On Dirac equations on phase spaces

Abstract

We consider Dirac equations on relativistic phase spaces $T^{\ast }{\mathbb{R}}^{p-1,1}$, 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 $L_{\mathsf{v}}$ over $T^{\ast }{\mathbb{R}}^{p-1,1}$. The covariant derivatives with connection $A_{\mathsf{vac}}$ in this bundle define canonical commutation relations. Fermions are charged with respect to the field $A_{\mathsf{vac}}$, so lifting the Dirac equations from space-time ${\mathbb{R}}^{p-1,1}$ to phase space $T^{\ast }{\mathbb{R}}^{p-1,1}$ 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.

32 pages; v3: clarifications and shortening