Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials

@article{Aquilanti2013HamiltonianDO,
  title={Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials},
  author={Vincenzo Aquilanti and Dimitri Marinelli and Annalisa Marzuoli},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2013},
  volume={46}
}
The action of the quantum mechanical volume operator, introduced in connection with a symmetric representation of the three-body problem and recently recognized to play a fundamental role in discretized quantum gravity models, can be given as a second-order difference equation which, by a complex phase change, we turn into a discrete Schrödinger-like equation. The introduction of discrete potential-like functions reveals the surprising crucial role here of hidden symmetries, first discovered by… 
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61 References

Discrete quantum mechanics

A comprehensive review of the discrete quantum mechanics with the pure imaginary shifts and the real shifts is presented in parallel with the corresponding results in the ordinary quantum mechanics.…

Discreteness of area and volume in quantum gravity [Nucl. Phys. B 442 (1995) 593]

Simplification of the spectral analysis of the volume operator in loop quantum gravity

The volume operator plays a crucial role in the definition of the quantum dynamics of loop quantum gravity (LQG). Efficient calculations for dynamical problems of LQG can therefore be performed only…

Quantum states of elementary three-geometry

We introduce a quantum volume operator K—which could play a significant role in discretized quantum gravity models—by taking into account a symmetrical coupling scheme of three SU(2) angular momenta.…

Properties of the volume operator in loop quantum gravity: I. Results

We analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in loop quantum gravity, which is the quantum analog of the classical volume expression for regions in…

Properties of the volume operator in loop quantum gravity: II. Detailed presentation

The properties of the volume operator in loop quantum gravity, as constructed by Ashtekar and Lewandowski, are analyzed for the first time at generic vertices of valence greater than four. We find…

Quantum and semiclassical spin networks: from atomic and molecular physics to quantum computing and gravity

The mathematical apparatus of quantum-mechanical angular momentum (re)coupling, developed originally to describe spectroscopic phenomena in atomic, molecular, optical and nuclear physics, is embedded…

Exact and Asymptotic Computations of Elementary Spin Networks: Classification of the Quantum-Classical Boundaries

This paper provides a contribution to the understanding of the transition between two extreme modes of the 6j, corresponding to the nearly classical and the fully quantum regimes, by studying the boundary lines (caustics) in the plane of the two matrix labels.

Combinatorics of angular momentum recoupling theory: spin networks, their asymptotics and applications

The quantum theory of angular momentum and the associated Racah–Wigner algebra of the Lie group SU(2) have been widely used in many branches of theoretical and applied physics, chemical physics, and…

Volume operator in discretized quantum gravity.

    Loll
    Physics
    Physical review letters
  • 1995
The spectral properties of the volume operator in quantum gravity are investigated in the framework of a previously introduced lattice discretization and the presence of a well-defined scalar product permits to make definite statements about the hermiticity of quantum operators.
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