Band structure in the polymer quantization of the harmonic oscillator

@article{JFernandoBarbero2013BandSI,
  title={Band structure in the polymer quantization of the harmonic oscillator},
  author={G. J.FernandoBarbero and Jorge Prieto and Eduardo J S Villase{\~n}or},
  journal={Classical and Quantum Gravity},
  year={2013},
  volume={30},
  pages={165011}
}
We discuss the detailed structure of the spectrum of the Hamiltonian for the polymerized harmonic oscillator and compare it with the spectrum in the standard quantization. As we will see the non-separability of the Hilbert space implies that the point spectrum consists of bands similar to the ones appearing in the treatment of periodic potentials. This feature of the spectrum of the polymeric harmonic oscillator may be relevant for the discussion of the polymer quantization of the scalar field… Expand

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References

SHOWING 1-10 OF 38 REFERENCES
The quantum harmonic oscillator on a lattice
The authors find the eigenvalue spectrum of a particle constrained to 'hop' between the sites of a simple cubic lattice in the presence of a spherically asymmetric 'harmonic oscillator' potential.Expand
Statistical Thermodynamics of Polymer Quantum Systems
Polymer quantum systems are mechanical models quantized similarly as loop quantum gravity. It is actually in quantizing gravity that the polymer term holds proper as the quantum geometry excitationsExpand
Hamiltonian and physical Hilbert space in polymer quantum mechanics
In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally testedExpand
Polymer Quantum Mechanics and its Continuum Limit
A rather nonstandard quantum representation of the canonical commutation relations of quantum mechanics systems, known as the polymer representation, has gained some attention in recent years, due toExpand
The thermodynamic limit and black hole entropy in the area ensemble
We discuss the thermodynamic limit in the canonical area ensemble used in loop quantum gravity to model quantum black holes. The computation of the thermodynamic limit is the rigorous way to obtain aExpand
Polymer quantization of the free scalar field and its classical limit
Building on prior work, a generally covariant reformulation of a free scalar field theory on the flat Lorentzian cylinder is quantized using loop quantum gravity (LQG)-type 'polymer' representations.Expand
On the choice of time in the continuum limit of polymeric effective theories
In polymeric quantum theories, a natural question pertains to the so called continuum limit, corresponding to the limit where the 'discreteness parameter' λ approaches zero. In particular one mightExpand
Propagator in polymer quantum field theory
We study free scalar field theory on flat spacetime using a background independent (polymer) quantization procedure. Specifically we compute the propagator using a method that takes the energyExpand
Background independent quantizations: the scalar field I
We are concerned with the issue of quantization of a scalar field in a diffeomorphism invariant manner. We apply the method used in Loop Quantum Gravity. It relies on the specific choice of scalarExpand
Background independent quantizations : the scalar field II
We are concerned with the issue of quantization of a scalar field in a diffeomorphism invariant manner. We apply the method used in Loop Quantum Gravity. It relies on the specific choice of scalarExpand
...
1
2
3
4
...