Aronszajn's reproducing kernels and Feynman propagators

TL;DR

This paper explores how Aronszajn's reproducing kernel theory can be applied to construct Hilbert spaces in quantum mechanics, providing a functional approach to Feynman propagators and space-time granularity.

Contribution

It introduces a novel application of reproducing kernels to quantum Hilbert spaces, linking Lagrangians to Hilbert space construction without prior assumptions.

Findings

Feynman propagator as a reproducing kernel under boundedness conditions

Hilbert spaces derived from Lagrangians without a priori assumptions

Mathematical justification of space-time granularity for free particles

Abstract

This study shows how Aronszajn's theory of reproducing kernels can be of use for the construction the Hilbert spaces of quantum theory. We show that the Feynman propagator is an example of a reproducing kernel under a boundedness condition. To every Lagrangian thus corresponds a Hilbert space that does not need to be postulated \emph{a priori}. For the free non-relativistic particle, we justify mathematically the concept of space-time granularity. Reproducing kernels allow for a functional, rather than distributional, description of the Hilbert spaces of quantum theory, including the Fock space.