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Path integrals in relativistic quantum mechanics

Gian Fabrizio De Angelis, Maurizio Serva

Published Jan 1, 1994• 0 citations
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Abstract

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.

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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 .
Three magnetic relativistic Schrödinger operators are considered, corresponding to the classical relativistic Hamiltonian symbol with both magnetic vector and electric scalar potentials. Path integral representations for the solutions of their respective imaginary-time relativistic Schrödinger equations, i.e. heat equations are given in two ways. The one is by means of the probability path space measure coming from the Lévy process concerned, and the other is through time-sliced approximation with Chernoff's theorem.
There are two stochastic descriptions of relativistic quantum spinless particles. In the first one, suggested by Feynman's path integral approach to Klein-Gordon propagator, the main ingredients are diffusions in Minkowski space parametrized by some kind of proper time. The second description, on the contrary, is based upon jumb Markov processes giving the space position of the particle in the sense of Newton and Wigner with the time as parameter. In this paper we bridge the gap between these different probabilistic scenarios by constructing the space jump processes by means of space-time diffusions. We give some aplications to the nonrelativistic limit of the hamiltonian semigroup t→exp−(t/¬h)H where H=√−c 2 ¬h 2 Δ+(M 2 c 4 /¬h 2 )−Mc 2 +V(.) Il y a deux descriptions stochastiques de la mecanique quantique relativiste pour une particule sans spin. Dans la premiere, suggere par la theorie de Feynman, les ingredients principaux sont des diffusions dans l'espace de Minkowski parametrise par une sorte de temps propre. La deuxieme description, par contre, emploie des processus de Markov a sauts donnant la position de la particule dans l'espace au sens de Newton et Wigner avec le temps comme parametre. Notre papier va relier les deux prescriptions probabilistes car nous construisons les processus a sauts dans l'espace par les diffusions dans l'espace-temps. Nous donnons aussi des applications a l'etude de la limite non relativiste du semi-groupe hamiltonien t→exp−(t/¬h)H ou H=√−c 2 ¬h 2 Δ+(M 2 c 4 /¬h 2 )−Mc 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.