Hilbert Spaces from Path Integrals

TL;DR

This paper demonstrates how a Hilbert space can be constructed from a histories-based framework in quantum mechanics, with the Decoherence Functional defining the inner product, aligning with the standard Hilbert space in non-relativistic cases.

Contribution

It introduces a method to derive the Hilbert space structure from histories, connecting path integral approaches with conventional quantum formalism.

Findings

History Hilbert space matches standard Hilbert space in non-relativistic quantum mechanics

Decoherence Functional acts as the inner product on the History Hilbert space

Provides a foundation for histories-based quantum theories

Abstract

It is shown that a Hilbert space can be constructed for a quantum system starting from a framework in which histories are fundamental. The Decoherence Functional provides the inner product on this "History Hilbert space". It is also shown that the History Hilbert space is the standard Hilbert space in the case of non-relativistic quantum mechanics.