Spin foam models with finite groups

TL;DR

Spin foam models with finite groups are explored as tractable laboratories for quantum gravity concepts, recasting lattice gauge theories, BF theory, and group field theory in a finite-group setting. The work derives spin foam representations, dual models, high-temperature expansions, and topological-sector expansions, and develops transfer operators, constraint structures, and constrained edge intertwiners inspired by Barrett–Crane/EPRL-type constructions. It then develops finite-group group field theories, including matrix-model analogues in $2$D and Boulatov–Ooguri-type higher-dimensional models, illustrating how spin foams and topological amplitudes arise from GFT Feynman diagrams. The discussion extends to nonlocal spin foams and outlines an outlook on coarse graining, renormalization, and gravity-like dynamics in finite-group frameworks, underscoring their utility as test beds for emergent phenomena and potential connections to gravity.

Abstract

Spin foam models, loop quantum gravity and group field theory are discussed as quantum gravity candidate theories and usually involve a continuous Lie group. We advocate here to consider quantum gravity inspired models with finite groups, firstly as a test bed for the full theory and secondly as a class of new lattice theories possibly featuring an analogue diffeomorphism symmetry. To make these notes accessible to readers outside the quantum gravity community we provide an introduction to some essential concepts in the loop quantum gravity, spin foam and group field theory approach and point out the many connections to lattice field theory and condensed matter systems.

Figures (6)

• Figure 1: A face $f$, bordered by edges $e_1,\ldots, e_5$. The curvature is given by $h_f=g_{e_5}^{-1}g_{e_4}g_{e_3}^{-1}g_{e_2}g_{e_1}$ (note the ordering).
• Figure 2: The relative orientations of $e_1$ and $f$, and $e_2$ and $f$ agree, so in both the edge Hilbert spaces $\mathcal{H}_{e_1}$ and $\mathcal{H}_{e_2}$ the factor $V_{\rho_f}$. Therefore $|\iota_{e_1}\rangle$ and $\langle\iota_{e_2}|$ have opposite indices belonging to $V_{\rho_f}$, which therefore can be contracted.
• Figure 3: A time like plaquette in a cubical lattice. The group elements at the time like edges can be understood to act as gauge transformations on the vertices either at time steps $n$ or $n+1$. Integration over the group elements assoicated to the time like edges enforces therefore (via group averaging) a projector onto the gauge invariant Hilbert space.
• Figure 4: Two states in the spin network basis for ${\hbox{$\rm Z$}}_2$ which are equivalent under the physical inner product. The thick edges carry non--trivial representations. The right state can be obtained from the left state by applying a plaquette stabilizer.
• Figure 5: For the case $n=3$, we have drawn the fundamental components for the Feynman diagram $\Gamma$, the strand diagram $s(\Gamma)$ and the dual n-simplex $\Delta$.