Coherent States and Coordinate-Free Quantization (original) (raw)
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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.
Journal of Mathematical Physics, 2000
Affine variables, which have the virtue of preserving the positivedefinite character of matrix-like objects, have been suggested as replacements for the canonical variables of standard quantization schemes, especially in the context of quantum gravity. We develop the kinematics of such variables, discussing suitable coherent states, their associated resolution of unity, polarizations, and finally the realization of the coherent-state overlap function in terms of suitable path-integral formulations.
Lecture Notes in Physics, 1999
Canonical quantization may be approached from several different starting points. The usual approaches involve promotion of c-numbers to q-numbers, or path integral constructs, each of which generally succeeds only in Cartesian coordinates. All quantization schemes that lead to Hilbert space vectors and Weyl operators-even those that eschew Cartesian coordinates-implicitly contain a metric on a flat phase space. This feature is demonstrated by studying the classical and quantum "aggregations", namely, the set of all facts and properties resident in all classical and quantum theories, respectively. Metrical quantization is an approach that elevates the flat phase space metric inherent in any canonical quantization to the level of a postulate. Far from being an unwanted structure, the flat phase space metric carries essential physical information. It is shown how the metric, when employed within a continuous-time regularization scheme, gives rise to an unambiguous quantization procedure that automatically leads to a canonical coherent state representation. Although attention in this paper is confined to canonical quantization we note that alternative, nonflat metrics may also be used, and they generally give rise to qualitatively different, noncanonical quantization schemes.
Coherent states and related quantizations for unbounded motions
Journal of Physics A: Mathematical and Theoretical, 2012
We build coherent states (CS) for unbounded motions along two different procedures. In the first one we adapt the Malkin-Manko construction for quadratic Hamiltonians to the motion of a particle in a linear potential. A generalization to arbitrary potentials is discussed. The second one extends to continuous spectrum previous constructions of action-angle coherent states in view of a consistent energy quantization.
To what extent are canonical and coherent state quantizations physically equivalent?
We investigate the consistency of coherent state (or Berezin-Klauder-Toeplitz, or anti-Wick) quantization in regard to physical observations in the non- relativistic (or Galilean) regime. We compare this procedure with the canonical quantization (on both mathematical and physical levels) and examine whether they are or not equivalent in their predictions: is it possible to dif- ferentiate them on a strictly physical level? As far as only usual dynamical observables (position, momentum, energy, ...) are concerned, the quantization through coherent states is proved to be a perfectly valid alternative. We successfully put to the test the validity of CS quantization in the case of data obtained from vibrational spectroscopy (data that allowed to validate canonical quantization in the early period of Quantum Mechanics).
On the quantization of mechanical systems
arXiv (Cornell University), 2017
We show what seems to be the key for quantization of classical systems. Given a manifold M , each riemannian metric (nondegenerate, of arbitrary signature) canonically determines a quantization rule or "Correspondence Principle", which assigns to each classical magnitude (function in T M , subject to certain conditions) a differential operator in C ∞ (M). The issue about the order in which the p' and q' are to be taken in quantization loses all meaning, once the general rule has been fixed. Specified the Correspondence Principle, each "classical state" of the system, understood as a vector field on M , determines a wave equation for each magnitude. The Schrödinger equation is a particular example of these wave equations.
Geometric quantization, complex structures and the coherent state transform
Journal of Functional Analysis, 2005
It is shown that the heat operator in the Hall coherent state transform for a compact Lie group K (J. Funct. Anal. 122 (1994) 103–151) is related with a Hermitian connection associated to a natural one-parameter family of complex structures on T*KT*K. The unitary parallel transport of this connection establishes the equivalence of (geometric) quantizations of T*KT*K for different choices of complex structures within the given family. In particular, these results establish a link between coherent state transforms for Lie groups and results of Hitchin (Comm. Math. Phys. 131 (1990) 347–380) and Axelrod et al. (J. Differential Geom. 33 (1991) 787–902).
Quantization of mechanical systems
Journal of Physics Communications, 2018
We show what seems to be the key for quantization of classical systems. Given a manifold M , each riemannian metric (nondegenerate, of arbitrary signature) canonically determines a quantization rule or "Correspondence Principle", which assigns to each classical magnitude (function in T M , subject to certain conditions) a differential operator in C ∞ (M). The issue about the order in which the p' and q' are to be taken in quantization loses all meaning, once the general rule has been fixed. Specified the Correspondence Principle, each "classical state" of the system, understood as a vector field on M , determines a wave equation for each magnitude. The Schrödinger equation is a particular example of these wave equations.