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

TL;DR

This paper introduces tagged vector spaces to formalize Dirac notation in quantum mechanics, enabling a more accurate mathematical representation of kets, bras, and operators, and deriving key quantum relations from axioms.

Contribution

It proposes a new axiomatic framework for tagged vector spaces that better captures the structure of Dirac notation in quantum physics.

Findings

Provides a one-to-one mapping between kets and bras.

Derives canonical commutation relations from axioms.

Naturally introduces symplectic phase space, Wigner function, and Weyl transform.

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.