On Dirac equations on phase spaces

TL;DR

This work develops a geometric quantization framework for Dirac-type equations on relativistic phase spaces $T^*\mathbb{R}^{p-1,1}$ ($p=2,4$), modeling wavefunctions as polarized sections of a vacuum-connection line bundle $L_{\sf v}$ and deriving canonical commutation relations from the curvature $F_{\sf vac}$. By lifting Dirac (and Klein–Gordon) dynamics to phase space, the paper demonstrates explicit, oscillator-type spectra and localized fermionic states, both in space and time, and introduces extended Dirac equations on higher-dimensional phase spaces with explicit ladder-operator constructions. It then unifies quantum- and electric-charge structures via tensor products of quantum-bundle sectors with electromagnetic bundles, yielding four charge sectors and associated densities, currents, and commutation relations. The results provide a cohesive geometric picture of vacuum interactions in quantum mechanics, yielding concrete bound-state solutions (Dirac and Dirac–oscillator types) and a spectrum of localized fermionic states across multiple spacetime embeddings, with potential implications for phase-space quantization and the role of vacuum fields in relativistic quantum theory.

Abstract

We consider Dirac equations on relativistic phase spaces $T^*{\mathbb R}^{p-1,1}$, where ${\mathbb R}^{p-1,1}$ is Minkowski space with $p=2,4$. We use the geometric quantization approach in which the wave functions are polarized sections of a complex line bundle $L_{\sf{v}}$ over $T^*{\mathbb R}^{p-1,1}$. The covariant derivatives with connection $A_{\sf{vac}}$ in this bundle define canonical commutation relations. Fermions are charged with respect to the field $A_{\sf{vac}}$, so lifting the Dirac equations from space-time ${\mathbb R}^{p-1,1}$ to phase space $T^*{\mathbb R}^{p-1,1}$ results in their solutions being localized in the space ${\mathbb R}^{p-1}$ or in space-time ${\mathbb R}^{p-1,1}$. We describe the explicit form of these solutions.