The Quantum Structure of Space and Time (293 pages)

Emergent Spacetime 187

and this implies some kind of hybrid model, since the scale of inflation must be
quite low. In this regime there are no observable tensor fluctuations. Even in this
regime, holographic cosmology is an advance over standard inflation, because the
primordial p = p regime sets up the right initial conditions for inflation, starting

from a fairly generic primordial state. Indeed, Penrose[lO] and others have argued

that conventional inflation models do not resolve the question of why the universe
began in a low entropy state. In holographic cosmology, this might be resolved by
the following line of argument: the most generic initial condition is the uniform
p = p fluid. The more normal universe, initially consists of defects in this fluid:
regions where not all of the degrees of freedom in a horizon volume are excited.
This has lower entropy, but can evolve into a stable normal universe if the following
two conditions are met:

0 The initial matter density in the normal regions is a dilute fluid of black holes.

This fluid must be sufficiently homogeneous that black hole collisions do not

result in a recollapse to the p = p phase.

0 The initial normal region either is a finite volume fraction of the infinite p = p

system, or contains only a finite number of degrees of freedom in total. The

latter case, which is entropically favored, evolves to a de Sitter universe. Thus,

holographic cosmology predicts a de Sitter universe with the largest cosmolog-

ical constant compatible with the existence of observers. If the gross features

of the theory of a small A universe is uniquely determined by A, then this may

predict a universe with physics like our own.

5.3.1.3 de Sitter space

Holographic cosmology predicts that the asymptotic future is a de Sitter space, so it

behooves us to construct a quantum theory for that symmetric space-time. We will

restrict attention to 4 dimensions, which may be the only case where the quantum

theory of de Sitter space is defined. In four dimensions, the holographic screen of

the maximal causal diamond of any observer following a time-like geodesic, is a two

sphere of radius R. Our general formalism tells us that we must pixelate the surface

of this sphere in order to have a finite number of quantum states[3][11]. The most

elegant pixelation is given by the fuzzy sphere.

The spinor bundle over the fuzzy sphere consists of complex N x N + 1 matrices

$$, transforming in the [N] @ [N + 11 dimensional representation of SU(2). If these

are quantized as fermions:

then the Hilbert space of the system has entropy N(N + l)ln2, which will agree

with the area formula for large N if N IX R.

The natural Hamiltonian, H, for a geodesic observer in dS space would seem

to be the one which generates motion along the observer’s time-like Killing vector.