Generalized spinfoams

Abstract

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.1 The enlargement is not an effect from the generalization to arbitrary 2-cell complexes. The Hilbert space is enlarged also in the simplicial case, compared with the state space defined in [9]. This additional quantum number was first noticed by Sergei Alexandrov [26].

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