Abstract
In this article we consider physical states in the hypercuboidal truncation of the EPRL-FK spin foam model for Euclidean quantum gravity. In particular, these states are defined on graphs which allow considering the entanglement entropy (EE) associated to the bipartition of space. We compute the EE numerically for some examples, and find that it depends on the coupling constant α within the theory, which has recently been introduced in the face amplitude. We also find that there appears a maximum of the EE within the region of the coupling constant containing the non-Gaussian fixed point of the RG flow of the truncated model. We discuss the relation of this behaviour with the restoration of diffeomorphism symmetry at the fixed point.
Highlights
Spin Foam models (SFM) are certain proposals for the construction of transition amplitudes for states defined on graphs
A prime example are the spin foam models for quantum gravity, which have been developed as expressions for the physical inner product of spin network states in loop quantum gravity (LQG), and topological BF theory, or even lattice gauge theory, can be formulated in terms of spin foam models. [1][2][3][4][5][6][7]
The set of spins kf distributed among the faces of the lattice, which comply to the hypercuboidal symmetry, contains many elements with non-metric interpretation, in the sense that they do not allow for a reconstruction of the 4d metric from the spins
Summary
Spin Foam models (SFM) are certain proposals for the construction of transition amplitudes for states defined on graphs. One of the most widely used models for the transition of LQG spin network states is the EPRL-FK model, which is defined on 2-complexes dual to 4d triangulations, and its KKL-extension to general 2-complexes, allowing the use of arbitrary polytopes [22][23][24] It relies on a specific implementation of the so-called simnplicity constraints on topological SO(4)-BF theory, building on a classical equivalence of GR with BF theory, in which the bivector field B is constrained to be simple. In particular the scaling behaviour of the EE is of interest: while generic states in the Hilbert space of a theory scale with the region volume, many ground states for interesting physical Hamiltonian operators scale with only the surface [33][32] It is this property which is used to identify and construct such states, for instance by a multiscale-entanglement renormalisation ansatz (MERA), or further developments building on this concept [34][35].
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