From su(2) gauge theory to spin 1/2 quantum mechanics

TL;DR

This paper explores how a specific gauge field ansatz in SU(2) gauge theory can lead to a reduction resembling spin 1/2 quantum mechanics, revealing spontaneous Lorentz symmetry breaking and Goldstone bosons.

Contribution

It introduces a gauge-invariant, non-Lorentz invariant ansatz that reduces SU(2) gauge fields to quantum mechanical spin 1/2 observables, linking gauge theory to quantum mechanics.

Findings

Reduction of SU(2) gauge fields to spin 1/2 observables

Identification of Goldstone bosons from Lorentz symmetry breaking

Geometrical interpretation via principal fiber bundles

Abstract

We consider a pure SU(2) gauge theory, and make an ansatz for the gauge field, which is gauge-invariant but manifestly non-Lorentz invariant. In a limit case of the ansatz, corresponding to a vacuum solution, the SU(2) gauge field reduces to a spin 1/2 observable times the generator of a global U(1). We find that the field equations written in terms of the ansatz make explicit the presence of an anomalous current which vanishes in the vacuum. This allows to interpret the components of the U(1) field as Goldstone bosons associated with the spontaneous breaking of Lorentz symmetry. Finally, we give an interpretation of the ansatz in the context of principal fiber bundles, which enlightens the geometrical aspects of the reduction of the gauge field theory to quantum mechanics.