Abstract
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.
Highlights
Our understanding of the nature through the physics has significantly developed since the advent of the quantum theory and its quantization techniques known as conventional quantization methods, which are essentially the canonical and the path integral quantizations
By taking into account the topological and geometrical concepts, a new approach of quantization based on a subclass of quantum states, the coherent states [6,7]
The self-adjoint character is crucial for the observables and von Neumann was the first to put this foward, by proposing a formalism for systematically constructing self-adjoint operators by extensions [8]. After such conventional quantization, the quantum action which leads to the Schrödinger equation has nothing to do with the classical action which provides the Euler-Lagrange equations of movement and one thinks that it’s one of the causes of inadequacy of conventional quantization procedures
Summary
Our understanding of the nature through the physics has significantly developed since the advent of the quantum theory and its quantization techniques known as conventional quantization methods, which are essentially the canonical and the path integral quantizations. By taking into account the topological and geometrical concepts, a new approach of quantization based on a subclass of quantum states, the coherent states [6,7] These quantum states express themselves by means of classical phase space variables and constitute suitable mediators to realize the link between classical and quantum theories. The self-adjoint character is crucial for the observables and von Neumann was the first to put this foward, by proposing a formalism for systematically constructing self-adjoint operators by extensions [8] After such conventional quantization, the quantum action which leads to the Schrödinger equation has nothing to do with the classical action which provides the Euler-Lagrange equations of movement and one thinks that it’s one of the causes of inadequacy of conventional quantization procedures.
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