Effective Hamiltonian Constraint from Group Field Theory

TL;DR

This paper proposes a method to derive a Hamiltonian constraint from group field theory by expanding around classical solutions, applied to 3D Riemannian gravity, with implications for GFT renormalization.

Contribution

It introduces a novel approach to relate GFT actions to Hamiltonian constraints via classical solution expansion, applied to Boulatov theory.

Findings

Derived a Hamiltonian operator from GFT for 3D gravity

Linked GFT action to canonical Hamiltonian constraint

Discussed spectrum relevance for GFT renormalization

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.