Group Field Theory: An Overview

Freidel's proposal uses tree-level group field theory amplitudes to define the physical inner product of quantum gravity.

“A prescription that avoids this last issue is the radical proposal of Freidel [179], where the tree-level GFT amplitudes are used to define the physical inner product of quantum gravity.”

The spin foam functional W|Γ5,j,i> can be expanded in a formal perturbative vertex expansion in the parameter λ of a Group Field Theory.

“The sum over two-complexes can be implemented in terms of a formal perturbative expansion in the parameter λ of a Group Field Theory [46]:”

The interaction vertex (10j symbol, or Barrett-Crane vertex amplitude) has been shown to include, beside the 'good' term approximating the exponential of the Einstein-Hilbert action, two 'bad' terms: an exponential with opposite sign giving the cosine of the Regge action, and a dominant term depending on the existence of degenerate four-simplices.

“The physical correctness of these theories has been questioned because in the low-energy limit their interaction vertex (10 j symbol, or Barrett-Crane vertex amplitude) has been shown to include -beside the 'good' term approximating the exponential of the Einstein-Hilbert action [19]- also two 'bad' terms: an exponential with opposite sign, giving the cosine of Regge action [19] (analogous to the”

In GFT, the initial big-bang singularity is replaced by a regular bounce, followed by an era of accelerated expansion.

“In particular, the initial big-bang singularity is replaced by a regular bounce (see Ref. [1, 2]), followed by an era of accelerated expansion [2].”

In the non-interacting GFT case, the Universe expands indefinitely and in the limit φ→∞ its dynamics follows the ordinary Friedmann equation for a flat Universe filled with a massless and minimally coupled scalar field.

“In that case one has that the Universe expands indefinitely and in the limit φ →∞ its dynamics follows the ordinary Friedmann equation for a flat Universe, filled with a massless and minimally coupled scalar field.”

The dynamical equations for the volume of the Universe can be recast in a form closer to the standard Friedmann equation.

“It is possible to recast the dynamical equations for the volume of the Universe in a form that bears a closer resemblance to the standard Friedmann equation, as shown in Ref. [2].”

The term in bracket in Eq. (38) can be interpreted as a dynamical effective gravitational constant.

“The term in bracket can thus be interpreted as a dynamical effective gravitational constant as in Ref. [2].”

Results already obtained in the free theory survive in the interacting case, in particular the occurrence of a bounce and an early epoch of accelerated expansion.

“Results that have already been obtained in the free theory (Refs. [1, 2]) survive in the interacting case, in particular for what concerns the occurrence of a bounce and an early epoch of accelerated expansion.”

Group Field Theory is a representative approach where the continuum geometric phase is identified with a condensate phase of the underlying quantum gravity system.

“A particular representative in this class of approaches where such a scenario has been proposed is Group Field Theory (GFT) [2] where one tries to identify the continuum geometric phase to a condensate phase of the underlying quantum gravity system [3]”

GFTs based on the Ooguri model may provide a covariant QFT formulation of the dynamics of LQG.

“One can then show, that GFTs based on the Ooguri model, may provide a covariant QFT formulation of the dynamics of LQG.”

GFT Feynman diagrams are dual to cellular complexes of arbitrary topology.

“This modelspecific combinatorial nonlocality implies that the GFT Feynman diagrams are dual to cellular complexes of arbitrary topology [2].”