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In physics, the topological structure of spinfoam or spin foam^[1] consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. These structures are employed in loop quantum gravity as a version of quantum foam.

In loop quantum gravity

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.

Definition

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 ${\displaystyle A_{f}(j_{f})=\dim(j_{f})}$

See also

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

References

External links

• 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].