Asymptotics of $$\mathrm {SL}(2,{{\mathbb {C}}})$$ coherent invariant tensors

Donà, Pietro; Fanizza, Marco; Martin-Dussaud, Pierre; Speziale, Simone

doi:10.1007/s00220-021-04154-3

Cited by 13 publications

(10 citation statements)

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“…It has more complicated critical point equations, and a richer geometric interpretation. In the simplest case of a product of generalized Clebsch-Gordan coefficients, with no graph structure, the critical point equations define a more elaborated map between the 3d vectors appearing in the boundary data and bivectors than the one appearing here: γ-simple bivectors instead of bivectors with vanishing magnetic part [66]. For a tensor invariant associated to a graph, it should be possible to combine the technique presented here with the ones of [66] to obtain asymptotics of all unitary SL(2, C) invariants on graphs.…”

Section: Discussionmentioning

confidence: 99%

“…In the simplest case of a product of generalized Clebsch-Gordan coefficients, with no graph structure, the critical point equations define a more elaborated map between the 3d vectors appearing in the boundary data and bivectors than the one appearing here: γ-simple bivectors instead of bivectors with vanishing magnetic part [66]. For a tensor invariant associated to a graph, it should be possible to combine the technique presented here with the ones of [66] to obtain asymptotics of all unitary SL(2, C) invariants on graphs. Secondly, it would be also interesting to extend our technique to more general settings with time-like or null faces, which involve the other little groups of SL(2, C) and typically use their associated representation basis instead of the canonical basis.…”

Section: Discussionmentioning

confidence: 99%

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Asymptotics of lowest unitary SL(2,C) invariants on graphs

Dona,

Speziale

2020

Preprint

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We describe a technique to study the asymptotics of SL(2, C) invariant tensors associated to graphs, with unitary irreps and lowest SU(2) spins, and apply it to the Lorentzian EPRL-KKL (Engle, Pereira, Rovelli, Livine; Kaminski, Kieselowski, Lewandowski) model of quantum gravity. We reproduce the known asymptotics of the 4-simplex graph with a different perspective on the geometric variables and introduce an algorithm valid for any graph. On general grounds, we find that critical configurations are not just Regge geometries, but a larger set corresponding to conformal twisted geometries. These can be either Euclidean or Lorentzian, and include curved and flat 4d polytopes as subsets. For modular graphs, we show that multiple pairs of critical points exist, and there exist critical configurations of mixed signature, Euclidean and Lorentzian in different subgraphs, with no 4d embedding possible. To Jurek Lewandowski, for his 60th birthday 8 Conclusions 39 A Vectorial representation of SL(2, C) 41 B Orientation conditions imply Euclidean geometry 41 C Evaluating the on-shell action 42 D Spherical and hyperbolic trigonometry 43 E Degrees of freedom of polytopes 43 1 Introduction The spin foam formalism provides transition amplitudes for loop quantum gravity. The state of the art is the Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) model [1, 2], which is based on the group SL(2, C) and its infinite-dimensional, unitary irreducible representations of the principal series. Support in favour of this model comes from the emergence of Regge geometries and the Regge action in the asymptotics of the 4-simplex vertex amplitude for large quantum numbers. This result was obtained by Barrett and collaborators [3], building on previous work [ 4,5,6,7,8], and has been used in a number of applications of the model, e.g. [9,10,11,12,13,14,15,16,17]. The 4-simplex vertex amplitude is sufficient if one restricts attention to spin foams which are dual to triangulations of spacetime, and is the building block for transition amplitudes of 4-valent, simplicial spin network states. But from a canonical perspective, more general vertex amplitudes are to be included in order to provide transition amplitudes to all spin network states, and not just simplicial ones. One such generalization has been proposed by Kaminski, Kieselowski and Lewandwski (KKL) [18], see also [19], and has been applied to cosmological models and studies of spin foam renormalization [20,21,22,23,24,25]. It is however not known what are the dominant geometric configurations and the asymptotic behaviour of the EPRL-KKL Lorentzian amplitude on general vertices. These are the open questions that we answer in this paper. To do so, we introduce a novel technique to study the saddle point approximation of the SL(2, C) amplitudes. The technique and results presented here, while motivated by the quantum gravity applications, are of more general interest for any situation in which asymptotics of unitary SL(2, C) Clebsch-Gordan coefficients are needed. The derivations of 4-simple...

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Asymptotics of lowest unitary SL(2,C) invariants on graphs

Dona,

Speziale

2020

Preprint

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“…The booster functions were first introduced in [31], numerically computed in [12,32], analytically evaluated in terms of complex gamma functions [33,34], and they have an interesting geometrical interpretation in terms of boosted tetrahedra [35]. The booster functions encode how the EPRL model imposes the quantum simplicity constraints and depend on the Immirzi parameter γ.…”

Section: How-to Compute the Eprl Vertex Amplitudesmentioning

confidence: 99%

How-To compute EPRL spin foam amplitudes

Dona,

Frisoni

2022

Preprint

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Spin foam theory is a concrete framework for quantum gravity where numerical calculations of transition amplitudes are possible. Recently, the field became very active, but the entry barrier is steep, mainly because of its unusual language and notions scattered around the literature. This paper is a pedagogical guide to spin foam transition amplitude calculations. We show how to write an EPRL-FK transition amplitude, from the definition of the 2-complex to its numerical implementation using sl2cfoam-next. We guide the reader using an explicit example balancing mathematical rigor with a practical approach. We discuss the approach's advantages and disadvantages and provide a novel look at a recently proposed approximation scheme.

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“…The booster functions were first introduced in [34], numerically computed in [12,35], analytically evaluated in terms of complex gamma functions [36,37], and they have an interesting geometrical interpretation in terms of boosted tetrahedra [38]. The booster functions encode how the EPRL model imposes the quantum simplicity constraints and depend on the Immirzi parameter γ.…”

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confidence: 99%

How-to Compute EPRL Spin Foam Amplitudes

Donà¹,

Frisoni

2022

Universe

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Spin foam theory is a concrete framework for quantum gravity where numerical calculations of transition amplitudes are possible. Recently, the field became very active, but the entry barrier is steep, mainly because of its unusual language and notions scattered around the literature. This paper is a pedagogical guide to spin foam transition amplitude calculations. We show how to write an EPRL-FK transition amplitude, from the definition of the 2-complex to its numerical implementation using sl2cfoam-next. We guide the reader using an explicit example balancing mathematical rigor with a practical approach. We discuss the advantages and disadvantages of our strategy and provide a novel look at a recently proposed approximation scheme.

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Asymptotics of $$\mathrm {SL}(2,{{\mathbb {C}}})$$ coherent invariant tensors

Cited by 13 publications

References 48 publications

Asymptotics of lowest unitary SL(2,C) invariants on graphs

Asymptotics of lowest unitary SL(2,C) invariants on graphs

How-To compute EPRL spin foam amplitudes

How-to Compute EPRL Spin Foam Amplitudes

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