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

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