The Quantum Structure of Space and Time (293 pages)

Emergent Spacetime 183

Fig. 5.1
Lattice of Overlapping Diamonds Defines Space-Time.

A Sequence of Causal Diamonds in a Big Bang Space-time Defines an Observer. A

The particle horizon condition guarantees that a certain set of degrees of free-

dom will interact only among themselves before a given fixed time. It makes the

apparent increase of spatial volume with cosmological time, compatible with quan-
tum unitarity and a Planck scale cutoff. In fact, the discretization of time implicit
in this formalism is not a simple cut-off. Subsequent Hilbert spaces in the sequence
have dimension increasing by a fixed factor, which we will specify below. This

implies a fixed area cutoff in space-time, which generically corresponds to a time

cut-off which goes to zero with the size of causal diamonds.

To model an entire space-time we need a collection of time-like observers, with

overlapping causal diamonds. We introduce a spatial lattice, which defines the

topology of the non-compact dimensions of space on the initial time-slice3. This

topology is conserved in time, and for the moment we will take it to be that of flat

3The compact dimensions will be dealt with in a completely different manner below.