A Metric-Deformed $q$-Gauge Dirac Equation

We construct a family of metric-deformed gauge theories based on a recently introduced $q$-Dirac operator $D_q = \gamma^\mu \sqrt{|g^{\mu\mu}|}\partial_\mu$, which arises from a deformed D'Alembertian $\Box_q = |g^{00}|\partial_t^2 - \sum_i |g^{ii}|\partial_i^2$. The deformation parameter $q$ is related to the metric components via $q_\mu = \sqrt{|g^{\mu\mu}|}$. By promoting $g^{\mu\mu}(x)$ to spacetime-dependent background fields, we define a deformed covariant derivative $D_\mu^{(q)} = \partial_\mu + ieA_\mu(x)/\sqrt{|g^{\mu\mu}(x)|}$ (no sum over $\mu$). The corresponding field strength $F_{\mu\nu}^{(q)} = [D_\mu^{(q)}, D_\nu^{(q)}]$ acquires new terms proportional to $\partial_\mu(1/\sqrt{|g^{\nu\nu}|})$, 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.