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Path integrals as analysis on path space by time slicing approximation
Author(s) -
Kumanogo Naoto,
Fujiwara Daisuke
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700757
Subject(s) - feynman diagram , path integral formulation , mathematics , functional integration , translation (biology) , semiclassical physics , path (computing) , transformation (genetics) , space (punctuation) , riemann integral , class (philosophy) , mathematical analysis , mathematical physics , physics , integral equation , quantum mechanics , computer science , biochemistry , chemistry , singular integral , artificial intelligence , messenger rna , quantum , gene , programming language , operating system
This is a survey of our papers [3, 4]. We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, translation, real linear transformation and functional differentiation. The invariance under translation and orthogonal transformation, the interchange of the order with Riemann‐Stieltjes integrals and some limits, the semiclassical approximation, the integration by parts and the Taylor expansion formula with respect to functional differentiation, and the fundamental theorem of calculus hold in Feynman path integral. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)