Dynamical aspects in the quantizer–dequantizer formalism (original) (raw)
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Eprint Arxiv Quant Ph 0603246, 2006
In this article we review some results obtained from a generalization of quantum mechanics obtained from modification of the canonical commutation relation [q, p] = ih. We present some new results concerning relativistic generalizations of previous works, and we calculate the energy spectrum of some simple quantum systems, using the position and momentum operators of this new formalism.
Coherent States and Coordinate-Free Quantization
Zeitschrift Fur Naturforschung a a Journal of Physical Sciences, 1997
The usual quantization procedures interpret canonical transformations in an active way linking them with unitary transformations, while the quantization procedure offered by coherent states completely separates classical canonical transformations and unitary operator transformations. By exploiting this property, along with a physically motivated shadow metric, it is seen how to realize the quantization process in as coordinate-free a form as holds in classical mechanics.
Graduate Texts in Physics, 2013
We propose a generalization of Heisenberg picture quantum mechanics in which a Lagrangian and Hamiltonian dynamics is formulated directly for dynamical systems on a manifold with non-commuting coordinates, which act as operators on an underlying Hilbert space. This is accomplished by defining the Lagrangian and Hamiltonian as the real part of a graded total trace over the underlying Hilbert space, permitting a consistent definition of the first variational derivative with respect to a general operator-valued coordinate. The Hamiltonian form of the equations is expressed in terms of a generalized bracket operation, which is conjectured to obey a Jacobi identity. The formalism permits the natural implementation of gauge invariance under operator-valued gauge transformations. When an operator Hamiltonian exists as well as a total trace Hamiltonian, as is generally the case in complex quantum mechanics, one can make an operator gauge transformation from the Heisenberg to the Schrödinger picture. When applied to complex quantum mechanical systems with one bosonic or fermionic degree of freedom, the formalism gives the usual operator equations of motion, with the canonical commutation relations emerging as constraints associated with the operator gauge invariance. More generally, our methods permit the formulation of quaternionic quantum field theories with operator-valued gauge invariance, in which we conjecture that the operator constraints act as a generalization of the usual canonical commutators.
Quantum mathematics: Backgrounds and some applications to nonlinear dynamical systems
Nonlinear Oscillations, 2008
UDC 517.9 The backgrounds of quantum mathematics, a new discipline in mathematical physics, are discussed and analyzed from both historical and analytical points of view. The magic properties of the second quantization method, invented by Fock in 1934, are demonstrated, and an impressive application to the theory of nonlinear dynamical systems is considered. Von Neumann first applied the spectral theory of self-adjoint operators on Hilbert spaces to explain the radiation spectra of atoms and the related matter stability [2] (1926); Fock was the first to introduce the notion of many-particle Hilbert space, named a Fock space, and introduced related creation and annihilation operators acting on it [3] (1932); Weyl understood the fundamental role of the notion of symmetry in physics and developed a physics-oriented group theory; moreover, he showed the importance of different representations of classical matrix groups for physics and studied unitary representations of the Heisenberg-Weyl group related to creation and annihilation operators on a Fock space [4] (1931). At the end of the 20th century, new developments were due to Faddeev with co-workers (quantum inverse spectral theory transform [5], 1978); Drinfeld, Donaldson, and Witten (quantum groups and algebras, quantum topology, and quantum superanalysis [6-8], 1982-1994);
Quantum Mechanics as a Classical Theory X: Quantization in Generalized Coordinates
arXiv: Quantum Physics, 1996
In this tenth paper of the series we aim at showing that our formalism, using the Wigner-Moyal Infinitesimal Transformation together with classical mechanics, endows us with the ways to quantize a system in any coordinate representation we wish. This result is necessary if one even think about making general relativistic extensions of the quantum formalism. Besides, physics shall not be dependent on the specific representation we use and this result is necessary to make quantum theory consistent and complete.
Conventional and Enhanced Canonical Quantizations, Application to Some Simple Manifolds
Journal of Modern Physics, 2013
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.
Quantization of non-standard Hamiltonian systems
Journal of Physics A: Mathematical and General, 1997
The quantization of classical theories that admit more than one Hamiltonian description is considered. This is done from a geometrical viewpoint, both at the quantization level (geometric quantization) and at the level of the dynamics of the quantum theory. A spin-1/2 system is taken as an example in which all the steps can be completed. It is shown that the geometry of the quantum theory imposes restrictions on the physically allowed nonstandard quantum theories.
Alternative Linear Structures for Classical and Quantum Systems
International Journal of Modern Physics A, 2007
The possibility of deforming the (associative or Lie) product to obtain alternative descriptions for a given classical or quantum system has been considered in many papers. Here we discuss the possibility of obtaining some novel alternative descriptions by changing the linear structure instead. In particular we show how it is possible to construct alternative linear structures on the tangent bundle TQ of some classical configuration space Q that can be considered as "adapted" to the given dynamical system. This fact opens the possibility to use the Weyl scheme to quantize the system in different nonequivalent ways, "evading," so to speak, the von Neumann uniqueness theorem.