Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Observables of Angular Momentum as Observables on the Fedosov Quantized Sphere

Philip C. Tillman, George A. J. Sparling

Published Sep 29, 2005• 3 citations
OpenAlex

Abstract

In this paper we construct the commutators of the Fedosov * (a generalization of the Moyal star product) on the phase space of S2. It is shown that this product obeys the standard angular momentum commutation relations in ordinary nonrelativistic quantum mechanics. The purpose of this paper is three-fold. One is to find an exact, non-perturbative solution to the commutators of the Fedosov *. The other is to verify that these commutation relations correspond to angular momentum commutation relations as in ordinary nonrelativistic quantum mechanics. The last is to show a more general computation of the Fedosov *; essentially undeforming Fedosov's algorithm.

Related papers

Observables of angular momentum as observables on the Fedosov quantized sphere

Philip Tillman, George Sparling

In this paper we construct quantum mechanical observables of a single free particle that lives on the surface of the two-sphere S2 by implementing the Fedosov ∗-formalism. The Fedosov ∗ is a generalization of the Moyal star product on an arbitrary symplectic manifold. After their construction we show that they obey the standard angular momentum commutation relations in ordinary nonrelativistic quantum mechanics. The purpose of this paper is threefold. One is to find an exact, nonperturbative solution of these observables. The other is to verify that the commutation relations of these observables correspond to angular momentum commutation relations. The last is to show a more general computation of the observables in Fedosov ∗-formalism; essentially an undeformation of Fedosov’s algorithm.

arXiv:quant-ph/0509210•Journal of Mathematical Physics•May 1, 2006•1 citation

Fractional angular momentum in non-commutative spaces

Jian-zu Zhang A B

In noncommutative spaces to maintain Bose–Einstein statistics for identical particles at the non-perturbation level described by deformed annihilation-creation operators when the state vector space of identical bosons is constructed by generalizing one-particle quantum mechanics it is explored that the consistent ansatz of commutation relations of phase space variables should simultaneously include space–space non-commutativity and momentum–momentum non-commutativity, and a new type of boson commutation relations at the deformed level is obtained. Consistent perturbation expansions of deformed annihilation-creation operators are obtained. The influence of the new boson commutation relations on dynamics is discussed. The non-perturbation and perturbation property of the orbital angular momentum of two-dimensional system are investigated. Its spectrum possesses fractional eigen values and fractional intervals.

arXiv:hep-th/0405135•Physics Letters B•Feb 12, 2004•139 citations

Asymptotic Quantization of a Particle on a Sphere

José Luis Romero, A. B. Klimov

Quantum systems whose states are tightly distributed among several invariant subspaces (variable spin systems) can be described in terms of distributions in a four-dimensional phase-space T∗S2 in the limit of large average angular momentum. The cotangent bundle T∗S2 is also the classical manifold for systems with E(3) symmetry group with appropriately fixed Casimir operators. This allows us to employ the asymptotic form of the star-product proper for variable (integer) spin systems to develop a deformation quantization scheme for a particle moving on the two-dimensional sphere, whose observables are elements of e(3) algebra and the corresponding phase-space is T∗S2. We show that the standard commutation relations of the e(3) algebra are recovered from the corresponding classical Poisson brackets and the explicit expressions for the eigenvalues and eigenfunctions of some quantized classical observables (such as the angular momentum operators and their squares) are obtained.

Quantum Reports•Mar 21, 2023•0 citations

SU(2) quantum kinematics: Rotation-observable versus angular-momentum generalized commutation relations

Jorge Krause

The canonical commutation relations of quantum mechanics are generalized to the case where appropriate dynamical variables are angular-momentum, rotation-angle, and rotation-axis observables. To this end, SU(2) is “quantized” on the compact group manifold, according to the standard procedure of non-Abelian quantum kinematics. Quantum-kinematic invariant operators are introduced, and their commutation relations with the rotation variables are found in an explicit manner. The quantum-kinematic invariants yield superselection rules in the form of eigenvalue equations of an isotopic structure (which one should solve in the applications, in order to get multiplets that carry the irreducible representations of the underlying quantum kinematic models). A wide range of applicability of SU(2) quantum kinematics is suggested.

Journal of Mathematical Physics•Nov 1, 1998•1 citation

Some remarks on the commutation relations for angle-angular momentum variables in quantum mechanics

Colm T. Whelan

A commutation relation for the observables corresponding to the azimuthal angle about the z axis and the z component of the orbital angular momentum in wave mechanics is presented which will be valid on the entire Hilbert space of states.

Journal of Physics A Mathematical and General•Jun 1, 1980•1 citation

A Dyadic Calculation of the Vector and Scalar Products of Two Angular Momentum Operators in Quantum Mechanics

Zhao Ru-shun

The algebraic relations between the components of an augular momentum operator are of key importance in solving Schrodinger.In this paper we have exploited the dyadic product between vector products to deduce the coordinate-independent formula of scalar product and vector product between angular momentum operators.Then the formula for the scalar product of angular momentum operators is used to deduce the Schrodinger equation for one particle in spherical coordinate.In the final,we make a interest contrast among vector product,commutation relation and classical Poisson brackets,and find out their common properties that they all have the anti-commutative relationship and the Jacohi identity.

Journal of Liaoning Normal University•Jan 1, 2006•0 citations
In this paper we construct the commutators of the Fedosov * (a generalization of the Moyal star product) on the phase space of S2. It is shown that this product obeys the standard angular momentum commutation relations in ordinary nonrelativistic quantum mechanics. The purpose of this paper is three-fold. One is to find an exact, non-perturbative solution to the commutators of the Fedosov *. The other is to verify that these commutation relations correspond to angular momentum commutation relations as in ordinary nonrelativistic quantum mechanics. The last is to show a more general computation of the Fedosov *; essentially undeforming Fedosov's algorithm.
In this paper we construct quantum mechanical observables of a single free particle that lives on the surface of the two-sphere S2 by implementing the Fedosov ∗-formalism. The Fedosov ∗ is a generalization of the Moyal star product on an arbitrary symplectic manifold. After their construction we show that they obey the standard angular momentum commutation relations in ordinary nonrelativistic quantum mechanics. The purpose of this paper is threefold. One is to find an exact, nonperturbative solution of these observables. The other is to verify that the commutation relations of these observables correspond to angular momentum commutation relations. The last is to show a more general computation of the observables in Fedosov ∗-formalism; essentially an undeformation of Fedosov’s algorithm.
In noncommutative spaces to maintain Bose–Einstein statistics for identical particles at the non-perturbation level described by deformed annihilation-creation operators when the state vector space of identical bosons is constructed by generalizing one-particle quantum mechanics it is explored that the consistent ansatz of commutation relations of phase space variables should simultaneously include space–space non-commutativity and momentum–momentum non-commutativity, and a new type of boson commutation relations at the deformed level is obtained. Consistent perturbation expansions of deformed annihilation-creation operators are obtained. The influence of the new boson commutation relations on dynamics is discussed. The non-perturbation and perturbation property of the orbital angular momentum of two-dimensional system are investigated. Its spectrum possesses fractional eigen values and fractional intervals.
Quantum systems whose states are tightly distributed among several invariant subspaces (variable spin systems) can be described in terms of distributions in a four-dimensional phase-space T∗S2 in the limit of large average angular momentum. The cotangent bundle T∗S2 is also the classical manifold for systems with E(3) symmetry group with appropriately fixed Casimir operators. This allows us to employ the asymptotic form of the star-product proper for variable (integer) spin systems to develop a deformation quantization scheme for a particle moving on the two-dimensional sphere, whose observables are elements of e(3) algebra and the corresponding phase-space is T∗S2. We show that the standard commutation relations of the e(3) algebra are recovered from the corresponding classical Poisson brackets and the explicit expressions for the eigenvalues and eigenfunctions of some quantized classical observables (such as the angular momentum operators and their squares) are obtained.
The canonical commutation relations of quantum mechanics are generalized to the case where appropriate dynamical variables are angular-momentum, rotation-angle, and rotation-axis observables. To this end, SU(2) is “quantized” on the compact group manifold, according to the standard procedure of non-Abelian quantum kinematics. Quantum-kinematic invariant operators are introduced, and their commutation relations with the rotation variables are found in an explicit manner. The quantum-kinematic invariants yield superselection rules in the form of eigenvalue equations of an isotopic structure (which one should solve in the applications, in order to get multiplets that carry the irreducible representations of the underlying quantum kinematic models). A wide range of applicability of SU(2) quantum kinematics is suggested.
The algebraic relations between the components of an augular momentum operator are of key importance in solving Schrodinger.In this paper we have exploited the dyadic product between vector products to deduce the coordinate-independent formula of scalar product and vector product between angular momentum operators.Then the formula for the scalar product of angular momentum operators is used to deduce the Schrodinger equation for one particle in spherical coordinate.In the final,we make a interest contrast among vector product,commutation relation and classical Poisson brackets,and find out their common properties that they all have the anti-commutative relationship and the Jacohi identity.