On Dynamical Quantization (original) (raw)
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In sections of the literature it is assumed, following early comments by Dirac, that a desirable quantization operation should preserve the Possion bracket (rather than merely agree in the limit h → 0). A celebrated mathematical theorem establishes that this is impossible; hence a consistent quantization is often considered to be unattainable. Here we question whether the premise of exact correspondence is sensible from a physical viewpoint. In particular, we give an exact quantum mechanical version of classical dynamics in which the commutator of any two quantized operators is proportional to the Poisson bracket of the corresponding functions (in a manner which subtly evades the aforementioned theorem). We then relate this novel dynamical system to the standard quantum theory, and identify the bracket structure for the case of symmetric ordering. Our conclusion is that a consistent quantization should not be expected to preserve the Poisson bracket.
Dynamical aspects in the quantizer–dequantizer formalism
Annals of Physics, 2017
The use of the quantizer-dequantizer formalism to describe the evolution of a quantum system is reconsidered. We show that it is possible to embed a manifold in the space of quantum states of a given auxiliary system by means of an appropriate quantizerdequantizer system. If this manifold of states is invariant with respect to some unitary evolution, the quantizer-dequantizer system provides a classical-like realization of such dynamics, which in general is non linear. Integrability properties are also discussed. Weyl systems and generalized coherente states are used as a simple illustration of these ideas.
Quantum Mechanics as a Classical Theory VIII: Second Quantization
2008
We continue in this paper our program of rederiving all quantum mechanical formalism from the classical one. We now turn our attention to the derivation of the second quantized equations, both for integral and half-integral spins. We then show that all the quantum results may be derived using our approach and also show the interpretation suggested by this derivation. This paper may be considered as a first approach to the study of the quantum field theory beginning by the same classical ideas we are supporting since the first paper of this series. 1
On Quantization, the Generalized Schrödinger Equation and Classical Mechanics
UM-P-91/47, 1991
Using a new state-dependent, λ-deformable, linear functional operator, Q λ ψ , which presents a natural C ∞ deformation of quantization, we obtain a uniquely selected non-linear, integro-differential Generalized Schrödinger equation. The case Q 1 ψ reproduces linear quantum mechanics, whereas Q 0 ψ admits an exact dynamic, energetic and measurement theoretic reproduction of classical mechanics. All solutions to the resulting classical wave equation are given and we show that functionally chaotic dynamics exists.
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.
A Classical Explanation of Quantization
Foundations of Physics, 2011
In the context of our recently developed emergent quantum mechanics, and, in particular, based on an assumed sub-quantum thermodynamics, the necessity of energy quantization as originally postulated by Max Planck is explained by means of purely classical physics. Moreover, under the same premises, also the energy spectrum of the quantum mechanical harmonic oscillator is derived. Essentially, Planck’s constant h is shown to be indicative of a particle’s “zitterbewegung” and thus of a fundamental angular momentum. The latter is identified with quantum mechanical spin, a residue of which is thus present even in the non-relativistic Schrödinger theory.
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);
Some reflections on mathematicians’ views of quantization
Journal of Mathematical Sciences, 2007
We start with a short presentation of the difference in attitude between mathematicians and physicists even in their treatment of physical reality, and look at the paradigm of quantization as an illustration. In particular, we stress the differences in motivation and realization between the Berezin and deformation quantization approaches, exposing briefly Berezin’s view of quantization as a functor. We continue with a schematic overview of deformation quantization and of its developments in contrast with the latter and discuss related issues, in particular, the spectrality question. We end by a very short survey of two main avatars of deformation quantization, quantum groups and quantum spaces (especially noncommutative geometry) presented in that perspective. Bibliography: 74 titles.
Quantum Mechanics As A Classical Theory I: Non-relativistic Theory, quant-ph/9503020
2012
The objective of this series of three papers is to axiomatically derive quantum mechanics from classical mechanics and two other basic axioms. In this first paper, Schroendiger’s equation for the density matrix is fist obtained and from it Schroedinger’s equation for the wave functions is derived. The momentum and position operators acting upon the density matrix are defined and it is then demonstrated that they commute. Pauli’s equation for the density matrix is also obtained. A statistical potential formally identical to the quantum potential of Bohm’s hidden variable theory is introduced, and this quantum potential is reinterpreted through the formalism here proposed. It is shown that, for dispersion free ensembles, Schroedinger’s equation for the density matrix is equivalent to Newton’s equations. A general non-ambiguous procedure for the construction of operators which act upon the density matrix is presented. It is also shown how these operators can be reduced to those which a...
Time and energy operators in the canonical quantization of special relativity
European Journal of Physics, 2020
Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) is shown to provide a unified origin for: i) the complex vector space formulation of Quantum Mechanics (QM); ii) the momentum and space commutation relations and the corresponding representations; iii) the Dirac Hamiltonian in the formulation of Relativistic Quantum Mechanics (RQM); iv) the existence of a self adjoint Time Operator that circumvents Pauli's objection.