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Discrete Space-Time Volume for 3-Dimensional BF Theory and Quantum Gravity
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The Turaev-Viro state sum invariant is known to give the transition amplitude for the three dimensional BF theory with cosmological term, and its deformation parameter hbar is related with the cosmological constant via hbar=sqrt{Lambda}. This suggests a way to find the expectation value of the spacetime volume by differentiating the Turaev-Viro amplitude with respect to the cosmological constant. Using this idea, we find an explicit expression for the spacetime volume in BF theory. According to our results, each labelled triangulation carries a volume that depends on the labelling spins. This volume is explicitly discrete. We also show how the Turaev-Viro model can be used to obtain the spacetime volume for (2+1) dimensional quantum gravity.
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Les Houches lectures on Spinfoam Path Integrals
A pedagogical review of spinfoam path integrals, from 1d quantum mechanics and 2d BF theory through Ponzano-Regge/Turaev-Viro to the 4d EPRL model, accurate but with no new results.
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