Deriving Gauge Symmetry and Spontaneous Lorentz Violation

TL;DR

This paper shows that in certain field theories with a four-vector field, spontaneous Lorentz violation due to a non-zero vacuum expectation value is unobservable under mild conditions, effectively maintaining gauge invariance.

Contribution

It proves that spontaneous Lorentz violation in theories with partial gauge symmetry is unobservable if initial conditions resemble a vacuum state, linking Lorentz violation to gauge constraints.

Findings

Lorentz violation is unobservable under mild assumptions.

Initial vacuum-like conditions preserve gauge invariance.

Observable Lorentz violation requires specific primordial conditions.

Abstract

We consider a class of field theories with a four-vector field $A_{\mu}(x)$ in addition to other fields supplied with a global charge symmetry - theories which have partial gauge symmetry in the sense of only imposing it on those terms in the Lagrangian density which have derivatives as factors in them. We suppose that spontaneous Lorentz invariance breaking occurs in such a theory due to the four-vector field taking a non-zero vacuum expectation value. Under some very mild assumptions, we show that this Lorentz violation is not observable and the whole theory is practically gauge invariant. A very important presupposition for this theorem is that an initial condition is imposed on the no-derivative expressions corresponding to the early Universe being essentially in a vacuum state. This condition then remains true forever and can be interpreted as a gauge constraint. We formulate the conditions under which the spontaneous Lorentz violation becomes observable. Spontaneously broken Lorentz invariance could be seen by some primordially existing or created "fossil" charges with the property of moving through the Universe with a fixed velocity.