# Can the CPT theorem be valid if Lorentz invariance is only spontaneously broken?

Earlier, I asked here whether one can have spontaneous breaking of the Lorentz symmetry and was shown a Lorentz invariant term that can drive the vacuum to not be Lorentz invariant. How relaxed are the assumptions in the CPT theorem? Can one have the theorem to be valid for a Lorentz invariant theory with a vacuum that breaks it? Can someone point me to the original and alternate proofs of this theorem?

Also, is there any connection between CPT invariance and renormalizability?

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Original Article: arxiv.org/abs/hep-ph/0201258 – Benjamin Horowitz Oct 4 '11 at 22:20
@dbrane if Lorentz invariance is only spontaneously broken if, maybe, perhaps... this is not physics sounds like wish – user8784 May 24 '12 at 21:10

The CPT theorem is a consequence of the statement that the Euclidean theory is invariant under 180 degree rotations in a plane involving (the analytic continuation of) the time axis. The result of this rotation flips the sense of Euclidean time, and flips one space axis. The analytic continuation that defines the Minkowski theory is a PT transformation (it reverses T and also P), and it maps each field to it's reflection positivity conjugate.

So any theory will have a CPT whenever the Lorentz breaking leaves a Euclidean subgroup with at least one 180 degree rotation involving time. For an explicit but somewhat trivial example, consider an effective field theory of a massive vector field with a field potential $-m^2|V|^2 + \lambda|V|^4$. The field acquires a spacelike expectation value, breaking the SO(3,1) symmetry down to SO(2,1). so the Euclidean symmetry is SO(3) and you have a 180 degree rotation. In this case, the CPT invariance is obvious, because the remaining SO(2,1) symmetry is enough to show it directly.

Other examples of this sort are string reductions, where you compactify a large dimensional space to a smaller dimension space, leaving the Lorentz group on the smaller space. The result is automatically CPT invariant, because of the lower dimensional Lorentz invariance.

For a less trivial but purely mathematical example, consider an effective symmetric tensor condensate T which has a vacuum value with no special symmetry. You can choose orthogonal coordinates to diagonalize T, and this determines a preferred frame. The Lorentz group is broken to a discrete subgroup, but this subgroup includes 180 degree Euclidean rotations, so that the result is a CPT invariant theory.

For another example, completely physical, and breaking translation invariance, consider a zero temperature crystal made out of antimatter together with the same crystal at another position, reflected along the line of separation. You can ask if the theory of this double-crystal system has CPT. The 180 rotation in this case is around the center of mass position, with one axis time, and the other axis the line of separation between the crystals. The system is obviously symmetric under this transformation, so there is a CPT. In this case, the CPT reflection is physically obvious: a quasi-particle in the crystal is a p-reversed quasi-particle in the reflected crystal. This doesn't work if the two crystals are not exactly reflected in both the matter/antimatter sense, and in the spatial sense--- if you rotate one of the crystal, but not the other, the CPT invariance will be broken by electromagnetic interactions between the two crystals.

For another physical example, this one more speculative in nature but perhaps realized, consider a gravitationally bound cluster of massive neutrinos, at zero temperature. The neutrinos are widely believed to be their own antiparticle, and they are the only known massive particle with this property. If the resulting configuration is symmetric under rotations (or even in the ordinarily absurd case, perhaps realizable at extremely high densities, that the neutrinos form a crystal which is symmetric under a crystal group that includes a reflection) then a time rotation by 180 degrees preserves the Euclidean theory in this state, so all excitations of this medium, despite breaking Lorentz symmetry, have a CPT invariance. So an electron moving or scattering off this crystal behaves in exactly the time reversed way as a positron with a reflected wavefunction moving the opposite way.

The general rule is that, any condensate which is CPT invariant with respect to at least one reflection plane for P preserves CPT. If a condensate is not invariant with respect to any P plane, it's interactions break CPT.

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