Banach spaces for the Feynman integral (Q1047470)

The objective of this paper is to provide a survey of progress on the general theory for path integrals as envisioned by Feynman. The authors construct a new class of spaces \(\mathbf{KS}^p({\mathbb R}^n)\), for \(1\leq p\leq\infty\) and \(n\in{\mathbb N}\), and their Sobolev counterparts \(\mathbf{KS}^{m,p}({\mathbb R}^n)\), for \(1\leq p\leq\infty\) and \(m\in{\mathbb N}\). These are separable Banach spaces which contain all of the standard \(\mathbf{L}^p\) spaces, as well as the space of finitely additive measures, as dense, continuous, compact embeddings. The authors prove that the Fourier transform and convolution operators have bounded extensions to \(\mathbf{KS}^2\). These results are applied to give the construction of the elementary path integral in the manner originally intended by Feynman. In particular, a suggestion is given that \(\mathbf{KS}^2\) is a more natural Hilbert space for quantum theory as the \(\mathbf{L}^2\).