Metric-Deformed Heisenberg Algebras and the $q$-Dirac Operator

TL;DR

This paper introduces metric-deformed Heisenberg algebras that unify various $q$-deformed algebras, and constructs a $q$-Dirac operator linking spacetime geometry with quantum algebra deformations.

Contribution

It presents a new family of metric-dependent $q$-Heisenberg algebras and a $q$-Dirac operator that generalizes previous constructions, connecting geometry and quantum algebra.

Findings

Unified several known $q$-deformed Heisenberg algebras within a metric framework.

Established a link between metric signature and deformation parameters using Sylvester's theorem.

Constructed a $q$-Dirac operator whose square yields the deformed Klein-Gordon operator.

Abstract

We introduce a family of metric-deformed Heisenberg algebras $M_1$ and $M_2$, where the commutation relations are expressed directly in terms of the components of a diagonal Lorentzian metric. We show that these algebras unify several known $q$-deformed Heisenberg algebras, including the $q$-$\hbar$ algebra, the new $q$-Heisenberg algebra, and the $q$-generalized Heisenberg algebra, which embed as special cases. Using Sylvester's theorem of inertia, we establish a connection between the metric signature and the deformation parameters. We construct a $q$-Dirac operator $D_q$ from the deformed D'Alembertian and prove that $D_q^2$ recovers the deformed Klein-Gordon operator. Furthermore, we relate this construction to the quadratic $q$-Dirac operator previously introduced by the author, providing a unified framework that bridges spacetime geometry and $q$-deformed quantum algebras.