Article,
Spin Foam Models and the Classical Action Principle
(1998)cite arxiv:hep-th/9807092 Comment: 65 pages, many figures (published version).
Abstract
We propose a new systematic approach that allows one to derive the spin foam
(state sum) model of a theory starting from the corresponding classical action
functional. It can be applied to any theory whose action can be written as that
of the BF theory plus a functional of the B field. Examples of such theories
include BF theories with or without cosmological term, Yang-Mills theories and
gravity in various spacetime dimensions. Our main idea is two-fold. First, we
propose to take into account in the path integral certain distributional
configurations of the B field in which it is concentrated along lower
dimensional hypersurfaces in spacetime. Second, using the notion of generating
functional we develop perturbation expansion techniques, with the role of the
free theory played by the BF theory. We test our approach on various theories
for which the corresponding spin foam (state sum) models are known. We find
that it exactly reproduces the known models for BF and 2D Yang-Mills theories.
For the BF theory with cosmological term in 3 and 4 dimensions we calculate the
terms of the transition amplitude that are of the first order in the
cosmological constant, and find an agreement with the corresponding first order
terms of the known state sum models. We discuss implications of our results for
existing quantum gravity models.
