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Noncommutative Fourier transform for the Lorentz group via the Duflo map
https://repo.scoap3.org/record/32928
We defined a noncommutative algebra representation for quantum systems whose phase space is the cotangent bundle of the Lorentz group, and the noncommutative Fourier transform ensuring the unitary equivalence with the standard group representation. Our construction is from first principles in the sense that all structures are derived from the choice of quantization map for the classical system, the Duflo quantization map.Oriti, DanieleTue, 28 May 2019 09:39:31 GMThttps://repo.scoap3.org/record/32928urn:ISSN:2470-0029APS2019-05-13Functional renormalization group analysis of rank-3 tensorial group field theory: The full quartic invariant truncation
https://repo.scoap3.org/record/26447
In this paper, we consider the complete momentum-independent quartic order truncation for the effective average action of a real Abelian rank-3 tensorial group field theory. This complete truncation includes nonmelonic as well as double-trace interactions. In the usual functional renormalization group perspective, the inclusion of more operators that belong to the underlying theory space corresponds to an improvement of the truncation of the effective average action. We show that the inclusion of nonmelonic and double-trace operators in the truncation brings subtleties. In particular, we discuss the assignment of scaling dimensions to the nonmelonic sector and how the inclusion of double-trace operators considerably changes the results for critical exponents with respect to those obtained when they are not included. We argue that this is not a particular problem of the present model by comparing the results with a pure tensor model. We discuss how these issues should be investigated in future work.Geloun, Joseph BenFri, 29 Jun 2018 16:50:41 GMThttps://repo.scoap3.org/record/26447urn:ISSN:2470-0029APS2018-06-29Ryu-Takayanagi formula for symmetric random tensor networks
https://repo.scoap3.org/record/26085
We consider the special case of random tensor networks (RTNs) endowed with gauge symmetry constraints on each tensor. We compute the Rényi entropy for such states and recover the Ryu-Takayanagi (RT) formula in the large-bond regime. The result provides first of all an interesting new extension of the existing derivations of the RT formula for RTNs. Moreover, this extension of the RTN formalism brings it in direct relation with (tensorial) group field theories (and spin networks), and thus provides new tools for realizing the tensor network/geometry duality in the context of background-independent quantum gravity, and for importing quantum gravity tools into tensor network research.Chirco, GoffredoFri, 08 Jun 2018 20:20:35 GMThttps://repo.scoap3.org/record/26085urn:ISSN:2470-0029APS2018-06-08Renormalization of an Abelian tensor group field theory: solution at leading order
https://repo.scoap3.org/record/10033
We study a just-renormalizable tensorial group field theory of rank six with quartic melonic interactions and Abelian group U(1). We introduce the formalism of the intermediate field, which allows a precise characterization of the leading order Feynman graphs. We define the renormalization of the model, compute its (perturbative) renormalization group flow and write its expansion in terms of effective couplings. We then establish closed equations for the two point and four point functions at leading (melonic) order. Using the effective expansion and its uniform exponential bounds we prove that these equations admit a unique solution at small renormalized coupling.Lahoche, VincentSat, 25 Apr 2015 00:52:42 GMThttps://repo.scoap3.org/record/10033urn:ISSN:1029-8479Springer/SISSA2015-04-20Functional renormalisation group approach for tensorial group field theory: a rank-3 model
https://repo.scoap3.org/record/9644
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.Benedetti, DarioThu, 19 Mar 2015 20:36:58 GMThttps://repo.scoap3.org/record/9644urn:ISSN:1029-8479Springer/SISSA2015-03-17Homogeneous cosmologies as group field theory condensates
https://repo.scoap3.org/record/2725
We give a general procedure, in the group field theory (GFT) formalism for quantum gravity, for constructing states that describe macroscopic, spatially homogeneous universes. These states are close to coherent (condensate) states used in the description of Bose-Einstein condensates. The condition on such states to be (approximate) solutions to the quantum equations of motion of GFT is used to extract an effective dynamics for homogeneous cosmologies directly from the underlying quantum theory. The resulting description in general gives nonlinear and nonlocal equations for the ‘condensate wavefunction’ which are analogous to the Gross-Pitaevskii equation in Bose-Einstein condensates. We show the general form of the effective equations for current quantum gravity models, as well as some concrete examples. We identify conditions under which the dynamics becomes linear, admitting an interpretation as a quantum-cosmological Wheeler-DeWitt equation, and give its semiclassical (WKB) approximation in the case of a kinetic term that includes a Laplace-Beltrami operator. For isotropic states, this approximation reproduces the classical Friedmann equation in vacuum with positive spatial curvature. We show how the formalism can be consistently extended from Riemannian signature to Lorentzian signature models, and discuss the addition of matter fields, obtaining the correct coupling of a massless scalar in the Friedmann equation from the most natural extension of the GFT action. We also outline the procedure for extending our condensate states to include cosmological perturbations. Our results form the basis of a general programme for extracting effective cosmological dynamics directly from a microscopic non-perturbative theory of quantum gravity.Steffen GielenWed, 04 Jun 2014 17:08:42 GMThttps://repo.scoap3.org/record/2725urn:ISSN:1029-8479Springer/SISSA2014-06-03