A Metric-Deformed $q$-Gauge Dirac Equation

Abstract

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 ${\square }_q=|g^{00}|{\partial }_t^2-{\sum }_i|g^{ii}|{\partial }_i^2$. 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 $D_{\mu }^{(q)}={\partial }_{\mu }+ie{A}_{\mu }(x)/\sqrt{|g^{\mu \mu }(x)|}$ (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.

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