Feynman Path Integral Formulation

Chapter 7
Analytical Lattice Expansion Methods
7.1 Motivation
The following sections will discuss a number of instances where the lattice theory
of quantum gravity can be investigated analytically, subject to some necessary simplifying
assumptions.
The first problem deals with the lattice weak field expansion about a flat background.
It will be shown that in this case the relevant modes are the lattice analogs
of transverse-traceless deformations.
The second problem involves the strong coupling (large G) expansion, where the
weight factor in the path integral is expanded in powers of 1/G. The domain of
validity of this expansion can be regarded as somewhat complementary to the weak
field limit.
The third case to be discussed is what happens in lattice gravity in the limit of
large dimensions d, which formally is similar in some ways to the large-N expansion
discussed previously in this review. In this limit one can derive exact estimates for
the phase transition point and for the scaling dimensions.
7.2 Lattice Weak Field Expansion and Transverse-Traceless
Modes
One of the simplest possible problems that can be treated in quantum Regge calculus
is the analysis of small fluctuations about a fixed flat Euclidean simplicial
background (Roček and Williams, 1981; 1984). In this case one finds that the lattice
graviton propagator in a De Donder-like gauge is precisely analogous to the
continuum expression.
To compute an expansion of the lattice Regge action
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