Lagrangian approach to the Barrett-Crane spin foam model

We provide the Barrett-Crane spin foam model for quantum gravity with a discrete action principle, consisting in the usual BF term with discretized simplicity constraints which in the continuum turn topological BF theory into gravity. The setting is the same as usually considered in the literature: space-time is cut into 4-simplices, the connection describes how to glue these 4-simplices together and the action is a sum of terms depending on the holonomies around each triangle. We impose the discretized simplicity constraints on disjoint tetrahedra and we show how the Lagrange multipliers distort the parallel transport and the correlations between neighboring simplices. We then construct the discretized BF action using a noncommutative $\star$ product between SU(2) plane waves. We show how this naturally leads to the Barrett-Crane model. This clears up the geometrical meaning of the model. We discuss the natural generalization of this action principle and the spin foam models it leads to. We show how the recently introduced spin foam fusion coefficients emerge with a nontrivial measure. In particular, we recover the Engle-Pereira-Rovelli spin foam model by weakening the discretized simplicity constraints. Finally, we identify the two sectors of Plebanski’s theory and we give the analog of the Barrett-Crane model in the nongeometric sector.

Figure 1

The dual face $f$ is equipped with an orientation and a base point. The bivector $B_f(v_0)$ is defined in the frame of this base point, and then parallelly transported along the boundary of $f$ until $B_f({\tau }_n)$ is reached. This ensures that the action for BF theory does not depend on the choice of the base points.

Figure 2

The left picture represents a dual face, whose edges, denoted $\tau$, are dual to tetrahedra and vertices dual to 4-simplices. The source and target vertices of $\tau$ are, respectively, denoted $s(\tau )$ and $t(\tau )$. The dashed lines delimit the pairs $(f,\tau )$ on which we define the bivectors $B_{f\tau }$ and which correspond to the triangles of a triangulation initially broken up into tetrahedra. The “holonomies” around the pairs $(f,\tau )$ are defined on the right picture, the base-points being the tetrahedra. The internal links carry group elements $L_{fv}$, each shared by exactly two tetrahedra. They can be seen as boundary connection variables for each $\tau$, and integrating them is thought of as a regluing of tetrahedra all together. We have: $G_{f\tau }=G_{\tau t(\tau )}{L}_{ft(\tau )}^{-1}{L}_{fs(\tau )}{G}_{s(\tau )\tau }$.

Figure 3

It represents a dual face, whose edges, denoted $\tau$, are dual to tetrahedra and vertices dual to 4-simplices. The dotted lines delimit the wedges that are pairs $(f,v)$ and which correspond to the triangles of a triangulation initially broken up into 4-simplices. The dashed lines represent the breaking up of each 4-simplex into disjoint tetrahedra and form the half-wedges, consisting in triplets $(f,\tau ,v)$. The standard spin foam model for BF theory can be defined with these half-wedges, with the help of holonomies $G_{\tau v}$ and of “gluing” variables $L_{f\tau }$ and $L_{fv}$ living on the internal links.

Figure 4

The left picture depicts the sequence of couplings for a given face. The links carrying the measure representations $k_{f,\tau }$ are the boundary edges of $f$, dual to tetrahedra. It is a very natural generalization of the BC model in which all $k_{f,\tau }$ are taken to be zero. The right picture represents the vertex of the model. The faces (the links in the picture) are labeled with irreducible representations $(j_{+f},j_{-f})$ of Spin(4). These are intertwined at each tetrahedron (node in the picture) with the representations $k_{f,\tau }$ which are shared by others vertices.