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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?


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People definitely investigate this sort of thing.There will be goldstone bosons in general. Other than that its outside my area. See – DJBunk May 18 '12 at 17:58
Yes, any privileged medium, such as a lattice, a superfluid, etc... breaks Lorentz or rotational invariance. The corresponding Goldstone theorem is similar to, but messier than in the Lorentz-invariant case. Phonons are the usual Goldstone modes of such situations. – Cosmas Zachos Jun 24 at 0:36

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