Effective Hamiltonian Constraint from Group Field Theory

Daniele Oriti; Etera R. Livine; James P. Ryan

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Effective Hamiltonian Constraint from Group Field Theory

Etera R. Livine, Daniele Oriti, James P. Ryan

General Relativity and Quantum Cosmology · Submitted 28 Apr 2011

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The authors' main claim 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.

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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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Spinfoam models provide a covariant formulation of the dynamics of loop quantum gravity.	background 1.00	0.04	0.02	asserted 1.00
s1	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.	background 1.00	0.07	0.03	asserted 1.00
s2	The issue remains, however, of explicitly relating the specific form of the group field theory action and the canonical Hamiltonian constraint.	gap 0.99	0.05	0.03	asserted 0.99
s3	Here, we suggest an avenue for addressing this issue.	contribution 0.89	0.40	0.03	hedged possibility 0.68
s4	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.	method 0.95	0.69	0.44	asserted 0.69
s5	We apply our procedure to Boulatov group field theory for 3d Riemannian gravity.	contribution 0.61	0.92	0.06	asserted 0.99
s6	Finally, we discuss the relevance of understanding the spectrum of this Hamiltonian operator for the renormalization of group field theories.	interpretation 0.50	0.40	0.09	unclear 0.57

What kind of paper the classifier reports this is: method or tool 0.65 · evidence from proof or derivation 0.81.

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Original 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.

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Cited by 13 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Constraining quantum-gravity predictions for evolving dark energy
astro-ph.CO 2026-09 unverdicted novelty 7.0 of 10

Group field theory quantum gravity predicts dark energy equations of state that are constrained by BAO and supernova data, with the logarithmic branch resembling a cosmological constant and oscillatory solutions allow...
Renormalization of an SU(2) Tensorial Group Field Theory in Three Dimensions
hep-th 2013-03 conditional novelty 7.0 of 10

The SU(2) tensorial group field theory in three dimensions with tensor-invariant interactions up to sixth order is just renormalizable, with all divergences melonic and absorbable into a finite set of counterterms.
Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions
hep-th 2012-07 conditional novelty 7.0 of 10

Four-dimensional Abelian tensorial group field theories with gauge invariance conditions are proven super-renormalizable for any polynomial interaction, with divergences confined to melonic tadpoles removable by a new...
Critical behavior of colored tensor models in the large N limit
hep-th 2011-05 accept novelty 7.0 of 10

The leading order of colored tensor models is in bijection with colored trees, so it is exactly summable and has a critical exponent of 1/2 in every dimension.
Tensorial methods and renormalization in Group Field Theories
hep-th 2013-10 conditional novelty 6.0 of 10

Group Field Theories with tensorial interactions and a closure constraint admit a consistent multiscale renormalization analysis, and a 3d SU(2) model is shown just-renormalizable at the perturbative level.

Show 8 more citing papers

The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders
hep-th 2012-03 conditional novelty 6.0 of 10

The Schwinger-Dyson equations of random tensor models close a Lie algebra of constraints at all orders in 1/N, with the leading-order tree algebra sitting inside it as a subalgebra.
The universe as a quantum gravity condensate
gr-qc 2016-12 accept novelty 3.0 of 10

Group field theory condensates, treated in a Gross-Pitaevskii mean-field approximation, yield generalised Friedmann equations whose classical limit is standard cosmology and whose solutions bounce instead of hitting a...
Group field theories
gr-qc 2012-10 accept novelty 3.0 of 10

Group field theory is presented as the generating functional for spin foam amplitudes, with a new explicit construction for the Lorentzian EPRL/FK model in four dimensions.
Quantum Cosmology from Group Field Theory Condensates: a Review
gr-qc 2016-02 accept novelty 2.0 of 10

The review argues that a macroscopic universe can be modeled as a group field theory condensate whose effective dynamics reproduces general-relativistic and loop-quantum-cosmology equations in suitable limits.
Group Field Theory and Loop Quantum Gravity
gr-qc 2014-08 accept novelty 2.0 of 10

Group field theory reformulates loop quantum gravity as a quantum field theory of spin network vertices, giving spin foam models a generating functional and a route to continuum and cosmological limits.
The Spin Foam Approach to Quantum Gravity
gr-qc 2012-05 accept novelty 2.0 of 10

The review presents the spin foam quantization of gravity as a constrained BF path integral, focusing on the EPRL and Freidel-Krasnov models and on 2+1 gravity as a solvable guide.
The microscopic dynamics of quantum space as a group field theory
hep-th 2011-10 accept novelty 2.0 of 10

Group field theory gives a discrete third quantization of 3d Riemannian gravity whose Feynman amplitudes equal Ponzano-Regge spin foam and simplicial BF path integrals, while 4d extensions remain tentative.
Colored Tensor Models - a Review
hep-th 2011-09 accept novelty 2.0 of 10

Colored tensor models provide a controlled large-N framework for random geometry in D at least 3, with melonic graphs dominating the leading order and exhibiting a critical continuum limit.

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22 arXiv papers cite this in 26 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Cites it as background 6 papers

The success of matrix models in describing random two dimensional surfaces inspired their generalization in higher dimensions to random tensor models [ref] (see also [ref] for more recent developments).

The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders · Introduction · 0.96
Then, to guide experimental efforts, bottom-up approaches have been proposed [ref] .

First Combined Study on LIV from Observations of Energy-dependent Time Delays from Multiple-type Gamma-ray Sources -- Part I. Motivation, Method Description and Validation through Simulations of H.E.S.S., MAGIC and VERITAS Datasets · Introduction · 0.86
In the context of group field theories this issue is explored in [ref] .

The Spin Foam Approach to Quantum Gravity · Relation with the canonical formulation: the scalar constraint · 0.84

What it cites · 43 sentences without a confident label

Uses the method or tool of 1 paper

We will expand it around the flat solutions introduced in [ref] and analyze the spectrum of the induced Hamiltonian constraint.

3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory · Introduction · 0.80

Cites as support 3 papers

This not only supports the physical relevance of our procedure but allows supports the idea that group field theory can be undertood (at least in certain phases, or for certain perturbation fields) as the momentum representation of a field theory on an actual space-time manifold, which would be obtained through a Fourier transform [ref] .

Matrix Models as Non-commutative Field Theories on R^3 · Conclusion & Outlook · 0.97
This not only supports the physical relevance of our procedure but allows supports the idea that group field theory can be undertood (at least in certain phases, or for certain perturbation fields) as the momentum representation of a field theory on an actual space-time manifold, which would be obtained through a Fourier transform [ref] .

A Deformed Poincare Invariance for Group Field Theories · Conclusion & Outlook · 0.97
This not only supports the physical relevance of our procedure but allows supports the idea that group field theory can be undertood (at least in certain phases, or for certain perturbation fields) as the momentum representation of a field theory on an actual space-time manifold, which would be obtained through a Fourier transform [ref] .

Group field theory with non-commutative metric variables · Conclusion & Outlook · 0.97

Cites as background 13 papers

The Spinfoam program [ref] has been originally developed in order to implement, in a covariant, sum-over-histories form, the dynamics of loop quantum gravity [ref] and compute transition amplitudes between its spin network states of quantum geometry.

Spin Foam Models for Quantum Gravity · Introduction · 1.00
Another possibility being explored [ref] was to obtain effective equations for geometry from the GFT equations of motion, as conditions for a given background configuration to define a solution of the same, in the spirit of mean field theory in Bose condensates.

Towards classical geometrodynamics from Group Field Theory hydrodynamics · Introduction · 0.85
A standard prescription for group field theories is to start with a trivial propagator (but see the last reference of [ref] ) and to encode all the dynamics (of both geometry, thus the Hamiltonian constraint, and topology) in the interaction term.

Towards Renormalizing Group Field Theory · Conclusion & Outlook · 0.86

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