Musings

De Sitter on My Mind

Tom Banks and Willy Fischler have been pushing a peculiar view of what quantum gravity in de Sitter space should look like. Taking off from the observation that de Sitter has a finite entropy, they argue that the Hilbert space of quantum gravity in de Sitter ought to be finite-dimensional.

Remember that a single harmonic oscillator has an infinite-dimensional Hilbert Space, and a single free scalar field corresponds to an infinite number of harmonic oscillators, and you see just how weird that statement is. Moreover, their recent paper highlights (in yet another way) the familiar fact that systems with finite-dimensional Hilbert spaces do not have classical limits.

One thing that has always bothered me about their proposal is the question of de Sitter invariance. Recall that in asymptotically flat space, we mod out only by those diffeomorphisms (“gauge transformations”) which go to the identity at infinity. Call this subgroup of diffeomorphisms Diff_c. The quotient Diff/Diff_c is a finite-dimensional group isomorphic to the Poincaré group. The elements of this group act as global symmetries of our theory, and we can decompose our Hilbert space into representations of Poincaré. And, indeed, the gauge-invariant observables of quantum gravity in asymptotically flat space are S-matrix elements, representing the scattering of quanta that make it off to infinity.

Similarly in asymptotically anti-de Sitter space. Here “infinity” is off in a spacelike direction (it was off in a null direction in the asymptotically flat case). Again, if we take as our gauge group Diff_c, the diffeomorphisms that go to the identity on the boundary, Diff/Diff_c is isomorphic to the anti-de Sitter group, which is also the conformal group of the conformally-rescaled boundary. The observables of quantum gravity in this case turn out to be the correlation functions of a quantum field theory living on this conformal boundary.

De Sitter, too, has a notion of “infinity”, but this time it lies in the time-like direction. If we do the same construction as before, we end up with the de Sitter group, SO(d,1), acting as global symmetries in our quantum theory. If this were right, it would fly in the face of the claim that the Hilbert space is finite-dimensional, as SO(d,1) has no finite-dimensional unitary representations.

OK, but Banks and Fischler would object that there is no super-observer who has access to the hypersurface at timelike infinity. It’s not so clear what behaviour one should demand of diffeomorphisms as one approaches timelike infinity. The global symmetry in their formulation should be SO(d-1)×R. Susskind et al concur . However, the reasoning, in both cases — that this is the subgroup of SO(d,1) which preserves the horizon does not make sense in a theory of quantum gravity where the metric (and hence the horizon) fluctuates.

But how do we get SO(d-1)×R? If we mod out by all diffeomorphisms, we get no global symmetries. If we mod out by those which preserve timelike infinity, we get SO(d,1). How to get something in between?

One obvious answer is the following. Pick a timelike worldline, γ. Recall that we are in asymptotically de Sitter space, that is, we consider only metrics which approach the de Sitter metric at timelike-infinity. So we will demand that γ approach a geodesic
(of the de Sitter metric) at timelike-infinity. Let Diff_γ be those diffeomorphisms which carry this “asymptotic geodesic” into itself (i.e. φ: γ → γ', with γ' asymptoting to γ as we approach infinity). As before, our gauge group is Diff_c, the diffeomorphisms which go to the identity at infinity. The quotient Diff_γ/Diff_c = SO(d-1)×R.

Note that this depends on a choice of worldline γ. This makes sense from the point of view of Banks et al. In de Sitter space, local observers fall out of causal contact if you wait long enough, so the formulation of the quantum theory ought to involve a choice of a particular local observer. If he waits long enough, he will not be able to compare notes with anyone else.

What about the other diffeomorphisms, which do not preserve the asymptotic form of our chosen worldline, γ? In this conjectured way of formulating quantum gravity in asymptotically de Sitter space, they simply don’t act as operators on the Hilbert space, as the Hilbert space is somehow based on the physics as seen by a local observer following the worldline γ.

It is still mysterious, however, how this setup is supposed to become isomorphic to the usual flat-space S-matrix formulation as Λ→0. In particular, the seemingly extraneous choice of γ and the restriction that only diffeomorphisms in Diff_γ and not all of Diff act on the Hilbert space look very odd in that context.