U(2,2) Spin-Gauge Theory Simplification by use of the Dirac Algebra

Abstract

Exponentials of elements of the Lorentz covariant Dirac algebra with l6 parameters define local U(2,2) spin-gauge transformations of bispinor fields $\psi \left( x \right),\overline \psi \left( x \right)$ over “flat” Minkowski space-time. From general algebraic identities, 12 vector gauge fields are shown to be sufficient to define local U(2,2) covariant derivatives ${\gamma ^\mu }{\overrightarrow D _\mu }\psi {,^ \leftarrow }{\overline \psi ^ \leftarrow }_\mu {\gamma ^\mu }$. Only one scalar, two vector and one anti-symmetric (rank 2) tensor gauge fields couple with $\psi ,\overline \psi$ in the hermitean bilinear Lorentz and local U(2,2) invariant $\frac{1}{2}i\overline \psi \left( {{{\overleftarrow D }_\mu }{\gamma ^\mu } - {\gamma ^\mu }{{\overrightarrow D }_\mu }} \right)\psi$

Physical interpretations, a local U(2,2) “breaking” mechanism and further generalisations are suggested.