Elementary formulas for a hyperbolic tetrahedron

Mednykh, Alexander; Pashkevich, M. G.

doi:10.1007/s11202-006-0079-5

Cited by 10 publications

(11 citation statements)

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“…Upon the substitution z := z ± , we see that a 2 , a 3 , a 4 , a 5 , a 6 , z ± ) = 0, by taking the respective derivative on both sides of the identity e z ± ∂U ∂z (a 1 ,a 2 ,a 3 ,a 4 ,a 5 ,a 6 ,z ± ) = 1, c.f. the definition of z ± and formula (6). Finally, we get The real part of the above expression is…”

Section: Lemmamentioning

confidence: 94%

“…Let G denote the Gram matrix for the normals G = n i , n j 4 i,j=1 . The conditions under which G describes a generalised mildly truncated tetrahedron are given by [6] and [13]. For a prism truncated tetrahedron, its existence and geometry are determined by the vectorsñ i , i = 1, 6, wherẽ n i = n i for i = 1, 4 andñ 5 = v 1 ,ñ 6 = v 2 .…”

Section: Preliminariesmentioning

confidence: 99%

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Volume of a doubly truncated hyperbolic tetrahedron

Kolpakov

Murakami

2012

Aequat. Math.

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Section: Lemmamentioning

confidence: 94%

Section: Preliminariesmentioning

confidence: 99%

Volume of a doubly truncated hyperbolic tetrahedron

Kolpakov

Murakami

2012

Aequat. Math.

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“…By using these equations and after straightforward calculations , we obtain (iii). The different proof of this theorem is given in [26]. As an immediate fundamental consequence of this Theorem which will be used in our future calculations, we have the following result.…”

Section: Edge Matrix For Hyperbolic Tetrahedronmentioning

confidence: 74%

On the Schlafli differential formula based on edge lengths of tetrahedron in H 3 and S 3

2008

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“…Indeed, the area of an hyperbolic triangle is given in terms of the triangle angles (97). There are various ways to express functions of the area in terms of the edge lengths [52]. A convenient one will be…”

Section: Rmentioning

confidence: 99%

Observables in loop quantum gravity with a cosmological constant

Dupuis¹,

Girelli

2014

Phys. Rev. D

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An open issue in loop quantum gravity (LQG) is the introduction of a non-vanishing cosmological constant Λ. In 3d, Chern-Simons theory provides some guiding lines: Λ appears in the quantum deformation of the gauge group. The Turaev-Viro model, which is an example of spin foam model is also defined in terms of a quantum group. By extension, it is believed that in 4d, a quantum group structure could encode the presence of Λ = 0. In this article, we introduce by hand the quantum group Uq(su(2)) into the LQG framework, that is we deal with Uq(su(2))-spin networks. We explore some of the consequences, focusing in particular on the structure of the observables. Our fundamental tools are tensor operators for Uq(su(2)). We review their properties and give an explicit realization of the spinorial and vectorial ones. We construct the generalization of the U(N ) formalism in this deformed case, which is given by the quantum group Uq(u(n)). We are then able to build geometrical observables, such as the length, area or angle operators ... We show that these operators characterize a quantum discrete hyperbolic geometry in the 3d LQG case. Our results confirm that the use of quantum group in LQG can be a tool to introduce a non-zero cosmological constant into the theory. ContentsIntroduction 2 I. U q (su(2)) in a nutshell 4 A. Definition of U q (su(2)) 4 B. q-harmonic oscillators and the Schwinger-Jordan trick 7II. Tensor operators for U q (su(2)) 7 A. Definition and Wigner-Eckart theorem 7 B. Product of tensor operators: scalar product, vector product and triple product 8 1. Scalar product 9 2. Vector product 10 C. Tensor products of tensor operators 10 III. Realization of tensor operators of rank 1/2 and 1 for U q (su(2)) 11 A. Rank 1/2 tensor operators 11 B. Rank 1 tensor operators 13IV. Observables for the intertwiner space 16 A. General construction and properties of intertwiner observables 16 B. U q (u(n)) formalism for LQG defined over U q (su (2)) 18 [6,7]. Indeed, in a 3d space-time, one can rewrite General Relativity with a (possibly zero) cosmological constant as a Chern-Simons gauge theory 1 . The general phase space structure of the theory for any metric signature and sign of Λ can be treated in a nice unified way [8], using Poisson-Lie groups [9], the classical counterparts of quantum groups. The quantization procedure leads explicitly to a quantum group structure. The full construction, from phase space to quantum group is usually called combinatorial quantization [5][6][7]. We can also quantize 3d gravity using the spinfoam approach. In this approach, 3d gravity is formulated as a BF theory. When Λ = 0, this is the well-known Ponzano-Regge model (both Euclidian or Lorentzian), based on the irreducible unitary representations of the relevant gauge group. When Λ = 0, the quantum group structure is introduced by hand. The Ponzano-Regge model is deformed, using irreducible unitary representations of the relevant quantum deformation of the gauge group. This is then called the Turaev-Viro model [10]. The argument co...

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Elementary formulas for a hyperbolic tetrahedron

Abstract: We derive some elementary formulas expressing the relation between the dihedral angles and edge lengths of a tetrahedron in hyperbolic space.

Cited by 10 publications

References 10 publications

Volume of a doubly truncated hyperbolic tetrahedron

Volume of a doubly truncated hyperbolic tetrahedron

On the Schlafli differential formula based on edge lengths of tetrahedron in H 3 and S 3

Observables in loop quantum gravity with a cosmological constant

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