Quantization of the Sphere with Coherent States

Current views link quantization with dynamics. The reason is that quantum mechanics or quantum field theories address to dynamical systems, i.e., particles or fields. Our point of view here breaks the link between quantization and dynamics: any (classical) physical system can be quantized. Only dynamical systems lead to dynamical quantum theories, which appear to result from the quantization of symplectic structures.     

Quantum Dynamical Yang–Baxter Equation¶Over a Nonabelian Base

In this paper we consider dynamical r-matrices over a nonabelian base. There are two main results. First, corresponding to a fat reductive decomposition of a Lie algebra ?=?⊕?, we construct geometrically a non-degenerate triangular dynamical r-matrix using symplectic fibrations. Second, we prove that a triangular dynamical r-matrix naturally corresponds to a Poisson manifold ?^⋆×G. A special type of quantization of this Poisson manifold, called compatible star products in this paper, yields a generalized version of the quantum dynamical Yang–Baxter equation (or Gervais–Neveu–Felder equation). As a result, the quantization problem of a general dynamical r-matrix is proposed. Received: 19 May 2001 / Accepted: 19 November 2001     

Algebraic quantization of the SU(3) skyrmion

The algebraic quantization of the SU(3) skyrmion is presented. We discuss the role of the number of colors (N_c) in QCD and the Wess-Zumino term in the quantization. Two distinct dynamical group chains are shown to correspond to different methods of semiclassical quantization. We obtain most of the SU(6) quark model predictions for the skyrmion quantized with N_c=3.     

Unambiguous Quantization from the Maximum Classical Correspondence that Is Self-consistent: The Slightly Stronger Canonical Commutation Rule Dirac Missed

Dirac’s identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac’s canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of Poisson brackets between any linear function of phase space and the sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E.H. Kerner. Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical “shadow world” which can manifest negative densities in phase space.     

Are the Weyl and coherent state descriptions physically equivalent?

We investigate the consistency of coherent state quantization in regard to physical observations in the non-relativistic (or Galilean) regime. We compare this particular type of quantization of the complex plane with the canonical (Weyl) quantization and examine whether they are or not equivalent in their predictions. 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.     

The Gervais-Neveu-Felder equation and the quantum Calogero-Moser systems

We quantize the spin Calogero-Moser model in theR-matrix formalism. The quantumR-matrix of the model is dynamical. ThisR-matrix has already appeared in Gervais-Neveu's quantization of Toda field theory and in Felder's quantization of the Knizhnik-Zamolodchikov-Bernard equation.     

Quantization of Symplectic Dynamical <Emphasis Type="Italic">r</Emphasis>-Matrices and the Quantum Composition Formula

In this paper we quantize symplectic dynamical r-matrices over a possibly nonabelian base. The proof is based on the fact that the existence of a star-product with a nice property (called strong invariance) is sufficient for the existence of a quantization. We also classify such quantizations and prove a quantum analogue of the classical composition formula for coboundary dynamical r-matrices.     

Number-phase quantization of a mesoscopic RLC circuit

With the help of the time-dependent Lagrangian for a damped harmonic oscillator, the quantization of mesoscopic RLC circuit in the context of a number-phase quantization scheme is realized and the corresponding Hamiltonian operator is obtained. Then the evolution of the charge number and phase difference across the capacity are obtained. It is shown that the number-phase analysis is useful to tackle the quantization of some mesoscopic circuits and dynamical equations of the corresponding operators.     

Dynamical Cremmer-Gervais R matrix

SL-type zero-graded solutions of the dynamical Yang-Baxter equation in dimension 3 are classified. In addition to the well-known Drinfeld-Jimbo-type dynamical R matrices, the classification of so-called “regular” cases includes a quantization of the classical dynamical r matrix found by O. Schiffmann and a dynamical partner of the constant Cremmer-Gervais R matrix. Nonperturbative effects are exhibited.     

Equivariant quantization of orbifolds

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces, etc. In this work, we prove the existence of an equivariant quantization for orbifolds. Our construction combines an appropriate desingularization of any Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in Poncin et al. (2009) [19]. Further, we suggest definitions of the common geometric objects on orbifolds, which capture the nature of these spaces and guarantee, together with the properties of the mentioned foliated resolution, the needed correspondences between singular objects of the orbifold and the respective foliated objects of its desingularization.     

Reductions and quantization

The Interchangeability of the Marsden-Weinstein reduction procedure and the Kostant-Souriau geometric quantization is studied by detailed examination of a concrete dynamical system-the so-called MIC-Kepler problem. It is proved that some stages of reduction plus geometric quantization technique produce the complete quantum spectrum of the system, while others give part of it or nothing.     

Quantum mechanics for totally constrained dynamical systems and evolving hilbert spaces

We analyze the quantization of dynamical systems that do not involve any background notion of space and time. We give a set of conditions for the introduction of an intrinsic time in quantum mechanics. We show that these conditions are a generalization of the usual procedure of deparametrization of relational theories with Hamiltonian constraint that allow one to include systems with an evolving Hilbert space. We apply our quantization procedure to the parametrized free particle and to some explicit examples of dynamical systems with an evolving Hilbert space. Finally, we conclude with some considerations concerning the quantum gravity case.     

Classical quantization condition in the generalized coherent-state representation with application to many-body systems

On the basis of the ansatz of the single-valuedness for the classically approximated wave function, a classical quantization rule is derived for a dynamical system described by a general class of coherent states. The classical wave function, which is defined on the manifold parametrizing the coherent states, is constructed by an application of the stationary phase approximation to the coherent-state path integral. The general form of the quantization rule for the time-dependent Hartree-Fock solutions is obtained as a particular case.     

Entangled State in Quantization of Magnetic Flux Qubits with Mutual Inductance Coupling

Via the Hamilton dynamical approach we have constructed Hamiltonian for the mutual inductance coupling magnetic flux qubits. The entangled state representation is used to propose Cooper-pair number-phase quantization and the Hamiltonian operator for the whole system. The dynamical evolution of the phase difference operator and the Cooper-pairs number operator is investigated by virtue of Heisenberg equations. Project 10574060 supported by the National Natural Science Foundation of China and project X071045 supported by the Science Foundation of Liaocheng University.     

Classical foundations of quantum groups

The concept of classical r matrices is developed from a purely canonical standpoint. The final purpose of this work is to bring about a synthesis between recent developments in the theory of integrable systems and the general theory of quantization as a deformation of classical mechanics. The concept of quantization algebra is here dominant; in integrable systems this is the set of dynamical variables that appear in the Lax pair. The nature of this algebra, a solvable Lie algebra in such models as the Sine-Gordon and Toda field theories but semisimple in the case of spin systems, provides a useful scheme for the classification of integrable models. A completely different classification is obtained by the nature of the r matrix employed; there are three kinds: rational, trigonometric, and elliptic. All cases are studied in detail, with numerous examples. Some of the problems connected with quantization are discussed.This paper is dedicated to my friend Asim Barut.     

On a dynamical quantization

A concept of a noncanonical quantization, called dynamical quantization, is defined in an axiomatic way. A dynamical quantization of a system of two nonrelativistic point particles interacting via a harmonic potential is considered in more details. The quantized system exhibits some new features. In particular, it has finite space dimensions. The distance between the particles is preserved in time and can have at most four different values. The position of any one of them cannot be localized, since the operators of the coordinates do not commute. The particles are smeared with a certain probability within a finite volume, which moves together with their centre of mass. The orbital momentum of the composite system is either one or zero.Presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

Spin-wave quantization in ferromagnetic nickel nanowires

The dynamical properties of uniform two-dimensional arrays of nickel nanowires have been investigated by inelastic light scattering. Multiple spin waves are observed that are in accordance with dipole-exchange theory predictions for the quantization of bulk spin waves. This first study of the spin-wave dynamics in ferromagnetic nanowire arrays reveals strong mode quantization effects and indications of a subtle magnetic interplay between nanowires. The results show that it is important to take proper account of these effects for the fundamental physics and future technological developments of magnetic nanowires.     

The Angular Momentum Dilemma and Born–Jordan Quantization

The rigorous equivalence of the Schrödinger and Heisenberg pictures requires that one uses Born–Jordan quantization in place of Weyl quantization. We confirm this by showing that the much discussed “ angular momentum dilemma” disappears if one uses Born–Jordan quantization. We argue that the latter is the only physically correct quantization procedure. We also briefly discuss a possible redefinition of phase space quantum mechanics, where the usual Wigner distribution has to be replaced with a new quasi-distribution associated with Born–Jordan quantization, and which has proven to be successful in time-frequency analysis.     

Classical Poisson Structure for a Hierarchy of One–Dimensional Particle Systems Separable in Parabolic Coordinates

Abstract

We consider a hierarchy of many-particle systems on the line with polynomial potentials separable in parabolic coordinates. The first non-trivial member of this hierarchy is a generalization of an integrable case of the Hénon-Heiles system. We give a Lax representation in terms of 2 × 2 matrices for the whole hierarchy and construct the associated linear r-matrix algebra with the r-matrix dependent on the dynamical variables. A Yang-Baxter equation of dynamical type is proposed. Classical integration in a particular case is carried out and quantization of the system is discussed with the help of separation variables.     

Equivalence of Constrained Models

We study two constrained scalar models. Whilethere seems to be equivalence when the partiallyintegrated Feynman path integral is expandedgraphically, the dynamical behaviors of the two modelsare different when quantization is done using Diracconstraint analysis.