Gauge fields and the algebra of polarizations

Abstract

Rigorousc-number solutions of the source-free (but non-linear, because of the curvature tensor) field equations of a gauge fieldBμ for the local groupG are found. The polarization matrices are forced thereby to satisfy a Lie algebra. If this is Abelian, one gets the usual gauge-field theory, with any Lie groupG desired. In the non-Abelian case, some curious new features emerge. The polarizations turn out to be the generators of the little group of a timelike momentum, andG is fixed as a group containing the homogeneous Lorentz groupL. If one uses these well-determined polarizations to build the interaction-picture quantized gauge field, and requiresG to be «internal» (i.e. to commute with the «external» Poincare group), then a continuous infinity of independent polarization states are required, even though as a group-theoretical objectBμ belongs to the mass >0, spin-1 representation space. InterpretingBμ asWμ, the intermediate boson, one gets an effective current-current interaction invariant against this internal Lorentz group which, sinceSU3⊅L, breaksSU3 in a specific way.