Spin foams as Feynman diagrams

TL;DR

The paper demonstrates that any spin foam model, including non-topological ones related to quantum gravity, can be obtained from the perturbative expansion of a field theory defined on a group. By constructing a real scalar field on $G^4$ with a carefully chosen interaction, the Feynman graphs of the theory are in one-to-one correspondence with 2-complexes, yielding a natural sum over 2-complexes that extends the usual triangulation-based models. It further provides a geometric interpretation as a lattice gauge theory on a dual lattice and shows that, for topological cases like TOCY, the continuum limit recovers BF theory, while non-topological cases require summing over 2-complexes to restore covariance. The framework connects covariant spin foam dynamics to canonical loop quantum gravity and opens avenues for applying quantum-field-theoretic tools, such as renormalization, to quantum gravity models.

Abstract

It has been recently shown that a certain non-topological spin foam model can be obtained from the Feynman expansion of a field theory over a group. The field theory defines a natural ``sum over triangulations'', which removes the cut off on the number of degrees of freedom and restores full covariance. The resulting formulation is completely background independent: spacetime emerges as a Feynman diagram, as it did in the old two-dimensional matrix models. We show here that any spin foam model can be obtained from a field theory in this manner. We give the explicit form of the field theory action for an arbitrary spin foam model. In this way, any model can be naturally extended to a sum over triangulations. More precisely, it is extended to a sum over 2-complexes.

Figures (4)

• Figure 1: Structure of the vertex function $A^{TOCY}(a_{ij},b_{k})$: the graph $\Gamma_{5}$.
• Figure 2: The structure of the vertex function $V(h^{i}_{j})$: the graph $\tilde{\Gamma}_{5}$ .
• Figure 3: The structure of the deltas in the propagator and in the vertex.
• Figure 4: The n-simplex and the elementary lattice $\tilde{\Gamma}_{n+1}$ on its boundary, here illustrated in the $n=3$ case.