Locality in Quantum Gravity

Today I argue that quantum gravity must be a local theory. However, a well-known theorem asserts that there are no local observables in quantum gravity. How can these statements be reconciled?

The thing to notice is that theorems have axioms. If one of the axioms is violated, the theorem does not apply. The relevant axiom in this case is the assumption that classical and quantum gravity have the same sets of gauge symmetries. If the group of spacetime diffeomorphisms acquires an extension upon quantization, that assumption fails. The multi-dimensional Virasoro algebra is that extension on the Lie algebra level.

Hence a theory of quantum gravity with local observables can not be a QFT. It must be a theory with horizontal fuzziness, i.e. QJT.

The same argument applies to theories with infinite conformal symmetry, because the infinite conformal group is isomorphic to the diffeomorphism group in one complex dimension. Hence there can be no local observables in a theory with infinite conformal symmetry. But this is not a problem in conformal field theory, because the relevant symmetry is not infinite conformal symmetry but rather the Virasoro symmetry with a nonzero central charge. 

If there are local observables, the symmetry group acts in a non-trivial way on them. A necessary condition for locality is thus that the symmetry group has nontrivial unitary representations. Indeed, the centerless Virasoro algebra does not have any of those, and hence there can't be any local observables if the central charge vanishes. If the central charge is nonzero there are many nontrivial representations, and hence local observables are possible. The same thing happens with gravity. If the spacetime diffeomorphism group does not have an extension, it remains a gauge symmetry and there are no local observables. When it has extensions described by the multi-dimensional Virasoro algebra, it becomes an ordinary symmetry and local observables are no longer ruled out.

Another way to see the need for locality is that correlation functions diverge in the same way when points coalesce, whether gravity is present or not. Correlation functions in flat space (ignoring gravity) typically diverge when spacetime points \(x\) and \(y\) approach each other in the following manner:
\[
<\phi(x) \phi(y)> \ \cong\ |x-y|^{-2\Delta}.
\]
The distance between the two points depends on the metric \(g_{\mu\nu}\):
\[
|x-y|^2 = g_{\mu\nu} (x^\mu - y^\mu) (x^\nu - y^\nu)
\]
In flat space the metric is the constant Minkowski metric. In the presence of gravity the physical metric is the gravitational field, and it is unclear what the distance would mean. However, we don't have to use the physical metric to define this distance; any metric would do. Of course the distance and hence the correlation function depends on the choice of unphysical metric, but the leading singularity does not. This is because the scaling dimension is the eigenvalue of the dilatation operator
\[
D = x^\mu {\partial \over \partial x^\mu}
\]
This is the contraction of an upper and lower index and hence independent of the metric. The scaling dimension only depends on the smooth structure. Since the correlation functions diverge in flat space, they must diverge in the presence of gravity as well.

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