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A Feynman-Kac formula with a Lebesgue-Stieltjes measure for the one-dimensional Dirac equation, associated Dyson series, and Feynman's operational calculus

Troy D. Riggs

Published Jan 1, 1993• 15 citations
OpenAlex

Abstract

The Dirac equation in one (space) dimension has a solution in the form of a path integral. If the equation includes the influence of an external potential, then this solution takes the form of a Feynman-Kac style representation. We consider the Feynman-Kac functional, where a Borel measure $\eta$ replaces the Lebesgue measure l in the time integration. Let Q(t) be the evolution operator associated with this functional through path integration. We obtain a series representation for this operator, adapting a construction given by T. Zastawniak to this setting. Using a technique we call operator time-reordering, we see that this series can be interpreted as a perturbation series. Restricting ourselves to the case where the discrete part $\nu$ of $\eta$ has finite support, we use the series to show that Q(t), considered as a function of time t, satisfies a certain Volterra-Stieltjes integral equation. We then deduce from the integral equation that Q(t) satisfies a differential equation associated with the continuous part $\mu$ of $\eta$. When $\eta = \mu = l$ this differential equation reduces to the Dirac equation with a potential. Thus we establish a "Feynman-Kac formula with a Lebesgue-Stieltjes measure $\eta$." Moreover, we show that the function Q(t) suffers a jump discontinuity at every point in the support of $\nu.$ Although the development here is quite different, the results obtained are similar those given by M. L. Lapidus for the nonrelativistic setting. Again, restricting ourselves to the case where the discrete part $\nu$ of $\eta$ has finite support, we obtain a "generalized Dyson series" for Q(t) which is analogous to the series obtained by G. W. Johnson and Lapidus in the nonrelativistic setting. The theory of semigroups of linear operators, and the theory of strong product integration for bounded linear operators play an important role in this development. The various series representations obtained for Q(t), and the method of operator time-reordering provide a rigorous means of carrying out the "disentangling" which is a central element of Feynman's time-ordered operational calculus for noncommuting operators.

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Operatorcalculus

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Cambridge University Press eBooks•Jul 13, 2023•0 citations
The Dirac equation in one (space) dimension has a solution in the form of a path integral. If the equation includes the influence of an external potential, then this solution takes the form of a Feynman-Kac style representation. We consider the Feynman-Kac functional, where a Borel measure η\etaη replaces the Lebesgue measure l in the time integration. Let Q(t) be the evolution operator associated with this functional through path integration. We obtain a series representation for this operator, adapting a construction given by T. Zastawniak to this setting. Using a technique we call operator time-reordering, we see that this series can be interpreted as a perturbation series. Restricting ourselves to the case where the discrete part ν\nuν of η\etaη has finite support, we use the series to show that Q(t), considered as a function of time t, satisfies a certain Volterra-Stieltjes integral equation. We then deduce from the integral equation that Q(t) satisfies a differential equation associated with the continuous part μ\muμ of η\etaη. When η=μ=l\eta = \mu = lη=μ=l this differential equation reduces to the Dirac equation with a potential. Thus we establish a "Feynman-Kac formula with a Lebesgue-Stieltjes measure η\etaη." Moreover, we show that the function Q(t) suffers a jump discontinuity at every point in the support of ν.\nu.ν. Although the development here is quite different, the results obtained are similar those given by M. L. Lapidus for the nonrelativistic setting. Again, restricting ourselves to the case where the discrete part ν\nuν of η\etaη has finite support, we obtain a "generalized Dyson series" for Q(t) which is analogous to the series obtained by G. W. Johnson and Lapidus in the nonrelativistic setting. The theory of semigroups of linear operators, and the theory of strong product integration for bounded linear operators play an important role in this development. The various series representations obtained for Q(t), and the method of operator time-reordering provide a rigorous means of carrying out the "disentangling" which is a central element of Feynman's time-ordered operational calculus for noncommuting operators.
We investigate what happens if in the Feynman‐Kac functional, we perform the time integration with respect to a Borel measureηrather than ordinary Lebesgue measurel. Letu(t) be the operator associated with this functional through path integration. We show thatu(t), considered as a function of timet, satisfies a certain Volterra‐Stieltjes integral equation. This result establishes a “FeynmanKac formula with Lebesgue‐Stieltjes measureη.” One recovers the classical Feynman‐Kac formula by lettingη=l. We deduce from the integral equation thatu(t) satisfies a differential equation associated with the continuous partμofηwhenη=μ=l, this differential equation reduces to the heat or the Schrödinger equation in the probabilistic or quantum‐mechanical case, respectively. Moreover, we observe a new phenomenon, due to the discrete partvofη: the functionu(t) undergoes a discontinuity at every point in the support ofv, assumed here to be finite. Further, one obtains an explicit expression foru(t) in terms of operators alternatively associated withμandv. Our results are new even in the probabilistic or “imaginary time” case and allow us to unify various concepts. The derivation of our integral equation has an interesting combinatorial structure and makes essential use of the “generalized Dyson series”— recently introduced by G. W. Johnson and the author—that “disentangle” the operatoru(t). We provide natural physical interpretations of our results in both the diffusion and quantum‐mechanical cases. We also suggest further connections with FeynmanȈs operational calculus for noncommuting operators.
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.
An approach to time-ordered operators based upon von Neumann’s infinite tensor product Hilbert spaces is used to define Feynman–Dyson algebras. This theory is used to show that a one-to-one correspondence exists between path integrals and semigroups, which are integral operators defined by a kernel, the reproducing property of the kernel being a consequence of the semigroup property. For path integrals constructed from two semigroups, the results are more general than those obtained by the use of the Trotter–Kato formula. Perturbation series for the Feynman–Dyson operator calculus for time evolution and scattering operators are discussed, and it is pointed out that they are ‘‘asymptotic in the sense of Poincaré’’ as defined in the theory of semigroups, thereby giving a precise formulation to a well-known conjecture of Dyson stated many years ago in the context of quantum electrodynamics. Moreover, the series converge when these operators possess suitable holomorphy properties.
Evolution semigroups generated by pseudo-differential operators are considered. These operators are obtained by different (parameterized by a number τ) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains Hamilton functions of particles with variable mass in magnetic and potential fields and more general symbols given by the Lévy-Khintchine formula. The considered semigroups are represented as limits of n-fold iterated integrals when n tends to infinity. Such representations are called Feynman formulae. Some of these representations are constructed with the help of another pseudo-differential operator, obtained by the same procedure of quantization; such representations are called Hamiltonian Feynman formulae. Some representations are based on integral operators with elementary kernels; these are called Lagrangian Feynman formulae. Langrangian Feynman formulae provide approximations of evolution semigroups, suitable for direct computations and numerical modeling of the corresponding dynamics. Hamiltonian Feynman formulae allow to represent the considered semigroups by means of Feynman path integrals. In the article, a family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented as phase space Feynman path integrals with respect to these Feynman pseudomeasures, i.e., different quantizations correspond to Feynman path integrals with the same integrand but with respect to different pseudomeasures. This answers Berezin’s problem of distinguishing a procedure of quantization on the language of Feynman path integrals. Moreover, the obtained Lagrangian Feynman formulae allow also to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures.
Evolution semigroups generated by pseudo-differential operators are considered. These operators are obtained by different (parameterized by a number τ) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains Hamilton functions of particles with variable mass in magnetic and potential fields and more general symbols given by the Lévy-Khintchine formula. The considered semigroups are represented as limits of n-fold iterated integrals when n tends to infinity. Such representations are called Feynman formulae. Some of these representations are constructed with the help of another pseudo-differential operator, obtained by the same procedure of quantization; such representations are called Hamiltonian Feynman formulae. Some representations are based on integral operators with elementary kernels; these are called Lagrangian Feynman formulae. Langrangian Feynman formulae provide approximations of evolution semigroups, suitable for direct computations and numerical modeling of the corresponding dynamics. Hamiltonian Feynman formulae allow to represent the considered semigroups by means of Feynman path integrals. In the article, a family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented as phase space Feynman path integrals with respect to these Feynman pseudomeasures, i.e., different quantizations correspond to Feynman path integrals with the same integrand but with respect to different pseudomeasures. This answers Berezin’s problem of distinguishing a procedure of quantization on the language of Feynman path integrals. Moreover, the obtained Lagrangian Feynman formulae allow also to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures.
Operator calculusThe operator calculus developed by Feynman [Fey51] makes it possible to represent functions of (noncommuting) operators as path integrals, with the integrand being the path-ordered exponential of operators, the order of which is controlled by a parameter that varies along the trajectory.This procedure is termed Feynman disentangling.It is also applicable to functions of matrices (say, γ-matrices which are associated with a spinor particle).When applied to the evolution operator, this procedure results in the standard path-integral representation of quantum mechanics.In this chapter we first demonstrate the general technique using the simplest example, a free propagator in Euclidean space, and then consider the path-integral representation of quantum mechanics, as well as propagators in an external electromagnetic field. Free propagatorLet us first consider the simplest propagator of a free scalar field which is given in the operator formalism by the vacuum expectation value of thewith ϕ being the field-operator.The T -product (1.1) obeys the equationwhere d = 4 is the dimension of space-time, however the formulas are applicable at any value of d.In the operator formalism, Eq. (1.2) is a * The ordered products of operators were introduced by Dyson [Dys49].This paper and other classical papers on quantum electrodynamics are collected in the book edited by Schwinger [Sch58].