QFT: Gauge Invariance, Ghosts, Symmetry & Lorentz Invariance

[RedX]

When quantizing boson fields, ghosts and gauge-fixing terms seem to break gauge invariance. The unitary gauge (where there are no ghosts or gauge-fixing terms) respects gauge invariance however. So which is correct - is the Standard Model a gauge theory or not?

Sometimes I hear people speak about breaking Lorentz invariance. Does anyone have any idea what they mean? I think it has to do with CPT symmetry. There is a close relationship between CPT symmetry and Lorentz symmetry - one practically implies the other. So if CPT is broken, then I think that's where they are saying Lorentz symmetry is broken. Does this sound like pseudo-science, because Lorentz symmetry seems to be sacred?

 

[Haelfix]

Any local quantum field theory that respects lorentz invariance and that has a bounded hamiltonian automatically respects CPT. Does the converse hold true? Certainly failure of CPT implies that you don't have a field theory and any such theory probably breaks lorentz invariance so that much I think is correct.

But does lorentz breaking imply a corresponding failure of CPT? I don't think so.

As for ghosts. We require that ghosts cancel at the end of calculation in order to have a consistent theory. When this fails to happen, it implies the theory is unstable and breaks unitarity at some point (probabilities don't add up to 1).

 

[nrqed]

Any local quantum field theory that respects lorentz invariance and that has a bounded hamiltonian automatically respects CPT.

Let's say we have an interaction phi^\dagger phi^2. This respects Lorentz invariance. Does this respect CPT?

 

[RedX]

Let's say we have an interaction phi^\dagger phi^2. This respects Lorentz invariance. Does this respect CPT?

The condition is Lorentz invariance and hermicity, so if you add its hermitian conjugate than it should. The bounded Hamiltonian I think is needed for thermodynamical reasons, but not quantum reasons.

Anyways, there is a particular gauge called the background field gauge that respects gauge invariance, but in order for that to happen, you have to set the boson field equal to the background field, and that is confusing to me as to why you can do that, so I thought I'd start by asking some simple questions about gauge invariance in general.

 

[Haelfix]

Thats a good catch, you definitely need hermiticity in the interaction hamiltonian in there (although that probably follows from other conditions if we shift the axioms around a bit, eg that the field theory is unitarity, that its built up from free fields as well as being bounded from below).

I think I read something about the possibility of using a nonhermitian hamiltonian once, but it was a little ackward and I didn't understand it.