Integral Quantization for the Discrete Cylinder
Abstract
:1. Introduction
- (i)
- What is the appropriate quantum position operator?
- (ii)
- What is the appropriate quantum momentum operator?
- (iii)
- What are the states (coherent states) that saturate some uncertainty relations?
- (iv)
- Do these operators explicitly display the topology of the circle or the fact that the circle is a curved manifold or a group?
2. (Covariant) Integral Quantization: A Survey
3. Overview: Weyl Operator Acting on
- It is an isometry:
- It can be inverted on its range:
- Its range is a reproducing kernel space:
- Periodized ( normalized ) gaussian kernel and theta function.A number of authors ([9,12]) have used variants of this function as a fiducial vector for coherent states on the circle.The corresponding reproducing kernel for is:
- Periodized ( normalized ) Poisson kernel.
- Dirichlet fiducial vector ( normalized ).The corresponding reproducing kernel is:
- Fejér fiducial vector.
- Von Mises fiducial vector ( normalized)
4. Quantization Operators and the Quantization Map
- (i)
- The operator is the integral operator:where the kernel is given by:Here, is the inverse discrete Fourier transform of ϖ with respect to the first variable.
- (ii)
- The operator is symmetric if—and only if—the weight satisfies:
- (iii)
- The operator is trace class, and its trace is given by
- (i)
- The action of on is given by:
- (ii)
- The condition that be symmetric implies the following condition on the kernel:
- (iii)
- Therefore, the trace of corresponds the integral of the kernel over its diagonal; that is:
5. Covariant Affine Integral Quantization from Weight Function
5.1. General Results
5.2. Quantization of Separable Functions
5.3. Quantization of Functions of Momentum Only
- Angular momentum , through integrating parts and appropriate derivability properties of the weight function,Hence, with a weight function obeying , we retrieve the usual angular momentum operator .
- Square angular momentum . Using similar methods and assumptions on the weight function, we findFinally,
5.4. Quantization of Function of Angular Position Only
6. Semi-Classical Portraits
7. Quantization with Various Weights
7.1. Weight Related to Coherent States for
7.2. Weight Related to the Angular Parity Operator
- , ,
- , ,
- ,
8. Conclusions
- With the parity weight, the quantization of the momentum is the expected angular momentum operator L.This is, of course, unacceptable. Alternatively, one can quantize the periodized angle function . One finds the multiplication operator defined by the same discontinuous . There is no regularization.
- With the coherent state weight, one obtains the quantization of the momentum as the usual L plus an additional term, i.e., a kind of covariant derivative on the circle whose topology is now taken into account,
- Analyzing circular data (see, for instance, [28,29,30]). We expect that the formalism we have developed above will be useful for circular data or circular statistics. Some of the fiducial vectors we have considered here are probability densities in these areas, namely the uniform distribution, the shifted gaussian, and the Von Mises, Fejér and Poisson.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
POVM | Positive operator-valued measure |
UIR | Unitary irreducible representation |
CS | Coherent state |
Appendix A. Some Fiducial Vectors & Reproducing Kernels
Fiducial Vector | Reproducing Kernel | |
General | ||
Constant | ||
Basis | ||
Shifted gaussian | ||
Dirichlet | ||
Fejer | ||
Von Mises |
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Gazeau, J.-P.; Murenzi, R. Integral Quantization for the Discrete Cylinder. Quantum Rep. 2022, 4, 362-379. https://doi.org/10.3390/quantum4040026
Gazeau J-P, Murenzi R. Integral Quantization for the Discrete Cylinder. Quantum Reports. 2022; 4(4):362-379. https://doi.org/10.3390/quantum4040026
Chicago/Turabian StyleGazeau, Jean-Pierre, and Romain Murenzi. 2022. "Integral Quantization for the Discrete Cylinder" Quantum Reports 4, no. 4: 362-379. https://doi.org/10.3390/quantum4040026