The Quantum Structure of Space and Time (293 pages)

188 The Quantum Structure of Space and Time

However, once quantum mechanics is taken into account, there are no stable local-
ized states. Everything decays back into the dS vacuum. Classically, the vacuum
has zero eigenvalue of H, but the semiclassical results of Gibbons and Hawking can
be explained if we assume instead that the spectrum of H is spread more or less
randomly between 0 and something of order the dS temperature, Tds = &R, with
level density e-T(RMp)2. The vacuum state corresponding to empty dS space is
the thermal density matrix p = e TdS. This ansatz explains the thermal nature of

dS space, as well as its entropy, which for large R is very close to the log of the

dimension of the Hilbert space.

However, the spectrum of H has no relation to our familiar notions of energy.

In [12] the latter concept was argued to emerge only in the large R limit. It is an

operator PO which converges to the Poincark Hamiltonian of the limiting asymptoti-

cally flat space-time, in the reference frame of the static observer. The conventional

argument that H + PO is wrong. This is true for the mathematical action of Killing

vectors on finite points in dS space. However, physical generators in GR are defined

on boundaries of space-time. The cosmological horizon of dS space converges to null

infinity in asymptotically flat space, and the two generators act differently on the

boundary. The boundary action motivates the approximate commutation relation:

-- H

which says that eigenvalues of PO much smaller than the maximal black hole mass,

are approximately conserved quantum numbers, which resolve the degeneracy of the

spectrum of H. The corresponding eigenstates of PO are localized in the observer’s

horizon volume. Semiclassical physics indicates that

tr ~-HIT~S~(P~ - E) w e-&tr ~--HIT~s.

This relation can be explained if, for small PO eigenvalue,E, the entropy deficit

of the corresponding eigenspace, relative to the dS vacuum is equal to (27rR)E.

This formula can be explicitly verified for black hole states, if we identify the mass

parameter in the Kerr-Newman- de Sitter black hole with the Poincare eigenvalue.

Based on this picture it is relatively easy to identify black hole states in terms

of the fermionic pixel operators. We work in the approximation in which all states

of the vacuum ensemble are exactly degenerate, as well as all black hole eigenstates

corresponding to the same classical solution. The vacuum is, in this approximation,

just the unit density matrix. A black hole of radius K is the density matrix of all

states satisfying

+~IBH >= 0,

for 0 5 i 5 K and 0 5 A 5 K + 1. This satisfies the geometrical relation between

radius and entropy. For K << N we can define the Hamiltonian PO (in Tds units)

to be the entropy deficit, as explained above. The choice of which K and K + 1