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

arXiv:1608.04707 · Physical review. D/Physical review. D.

Dirac’s monopole, quaternions, and the Zassenhaus formula

Michael Alexandrovich Soloviev

Published Nov 23, 2016• 9 citations
arXivHTMLPDF

Abstract

Starting from the quaternionic quantization scheme proposed by Emch and Jadczyk for describing the motion of a quantum particle in the magnetic monopole field, we derive an algorithm for finding the differential representation of the star product generated by the quaternionic Weyl correspondence on phase-space functions. This procedure is illustrated by the explicit calculation of the star product up to the second order in the Planck constant $\ensuremath{\hbar}$. Our main tools are an operator analog of the twisted convolution and the Zassenhaus formula for the products of exponentials of noncommuting operators.

DOI
10.1103/physrevd.94.105021

Related papers

Monopole-based quantization: a programme

José Fernando Cariñena Marzo, José M. Gracia Bondía, Fedele Lizzi +2 more

We describe a programme to quantize a particle in the field of a (three dimensional) magnetic monopole using a Weyl system. We propose using the mapping of position and momenta as operators on a quaternionic Hilbert module following the work of Emch and Jadczyk.

•Jan 1, 2009•1 citation

Monopole-based quantization: a programme

José F. Cariñena, José M. Gracia-Bondı́a, Fedele Lizzi +2 more

We describe a programme to quantize a particle in the field of a (three dimensional) magnetic monopole using a Weyl system. We propose using the mapping of position and momenta as operators on a quaternionic Hilbert module following the work of Emch and Jadczyk.

arXiv:0912.2188•Dec 11, 2009•5 citations

Star product for qubit states in phase space and star exponentials

Jasel Berra-Montiel, Alberto Molgado, Mar Sánchez-Córdova

In this paper, we formulate the phase space description of qubit systems using coadjoint orbits of $SU(2)$ and the Stratonovich-Weyl correspondence, yielding a deformation quantization on the sphere. The resulting star product reproduces the operator algebra of complexified quaternions and its antisymmetric part induces the Lie-Poisson structure associated with the Kirillov-Kostant-Souriau symplectic form. We show that quantum dynamics can be expressed entirely in phase space through star exponentials of Hamiltonian symbols, leading to an explicit representation of the propagator. Further, we establish the equivalence between the coherent-state path integral formulation on $S^2$ and the algebraic description in terms of star exponentials. Some examples illustrating the construction of the star-exponential functions and the resulting Poisson structure are included.

quant-pharXiv:2604.05170•Apr 8, 2026•0 citations

On Quaternions and Monopoles

Gérard G. Emch, Arkadiusz Jadczyk

It is shown that the quaternionic Hilbert space formulation of quantum mechanics allows a quantization, based on a generalized system of imprimitivity, that leads to a description of the motion of a quantum particle in the field of a magnetic monopole. The corresponding Hamilton operator is linked to the theory of projective representations in the weakened form proposed by Adler.

arXiv:quant-ph/9803002•Feb 28, 1998•8 citations

Quantum magnetic algebra and magnetic curvature

Mihail Vladimirovich Karasev, T. A. Osborn

The symplectic geometry of the phase space associated with a charged particle is determined by the addition of the Faraday 2-form to the standard d p ∧ d q structure on . In this paper, we describe the corresponding algebra of Weyl-symmetrized functions in operators satisfying nonlinear commutation relations. The multiplication in this algebra generates an associative ∗ product of functions on the phase space. This ∗ product is given by an integral kernel whose phase is the symplectic area of a groupoid-consistent membrane. A symplectic phase space connection with non-trivial curvature is extracted from the magnetic reflections associated with the Stratonovich quantizer. Zero and constant curvature cases are considered as examples. The quantization with both static and time-dependent electromagnetic fields is obtained. The expansion of the ∗ product by the deformation parameter ℏ, written in the covariant form, is compared with the known deformation quantization formulae.

arXiv:quant-ph/0311053•Journal of Physics A Mathematical and General•Jan 28, 2004•32 citations

Magnetic quantization over Riemannian manifolds

Mihail Vladimirovich Karasev, T. A. Osborn

We demonstrate that Weyl's pioneering idea (1918) to intertwine metric and magnetic fields into a single joint connection can be naturally realized, on the phase space level, by the gauge-invariant quantization of the cotangent bundle with magnetic symplectic form. Quantization, for systems over a noncompact Riemannian configuration manifold, may be achieved by the introduction of a magneto-metric analog of the Stratonovich quantizer — a family of invertible, selfadjoint operators representing quantum δ functions. Based on the quantizer, we construct a generalized Wigner transform that maps Hilbert–Schmidt operators into L2 phase-space functions. The algebraic properties of the quantizer allow one to extract a family of symplectic reflections, which are then used to (i) derive a simple, explicit, and geometrically invariant formula for the noncommutative product of functions on phase space, and (ii) construct a magneto-metric connection on phase space. The classical limit of this product is given by the usual multiplication of functions (zeroth-order term), the magnetic Poisson bracket (first-order term), and by the magneto-metric connection (second-order term).PACS Nos.: 02.40.-k, 11.10.Nx

Canadian Journal of Physics•Jan 15, 2006•0 citations
Starting from the quaternionic quantization scheme proposed by Emch and Jadczyk for describing the motion of a quantum particle in the magnetic monopole field, we derive an algorithm for finding the differential representation of the star product generated by the quaternionic Weyl correspondence on phase-space functions. This procedure is illustrated by the explicit calculation of the star product up to the second order in the Planck constant \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ. Our main tools are an operator analog of the twisted convolution and the Zassenhaus formula for the products of exponentials of noncommuting operators.
In this paper, we formulate the phase space description of qubit systems using coadjoint orbits of SU(2)SU(2)SU(2) and the Stratonovich-Weyl correspondence, yielding a deformation quantization on the sphere. The resulting star product reproduces the operator algebra of complexified quaternions and its antisymmetric part induces the Lie-Poisson structure associated with the Kirillov-Kostant-Souriau symplectic form. We show that quantum dynamics can be expressed entirely in phase space through star exponentials of Hamiltonian symbols, leading to an explicit representation of the propagator. Further, we establish the equivalence between the coherent-state path integral formulation on S2S^2S2 and the algebraic description in terms of star exponentials. Some examples illustrating the construction of the star-exponential functions and the resulting Poisson structure are included.
The symplectic geometry of the phase space associated with a charged particle is determined by the addition of the Faraday 2-form to the standard d p ∧ d q structure on . In this paper, we describe the corresponding algebra of Weyl-symmetrized functions in operators satisfying nonlinear commutation relations. The multiplication in this algebra generates an associative ∗ product of functions on the phase space. This ∗ product is given by an integral kernel whose phase is the symplectic area of a groupoid-consistent membrane. A symplectic phase space connection with non-trivial curvature is extracted from the magnetic reflections associated with the Stratonovich quantizer. Zero and constant curvature cases are considered as examples. The quantization with both static and time-dependent electromagnetic fields is obtained. The expansion of the ∗ product by the deformation parameter ℏ, written in the covariant form, is compared with the known deformation quantization formulae.
We demonstrate that Weyl's pioneering idea (1918) to intertwine metric and magnetic fields into a single joint connection can be naturally realized, on the phase space level, by the gauge-invariant quantization of the cotangent bundle with magnetic symplectic form. Quantization, for systems over a noncompact Riemannian configuration manifold, may be achieved by the introduction of a magneto-metric analog of the Stratonovich quantizer — a family of invertible, selfadjoint operators representing quantum δ functions. Based on the quantizer, we construct a generalized Wigner transform that maps Hilbert–Schmidt operators into L2 phase-space functions. The algebraic properties of the quantizer allow one to extract a family of symplectic reflections, which are then used to (i) derive a simple, explicit, and geometrically invariant formula for the noncommutative product of functions on phase space, and (ii) construct a magneto-metric connection on phase space. The classical limit of this product is given by the usual multiplication of functions (zeroth-order term), the magnetic Poisson bracket (first-order term), and by the magneto-metric connection (second-order term).PACS Nos.: 02.40.-k, 11.10.Nx