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Birkhäuser Basel eBooks

Path Integral for the Relativistic Schrödinger Semigroup

Takashi Ichinose

Published Jan 1, 1995• 2 citations
Publisher

Abstract

We consider the Weyl quantized relativistic Hamiltonian H = H A ∔Ф defined through the quadratic form corresponding to the classical symbol \( \sqrt {{{{{\left( {p - A\left( x \right)} \right)}}^{2}} + {{m}^{2}}}} + \Phi \left( x \right),{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \left( {p,x} \right) \in {{\mathbb{R}}^{d}} \times {{\mathbb{R}}^{d}}, \) (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.

DOI
10.1007/978-3-0348-9092-2_21

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We consider the Weyl quantized relativistic Hamiltonian H = H A ∔Ф defined through the quadratic form corresponding to the classical symbol ( \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}}, ) (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.
The method of Weyl transforms is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time-dependent Schrödinger equation for arbitrary Hermitian Hamiltonians. It is found that all paths in phase space contribute equally in magnitude, but that each path has a different phase, equal to 1/h/ times an ’’effective action’’ taken along it. The latter is the time integral of p⋅q̇−h (p,q), h (p,q) being the Weyl transform of the Hamiltonian operator H, which differs from the classical Hamiltonian function by terms of order h/2, vanishing in the classical limit. These terms, which can be explicitly computed, are zero for relatively simple Hamiltonians, such as (1/2M)[P−eA (Q)]2+V (Q), but appear when the coupling of the position and momentum operators is stronger, such as for a relativistic spinless particle in an electromagnetic field, or when configuration space is curved. They are always zero if one opts for Weyl’s rule for forming the quantum operator corresponding to a given classical Hamiltonian. The transition amplitude between two position states is found to be expressible as a path integral in configuration space alone only in very special cases, such as when the Hamiltonian is quadratic in the momenta.
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.