Approaches to Quantum Gravity

100 O. Dreyer
Is
spacetime
fundamental
?
Are
No Uses No Einstein's No
background
Equations
time
used ?
?
Internal
Relativity
Yes Yes Yes
Strings,
LQG, etc.
Volovik
Lloyd
Fig. 7.1. Choices on the road to Quantum Gravity.
uses the Einstein equations to formulate the theory. The other possibility is to argue
for why the Einstein equations hold true. In section 7.3 we will show how such an
argument can be made. We call this approach Internal Relativity.
7.2 Two views of time
In this section we review two approaches to Quantum Gravity that differ in the
way they view time. The first approach comes from solid state physics; the second
comes from quantum information theory.
7.2.1 Fermi points
In this section we are interested in the low energy behavior of quantum mechanical
Fermi liquids. It turns out that this behavior does not depend on the details of the
model but is rather described by a small number of universality classes. Which
universality class a given model falls into is determined by the topology of the
energy spectrum in momentum space. The best known class is that of a simple
Fermi surface (see figure 7.2Aa). In an ideal free Fermi gas the Fermi surface is
the boundary in momentum space between the occupied and unoccupied states. If
p F is the corresponding momentum then the energy spectrum is given by
E( p) = v F (| p|−p F ). (7.1)
In addition to these fermionic degrees of freedom there are also bosonic excitations
given by oscillations of the Fermi surface itself. The dynamics of the fermionic and
bosonic degrees of freedom is described by the Landau theory of Fermi liquids.
The other well known situation is that of a fully gapped system (see figure
7.2Ab). In this case the next available energy level above the Fermi surface is everywhere
separated from it by a non-zero amount . This situation is encountered in
superfluids and superconductors.