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.
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 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 of
η 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 of
η. When
η=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." Moreover, we show that the function Q(t) suffers a jump discontinuity at every point in the support of
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 of
η 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.
Abstract In this paper, we give a probabilistic representation of the heat equation associated with the quantum K-Gross Laplacian using infinite-dimensional stochastic calculus in two variables. Applying the heat semigroup to the particular case where the operator is the multiplication one, we establish a relation between the classical and the quantum heat semigroup. Finally, using a combination between convolution calculus and the generalized stochastic calculus, we give a generalization of the Feynman–Kac formula.
The Feynman formula is a representation of the solution of an evolutionary equation by a limit of some multiple integrals over Cartesian products of the classical configuration space, or of the classical phase space, when the multiplicity of integrals tends to infinity. From the Feynman formula one can deduce the Feynman–Kac formula, i.e. a representation of the solution by an integral over trajectories. In this paper we consider representations, by the Feynman formula, of solutions both of some Schrödinger type equations and of the corresponding diffusion equations. Introduction The first of Feynman's papers [7], on what one now calls Feynman path integrals over trajectories in the configuration space, contains three main observations. Firstly, it is shown that the solution of the Cauchy problem for a Schrödinger equation can be represented by a limit of a sequence of integrals over Cartesian products of the classical configuration space when the multiplicity of integrals tends to infinity. Secondly, the limit is interpreted as an integral over trajectories in the configuration space. And finally, it is noticed that the integrand contains the exponent of the classical action. Feynman's definition of the Feynman path integrals over trajectories in the phase space, which is formulated in his second paper on the subject, has a similar structure but, in contrast to the preceding definition, the Lagrangian in the classical action is expressed through the Hamiltonian function.
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.
Techniques for integration in function spaces which are not necessarily vector spaces are presented in the light of DeWitt-Morette’s recent redefinition of path integrals, which does not involve the usual limiting process, and is therefore free from the ambiguities inherent in this approach. General translated Gaussian measures defined by their Fourier transforms are introduced on various path spaces and used to derive generalized moments formulas, and what is essentially the Feynman–Kac formula for the expression of solutions of the Schrödinger equation as functional integrals.