Feynman Path Integral Formulation

Chapter 6
Lattice Regularized Quantum Gravity
6.1 The Lattice Theory
The following sections are based on the lattice discretized description of gravity
known as Regge calculus, where the Einstein theory is expressed in terms of a
simplicial decomposition of space-time manifolds. Its use in quantum gravity is
prompted by the desire to make use of techniques developed in lattice gauge theories
(Wilson, 1973), 1 but with a lattice which reflects the structure of space-time
rather than just providing a flat passive background (Regge, 1961). It also allows
one to use powerful nonperturbative analytical techniques of statistical mechanics
as well as numerical methods. A regularized lattice version of the continuum field
theory is also usually perceived as a necessary prerequisite for a rigorous study of
the latter.
In Regge gravity the infinite number of degrees of freedom in the continuum is
restricted by considering Riemannian spaces described by only a finite number of
variables, the geodesic distances between neighboring points. Such spaces are taken
to be flat almost everywhere and are called piecewise linear (Singer and Thorpe,
1967). The elementary building blocks for d-dimensional space-time are simplices
of dimension d. A 0-simplex is a point, a 1-simplex is an edge, a 2-simplex is a
triangle, a 3-simplex is a tetrahedron. A d-simplex is a d-dimensional object with
d + 1 vertices and d(d + 1)/2 edges connecting them. It has the important property
that the values of its edge lengths specify the shape, and therefore the relative angles,
uniquely.
A simplicial complex can be viewed as a set of simplices glued together in such
a way that either two simplices are disjoint or they touch at a common face. The
relative position of points on the lattice is thus completely specified by the incidence
matrix (it tells which point is next to which) and the edge lengths, and this in turn
induces a metric structure on the piecewise linear space. Finally the polyhedron
constituting the union of all the simplices of dimension d is called a geometrical
1 As an example of a state-of-the-art calculation of hadron properties in the lattice formulation of
SU(3) QCD see (Aoki et al, 2003).
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