Implementing causality in the spin foam quantum geometry

Abstract

We analyse the classical and quantum geometry of the Barrett-Crane spin foam model for four dimensional quantum gravity, explaining why it has to be considering as a covariant realization of the projector operator onto physical quantum gravity states. We discuss how causality requirements can be consistently implemented in this framework, and construct causal transiton amplitudes between quantum gravity states, i.e. realising in the spin foam context the Feynman propagator between states. The resulting causal spin foam model can be seen as a path integral quantization of Lorentzian first order Regge calculus, and represents a link between several approaches to quantum gravity as canonical loop quantum gravity, sum-over-histories formulations, dynamical triangulations and causal sets. In particular, we show how the resulting model can be rephrased within the framework of quantum causal sets (or histories).2

Disappearance and emergence of space and time in quantum gravity

“…This is clearly true to some extent, but some similarities can be identified. More precisely, Lorentzian spin foam models can be constructed [25] in such a way that the 1-skeleton of the spin foam 2-complex is understood as a discrete set of events (the vertices) with their causal relations, restricted to five per vertex (in four dimensions), i.e. to a sort of "causal nearest neighbors", and the spin foam amplitudes can be defined so to reflect this underlying causal structure.…”

Group field theory and simplicial quantum gravity

“…More details on the new 4D gravity model, together with the construction and thorough analysis of the simpler 3D gravity model, can instead be found in [21]. This work can also be understood in the context of the line of research on the issue of causality in spin foam quantum gravity and GFT, and on the construction of a unified GFT framework for loop quantum gravity and simplicial quantum gravity, developed in [16][17][18]20]. …”

Coupling of spacetime atoms in 4D spin foam models from group field theory

“…un-coupled. This is not necessarily a problem and in fact it has a nice geometric interpretation (see [6] and references therein), but it represents a sort of "ultra-locality"of the model that one may want to relax (on the ultra-locality of the Barrett-Crane model seen from the point of view of group field theories, see [20]). …”

“…were already discussed in [6]. Mathematically, it is the evaluation of the eye diagram labeled by the four representations J i on the group element G. It can be interpreted as the probability amplitude of the quantum tetrahedron, with triangles defined by the four representations J i , carrying a curvature defined by the parallel transport G between the two 4-simplices sharing that tetrahedron.…”

“…We call this model the flat model since it amounts to imposing a trivial parallel transport between 4-simplices, i.e. constraining the (boost part of the) connection to be flat (see again [6] for the role of the connection in relating the normals to the tetrahedra). Here we stress that we constrain the fifth group variables G, G to be equal and not the normals Λ, Λ , so the coupling is simply the eye diagram evaluation:…”