Spin foam

The covariant formulation of loop quantum gravity provides the best formulation of the dynamics of the theory of quantum gravity – a quantum field theory where the invariance under diffeomorphisms of general relativity applies. The resulting path integral represents a sum over all the possible configurations of spin foam.^[how?]

A spin network is a two-dimensional graph, together with labels on its vertices and edges which encode aspects of a spatial geometry.

A spin network is defined as a diagram like the Feynman diagram which makes a basis of connections between the elements of a differentiable manifold for the Hilbert spaces defined over them, and for computations of amplitudes between two different hypersurfaces of the manifold. Any evolution of the spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network.^{[clarification needed]} A spin foam is analogous to quantum history.^[why?]

Spin networks provide a language to describe the quantum geometry of space. Spin foam does the same job for spacetime.

Spacetime can be defined as a superposition of spin foams, which is a generalized Feynman diagram where instead of a graph, a higher-dimensional complex is used. In topology this sort of space is called a 2-complex. A spin foam is a particular type of 2-complex, 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.

In loop quantum gravity, the present spin foam theory has been inspired by the work of Ponzano–Regge model. The idea was introduced by Reisenberger and Rovelli in 1997,^[2] and later developed into the Barrett–Crane model. The formulation that is used nowadays is commonly called EPRL after the names of the authors of a series of seminal papers,^[3] but the theory has also seen fundamental contributions from the work of many others, such as Laurent Freidel (FK model) and Jerzy Lewandowski (KKL model).

The summary partition function for a spin foam model is

${\displaystyle Z:=\sum _{\Gamma }w(\Gamma )\left[\sum _{j_{f},i_{e}}\prod _{f}A_{f}(j_{f})\prod _{e}A_{e}(j_{f},i_{e})\prod _{v}A_{v}(j_{f},i_{e})\right]}$ 

with:

• a set of 2-complexes ${\displaystyle \Gamma }$  each consisting out of faces ${\displaystyle f}$ , edges ${\displaystyle e}$  and vertices ${\displaystyle v}$ . Associated to each 2-complex ${\displaystyle \Gamma }$  is a weight ${\displaystyle w(\Gamma )}$ 
• a set of irreducible representations ${\displaystyle j}$  which label the faces and intertwiners ${\displaystyle i}$  which label the edges.
• a vertex amplitude ${\displaystyle A_{v}(j_{f},i_{e})}$  and an edge amplitude ${\displaystyle A_{e}(j_{f},i_{e})}$ 
• a face amplitude ${\displaystyle A_{f}(j_{f})}$ , for which we almost always have ${\displaystyle A_{f}(j_{f})=\dim(j_{f})}$ 

• Group field theory
• Loop quantum gravity
• Lorentz invariance in loop quantum gravity
• String-net liquid

• Baez, John C. (1998). "Spin foam models". Classical and Quantum Gravity. 15 (7): 1827–1858. arXiv:gr-qc/9709052. Bibcode:1998CQGra..15.1827B. doi:10.1088/0264-9381/15/7/004. S2CID 6449360.
• Perez, Alejandro (2003). "Spin Foam Models for Quantum Gravity". Classical and Quantum Gravity. 20 (6): R43–R104. arXiv:gr-qc/0301113. doi:10.1088/0264-9381/20/6/202. S2CID 13891330.
• Rovelli, Carlo (2011). "Zakopane lectures on loop gravity". arXiv:1102.3660 [gr-qc].