Tagged vector space, Part II: function space index and the implied functional integration measure

TL;DR

This paper extends the concept of tagged vector spaces to include function space indices, deriving axioms and a generating functional for the associated functional integration measure, with applications to Gaussian functionals.

Contribution

It introduces a new axiomatic framework for functional integration in tagged vector spaces, generalizing quantum state representations to include spatiotemporal and spin degrees of freedom.

Findings

Derived axioms for functional integration measure

Obtained a generating functional for moments of the measure

Proved measure uniqueness for Gaussian functionals

Abstract

The definition of quantum states in terms of tagged vector spaces is generalized to incorporate the spatiotemporal and spin degrees of freedom. Considering a tagged vector space where the index space is a function space, representing the additional degrees of freedom, we obtained axioms for the tags that include a completeness condition expressed in terms of a functional integral with an abstract functional integration measure. Using these axioms, we derive a generating functional for the moments of this functional integration measure. These moments are then used to evaluate the functional integrals of Gaussian functionals, leading to expressions in accordance with those obtained as generalizations of equivalent integrals over a finite number of integration variables. For a Gaussian functional used as a probability distributions, we show that its moments, obtained with this functional integration measure, satisfy Carleman's condition, indicating that the measure is unique.