Journal ArticleDOI
Quantization as a consequence of the symmetry group: An approach to geometric quantization
TLDR
In this article, a method is proposed to obtain the dynamics of a system which only makes use of the group law. But the method is applied to the free-particle dynamics and the harmonic oscillator.Abstract:
A method is proposed to obtain the dynamics of a system which only makes use of the group law. It incorporates many features of the traditional geometric quantization program as well as the possibility of obtaining the classical dynamics: The classical or quantum character of the theory is related to the choice of the group, avoiding thus the need of quantizing preexisting classical systems and providing a group connection between the quantum and classical systems, i.e., the classical limit. The method is applied to the free‐particle dynamics and the harmonic oscillator.read more
Citations
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Journal ArticleDOI
Obstruction Results in Quantization Theory
TL;DR: In this article, the authors conjecture a generalized Groenewold-Van Hove theorem and determine the maximal subalgebras of observables which can be consistently quantized.
Journal ArticleDOI
Algebraic quantization on a group and nonabelian constraints
TL;DR: In this article, a generalization of a previous group manifold quantization formalism is proposed, in which discrete transformations in the group are allowed, and a nonabelian group replaces the ordinary (central)U(1) subgroup of the Heisenberg-Weyl-like quantum group.
Journal ArticleDOI
Cohomology, central extensions, and (dynamical) groups
V. Aldaya,J. A. de Azcárraga +1 more
TL;DR: In this paper, the authors analyzed the process of group contraction which allows the transition from the Einstenian quantum dynamics to the Galilean one in terms of the cohomology of the Poincare and Galilei groups.
Book ChapterDOI
Obstructions to Quantization
TL;DR: In this paper, it has been shown that there are no obstructions to quantizing T 2 or T*R + for any polynomial observable on the phase space R 2n in a physically meaningful way.
Journal ArticleDOI
The quantum Arnold transformation
Victor Aldaya,Francisco Cossio,Julio Becerra Guerrero,Julio Becerra Guerrero,Francisco F. López-Ruiz +4 more
TL;DR: Using a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are nonhomogeneous linear second-order ordinary differential equations, including systems with friction linear in velocity, can be related to the quantum free-particle dynamical system as mentioned in this paper.
References
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Book
Foundations of mechanics
TL;DR: In this article, Ratiu and Cushman introduce differential theory calculus on manifolds and derive an overview of qualitative and topological properties of differentiable properties of topological dynamics.
Book
The theory of groups
TL;DR: The theory of normal subgroups and homomorphisms was introduced in this article, along with the theory of $p$-groups regular $p-groups and their relation to abelian groups.