Approaches to Quantum Gravity

202 T. Banks
of the dS–Schwarzschild metric with the P 0 eigenvalue. Of course, extant quantum
field theory calculations which demonstrate that we have a thermal ensemble of
ordinary energies in de Sitter space, refer only to energies much smaller than the
maximal black hole mass. We are led to conjecture the above relation between the
entropy deficit (relative to the vacuum) of a p 0 eigenspace, and the eigenvalue, only
to leading order in the ratio of the black hole mass to the Nariai mass. Remarkably,
this prediction is valid [26]!
It is easy to construct a Hamiltonian out of the fermionic pixel operators introduced
above, which reproduces the spectrum of black holes in dS space. One works
in the approximation where the vacuum eigenstates are all exactly degenerate, so
that black holes are stable. The vacuum density matrix is just the unit matrix.
Black hole states are simply states in which we break the fermionic matrix ψi
A
into four blocks, and insist that ψm D |BH〉=0, for matrix elements in the lower
off diagonal block. A clumsy but explicit formula for the Hamiltonian P 0 can be
constructed [26].
Some insight into the Hamiltonian P 0 is gained by remembering that global
symmetry generators in General Relativity are defined on space-like or null boundaries.
The way in which dS space converges to Minkowski space is that the causal
diamond of a single observer approaches the full Minkowski geometry. The future
and past cosmological horizons of the observer converge to future and past infinity
in asymptotically flat space. Our basic proposal for the definition of observables in
de Sitter space [27] is that there is an approximate S-matrix, S R which, as R →∞,
approaches the S-matrix of asymptotically flat space. S R refers only to localizable
processes in a single horizon volume. As in any such limiting situation, we may
expect that S R is not unique, and it is important to understand what aspects of it
are universal for large R. We will argue later that for scattering processes whose
center of mass energy is fixed as R →∞, the non-universal features may fall off
like e −(RM P ) 3/2 .
The geometry of the future cosmological horizon is the v → 0 limit of:
ds 2 = R 2 (−dudv + d 2 ),
and the static Hamiltonian is associated with the Killing vector
(u∂ u − v∂ v ).
Here, d 2 is the round metric on the 2-sphere. By contrast, future infinity in
asymptotically flat space, is the v → 0 limit of
ds 2 =
−dudv + d2
v 2 .