Noncommutative operations on Wiener functionals and Feynman's operational calculus
type: Item
instance of: scholarly article
title: Noncommutative operations on Wiener functionals and Feynman's operational calculus
zbMATH Open document ID: 0663.47016
DOI: 10.1016/0022-1236(88)90113-9
author: Michel L. Lapidus
author: G. W. Johnson
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: The article [Mem. Am. Math. Soc. 351, 78 p. (1986; Zbl 0638.28009)] gave a rigorous formulation of the heuristic operational calculus due to \textit{R. P. Feynman} [Phys. Rev. 84, 108-128 (1951; Zbl 0044.233)]. The present paper pursues further Feynman's ideas by introducing new noncommutative operations \({\dot+}\) and \(*\) on Wiener functionals, and show, in particular, two important equalities, \[ K_{\lambda}^{s+t}(F*G)=K^ s_{\lambda}(F)K^ t_{\lambda}(G)\quad and\quad \exp (F\dot +G)=\exp (F)*\exp (G), \] where the function \(K^ t_{\lambda}\) is defined by \[ (K^ t_{\lambda}(F)\psi)\xi)=\int_{C^ t_ 0}F(\lambda^{-1/2} x+\xi)\psi (\lambda^{-1/2}x(t)+\xi)dm_ t(x) \] and \(m_ t\) means Wiener measure on Wiener space \(C^ t_ 0\). A shortened form of this paper appears in C. R. Acad. Sci., Paris, Sér. I 304, 523-526 (1987; Zbl 0609.46024).
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 46H30
Mathematics Subject Classification ID: 81S40
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 58D30
zbMATH DE Number: 4084404
zbMATH Keywords: Feynman's operational calculus
zbMATH Keywords: noncommutative operations
zbMATH Keywords: Wiener functionals
zbMATH Keywords: Wiener measure on Wiener space
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90113-9
OpenAlex ID: W1994892778
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description: scientific article; zbMATH DE number 4084404