Lattice Field Theory
Lattice techniques are now in widespread use as tools for
nonperturbative studies of quantum field theories. The application of
similar techniques to the quantization of the gravitational field is,
however, hampered by the lack of a consistent lattice formulation of
differential geometry, on which the continuum theory of General
Relativity rests. While one may write lattice analogs of continuum
quantities, the absence of a systematic formalism for lattice
``difference geometry'' prevents the study of the lattice system as a
theory in its own right.
Lattice Geometry
I am currently developing such a formalism by generalizing the ideas of continuum differential geometry to the realm of lattice physics. The ultimate goal of this work is to achieve a consistent lattice theory of gravity which retains a lattice version of the continuum diffeomorphism invariance. Emphasis is placed on the gauge invariance, because gauge symmetries generally play an important role in the renormalization of quantum field theories. This is likely to be doubly true of gravity, since it is not a perturbatively renormalizable theory. Ideally then, any attempt to quantize General Relativity numerically should begin with a gauge invariant lattice theory.
While lattice formulations of General Relativity exist, none currently retain a lattice analog of the diffeomorphism invariance. The Regge calculus studies the geometry of piecewise linear continuous spacetimes, but fails to retain the desired symmetry. One may also mimic Wilson's formulation of lattice gauge theories by defining lattice versions of parallel transport, covariant derivatives, and curvature. Unfortunately, the most obvious choices for lattice actions in this approach have no gauge invariance. Indeed, it is unclear what might be meant by ``diffeomorphism'' on a lattice. In addition, Wilson-like formulations suffer from a ``doubling'' problem, in which the lattice theory contains degrees of freedom that the continuum theory does not.
If invariants are to be found, it seems that the lattice theory must be set on its own footing. Then analogies with the continuum can be replaced by serious investigations using a coherent mathematical formalism. Much of this lattice formalism is straightforward. There are simple lattice operators corresponding to covariant, exterior, and Lie derivatives, all of which satisfy a lattice version of Leibnitz rule. Thus, one can discuss ``difference geometry'' on a lattice in much the same fashion as in the continuum. Each of these lattice objects reduces to the corresponding continuum object as the lattice spacing approaches zero.
A lattice analog for integration is a bit more difficult. While one can
define a volume form and obtain a lattice Stoke's theorem, it is not
currently clear how to find a volume form which is compatible with the
lattice version of the connection. This compatibility would allow
Stoke's theorem to be written in the form of a Gauss' theorem using the
lattice covariant derivative. Since Gauss theorem plays an important
role in the continuum theory of General Relativity, it would be
extremely convenient to have a lattice version of this theorem.
Lattice Topology
Once this obstacle is overcome, however, most of the tools will be in place to study ``curved'' lattices in a systematic fashion. There is no reason that one could not extend investigations to lattice analogs of continuum topological notions. For example, one could define a lattice exterior derivative and a lattice hodge operator and use them to define something like a cohomology. This might be an interesting topic for future investigations.