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Relativistic quantum mechanics and path integral for Klein-Gordon equation

Gian Fabrizio De Angelis, Maurizio Serva

Published Jan 1, 1995• 0 citations
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Abstract

The subject of this paper is a path integral representations for the semigroup {exp(-t H_1)}{t≥0} generated by the quantum Hamiltonian H_1 of a relativistic spinless particle in an external electromagnetic field. The result is compared with the "Feynman-Kac" formula which holds for relativistic Schroedinger operators.

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This paper is a review of path integral representations for semigroups {exp(-t H_r/ħ)}{t≥0} where H_r's are relativistic quantum Hamiltonians. We consider three different cases: in the first one Hr is a relativistic Schrödinger operator, in the second is the Hamiltonian associated to Klein-Gordon equation and in the third is that coming from the Dirac one. The paths are trajectories of diffusion processes and, for the Dirac case, they differ from previous constructions based on jump Markov processes. All formulas describe one relativistic particle in an external static electromagnetic field therefore they are relativistic versions of the Feymann-Kac representation for Schrödinger semigroups.
The ordinary Feynman-Kac-Itô formula gives a path integral representation of the Schrödinger semi-groups. We discuss here an analogous probabilistic expression for the (positive energy) solutions of the imaginary-time Klein-Gordon equation in a static external electromagnetic field. When the external field is not purely magnetic, our result is different from the path integral associated to the semi-group t ∊ [0, + ∞ ) → exp[− t ( H − mc 2 )/ℏ] of a relativistic Schrödinger operator H = { c 2 [− i ℏ∇−( e/c ) A ] 2 + m 2 c 4 } 1/2 + V .
A generalized Feynman-Kac formula based on the Wiener measure is presented. Within the setting of a quantum particle in an electromagnetic field it yields the standard Feynman-Kac formula for the corresponding Schrödinger semigroup. In this case rigorous criteria for its validity are compiled. Finally, phase-space path-integral representations for more general quantum Hamiltonians are derived. These representations rely on a generalized Lie-Trotter formula which takes care of the operator-ordering multiplicity, but in general is not related to a path measure.
We consider the Weyl quantized relativistic Hamiltonian H = H A ∔Ф defined through the quadratic form corresponding to the classical symbol (p−A(x))2+m2+Φ(x),          (p,x)∈Rd×Rd, \sqrt {{{{{\left( {p - A\left( x \right)} \right)}}^{2}} + {{m}^{2}}}} + \Phi \left( x \right),{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \left( {p,x} \right) \in {{\mathbb{R}}^{d}} \times {{\mathbb{R}}^{d}}, (p−A(x))2+m2​+Φ(x),(p,x)∈Rd×Rd, (1.1)of a spinless particle with mass m ≥ 0 in an electromagnetic field (e.g. Landau-Lifschitz [9]), where the light velocity c is taken to be equal to 1.
Representations of propagators by means of path integrals over velocities are discussed both in nonrelativistic and relativistic quantum mechanics. It is shown that all the propagators can only be expressed through bosonic path integrals over velocities of space-time coordinates. In the representations the integration over velocities is not restricted by any boundary conditions; matrices, which have to be inverted in course of doing Gaussian integrals, do not contain any derivatives in time, and spinor and isospinor structures of the propagators are given explicitly. One can define universal Gaussian and quasi-Gaussian integrals over velocities and rules of handling them. Such a technique allows one effectively calculate propagators in external fields. Thus, Klein-Gordon propagator is found in a constant homogeneous electromagnetic field and its combination with a plane wave field.