Approaches to Quantum Gravity

String theory, holography and Quantum Gravity 201
causal diamond. There is a small but finite quantum tunneling amplitude for this
charge to materialize as a positron, and annihilate the electron. The decay products
will move out through the de Sitter horizon and the state will become identical
to the vacuum ensemble. Every localized system in de Sitter space has a finite
life-time.
Thus, all of the eigenstates of the static Hamiltonian must be states of the vacuum
ensemble. Classically these all have zero energy. Quantum mechanically we
envision them as being spread between 0 and something of order the dS temperature,
with a density e −π(RM P ) 2 . A random Hamiltonian with these spectral bounds,
acting on a generic initial state, will produce a state where correlation functions of
simple operators are practically indistinguishable from thermal correlation functions
at the de Sitter temperature. In other words, the hypothesized spectrum could
explain the thermal nature of dS space.
There is a further piece of semi-classical evidence for this picture of the static
spectrum. The Coleman–DeLucia instanton [25] for transitions between two dS
spaces with different radii, indicates that the ratio of transition probabilities is
P 1→2
∼ e −S .
P 2→1
This is in accord with the principle of detailed balance, if the free energy of both
these systems is dominated by their entropy. The condition for this is that the
overwhelming majority of states have energies below the de Sitter temperature.
Note that in this case, the thermal density matrix is essentially the unit matrix as
R →∞, and the Gibbons–Hawking ansatz agrees qualitatively with that of Banks
and Fischler.
There are still two peculiar points to be understood. If the static energy is
bounded by the dS temperature, then what are the energies we talk about in everyday
life? In addition to this, the semi-classical evidence that the vacuum of dS
space is thermal seems to suggest a thermal ensemble with precisely these everyday
energies in the exponent. The fact that, in the classical limit of R →∞,the
density matrix is actually proportional to the unit matrix, suggests an answer to
both questions. Imagine that there is an operator P 0 , whose eigenspaces all have
the form
|p 0 〉⊗|v p0 〉,
where |v p0 〉 is any vector in a certain tensor factor of the Hilbert space, associated
with the eigenvalue p 0 . Suppose further that the dimension of the tensor factor
is e −2π Rp 0. Then the probability of finding a given p 0 eigenvalue, with a density
matrix ρ ∼ 1, will be precisely a Boltzmann factor of p 0 .
There is a class of semi-classical eigenspaces of p 0 for which we can check both
the entropy and the energy. These are black holes, if we identify the mass parameter