3d Lorentzian loop quantum gravity and the spinor approach

Abstract

We consider the generalization of the "spinor approach" to the Lorentzian case, in the context of 3d loop quantum gravity with cosmological constant $\Lambda =0$. The key technical tool that allows this generalization is the recoupling theory between unitary infinite-dimensional representations and non-unitary finite-dimensional ones, obtained in the process of generalizing the Wigner-Eckart theorem to SU(1,1). We use SU(1,1) tensor operators to build observables and a solvable quantum Hamiltonian constraint, analogue of the one introduced by V. Bonzom and his collaborators in the Euclidean case (with both $\Lambda =0$ and $\Lambda \ne 0$). We show that the Lorentzian Ponzano-Regge amplitude is solution of the quantum Hamiltonian constraint by recovering the Biedenharn-Elliott relation (generalized to the case where unitary and non-unitary SU(1,1) representations are coupled to each other). Our formalism is sufficiently general that both the Lorentzian and the Euclidean case can be recovered (with $\Lambda =0$).

Fixed typos. 28 pages, 3 figures. To appear in Phys. Rev. D

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Cited by in corpus (9)

• Quasi-local holographic dualities in non-perturbative 3d quantum gravity I - Convergence of multiple approaches and examples of Ponzano-Regge statistical duals
• Quantum geometry from higher gauge theory
• Timelike twisted geometries
• SO*(2N) coherent states for loop quantum gravity
• Deformations of Lorentzian Polyhedra: Kapovich-Millson phase space and SU(1,1) Intertwiners
• The AdS^2_θ/CFT_1 Correspondence and Noncommutative Geometry I: A QM/NCG Correspondence
• 3D Quantum Gravity from Holomorphic Blocks
• The AdS^2_θ/CFT_1 Correspondence and Noncommutative Geometry II: Noncommutative Quantum Black Holes
• Wigner-Eckart theorem and Jordan-Schwinger representation for infinite-dimensional representations of the Lorentz group