Combining continuous and discrete phenomena for Feynman's operational calculus in the presence of a \((C_0)\) semigroup and Feynman-Kac formulas with Lebesgue-Stieltjes measures
In [Phys. Rev., II. Ser. 84, 108--128 (1951; Zbl 0044.23304)], \textit{R. P. Feynman} presented a heuristic formulation for an operational calculus for non-commuting operators and gave applications to non-relativistic quantum mechanics and to quantum electrodynamics where the problem of finding useful formulas for the exponential of the sum of non-commutating operators arises naturally. The three fundamental rules of his calculus are: (i)\,attach time indices to the operators to specify the order of operation in products, (ii)\, with the indices attached, form functions of the operators by treating them as if they were commuting, and (iii)\, restore the conventional ordering of the operators, that is, disentangle the resulting expressions. In [Stud. Appl. Math. 76, 93--132 (1987; Zbl 0626.28008)], \textit{M. L. Lapidus} provided a way of formulating and making rigorous some of Feynman's ideas. The author investigates what happens if, instead of using Lebesgue measure in the Feynman-Kac functional \(F(x)=\exp\left(\int^t_0\theta(x(s))ds\right)\), one uses a Volterra-Stieltjes measure. He studies the time evolution of the operator \(u(t)=K^t_\zeta(F)^*\), the Hilbert space adjoint of the operator \(K^t_\zeta(F)\) associated via path integration with the functional \(F_\eta(x)=\exp\left(\int^t_0\theta(x(s))\eta(ds)\right)\), where \(\eta=\mu+\nu\) is a Borel measure on an interval which contains \([0,t]\) and where \(\mu\) is continuous and \(\nu\) is discrete. In this paper, the author follows the abstract approach to the operational calculus and presents some results. First, the author considers the setting of the operational calculus where are used time-ordering measures with non-zero finitely supported discrete parts in conjunction with the presence of the generator of a \((C_0)\) contraction semigroup of linear operators. This approach differs somewhat from the above mentioned paper. The primary differences are, first, in the definition of the disentangling algebra, second, in the assumptions used when a strongly continuous semigroup of linear operators is present. The main result of this paper is the evolution equation satisfied by disentangled exponential functions, and a related generalized integral equation is also discussed. The author presents some detailed examples and derives the Volterra-Stieltjes integral equations with Lebesgue-Stieltjes measures.
scientific article; zbMATH DE number 6859651
scientific article