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Quantum gravity needs Faddeev-Popov ghosts. Feynman showed that. Take a black hole. Hawking pair production of ghost-antighost pair. One ghost falls into the hole and hits the singularity. The ghost number outside changes. Let the black hole evaporate away completely. Or even simpler, just throw a ghost into an evaporating black hole and let the black hole vanish. From no black hole initially to no black hole finally, with the global ghost number changed.

Now consider a state with no large black holes. Quantum fluctuations of the vacuum produce virtual black hole pairs. Ghosts also interact with virtual black holes. As they say, there are no global symmetries in quantum gravity. That includes total ghost number? Does this violate BRST?

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Sorry to not answer your question, but is there no global symmetries in QG? –  toot Jul 10 '12 at 11:33
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3 Answers 3

Your mechanism of "one ghost falling in" is a different way of talking about the "evaporation of FP ghosts by a black hole". So do black holes evaporate ghosts?

This is not a well-defined question because FP ghosts are unphysical, too. They're just a mathematical method to deal with gauge symmetries, in this case the diffeomorphism symmetry. In this sense, the question "whether there are FP ghosts included in the Hawking radiation" is analogous to the question "which gauge you should use to gauge-fix the redundancy in Yang-Mills symmetries". There is no physical answer to this question. It's up to you. It's really the point of gauge symmetries – and the point of the BRST formalism – that such things are up to you.

So you may always choose the state in the BRST equivalence class $$ |\psi\rangle \to |\psi\rangle + Q|\lambda\rangle $$ in such a way that the ghost number is conserved and many other additional constraints are satified, too.

The ghost number can't be considered a counterexample to the lore about the "non-existence of global symmetries in quantum gravity" because physical states don't transform nontrivially under this would-be symmetry. Because one always chooses the physical states to be states in a particular sector with a fixed ghost number $N_{gh}$, the action of the ghost number $U(1)$ symmetry is just a universal phase changing all state vectors, not a symmetry that changes the physical essence of the state.

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The OP's example is none other than a classic example of Gribov ambiguities. It reflects badly on the choice of BRST formalism more than anything else. It shouldn't be taken to have physical implications.

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If we have a superposition between a state with no black hole and a state with an evaporating black hole, the ghost number can't change for the former branch but it can(?) change for the latter branch. This means the matter sector can affect the ghost sector. Classically, if a sector affects another sector, it's still possible for there to be no backreaction of the latter sector on the former, but not so in quantum mechanics. So, if the ghost sector evolution depends upon the matter sector, then the matter sector evolution also depends upon the ghost sector evolution. There's no decoupling and the ghost sector becomes observable and becomes real!

Might it be the case that's because we have a superposition over different spacetime topologies, so a superposition between nonisomorphic diffeomorphism groups? That can't be the cause really.

Now consider a superposition of an evaporating black hole at location A and the black hole at a distant spacelike separated region B. We have the same spacetime topology in each case. Now suppose we have a ghost moving along in region A. In the branch wit the black hole at A, the ghost can be captured at A, but not in the other branch. This leads to ghost induced decoherence, an observable effect in principle, although decoherence by other matter would have occured anyway.

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