Banach Spaces for the Feynman Integral

Abstract

In this paper, we survey progress on the general theory for path integrals as envisioned by Feynman. We introduce a new class of spaces ${\bf{KS}}^p(\R^n)$ for $1 \le p \le \infty$ and $n \in \N$, and their Sobolev counterparts, ${\mathbf{KS}}^{m, p}(\R^n)$, for $1 \le p \le \infty,\; m \in \N$, which allow us to construct the path integral in the manner originally intended by Feynman. Each space contains all of the standard Lebesgue spaces, ${\bf{L}}^{ p}(\R^n)$ (respectively Sobolev spaces, ${\bf{W}}^{m, p}(\R^n)$), as compact dense embeddings. More importantly, these spaces all provide finite norms for nonabsolutely integrable functions. We show that both the convolution and Fourier transform extend as bounded linear operators. This allows us to construct the path integral of quantum mechanics in exactly the manner intended by Feynman. Finally, we then show how a minor change of view makes it possible to construct Lebesgue measure on (a version of) $\mathbb{R}^{\infty}$ which is no more difficult than the same construction on $\mathbb{R}^{n}$. This approach allows us to construct versions of both Lebesgue and Gaussian measure on every separable Banach space, which has a basis.