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