltonians, as well as Lagrangians, in their commutative and noncommutative regimes. The general procedure to compute Feynman's path integral on this noncommutative phase space with quadratic Lagrangians (Hamiltonians) is presented. Using this approach, a particle in a constant field, ordinary and inverted harmonic oscillators are elaborated in detail.
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In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
We present a consistent and comprehensive treatise on the foundations of the extended Hamilton–Lagrange formalism — where the dynamical system is parametrized along a general system evolution parameter s, and the time t is treated as a dependent variable t(s) on equal footing with all other configuration space variables qi(s). In the action principle, the conventional classical action L1d t is then replaced by the generalized action L1d s, with L and L1denoting the conventional and the extended Lagrangian, respectively. It is shown that a unique correlation of L1and L exists if we refrain from performing simultaneously a transformation of the dynamical variables. With the appropriate correlation of L1and L in place, the extension of the formalism preserves its canonical form. In the extended formalism, the dynamical system is described as a constrained motion within an extended space. We show that the value of the constraint and the parameter s constitutes an additional pair of canonically conjugate variables. In the corresponding quantum system, we thus encounter an additional uncertainty relation. As a consequence of the formal similarity of conventional and extended Hamilton–Lagrange formalisms, Feynman's nonrelativistic path integral approach can be converted on a general level into a form appropriate for relativistic quantum physics. In the emerging parametrized quantum description, the additional uncertainty relation serves as the means to incorporate the constraint and hence to finally eliminate the parametrization. We derive the extended Lagrangian L1of a classical relativistic point particle in an external electromagnetic field and show that the generalized path integral approach yields the Klein–Gordon equation as the corresponding quantum description. We furthermore derive the space–time propagator for a free relativistic particle from its extended Lagrangian L1. These results can be regarded as the proof of principle of the relativistic generalization of Feynman's path integral approach to quantum physics.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism. It is a generalization of the classical least action principle. The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state. Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral. We look in to the subject of two kinds of slit-experiments: The square slit, the single- and the double-Gaussian slit. Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics. It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral. Further applications of the path integral are discussed.