Effective Hamiltonian Constraint from Group Field Theory

TL;DR

The paper develops a program to extract an effective Hamiltonian constraint for loop quantum gravity from group field theory by expanding the GFT around non-trivial classical backgrounds. The resulting quadratic term defines a non-trivial kinetic operator $\mathcal{H}_{\phi_0}$ that acts as a Hamiltonian constraint on group fields and spin-network states, enabling a spectral analysis of physical states. Applied to Boulatov's 3d GFT for BF theory, the induced spectrum resembles a Klein-Gordon type operator with eigenvalues determined by a background function $f$ and a group element $G$, providing a concrete link between spinfoam amplitudes, renormalization in GFT, and canonical LQG dynamics. This framework suggests a path to leverage standard QFT tools in GFT phases and to explore how geometry and topology emerge from a non-trivial kinetic structure, with potential extensions to 4d models and their renormalization properties.

Abstract

Spinfoam models provide a covariant formulation of the dynamics of loop quantum gravity. They are non-perturbatively defined in the group field theory (GFT) framework: the GFT partition function defines the sum of spinfoam transition amplitudes over all possible (discretized) geometries and topologies. The issue remains, however, of explicitly relating the specific form of the group field theory action and the canonical Hamiltonian constraint. Here, we suggest an avenue for addressing this issue. Our strategy is to expand group field theories around non-trivial classical solutions and to interpret the induced quadratic kinematical term as defining a Hamiltonian constraint on the group field and thus on spin network wave functions. We apply our procedure to Boulatov group field theory for 3d Riemannian gravity. Finally, we discuss the relevance of understanding the spectrum of this Hamiltonian operator for the renormalization of group field theories.

Figures (4)

• Figure 1: The functional $\psi^{\{\phi_v\}}$ consisting in group fields $\varphi_v$ at unrelated vertices $v$ and the functional $\psi_{\Gamma}^{\{\phi_v\}}$ constructed by gluing these vertices along the edges of a graph $\Gamma$. For a given edge $e$, the two group field living at the source vertex $s(e)$ and target vertex $t(e)$ are glued by an intermediate (fiducial) vertex $i(e)$.
• Figure 2: The $\Theta$-graph with two vertices and three edges linking them: the two possibilities for defining the spin network functional made from $\phi_f$ group field insertions at the two vertices but with different choices of permutations at each vertex.
• Figure 3: The oriented tetrahedron graph with its four vertices and six edges and a particular choice of permutations at the four vertices in order to define the spin network functional.
• Figure 4: The tetrahedron graph (faithfully) embedded on the 2-torus with the edges 2 and 5 wrapped around the two cycles.