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Journal of Physics A Mathematical and General

Brownian path integral from Dirac equation: a probabilistic approach to the Foldy-Wouthuysen transformation

Gian Fabrizio De Angelis, Maurizio Serva

Published Dec 21, 1992• 2 citations
Publisher

Abstract

The spectrum of the Dirac Hamiltonian H D has a positive and a negative part, the first corresponds to the energy levels of the electron, the second to minus the energy levels of the positron. The Foldy-Wouthuysen transformation is a tool to obtain a decomposition of H D in a direct sum H D =H e- (+)-H e+ , where H e- is the Hamiltonian of the electron and H e+ is that of the positron. Unfortunately, this decomposition has been obtained in a closed form only when the external field is purely magnetic, while, in the presence of an electric field, only a perturbative expansion is available. The authors give a path integral representation of the semigroup phi (x, t)=exp(-(t/)H e- ) phi 0 (x) in an external electromagnetic field. Their formula is the relativistic version, for Dirac particles, of the well known Feynman-Kac-Ito formula.

DOI
10.1088/0305-4470/25/24/008

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The spectrum of the Dirac Hamiltonian H D has a positive and a negative part, the first corresponds to the energy levels of the electron, the second to minus the energy levels of the positron. The Foldy-Wouthuysen transformation is a tool to obtain a decomposition of H D in a direct sum H D =H e- (+)-H e+ , where H e- is the Hamiltonian of the electron and H e+ is that of the positron. Unfortunately, this decomposition has been obtained in a closed form only when the external field is purely magnetic, while, in the presence of an electric field, only a perturbative expansion is available. The authors give a path integral representation of the semigroup phi (x, t)=exp(-(t/)H e- ) phi 0 (x) in an external electromagnetic field. Their formula is the relativistic version, for Dirac particles, of the well known Feynman-Kac-Ito formula.
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 .
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.
We derive and solve the Hamiltonian flow equations for a Dirac particle in an external static potential. The method shows a general procedure for the set up of continuous unitary transformations to reduce the Hamiltonian to a quasidiagonal form. Supported by DFG Grant 436 RUS 113/205/1 on Russian German CooperationThe well known Foldy-Wouthuysen (FW) transformation [1] is a unitary transformation that separates big and small components in the Dirac equation for a particle interacting with an external electromagnetic field. If the external field is stationary then the FW transformation reduces to a unitary transformation of the Hamiltonian H of the Dirac particle H ′ = e iS (⃗α(⃗p − e ⃗ A(⃗x)) + βm + eA0(⃗x))e −iS (1) where S can be expanded in a sequence of hermitean operators in powers of 1/m. Recently a novel method to diagonalize Hamiltonians via continuous unitary transformations has been proposed [2], where the Hamiltonian H and the S are considered to be functions of a continuous variable, the flow parameter l. This new method has been applied to several topics like the n − orbital model [2], superconductivity and the electron phonon interaction [3, 4]. We want to apply it here to the Dirac particle in an external field in order to explore its possibilities with a well known problem. Thereby we can gain experience in solving problems like the elimination of negative energy states in near light cone QCD [5], where the solution has yet to be discovered. For infinitesimal steps exp(η(l)dl) a continuous unitary transformation of the Hamiltonian H leads to the so-called Hamiltonian flow equation dH(l) dl = [η(l), H(l)] (2) with initial condition H(0) = H. The transformation matrix η(l) is antihermitian and related to S as follows η(l) = ( d dl eiS(l))e −iS(l). The parameter l has dimension of energy −2 and characterizes the amount of diagonalization which has been achieved. In the energy representation the matrix elements of the Hamiltonian H will be grouped in a band of width