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Towards classical geometrodynamics from Group Field Theory hydrodynamics

Daniele Oriti, Lorenzo Sindoni

TL;DR

This work investigates how a hydrodynamic limit of Group Field Theories (GFTs) can reproduce aspects of classical geometrodynamics. By leveraging Loop Quantum Gravity coherent states as a source of macroscopic background data, the authors propose heat-kernel–based mean-field configurations and derive mean-field (geometrodynamics) equations in 2D BF-type and 3D Boulatov GFT models, including both ordinary and colored variants. They analyze the resulting EOM in representation space, identify approximate nontrivial backgrounds that correspond to flat connections with degenerate $B$-fields, and derive the effective dynamics for perturbations, highlighting renormalized couplings and potential instabilities. The paper also discusses how the chosen vacuum alters fundamental diffeomorphism-like symmetries in the emergent theory, revealing nonlinear realizations and possible phase structures, which collectively outline a path toward connecting microscopic GFT dynamics with classical GR through hydrodynamic approximations.

Abstract

We take the first steps towards identifying the hydrodynamics of group field theories (GFTs) and relating this hydrodynamic regime to classical geometrodynamics of continuum space. We apply to GFT mean field theory techniques borrowed from the theory of Bose condensates, alongside standard GFT and spin foam techniques. The mean field configuration we study is, in turn, obtained from loop quantum gravity coherent states. We work in the context of 2d and 3d GFT models, in euclidean signature, both ordinary and colored, as examples of a procedure that has a more general validity. We also extract the effective dynamics of the system around the mean field configurations, and discuss the role of GFT symmetries in going from microscopic to effective dynamics. In the process, we obtain additional insights on the GFT formalism itself.

Towards classical geometrodynamics from Group Field Theory hydrodynamics

TL;DR

This work investigates how a hydrodynamic limit of Group Field Theories (GFTs) can reproduce aspects of classical geometrodynamics. By leveraging Loop Quantum Gravity coherent states as a source of macroscopic background data, the authors propose heat-kernel–based mean-field configurations and derive mean-field (geometrodynamics) equations in 2D BF-type and 3D Boulatov GFT models, including both ordinary and colored variants. They analyze the resulting EOM in representation space, identify approximate nontrivial backgrounds that correspond to flat connections with degenerate -fields, and derive the effective dynamics for perturbations, highlighting renormalized couplings and potential instabilities. The paper also discusses how the chosen vacuum alters fundamental diffeomorphism-like symmetries in the emergent theory, revealing nonlinear realizations and possible phase structures, which collectively outline a path toward connecting microscopic GFT dynamics with classical GR through hydrodynamic approximations.

Abstract

We take the first steps towards identifying the hydrodynamics of group field theories (GFTs) and relating this hydrodynamic regime to classical geometrodynamics of continuum space. We apply to GFT mean field theory techniques borrowed from the theory of Bose condensates, alongside standard GFT and spin foam techniques. The mean field configuration we study is, in turn, obtained from loop quantum gravity coherent states. We work in the context of 2d and 3d GFT models, in euclidean signature, both ordinary and colored, as examples of a procedure that has a more general validity. We also extract the effective dynamics of the system around the mean field configurations, and discuss the role of GFT symmetries in going from microscopic to effective dynamics. In the process, we obtain additional insights on the GFT formalism itself.

Paper Structure

This paper contains 18 sections, 160 equations, 6 figures.

Table of Contents

  1. Mean field theory and effective hydrodynamics in Bose condensates: analogies and differences with the GFT case
  2. From LQG coherent states to an ansatz for for a GFT background configuration
  3. Coherent states on $\mathrm{SU}(2)$ and vertex wavefunctions
  4. Candidate GFT mean field configuration
  5. $2$-dimensional case
  6. Single-field GFT
  7. Colored GFT
  8. 3-dimensional GFT
  9. Single-field case
  10. Colored model
  11. Effective dynamics around the heat kernel configuration
  12. 2D - single-field model
  13. 2D - colored model
  14. 3D model(s)
  15. Fundamental vs effective symmetries
  16. ...and 3 more sections

Figures (6)

  • Figure 1: Entire oriented link from $a$ to $b$, with associated group variable $h_{ab}$ and peak group element $g_{ab}$ of the heat kernel.
  • Figure 2: Splitting the link $a$ to $b$. The group elements $h_{a},h_{b},g_{a},g_{b}$ refer to the semi-link from the vertices $a,b$ to $P$, with the appropriate orientation.
  • Figure 3: Gluing vertices
  • Figure 4: Basic propagator and vertex
  • Figure 5: The three contributions to the kinetic term: the gray rectangles correspond to insertions of heat kernels. It is understood that a sum is performed over the lines that are beginning and ending at one heat kernel insertion.
  • ...and 1 more figures