Connes distribution space is a ccertain analogue of the theory of generalized functions (distributions) for functional spaces.
Let , be a sequence of real strictly positive numbers and , the Sobolev space in the corresponding geometric setip (typically of certain sections of a Hermitean bundle on a compact Riemannian manifold ). Let
where . For every set
Banach space is the space of for which . The space of Connes functionals . The Connes distribution space will be its topological dual.
The Potthoff–Streit theorem allows to define flat Feynman path integrals as distributions.
Rémi Léandre, Path integrals in non-commutative geometry, in Encyclopedia of Mathematical Physics (Elsevier, 2006); Stochastic analysis without probability: study of some basic tools, J. Pseudo-Differ. Oper. Appl. 1 (2010), no. 4, 389–400, preprint version pdf; Theory of distributions in the sense of Connes-Hida and Feynman path integral on a manifold, ps; Connes-Hida calclus in index theory, ps
Ezra Getzler, Cyclic homology and the path integral of the Dirac operator, 1988, Unpublished Preprint.
J D S Jones, Rémi Léandre, Chen forms on loop spaces, In: Barlow M, Bingham N (eds.) Stochastic Analysis, 104–162, Cambridge Univ. Press 1991
A closely related approach is that of a Hida in the theory of white noise?.
T. Hida, H.-H. Kuo, J. Potthoff, L. Streit, White Noise. An infinite dimensional calculus, Kluwer 1993.
M. de Faria1, J. Potthoff, L. Streit, The Feynman integrand as a Hida distribution, J. Math. Phys. 32, 2123 (1991); doi
Gel'fand triples are often used in spectral analysis and distribution theory on infinite-dimensional spaces:
Yu.M. Berezansky, Yu.G. Kondratiev, Spectral Methods in Infinite-Dimensional Analysis, Kluwer 1995. Originally in Russian, Naukova Dumka, Kiev, 1988.
Sebastian Jung, d-dimensional Feynman Integrands as Hida Distributions, slides, pdf
Revised on January 30, 2012 19:21:28
by Zoran Škoda