Spin Foam Models and the Classical Action Principle

TL;DR

Freidel and Krasnov introduce a generating-functional framework that derives spin-foam (state-sum) structures from BF-like actions by inserting distributional $E$-fields on 2D polygons. The method, tested on BF theory and 2D YM, reproduces known spin-foam amplitudes exactly and yields first-order cosmological-term corrections in 3D and 4D BF theories that align with TV/CY models; it also exposes regularization ambiguities in gravity cases. The approach provides a unified perturbative scheme with BF as the free theory and interaction terms as derivatives with respect to a current, offering new insights into the link between classical action principles and quantum gravity candidates. It also highlights that existing 4D gravity models (Reisenberger, Barrett–Crane) may require incorporating additional intersection-type contributions to achieve full agreement with state-sum constructions. Overall, the work clarifies the strengths and limitations of current spin-foam models and points to directions for refining regularization and extending the formalism to capture the complete gravity dynamics.

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.

Figures (10)

• Figure 1: A typical quantum state of (2+1) dimensional theory is labelled by a graph in space. The corresponding functional of connection depends on $A$ only through holonomies along edges $e_i$. The electric field in such state is distributional and concentrated along the edges $e_i$.
• Figure 2: A face $\sigma$ of the dual triangulation. The portion of $\sigma$ indicated by bold lines is what is called wedge here. The point labelled by $C$ is the center of one of the d-simplices neighbored by $\sigma$.
• Figure 3: A wedge $w$ of the dual face and the group elements: holonomies along the edges of $w$.
• Figure 4: The product of the group elements is the holonomy around the dual face $\sigma$ with the insertions of the group elements $\exp{J}$ at each center of the corresponding d-simplex.
• Figure 5: Part of a triangulation of 2-dimensional spacetime manifold, with one face of the dual complex shown. Bold lines indicate a wedge of the dual face.