- #1

marcus

Science Advisor

Gold Member

Dearly Missed

- 24,757

- 789

In trying to get some perspective on the papers that appeared first quarter 2007, I've come to the opinion that this is potentially one of the more productive ones. It is not just about 3D quantum gravity-and-matter. It has also things to say about 4D gravity-and-matter.

http://arxiv.org/abs/gr-qc/0702125

Winston Fairbairn, Etera R. Livine

17 pages, 1 figure

"An effective field theory for matter coupled to three-dimensional quantum gravity was recently derived in the context of spinfoam models in hep-th/0512113. In this paper, we show how this relates to group field theories and generalized matrix models. In the first part, we realize that the effective field theory can be recasted as a matrix model where couplings between matrices of different sizes can occur. In a second part, we provide a family of classical solutions to the three-dimensional group field theory. By studying perturbations around these solutions, we generate the dynamics of the effective field theory. We identify a particular case which leads to the action of hep-th/0512113 for a massive field living in a flat non-commutative space-time. The most general solutions lead to field theories with non-linear redefinitions of the momentum which we propose to interpret as living on curved space-times. We conclude by discussing the possible extension to four-dimensional spinfoam models."

Livine is at Lyon, Fairbairn is in Rovelli and Perez' group at Marseille. One reason to take a careful look at the recent Fairbairn-Livine paper is

the work

[21]

[22] L. Freidel and S. Majid, Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity,

[arXiv:hep-th/0601004].

[23] G. ’tHooft, A Planar Diagram Theory for Strong Interactions, Nucl.Phys. B72 461 (1974);

E. Witten, Baryons In The 1/N Expansion, Nucl. Phys. B160 57 (1979)

[24] H. Ooguri, Topological lattice models in four-dimensions, Mod.Phys.Lett. A7 2799-2810 (1992), [arXiv:hep-th/9205090]

[25] D. Oriti, Boundary terms in the Barrett-Crane spin foam model and consistent gluing, Phys.Lett. B532 (2002) 363-372,

[arXiv:gr-qc/0201077]

[26]

[27]

[28]

We should be able to tell something from the future work that is cited here--also in connection to the Freidel-Girelli-Livine paper that appeared a couple of months ago. Here was our discussion of the F-G-L paper

https://www.physicsforums.com/showthread.php?t=151411

Here is the abstract

http://arxiv.org/abs/hep-th/0701113

"We analyze the algebra of Dirac observables of the relativistic particle in four space-time dimensions. We show that the position observables become non-commutative and the commutation relations lead to a structure very similar to the non-commutative geometry of Deformed Special Relativity (DSR). In this framework, it appears natural to consider the 4d relativistic particle as a five dimensional massless particle. We study its quantization in terms of wave functions on the 5d light cone. We introduce the corresponding five-dimensional action principle and analyze how it reproduces the physics of the 4d relativistic particle. The formalism is naturally subject to divergences and we show that DSR arises as a natural regularization: the 5d light cone is regularized as the de Sitter space. We interpret the fifth coordinate as the particle's proper time while the fifth moment can be understood as the mass. Finally, we show how to formulate the Feynman propagator and the Feynman amplitudes of quantum field theory in this context in terms of Dirac observables. This provides new insights for the construction of observables and scattering amplitudes in DSR."

http://arxiv.org/abs/gr-qc/0702125

**3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory**Winston Fairbairn, Etera R. Livine

17 pages, 1 figure

"An effective field theory for matter coupled to three-dimensional quantum gravity was recently derived in the context of spinfoam models in hep-th/0512113. In this paper, we show how this relates to group field theories and generalized matrix models. In the first part, we realize that the effective field theory can be recasted as a matrix model where couplings between matrices of different sizes can occur. In a second part, we provide a family of classical solutions to the three-dimensional group field theory. By studying perturbations around these solutions, we generate the dynamics of the effective field theory. We identify a particular case which leads to the action of hep-th/0512113 for a massive field living in a flat non-commutative space-time. The most general solutions lead to field theories with non-linear redefinitions of the momentum which we propose to interpret as living on curved space-times. We conclude by discussing the possible extension to four-dimensional spinfoam models."

Livine is at Lyon, Fairbairn is in Rovelli and Perez' group at Marseille. One reason to take a careful look at the recent Fairbairn-Livine paper is

the work

**in preparation**that it cites. To provide context, here is a sample chunk of the citations, including an interesting paper of Baez and Perez as well as work in preparation.[21]

**L. Freidel and E. R. Livine, Non-perturbative structures for Spinfoams: Instantons for the Group Field Theory**, in preparation[22] L. Freidel and S. Majid, Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity,

[arXiv:hep-th/0601004].

[23] G. ’tHooft, A Planar Diagram Theory for Strong Interactions, Nucl.Phys. B72 461 (1974);

E. Witten, Baryons In The 1/N Expansion, Nucl. Phys. B160 57 (1979)

[24] H. Ooguri, Topological lattice models in four-dimensions, Mod.Phys.Lett. A7 2799-2810 (1992), [arXiv:hep-th/9205090]

[25] D. Oriti, Boundary terms in the Barrett-Crane spin foam model and consistent gluing, Phys.Lett. B532 (2002) 363-372,

[arXiv:gr-qc/0201077]

[26]

**W. J. Fairbairn and A. Perez, Quantisation of string-like sources coupled to BF theory : Physical scalar product and topological invariance**, in preparation[27]

**J. C. Baez and A. Perez, Quantization of strings and branes coupled to BF theory**, [arXiv:gr-qc/0605087][28]

**W. J. Fairbairn and E. R. Livine, String theory as a phase of the four dimensional group field theory**, in preparationWe should be able to tell something from the future work that is cited here--also in connection to the Freidel-Girelli-Livine paper that appeared a couple of months ago. Here was our discussion of the F-G-L paper

https://www.physicsforums.com/showthread.php?t=151411

Here is the abstract

http://arxiv.org/abs/hep-th/0701113

**The Relativistic Particle: Dirac observables and Feynman propagator**"We analyze the algebra of Dirac observables of the relativistic particle in four space-time dimensions. We show that the position observables become non-commutative and the commutation relations lead to a structure very similar to the non-commutative geometry of Deformed Special Relativity (DSR). In this framework, it appears natural to consider the 4d relativistic particle as a five dimensional massless particle. We study its quantization in terms of wave functions on the 5d light cone. We introduce the corresponding five-dimensional action principle and analyze how it reproduces the physics of the 4d relativistic particle. The formalism is naturally subject to divergences and we show that DSR arises as a natural regularization: the 5d light cone is regularized as the de Sitter space. We interpret the fifth coordinate as the particle's proper time while the fifth moment can be understood as the mass. Finally, we show how to formulate the Feynman propagator and the Feynman amplitudes of quantum field theory in this context in terms of Dirac observables. This provides new insights for the construction of observables and scattering amplitudes in DSR."

Last edited: