Spin foam models and the classical action principle

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.e-print

Gauge field theory coherent states (GCS): III. Ehrenfest theorems

“…Spin foam models are in close connection with causal spin-network evolutions [30], state sum models [31] and topological quantum field theory, in particular BF theory [32]. To date most results are at a formal level and for the Euclidean version of the theory only but the programme is exciting since it may restore manifest four-dimensional diffeomorphism invariance which in the Hamiltonian formulation is somewhat hidden.…”

Group Field Theory: An Overview

“…We have argued that spin foam models generically appear in the context of background independent approach to quantum gravity [21], this result shows that GFT is an important and unexpected universal structure behind the dynamics of such models. A deeper understanding of this theory is clearly needed.…”

“…The main lesson is that spin foams are gives a very general framework which allows to address in a background independent manner the dynamical issues of a large class of diffeomorphism invariant models including gravity in any dimensions coupled to gauge fields [21]. This formulation naturally incorporated the fact that the kinematical Hilbert space of the theory is labelled by spin networks 6 .…”

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

“…A different strategy for the quantum theory would be to start from the state space for a small cosmological constant (large k) and to impose curvature constraints with a large cosmological constant (corresponding to a smaller k). See [81][82][83] for a perturbative ansatz based on path integral quantization.…”