Lorentz invariance and you

Where were we? Ah yes, spontaneous symmetry breaking. When some field takes on a nonzero value even in empty space, and that field is affected by some symmetry transformation, the resulting symmetry is said to be "spontaneously broken," and becomes hard for us to see directly. The classic example is the electroweak symmetry of the Standard Model, which is purportedly broken by a Higgs field that we have yet to directly detect. The fields that get expectation values and spontaneously break symmetries are generally taken to be "scalar" fields -- that is, they are single functions of spacetime, not something more complicated like a vector field. If a vector field did get a nonzero expectation value, it would have to point somewhere, thereby picking out a preferred direction in spacetime. That means that Lorentz invariance -- the physical symmetry corresponding to rotations and changes of velocity -- would be broken. ...