Gauge fixing, Lorentz invariance and positive definite metric of Hilbert space

It is difficult to extract precise statements from a couple of sentences in a talk. FWIW, we have the following comments:

  1. The standard Goldstone theorem assumes Lorentz covariance, e.g. to have a relativistic dispersion relation.

  2. A QFT must have a positive definite physical Hilbert space ${\cal H}_{\rm phys}$ to begin with in order to be consistent and have non-negative$^1$ probabilities. In other words this requirement is at a more fundamental level than Goldstone's theorem, and must in principle always be assumed whenever we discuss various aspects of QFT, such as, Goldstone's theorem.

    That said, when we consider the standard proof of Goldstone's theorem in a (possibly extended) Hilbert space ${\cal H}$ that is not necessarily positive definite, we still deduce a massless mode in ${\cal H}$. The caveat is of course that the massless mode could belong to an unphysical sector of the (extended) Hilbert space ${\cal H}$, cf. the BRST formalism.


$^1$ That only requires a semi-positive definite Hilbert space, but one can always take quotient with the seminorm kernel to get a positive definite Hilbert space.