A Metric-Deformed $q$-Gauge Dirac Equation

We construct a family of metric-deformed gauge theories based on a recently introduced $q$-Dirac operator $D_q={\gamma }^{\mu }\sqrt{\mathrm{\mid }{g}^{\mu \mu }\mathrm{\mid }}{\mathrm{\partial }}_{\mu }$, which arises from a deformed D’Alembertian ${\mathrm{\square }}_q=\mathrm{\mid }{g}^{00}\mathrm{\mid }{\mathrm{\partial }}_t^2-{\sum }_i\mathrm{\mid }{g}^{ii}\mathrm{\mid }{\mathrm{\partial }}_i^2$. The deformation parameter $q$ is related to the metric components via $q_{\mu }=\sqrt{\mathrm{\mid }{g}^{\mu \mu }\mathrm{\mid }}$. By promoting $g^{\mu \mu }(x)$ to spacetime-dependent background fields, we define a deformed covariant derivative $D_{\mu }^{(q)}={\mathrm{\partial }}_{\mu }+ie{A}_{\mu }(x)\mathrm{/}\sqrt{\mathrm{\mid }{g}^{\mu \mu }(x)\mathrm{\mid }}$ (no sum over $\mu$). The corresponding field strength $F_{\mu \nu }^{(q)}=[D_{\mu }^{(q)},D_{\nu }^{(q)}]$ acquires new terms proportional to ${\mathrm{\partial }}_{\mu }(1\mathrm{/}\sqrt{\mathrm{\mid }{g}^{\nu \nu }\mathrm{\mid }})$, which vanish for constant metrics. We write down gauge-invariant actions for deformed Yang-Mills theory and for fermions minimally coupled to $D_{\mu }^{(q)}$. This work provides a mathematical foundation for $q$-deformed gauge theories from a metric perspective.