The geometry of quantum spin networks

Borissov, Roumen; Major, Seth; Smolin, Lee

doi:10.1088/0264-9381/13/12/009

Cited by 58 publications

(50 citation statements)

References 49 publications

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“…In accordance with our last observation, a minimal length is introduced in most models (see, e.g., the Double Special Relativity 11,12 or spin-networks 13 ). The length is handled as a vector, subject to Lorentz contraction.…”

Section: Introductionsupporting

confidence: 55%

Pseudo-Complex Field Theory

Hess

Greiner

2007

Int. J. Mod. Phys. E

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A new formulation of field theory is presented, based on a pseudo-complex description. An extended group structure is introduced, implying a minimal scalar length, rendering the theory regularized a la Pauli-Villars. Cross sections are calculated for the scattering of an electron at an external Coulomb field and the Compton scattering. Deviations due to a smallest scalar length are determined. The theory also permits a modification of the minimal coupling scheme, resulting in a generalized dispersion relation. A shift of the Greisen-Zatsepin-Kuzmin (GZK) limit of the cosmic ray spectrum is the consequence.Pseudo-Complex Field Theory 1645 energy. The GZK cutoff gives us the opportunity of investigating such high energy events. If not observed, at least we can obtain an upper limit on the smallest length scale l.Consequently, the change in the dispersion relation is visible only at high energies, comparable to the GZK scale. At low energies, the dispersion relation is to very high approximation maintained. One may ask, however, if the smallest length l may also produce deviations at intermediate energies, for example, in the TeV range, accessible to experiment now. In order to be measurable, we look for differences in the cross section of a particular reaction, of the lowest power in l possible.The advantage of the proposed extended field theory is obvious: All symmetries are maintained and thus it permits the calculation of cross sections as we are used to. Still, an invariant length scale appears, rendering the theory regularized and reflecting the deviation of the space-time structure at distances of the order of the Planck length.The main objective of this paper is to formulate the pseudo-complex extension of the standard field theory (SFT). For the extension we propose the name Pseudo-Complex Field Theory (PCFT). First results are reported in Ref. 20.The structure of the paper is as follows: In Sec. 2 the pseudo-complex numbers are introduced and it is shown how to perform calculations, like differentiation and integration. This section serves as a quick reference guide to the reader unfamiliar with the concept of pseudo-complex numbers. In Sec. 3 the pseudo-complex Lorentz and Poincaré groups are discussed. The representations of the pseudocomplex Poincaré group are indicated. Section 4 introduces a modified variational procedure, required in order to obtain a new theory and not two separated old ones. The language is still classical. As examples, scalar and Dirac fields are discussed and an extraction procedure, on how to obtain physical observables, is constructed and at the end formally presented. Section 5 is dedicated to the symmetry properties of the PCFT. Finally, in Sec. 6 the quantization formalism is proposed. In Sec. 7 a couple of cross sections are calculated within the PCFT: (i) the dispersion of a charged particle at a Coulomb field and (ii) the Compton scattering. One could also consider high precision measurements, like the Lamb shift and the magnetic moment of the electron. These, however, re...

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Section: Introductionsupporting

confidence: 55%

Pseudo-Complex Field Theory

Hess

Greiner

2007

Int. J. Mod. Phys. E

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“…The kinematical state space consists of diffeomorphism classes of spin networks [11,12]. These are endowed with a geometrical interpretation by the fact that the spin-network basis makes possible the diagonalization of the operators that correspond to three dimensional geometrical quantities, such as area [8,12,14], volume [8,12,13,15,16,17,18,19] and length [20]. The spectra of all of these observables are discrete, which gives rise to a picture in which quantum geometry is discrete and combinatorial.…”

Section: Introductionmentioning

confidence: 99%

Causal evolution of spin networks

Markopoulou

Smolin

1997

Nuclear Physics B

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A new approach to quantum gravity is described which joins the loop representation formulation of the canonical theory to the causal set formulation of the path integral. The theory assigns quantum amplitudes to special classes of causal sets, which consist of spin networks representing quantum states of the gravitational field joined together by labeled null edges. The theory exists in 3+1, 2+1 and 1+1 dimensional versions, and may also be interepreted as a theory of labeled timelike surfaces. The dynamics is specified by a choice of functions of the labelings of d + 1 dimensional simplices,which represent elementary future light cones of events in these discrete spacetimes. The quantum dynamics thus respects the discrete causal structure of the causal sets. In the 1 + 1 dimensional case the theory is closely related to directed percolation models. In this case, at least, the theory may have critical behavior associated with percolation, leading to the existence of a classical limit.

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“…Not much work has been done in this context: LQG with a quantum group has only been explored using the loop variables by Major and Smolin [24][25][26].…”

Section: Chern-simonsmentioning

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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The geometry of quantum spin networks

Cited by 58 publications

References 49 publications

Pseudo-Complex Field Theory

Pseudo-Complex Field Theory

Causal evolution of spin networks

Observables in loop quantum gravity with a cosmological constant

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