Identifying metric spinor phase with axion field from axion-fermion coupling and Weyl-Peccei Quinn spin transformations

Metric spinor phase of the Infeld-van der Waerden {\gamma}-formalism and axion field were identified in Ref. [1], by using Maxwell theory. Since axion couples with fermions, we will investigate Dirac theory to extend the work given in [1], showing that is possible again to identify this phase with the axion. By searching an exact identification, we will work yet the spin transformations to adapt chiral rotations and so to show that the metric spinor phase behaves exactly as required for the axion field. We will study also the Dirac-Maxwell system which provides a minimal Lagrangian approach for the axion classical sector and, finally, we will obtain an explicit 2-spinor description of magnetic monopole and its charge.