.

In physics, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's path integral (functional integration) description of quantum gravity. It is closely related to loop quantum gravity.

Spin foam in loop quantum gravity

Loop Quantum Gravity has a covariant formulation that provides at present the best formulation of the dynamics of the theory. This is a Quantum Field Theory where the invariance under diffeomorphisms of general relativity is implemented. The resulting path integral represents a sum over all the possible configuration of the geometry, coded in the spinfoam. A spin network is defined as a diagram (like the Feynman diagram) that makes a basis of connections between the elements of a differentiable manifold for the Hilbert spaces defined over them. Spin networks provide a representation for computations of amplitudes between two different hypersurfaces of the manifold. Any evolution of spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network. A spin foam may be viewed as a quantum history.
The idea

Spin networks provide a language to describe quantum geometry of space. Spin foam does the same job on spacetime. A spin network is a one-dimensional graph, together with labels on its vertices and edges which encodes aspects of a spatial geometry.

Spacetime is considered as a superposition of spin foams, which is a generalized Feynman diagram where instead of a graph we use a higher-dimensional complex. In topology this sort of space is called a 2-complex. A spin foam is a particular type of 2-complex, together with labels for vertices, edges and faces. The boundary of a spin foam is a spin network, just as in the theory of manifolds, where the boundary of an n-manifold is an (n-1)-manifold.

The concept of a spin foam, although not called that at the time, was introduced in the paper "A Step Toward Pregeometry I: Ponzano-Regge Spin Networks and the Origin of Spacetime Structure in Four Dimensions" by Norman J. LaFave.[1] In this paper, the concept of creating sandwiches of 4-geometry (and local time scale) from spin networks is described, along with the connection of these spin 4-geometry sandwiches to form paths of spin networks connecting given spin-network boundaries (spin foams). Quantization of the structure leads to a generalized Feynman path integral over connected paths of spin networks between spin-network boundaries. This paper goes beyond much of the later work by showing how 4-geometry is already present in the seemingly three dimensional spin networks, how local time scales occur, and how the field equations and conservation laws are generated by simple consistency requirements.

In Loop Quantum Gravity, the present Spinfoam Theory has been inspired by the work of Ponzano-Regge Model. The idea was introduced in [2] and later developed into the Barret-Crane Model. The formulation that is used nowadays is commonly called EPRL by the name of the authors of a series of seminal papers[3], but the theory has seen also fundamental contributions coming from the work of many persons such as Laurent Freidel (FK model) and Jerzy Lewandowski (KKL model).
Definition

A spin foam model is:

\( Z:=\sum_{\Gamma}w(\Gamma)[ \sum_{j_f,i_e}\prod_f A_f(j_f) \prod_e A_e(j_f,i_e)\prod_v A_v(j_f,i_e)] \)

with:

a set of two complexes \( \Gamma \) each consisting out of faces f, edges e and vertices v. Associated to each two complexes\( \Gamma \) is a weight \( w(\Gamma) \)
a set of irreducible representations j which label the faces and intertwiners i which label the edges.
a vertex amplitude \( A_v(j_f,i_e) \) and an edge amplitude \( A_e(j_f,i_e) \)
a face amplitude \( A_f(j_f) \), for which we almost always have \( A_f(j_f)=\dim(j_f) \)