Functional renormalisation group approach for tensorial group field theory: a rank-3 model
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We set up the Functional Renormalisation Group formalism for Tensorial Group Field Theory in full generality. We then apply it to a rank-3 model over U(1)3, endowed with a kinetic term linear in the momenta and with nonlocal interactions. The system of FRG equations turns out to be non-autonomous in the RG flow parameter. This feature is explained by the existence of a hidden scale, the radius of the group manifold. We investigate in detail the opposite regimes of large cut-off (UV) and small cut-off (IR) of the FRG equations, where the system becomes autonomous, and we find, in both case, Gaussian and non-Gaussian fixed points. We derive and interpret the critical exponents and flow diagrams associated with these fixed points, and discuss how the UV and IR regimes are matched. Finally, we discuss the evidence for a phase transition from a symmetric phase to a broken or condensed phase, from an RG perspective, finding that this seems to exist only in the approximate regime of very large radius of the group manifold, as to be expected for systems on compact manifolds.
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The number operator N exists only in the zero-interaction representation, when all representations of the CCRs are equivalent to the Fock representation.
“Strictly speaking, N exists only in the zero-interaction representation which is when all representations of the CCRs are equivalent to the Fock representation, as is well known within the context local QFTs [34].”
From Impact of nonlinear effective interactions on group field theory quantum gravity condensates (2016) · cites this paper as [34]
The proper treatment of interactions necessitates the use of non-Fock representations for which N is infinite.
“Such behavior is not surprising, as it is well known within the context of local QFTs that the proper treatment of interactions necessitates the use of non-Fock representations for which N is infinite (cf. appendix B and [34]).”
From Impact of nonlinear effective interactions on group field theory quantum gravity condensates (2016) · cites this paper as [34]
It is only in the thermodynamic limit that inequivalent irreducible representations of the CCRs become available which is a prerequisite for the occurrence of nonunique equilibrium states, in turn essential to consistently describing phase transitions.
“It is only in this limit that inequivalent irreducible representations of the CCRs become available which is a prerequisite for the occurrence of nonunique equilibrium states, in turn essential to consistently describing phase transitions [34].”
From Impact of nonlinear effective interactions on group field theory quantum gravity condensates (2016) · cites this paper as [34]
For interacting fields, models with nonvanishing ground state expectation value and many body systems in the thermodynamic limit, the Fock representation is not allowed and N is not a good quantum number.
“However, this is different for interacting fields, models with nonvanishing ground state expectation value and many body systems in the thermodynamic limit, where the Fock representation is not allowed and N is not a good quantum number for the characterization of the system since ˆ N is unbounded from above.”
From Impact of nonlinear effective interactions on group field theory quantum gravity condensates (2016) · cites this paper as [34]
Non-Fock representations are also required in order to describe many particle systems in the thermodynamic limit.
“because non-Fock representations are also required in order to describe many particle systems in the thermodynamic limit.”
From Impact of nonlinear effective interactions on group field theory quantum gravity condensates (2016) · cites this paper as [34]
The loss of normalizability is a generic feature of massless or interacting (local) QFTs according to Haag's theorem which require the use of non-Fock representations.
“In fact, the loss of normalizability is not too surprising because it is a generic feature of massless or interacting (local) QFTs according to Haag's theorem which require the use of non-Fock representations [34].”
From Impact of nonlinear effective interactions on group field theory quantum gravity condensates (2016) · cites this paper as [34]
In local QFT, in finite dimensional and noninteracting infinite dimensional cases all irreducible Fock representations are unitarily equivalent and hence there is just one phase associated to the quantum system.
“Within the context of local QFT [34] it is well known, that in the finite dimensional and noninteracting infinite dimensional cases all irreducible Fock representations are unitarily equivalent and hence there is just one phase associated to the quantum system.”
From Impact of nonlinear effective interactions on group field theory quantum gravity condensates (2016) · cites this paper as [34]
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- Authors
- Benedetti, D., Geloun, J. B., & Oriti, D.
- Affiliations
- Max Planck Institute for Gravitational Physics (DE)
- Université d'Abomey-Calavi (BJ)
- Corresponding author
- Benedetti, Dario
- Venue
- Journal of High Energy Physics
- Published
- 2015-03
- Volume
- 2015
- Issue
- 3
- ISSN
- 1029-8479 (ISSN-L)
- OpenAlex
- W2111528318
- Cited by
- 104
- References
- 80
- Citations by year
- 2026: 4 · 2025: 1 · 2024: 7 · 2023: 3 · 2022: 3 · 2021: 5 · 2020: 10 · 2019: 9 · 2018: 16 · 2017: 10 · 2016: 31 · 2015: 5
- FWCI
- 13.04
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Licence
- cc-by
- Locations
- Journal of High Energy Physics (publishedVersion, cc-by)
- arXiv (Cornell University) (publishedVersion)
- arXiv (Cornell University)
- Funders
- Alexander von Humboldt-Stiftung
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
- Physical Sciences › Mathematics › Mathematical Physics › advanced mathematical theories
- Keywords
- functional renormalization group, tensorial group field theory, Gaussian fixed point, non-Gaussian fixed point, critical exponents, flowcharts, phase transitions, compact manifolds, nonlocal interactions
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