Quantum gravity states, entanglement graphs and second-quantized tensor networks

In recent years, the import of quantum information techniques in quantum gravity opened new perspectives in the study of the microscopic structure of spacetime. We contribute to such a program by establishing a precise correspondence between the quantum information formalism of tensor networks (TN), in the case of projected entangled-pair states (PEPS) generalised to a second-quantized framework, and group field theory (GFT) states, and by showing how, in this quantum gravity approach, discrete spatial manifolds arise as entanglement patterns among quanta of space, having a dual representation in terms of graphs and simplicial complexes. We devote special attention to the implementation and consequences of the label independence of the graphs/networks, corresponding to the indistinguishability of the space quanta and representing a discrete counterpart of the diffeomorphism invariance of a consistent quantum gravity formalism. We also outline a relational setting to recover distinguishability of graph/network vertices at an effective and physical level, in a partial semi-classical limit of the theory.

Quantum gravity, group field theory (GFT), and combinatorics

This chapter is the first chapter of the book dedicated to the study of the combinatorics of various quantum gravity approaches. After a brief introductory section to quantum gravity, we shortly mention the main candidates for a quantum theory of gravity: string theory, loop quantum gravity, and group field theory (GFT), causal dynamical triangulations, matrix models. The next sections introduce some GFT models such as the Boulatov model, the colourable and the multi-orientable model. The saddle point method for some specific GFT Feynman integrals is presented in the fifth section. Finally, some algebraic combinatorics results are presented: definition of an appropriate Conne–Kreimer Hopf algebra describing the combinatorics of the renormalization of a certain tensor GFT model (the so-called Ben Geloun–Rivasseau model) and the use of its Hochschild cohomology for the study of the combinatorial Dyson–Schwinger equation of this specific model.

Renormalization of Group Field Theories for Quantum Gravity: New Computations and Some Suggestions

We discuss motivation and goals of renormalization analyses of group field theory models of simplicial 4d quantum gravity, and review briefly the status of this research area. We present some new computations of perturbative Group field theories amplitudes, concerning in particular their scaling behavior, and the numerical techniques employed to obtain them. Finally, we suggest a number of research directions for further progress.

Thermofield double states in group field theory

Group field theories are higher-rank generalizations of matrix/tensor models, and encode the simplicial geometries of quantum gravity. In this paper, we study the thermofield double states in group field theories. The starting point is the equilibrium Gibbs states in group field theory recently found by Kotecha and Oriti, based on which we construct the thermofield double state as a “thermal” vacuum respecting the Kubo–Martin–Schwinger condition. We work with the Weyl [Formula: see text]-algebra of group fields, and a particular type of thermofield double states with single type of symmetry is obtained from the squeezed states on this Weyl algebra. The thermofield double states, when viewed as states on the group field theory Fock vacuum, are condensate states at finite flow parameter [Formula: see text]. We suggest that the equilibrium flow parameters [Formula: see text] of this type of thermofield double states in the group field theory condensate pictures of black hole horizon and quantum cosmology are related to the inverse temperatures in gravitational thermodynamics.

Phase transitions in TGFT: functional renormalization group in the cyclic-melonic potential approximation and equivalence to O(N) models

In the group field theory approach to quantum gravity, continuous spacetime geometry is expected to emerge via phase transition. However, understanding the phase diagram and finding fixed points under the renormalization group flow remains a major challenge. In this work we tackle the issue for a tensorial group field theory using the functional renormalization group method. We derive the flow equation for the effective potential at any order restricting to a subclass of tensorial interactions called cyclic melonic and projecting to a constant field in group space. For a tensor field of rank r on U(1) we explicitly calculate beta functions and find equivalence with those of O(N) models but with an effective dimension flowing from r − 1 to zero. In the r − 1 dimensional regime, the equivalence to O(N) models is modified by a tensor specific flow of the anomalous dimension with the consequence that the Wilson-Fisher type fixed point solution has two branches. However, due to the flow to dimension zero, fixed points describing a transition between a broken and unbroken phase do not persist and we find universal symmetry restoration. To overcome this limitation, it is necessary to go beyond compact configuration space.