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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
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.
Subject Area
Mathematics
Recommended Citation
Riggs, Troy Darin, "A Feynman-Kac formula with a Lebesgue-Stieltjes measure for the one-dimensional Dirac equation, associated Dyson series, and Feynman's operational calculus" (1993). ETD collection for University of Nebraska-Lincoln. AAI9402403.
https://digitalcommons.unl.edu/dissertations/AAI9402403