nLab Connes distribution

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

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

$$\sigma = \sum_n \sigma_n$$

where $\sigma_k \in H^{\otimes n}_k$. For every $C\gt 0$ set

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

Banach space $Co_{C,k}$ is the space of $\sigma$ for which $\|\sigma\|_{1,C,k}\lt \infty$. The space of Connes functionals $Co_{\infty -}:= \cap_{C\gt 0, k\gt 0} Co_{C,k}$. The Connes distribution space $Co_{-\infty}$ 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• Ezra Getzler, Cyclic homology and the path integral of the Dirac operator, 1988, Unpublished Preprint.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 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