In this paper we propose an algebraic formulation of group field theory and consider non-Fock representations based on coherent states. We show that we can construct representations with infinite number of degrees of freedom on compact base manifolds. We also show that these representations break translation symmetry. Since such representations can be regarded as quantum gravitational systems with an infinite number of fundamental pre-geometric building blocks, they may be more suitable for the description of effective geometrical phases of the theory.

We propose a geometrical treatment of symmetries in non-local field theories, where the nonlocality is due to a lack of identification of field arguments in the action. We show that the existence of a symmetry of the action leads to a generalised conservation law, in which the usual conserved current acquires an additional non-local correction term, obtaining a generalisation of the standard Noether theorem. We illustrate the general formalism by discussing the specific physical example of complex scalar field theory of the type describing the hydrodynamic approximation of Bose-Einstein condensates. We expect our analysis and results to be of particular interest for the group field theory formulation of quantum gravity.

We study the Euler-Lagrange equation of the dynamical Boulatov model which is a simplicial model for 3d Euclidean quantum gravity augmented by a Laplace-Beltrami operator. We provide all its solutions on the space of left and right invariant functions that render the interaction of the model an equilateral tetrahedron. Surprisingly, for a non-linear equation of motion, the solution space forms a vector space. This space distinguishes three classes of solutions: saddle points, global and local minima of the action. Our analysis shows that there exists one parameter region of coupling constants for which the action admits degenerate global minima. * Electronic address: jobengeloun@lipn.univ-paris13.fr † Electronic address: kegeles@aei.mpg.de ‡ Electronic address: andreas.pithis@kcl.ac.uk arXiv:1806.09961v2 [gr-qc] 7 Dec 2018Hence, non-trivial extrema are saddle points. For the trivial extremum the second variation of S m,λ reads for any f ∈ S EL S m,λ (0, f ) = J∈JEL f J 2 J 2 + m 2 ≥ 0, and the necessary condition is satisfied. Indeed, the trivial extremum is a local minimum. To prove this we first notice that the Peter-Weyl transform is a topological isomorphism from S EL to the space of rapidly decreasing sequences S (N) with topology given by the family of semi-norms [23, theorem 4],The action evaluated at f becomesSince the Wigner-6J-symbol is upper-bounded by 1, we can estimateUsing the Peter-Weyl decomposition for f , we can interchange the limit and the integral by the dominant convergence theorem (using the bound c) and obtain S m,λ (ϕ, f ) = lim N →∞ J f J S m,λ ϕ, X J = 0, for any f ∈ S (EL) , from which the statement follows.

We propose a geometrical treatment of symmetries in non-local field theories, where the nonlocality is due to a lack of identification of field arguments in the action. We show that the existence of a symmetry of the action leads to a generalised conservation law, in which the usual conserved current acquires an additional non-local correction term, obtaining a generalisation of the standard Noether theorem. We illustrate the general formalism by discussing the specific physical example of complex scalar field theory of the type describing the hydrodynamic approximation of Bose-Einstein condensates. We expect our analysis and results to be of particular interest for the group field theory formulation of quantum gravity.1 Notice that most such models are non-local in the sense that they involve an infinite number of field derivatives (see for example [6][7][8][9][10]).It is unclear to us if there is a general connection between this type of non-locality and the one we deal with in our analysis. At any rate, we will not deal with this type of field theories.