Approaches to Quantum Gravity

200 T. Banks
de Sitter horizon is the fuzzy sphere. The spinor bundle over the fuzzy sphere is
the set of complex N × N + 1 matrices, transforming in the [N] ⊗[N + 1] =
[2] ⊕ ···[2N] dimensional representation of SU(2). We postulate the invariant
commutation relations
[ψi A ,(ψ † ) j B ]=δ j
i δ B A .
The logarithm of the dimension of the Hilbert space of this system is N(N + 1)
ln2 → π(RM P ) 2 , which indicates that we should identify N √ ln2 = √ π RM P in
the large N limit.
To get a better idea of what the Hamiltonian for dS space should look like, we
use the semi-classical results of Gibbons and Hawking [23], and work that followed
it, as experimental data. The natural Hamiltonian, H, should be the generator of
static translations for a time-like geodesic observer. The density matrix is thermal
for this system, with inverse temperature β dS = 2π R. Note that, at first glance, this
seems to contradict the assumption of Banks and Fischler, that the density matrix is
proportional to the unit matrix. Indeed, finite entropy for a thermal density matrix
does not imply a finite number of states, unless the Hamiltonian is bounded from
above.
That the Hamiltonian is so bounded follows from the fact that black holes in de
Sitter space have a maximum mass, the Nariai mass [24]. The Schwarzschild–de
Sitter metric is
ds 2 = (1 − 2M r
− r 2
R 2 )dt2 +
dr 2
(1 − 2M − r 2
) + r 2 d 2 .
r R 2
The equations for cosmological and black hole horizons, R ± are
(r − R + )(r − R − )(r + R + + R − ) = 0,
R + R − (R + + R − ) = 2MR 2 ,
R 2 = (R + + R − ) 2 − R + R − .
√
2
These have a maximal solution when R + = R − = R. Note that as the black
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hole mass is increased, its entropy increases, but the total entropy decreases. We
interpret this as saying that states with entropy localized along the world line of the
static observer are states where the system is frozen into a special configuration.
The generic state of the system is the thermal de Sitter vacuum ensemble.
In fact, the Nariai estimate is a wild overestimate of the maximal eigenvalue
of the static Hamiltonian. This follows from the fact that black holes decay into
the vacuum. Indeed, even elementary charged particles decay in de Sitter space.
The solution of Maxwell’s equations corresponding to an electron in dS space has
a compensating positive charge density spread over the horizon of the electron’s