Tagged vector space, Part I: Dirac notation as originally intended

TL;DR

This work introduces tagged vector spaces as a generalization of knots of labels (tags) and extractors to embed the Dirac bra–ket formalism within a rigorous mathematical framework. By defining index space $\mathcal{I}$ and coefficient space $\mathcal{C}$ and enforcing orthogonality and completeness, the authors build mapped representations that support unitary invariance and left/right operator action. The formalism yields a natural construction of bras/kets, density operators, moments, and the canonical quadrature/ladder operators, while exposing their phase-space structure via Weyl transforms and Wigner functions. This approach preserves the intuitive Dirac notation used in quantum optics and paves the way for a consistent Moyal-/phase-space treatment, with Part II promising functional extensions. Overall, tagged vector spaces provide a rigorous foundation for Dirac notation compatible with both physics practice and mathematical formalism.

Abstract

A generalization is provided for the notion of tags, as used in various formulations of physical scenarios. It leads to the definition of tagged vector spaces, based on a set of axioms for tags and their extractors. As an application, such a tagged vector space is used to provide, in the context of quantum optics, a formal mathematical description for the Dirac notation that is closer to its intended usage compared to current mathematical formulations: it provides a one-to-one mapping between kets and bras and allows operators to operate either to the left or to the right. The canonical commutation relations for the quadrature and ladder operators are derived as consequences of the axioms of the tagged vector space. These axioms also lead to a symplectic phase space with the Wigner function and the Weyl transform emerging naturally.