Say you are trying to find the amplitude for a particular state $A$ to a particular state $B$ of the Universe. For the final state you pick a spacial slice given by a 3-manifold $M$ (say a space-like plane) and a configuration of strings $S$.

But according to General Covariance, this will be equivalent to picking a different slice through space-time, a 3-manifold $M'$ with a different configuration $S'$. (e.g. a superposition of flat space-time slices with strings could be equivalent to a curved space-time slice without strings).

So in a sense the physical states should be equivalence classes of 3-manifolds with (superpositions of) string configurations on them. (Ignoring extra dimensions).

LQG deals with this by not being defined on any background.

As far as I can see String Perturbation Theory doesn't address this equivalence of states. You sort of pretend you can pick $M$ and $S$ separately. Is this so? Doesn't this lead to too many states in string theory some of which ought to be equivalent?


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