A Lagrangian approach to the Barrett-Crane spin foam model

TL;DR

This work casts the Barrett-Crane spin-foam model as the outcome of a discretized BF action augmented by discretized simplicity constraints, implemented within a Lagrangian framework on disjoint tetrahedra. By carefully treating diagonal and cross-simplicity and introducing SU(2) Lagrange multipliers and a non-commutative plane-wave product, the BC model arises as a weak imposition of the constraints, and a natural non-geometric sector is identified. The authors further show how generalizations emerge: fusion coefficients arise from nontrivial measures, and EPR/FKLS-type vertices appear when the constraints are weakened through flexible measures $\rho$ or $\mu$, linking the BC model to more complete gravity spin-foam proposals. They also discuss the role of the Immirzi parameter and outline a coherent Lagrangian path-integral path to incorporate it, with the crucial insight that regluing across tetrahedra drives the correlations responsible for correct semiclassical behavior. Overall, the paper clarifies the geometric meaning of the BC construction, and provides a unifying lattice-action framework for BC and its extensions toward EPR/FKLS-type models.

Abstract

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 for the simplicity constraints distort the parallel transport and the correlations between neighbouring 4-simplices. We then construct the discretized BF action using a non-commutative 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 spinfoam fusion coefficients emerge with a non-trivial measure. In particular, we recover the Engle-Pereira-Rovelli spinfoam 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 non-geometric sector.

Figures (4)

• 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: $G_{f\tau} = G_{\tau\, t(\tau)}\, L^{-1}_{ft(\tau)}\, L_{fs(\tau)}\, G_{s(\tau)\, \tau}$.
• 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 generalisation 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 labelled 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.