Thinking about the concept of symmetry breaking led me to the following question: Let's say that I have a theory described by a Lorentz invariant Lagrangian, and the true vacuum of the theory is not invariant under Lorentz transformation, will there be massless Goldstone bosons (or fermions) similar to the breaking of a gauge symmetry? How would it be different from this kind of symmetry breaking? What would be some phenomenological consequences? Or would it be stupid (for some reasons) to look at this kind of vacuum from the beginning?
Thanks.
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