On knottings in the physical Hilbert space of LQG as given by the EPRL model

TL;DR

The paper shows that, for the Euclidean EPRL spin-foam model with amplitudes chosen to be invariant under trivial subdivisions, the physical Hilbert space obtained by summing over spin-foam histories contains no knotting information of boundary graphs. This is proved by demonstrating a large symmetry, called consistent deformation, under which the spin-foam amplitude $Z[\kappa]$ is invariant, and by constructing an unknotting spin foam $\kappa_0$ with unit amplitude that relates knotting classes. A key step is proving that the knotting-undoing move induces a bijection between foam equivalence classes, and that the physical inner product is unchanged under composition with $\kappa_0$ or $\kappa_0^{-1}$, hence any two graphs with the same combinatorics but different knotting project to the same physical state. The results hinge on the chosen boundary and interior amplitudes and raise questions about knotting sensitivity in more general spin-foam formulations and observables probing four-dimensional topology.

Abstract

We consider the EPRL spin foam amplitude for arbitrary embedded two-complexes. Choosing a definition of the face- and edge amplitudes which lead to spin foam amplitudes invariant under trivial subdivisions, we investigate invariance properties of the amplitude under consistent deformations, which are deformations of the embedded two-complex where faces are allowed to pass through each other in a controlled way. Using this surprising invariance, we are able to show that in the physical Hilbert space as defined by the sum over all spin foams contains no knotting classes of graphs anymore.

Figures (14)

• Figure 1: A spin network function consists of an embedded graph $\gamma\subset\Sigma$, spins $k_e$ along its edges and interwiners $\iota_v$ along its vertices.
• Figure 2: An edge $e$ with bordered by three faces. The intertwiner $\iota_e$ associated to it maps $\iota_e\;:\;V_{j_{f_1}^+,j_{f_1}^-}\;\longrightarrow\;V_{j_{f_2}^+,j_{f_2}^-}\otimes V_{j_{f_3}^+,j_{f_3}^-}$ and intertwines the action of $Spin(4)\simeq SU(2)\times SU(2)$ on both sides.
• Figure 3: Diagrammatic representation of the map $\phi$. The lines carry $SU(2)$ representations, the box denotes integration over $SU(2)\times SU(2)$, and each vertex carries an $SU(2)$ intertwiner. All face are outgoing, so there is no additional index $out,\,in$ to distinguish them.
• Figure 4: A vertex $v$ in a spin foam $\kappa$, and the associated vertex function $T_{\gamma_v,j^\pm_f,\iota_e}$, which is a $SU(2)\times SU(2)$-network function on the vertex graph $\gamma_v$ on a three sphere $S^3$, which results as a dimensional reduction of the neighbourhood of $v$. The edges and vertices of $\gamma_v$ correspond to faces and edges in $\kappa$ touching $v$.
• Figure 5: Trivial subdivision of an edge with a vertex.