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

TL;DR

This work generalizes quantum-state representation to a functional tagged vector space with a function-space index, enabling functional integrals over infinite degrees of freedom. It develops an axiomatic framework with orthogonality, completeness, and unbiasedness for tags, yielding a generating functional for the moments of the implied integration measure. The paper derives explicit Gaussian-functional integrals for real and complex fields, including cases with shifts and kernels, and proves uniqueness of the functional measure under Carleman’s condition for Gaussian functionals. These results underpin a coherent functional phase-space approach to quantum optics and related domains, providing closed-form expressions for Gaussian functionals and their moments. The formalism supports robust handling of infinities, measure invariance under shifts, and transformations to complex-variable representations with exact determinant factors, enabling practical calculations of Wigner-functionals and related observables in functional spaces.

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.