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- TL;DR Summary
- Could the Lorentz symmetry be theoretically broken in vacuum?

In this paper [1] which considers the possibility that the Lorentz symmetry could be broken, at page 4-5 the author says:

"We now introduce a Higgs sector into the Lagrangian density such that the gravitational vacuum symmetry, which we set equal to the Lagrangian symmetry at low temperatures, will break to a smaller symmetry at high temperature. The pattern of vacuum phase transition that emerges contains a symmetry anti-restoration5. This vacuum symmetry breaking leads to the interesting possibility that exact zero temperature conservation laws e.g. electric charge and baryon number are broken in the early Universe. In our case, we shall find that the spontaneous breaking of the Lorentz symmetry of the vacuum leads to a spontaneous violation of the exact, zero temperature conservation of energy."

I have three questions:

[1]: https://arxiv.org/pdf/gr-qc/9312017.pdf

"We now introduce a Higgs sector into the Lagrangian density such that the gravitational vacuum symmetry, which we set equal to the Lagrangian symmetry at low temperatures, will break to a smaller symmetry at high temperature. The pattern of vacuum phase transition that emerges contains a symmetry anti-restoration5. This vacuum symmetry breaking leads to the interesting possibility that exact zero temperature conservation laws e.g. electric charge and baryon number are broken in the early Universe. In our case, we shall find that the spontaneous breaking of the Lorentz symmetry of the vacuum leads to a spontaneous violation of the exact, zero temperature conservation of energy."

I have three questions:

- Does this mean that there could be certain vacua where the Lorentz symmetry (and other symmetries) would be broken?
- Does this mean that there could be a vacuum phase transition (a vacuum decay process) where the new vacuum would violate Lorentz symmetry (and other symmetries like time translational symmetry, leading to the violation of the conservation of energy as the author says)? Could there be a vacuum phase transition to another vacuum that would not have any symmetries at all?
- If there could be such vacua, are they mentioned or used in any theory? Would the reference #5 in the article be examples of models/theories that would allow such vacua?

[1]: https://arxiv.org/pdf/gr-qc/9312017.pdf