Generalized Dirac Equation in Metric-Affine Spacetimes: Consistency, Effective Mass, and Spinor Curvature

Abstract

We formulate a generalized Dirac equation in four-dimensional metric-affine spacetimes with curvature, torsion, and non-metricity. The spinor connection is expanded over the complete basis of the Clifford algebra $cl(1,3)$ and contains eight complex coupling constants. Requiring agreement between direct covariantization of the Dirac equation and variation of the corresponding symmetrized Lagrangian imposes eight real consistency conditions, reducing the number of independent real parameters from sixteen to eight. In the static isotropic symmetric teleparallel sector, formal Hermiticity of the Hamiltonian with respect to the curved-space Dirac inner product reproduces the corresponding restrictions on the non-metricity and coframe couplings. The coframe-dependent terms introduce scalar and pseudoscalar mass contributions that persist in Minkowski spacetime. For a single free Dirac field with constant coefficients, they yield one effective mass, $m_{\mathrm{eff}}=\sqrt{(m-4b_3{)}^2+16b_4^2}$, and a chiral phase removable by a constant chiral field redefinition. A nonzero effective mass is therefore possible even when the original mass parameter vanishes. We also derive the Clifford decomposition of the generalized spinor curvature and show that it can remain nonzero when affine curvature, torsion, and non-metricity all vanish. When $m=0$, the combination $b_3^2+b_4^2$ determines both the effective mass squared and the surviving spinor curvature, establishing an algebraic link between these contributions.

Added a discussion the Hermicity of Dirac Hamiltonian, section 6 on fermion mass and chiral structure, section 7 on spinor curvature