Spin Foam Models of Quantum Spacetime [thesis] by D. Oriti PDF

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We want to give here an outline of the two main simplicial approaches to quantum gravity, namely quantum Regge calculus and dynamical triangulations, since they both turn out to be very closely related to the spin foam approach. For more extensive review of both we refer to the literature [83, 84, 85, 42, 86, 87]. Quantum Regge calculus Consider a Riemannian simplicial manifold S, that may be thought of as an approximation of a continuum manifold M. More precisely, one may consider the simplicial complex to represent a piecewise flat manifold made up out of patches of flat 4dimensional space, the 4-simplices, glued together along the common tetrahedra.

E. that are not globally hyperbolic. Of course a sum-over-histories formulation should be completed by a rule for partitioning the whole set of spacetime histories into a set of exhaustive alternatives to which the theory can consistently assign probabilities, using some coarse-graining procedure, and this is where the mechanism of decoherence is supposed to be necessary; such a mechanism looks particularly handy in a quantum gravity context, at least to some people [46], since it provides a definition of physical measurements that does not involve directly any notion of observer, and can thus apply to closed system as the universe as a whole.

Dynamical triangulations are based on the opposite, and in a way complementary, approach. One still describes spacetime as a simplicial manifold and uses the Regge action as a discretization of the gravity action, but now one fixed the edges lengths to a fixed value, say the Planck length lp , and treats as variable the connectivity of the triangulations, for fixed topology. In other words, now in constructing a path integral for gravity, one sums over all the possible equilateral triangulations for a given topology (usually one deals with the S 4 topology).

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Spin Foam Models of Quantum Spacetime [thesis] by D. Oriti

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