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It is established that the only states in the Hilbert space of a gauge invariant theory are states which are gauge invariant. This seems like a redundant sentence, but it has nontrivial consequences for the spectrum of the theory.

Trying to use this as an analogy for a would-be quantum gravity theory, I would then replace "gauge invariant" with "diffeomorphism invariant".

I am aware of the fact that there is no diffeomorphism invariant way to define a particle state (that of definite momentum and spin). Think of the Unruh effect and related concepts.

Does this mean that a truly quantum theory of gravity would not have any particle states?

fewfew4
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  • String theories can embed the particle standard model, and are worked on because they can at the same time quantize gravity so maybe your rules are not relevant? – anna v Dec 23 '20 at 18:40
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    There are no local observables in QG. However, something like S-matrix is an observable. Look here https://physics.stackexchange.com/questions/4359/diffm-as-a-gauge-group-and-local-observables-in-theories-with-gravity – nwolijin Dec 23 '20 at 18:51
  • @nwolijin if there are no particle states, how would one define an S-matrix? The S-matrix must have some dependence on what coordinates we use, even if we are at asymptotically flat infinity. – fewfew4 Dec 23 '20 at 19:07
  • @annav while string theory does constitute a quantum theory of gravity, practical calculations always amount to assuming the gravitational field strength is weak, in which case a particle state can approximately be defined. But I am talking about it in its full nonperturbative glory. – fewfew4 Dec 23 '20 at 19:28
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    Should I open the can of worms about whether the group of diffeomorphisms is the gauge group of GR? –  Dec 23 '20 at 19:42
  • @fewfew4 the string is the particle, the excitations of the string carry the specific mass and quantum numbers. – anna v Dec 23 '20 at 19:55
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    @DvijD.C. In the link to ananswer by Motl given by nwolijin this is explained – anna v Dec 23 '20 at 20:00
  • @DvijD.C. I don't think that's necessary lol. Whatever the case may be regarding that, states in the Hilbert space should be invariant under diffeomorphisms. – fewfew4 Dec 23 '20 at 20:07
  • @annav I guess string theory avoids this lack of particle states because they are nonlocal degrees of freedom. – fewfew4 Dec 23 '20 at 20:10
  • @fewfew4 That the particle definition stops making sense at very large masses and energies is expected in the Big Bang model for example, in the inflation time and before the quark gluon plasma where there are no particles as known at low energies and masses. Maybe string theory goes smoothly through this transition (i have no reference) – anna v Dec 24 '20 at 04:32

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