The Symmetry of The Vacuum State
For a relativistic field theory, the vacuum is Poincaré invariant, which follows from Wightman axioms. Poincaré invariance implies that only scalar combinations of field operators have non-vanishing VEV's. The VEV may break some of the internal symmetries of the Lagrangian of the field theory. In this case the vacuum has less symmetry than the theory allows, and one says that spontaneous symmetry breaking has occurred. See Higgs mechanism, standard model and Woit.
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