nLab
Connes distribution

Connes distribution space is a ccertain analogue of the theory of generalized functions (distributions) for functional spaces.

Let ${\alpha }_{(}n)$, $n\in N$ be a sequence of real strictly positive numbers and $H_k$, $k>0$ the Sobolev space in the corresponding geometric setip (typically of certain sections of a Hermitean bundle $E$ on a compact Riemannian manifold $M$). Let

$$\sigma =\sum _n{\sigma }_n$$\sigma = \sum_n \sigma_n

where ${\sigma }_k\in H_k^{\otimes n}$. For every $C>0$ set

$$\parallel \sigma {\parallel }_{1,C,k}:=\sum C^n\alpha (n)\parallel {\sigma }_n{\parallel }_{H_k^{\otimes n}}$$\|\sigma\|_{1,C,k} := \sum C^n \alpha(n) \| \sigma_n \|_{H^{\otimes n}_k}

Banach space ${\mathrm{Co}}_{C,k}$ is the space of $\sigma$ for which $\parallel \sigma {\parallel }_{1,C,k}<\mathrm{\infty }$. The space of Connes functionals ${\mathrm{Co}}_{\mathrm{\infty }-}:={\cap }_{C>0,k>0}{\mathrm{Co}}_{C,k}$. The Connes distribution space ${\mathrm{Co}}_{-\mathrm{\infty }}$ will be its topological dual.

The Potthoff–Streit theorem allows to define flat Feynman path integrals as distributions.

• Alain Connes, On the Chern character of θ summable Fredholm modules, Comm. Math. Phys. 139 (1991), no. 1, 171–181, MR92i:19003, euclid
• 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• Ezra Getzler, Cyclic homology and the path integral of the Dirac operator, 1988, Unpublished Preprint.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, $L_p$ 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 triple?s 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