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arXiv:1206.1180 · Physica Scripta

Enhanced quantization on the circle

Joseph Ben Geloun, John R. Klauder

Published Feb 15, 2013• 6 citations
arXivHTMLPDF

Abstract

We apply the quantization scheme introduced in Klauder 2012 (arXiv:1204.2870) to a particle on a circle. We find that the quantum action functional restricted to appropriate coherent states can be expressed as the classical action plus ℏ-corrections. We succeed in proving this for a Hamiltonian endowed with a general periodic potential. This result extends the examples presented in the cited paper for a system on a nontrivial configuration space topology.

DOI
10.1088/0031-8949/87/03/035006

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We apply the quantization scheme introduced in Klauder 2012 (arXiv:1204.2870) to a particle on a circle. We find that the quantum action functional restricted to appropriate coherent states can be expressed as the classical action plus ℏ-corrections. We succeed in proving this for a Hamiltonian endowed with a general periodic potential. This result extends the examples presented in the cited paper for a system on a nontrivial configuration space topology.
It is well known that the representations over an arbitrary configuration space related to a physical system of the Heisenberg algebra allow to distinguish the simply and non simply-connected manifolds [arXiv:quant-ph/9908.014, arXiv:hep-th/0608.023]. In the light of this classification, the dynamics of a quantum particle on the line is studied in the framework of the conventional quantization scheme as well as that of the enhanced quantization recently introduced by J. R. Klauder [arXiv:quant-ph/1204.2870]. The quantum action functional restricted to the phase space coherent states is obtained from the enhanced quantization procedure, showing the coexistence of classical and quantum theories, a fundamental advantage offered by this new approach. The example of the one dimensional harmonic oscillator is given. Next, the spectrum of a free particle on the two-sphere is recognized from the covariant diffeomorphic representations of the momentum operator in the configuration space. Our results based on simple models also point out the already-known link between interaction and topology at quantum level.
Enhanced quantization offers a different classical/quantum connection than that of canonical quantization in which ℏ > 0 throughout. This result arises when the only allowed Hilbert space vectors allowed in the quantum action functional are coherent states, which leads to the classical action functional augmented by additional terms of order ℏ. Canonical coherent states are defined by unitary transformations of a fixed, fiducial vector. While Gaussian vectors are commonly used as fiducial vectors, they cannot be used for all systems. We focus on choosing fiducial vectors for several systems including bosons, fermions and anyons.
Enhanced quantization offers a different classical/quantum connection than that of canonical quantization in which > 0 throughout.This result arises when the only allowed Hilbert space vectors allowed in the quantum action functional are coherent states, which leads to the classical action functional augmented by additional terms of order .Canonical coherent states are defined by unitary transformations of a fixed, fiducial vector.While Gaussian vectors are commonly used as fiducial vectors, they cannot be used for all systems.We focus on choosing fiducial vectors for several systems including bosons, fermions, and anyons.
Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.