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Europhysics Letters (EPL)

Imaginary-Time Path Integral for a Relativistic Spin-(1/2) Particle in a Magnetic Field

Gian Fabrizio De Angelis, A Rinaldi, Maurizio Serva

Published Jan 15, 1991• 12 citations
Publisher

Abstract

We construct a Feynman-Kac-Itô formula for positive-energy spin-(1/2) relativistic particles in a purely magnetic external field. We also introduce a new class of quantum Hamiltonians, the "relativistic Pauli operators", along with a path integral representation of their semi-groups. These Hamiltonians differ from the relativistic Schrödinger operators of Carmona, Masters and Simon only when a magnetic field is present.

DOI
10.1209/0295-5075/14/2/001

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Physical Review•Apr 1, 1950•2,183 citations
We construct a Feynman-Kac-Itô formula for positive-energy spin-(1/2) relativistic particles in a purely magnetic external field. We also introduce a new class of quantum Hamiltonians, the "relativistic Pauli operators", along with a path integral representation of their semi-groups. These Hamiltonians differ from the relativistic Schrödinger operators of Carmona, Masters and Simon only when a magnetic field is present.
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.
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.
A Feynman-Kac-type formula for a Lévy and an infinite dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of $e^{-t\PF}$ generated by the Pauli-Fierz Hamiltonian with spin $\han$ in non-relativistic quantum electrodynamics is constructed. When no external potential is applied $\PF$ turns translation invariant and it is decomposed as a direct integral $\PF = \int_\BR^\oplus \PF(P) dP$. The functional integral representation of $e^{-t\PF(P)}$ is also given. Although all these Hamiltonians include spin, nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions. As an application of the functional integral representations energy comparison inequalities are derived.
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and ccc. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.