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Spin foam models and the classical action principle. (English) Zbl 0948.83024

Summary: 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. (e-print archive: http://xxx.lanl.gov/abs/hep-th/9807092).

MSC:

83C45 Quantization of the gravitational field
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
83C27 Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory
83C80 Analogues of general relativity in lower dimensions
83F05 Relativistic cosmology
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