Renormalized integrals and a path integral formula for the heat kernel on a manifold

  • We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an L^p function. We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold.

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Metadaten
Author:Christian Bär
URN:urn:nbn:de:kobv:517-opus-60052
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Potsdam (1(2012)21)
Document Type:Preprint
Language:English
Date of Publication (online):2012/07/06
Year of Completion:2012
Publishing Institution:Universität Potsdam
Release Date:2012/07/06
Tag:Feynman-Kac formula; Renormalized integral; Riemannian manifold; generalized Laplace operator; path integral
Source:arXiv:1202.3392v2 [math.DG] 11 Jun 2012
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC Classification:47-XX OPERATOR THEORY / 47Dxx Groups and semigroups of linear operators, their generalizations and applications / 47D08 Schrödinger and Feynman-Kac semigroups
58-XX GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx](For geometric integration theory, see 49Q15) / 58Jxx Partial differential equations on manifolds; differential operators [See also 32Wxx, 35-XX, 53Cxx] / 58J35 Heat and other parabolic equation methods
58-XX GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx](For geometric integration theory, see 49Q15) / 58Jxx Partial differential equations on manifolds; differential operators [See also 32Wxx, 35-XX, 53Cxx] / 58J65 Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]
Collections:Universität Potsdam / Schriftenreihen / Preprints des Instituts für Mathematik der Universität Potsdam, ISSN 2193-6943 / 2012
Licence (German):License LogoKeine Nutzungslizenz vergeben - es gilt das deutsche Urheberrecht
Notes extern:RVK-Klassifikation: SI 990