$q$-Differential Operators for $q$-Spinor Variables
Julio Cesar Jaramillo Quiceno
TL;DR
This work extends $q$-calculus to spinor structures by introducing a $q$-differential operator for $q$-spinor variables, a $q$-spinor chain rule, and a Dirac-type operator, together with integral formulas and contour techniques. The approach yields explicit factorization formulas $D^{q}\Psi = (\partial^{q}\Psi/\partial^{q}u^{\alpha}_{\dot{\beta}}) D^{q}u^{\alpha}_{\dot{\beta}}$ and a $q$-Dirac operator $D^{q}_{\mu} = \gamma_{\mu} \partial^{q}/\partial^{q}x_{\mu}$ that drive $q$-spinor differential equations. Contour-integral solutions provide analytic representations for equations of the form $D^{q}_{\mu}\Psi - b\Psi = 0$ and their Maxwell-coupled variants, highlighting a path toward a $q$-Dirac–Maxwell algebra and a broader $q$-real spinor calculus. The framework advances noncommutative and quantum-group–inspired formulations of spinor field equations with potential applications in quantum deformations of relativistic dynamics.
Abstract
We introduce a \emph{q}-differential operator adapted to \emph{q}-spinor variables, establishing a corresponding \emph{q}-spinor chain rule and defining both standard and Dirac-type \emph{q}-differential operators. Integral formulas in \emph{q}-spinor variables are derived, and applications to \emph{q}-deformed spinor differential equations are explored through explicit examples. The framework extends existing \emph{q}-calculus to spinorial structures, offering potential insights into quantum deformations of relativistic field equations. We conclude with suggestions for future developments, including a \emph{q}-analogue of the Dirac--Maxwell algebra.