Generalized spinfoams

We reconsider the spinfoam dynamics that has been recently introduced, in the generalized Kamiński-Kisielowski-Lewandowski (KKL) version where the foam is not dual to a triangulation. We study the Euclidean as well as the Lorentzian case. We show that this theory can still be obtained as a constrained BF theory satisfying the simplicity constraint, now discretized on a general oriented 2-cell complex. This constraint implies that boundary states admit a (quantum) geometrical interpretation in terms of polyhedra, generalizing the tetrahedral geometry of the simplicial case. We also point out that the general solution to this constraint (imposed weakly) depends on a quantum number $r_f$ in addition to those of loop quantum gravity. We compute the vertex amplitude and recover the KKL amplitude in the Euclidean theory when $r_f=0$. We comment on the eventual physical relevance of $r_f$, and the formal way to eliminate it.

Figure 1

A generalized spinfoam vertex.

Figure 2

An oriented 2-cell complex $\mathcal{K}≔(F(\mathcal{K}),E(\mathcal{K}),V(\mathcal{K}))$, where $F(\mathcal{K})=\{f_1,\cdots ,f_6\}$, $E(\mathcal{K})=\{e_1,\cdots ,e_{19}\}$, $V(\mathcal{K})=\{v_1,\cdots ,v_{14}\}$. $v_1$ is an internal vertex, and $e_1$, $e_2$, $e_3$, $e_4$ are internal edges, while all other edges and vertices belong to the boundary graph $\gamma =\partial \mathcal{K}$.

Figure 3

Assign $I_e$ to the begin point and assign $I_e^{†}$ to the end point of an internal edge $e$.