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:2607.20283

Angular Momentum Quantization of a Charge-flux Composite: Quantum Electrodynamic Approach

Kicheon Kang

cond-mat.mes-hallquant-phSubmitted Jul 22, 2026• Announced Jul 23, 2026
v1 · Jul 22, 2026v2 · Aug 7, 2026
arXivHTMLPDF

Abstract

The fractional angular momentum of a two-dimensional charge-flux composite is a well-established phenomenon usually derived from a semiclassical Hamiltonian. However, when the composite is treated as an isolated system in free two-dimensional space, the fundamental rotational and reflection symmetries of the $O(2)$ group demand that its total angular momentum is strictly quantized. We address this conceptual discrepancy by applying a full quantum electrodynamic (QED) approach combined with Noether's theorem. We demonstrate that the interaction between the charge and flux, mediated by the vacuum electromagnetic field, generates an intrinsic interaction angular momentum composed of both field momentum and hidden relativistic momentum. This gauge-invariant interaction angular momentum exactly compensates for the fractional part of the kinetic angular momentum. Consequently, the net angular momentum of the composite strictly follows the integer or half-integer quantization rule. Our formalism clarifies that the conventional fractional spin corresponds to the expectation value of the kinetic angular momentum within the perturbed QED ground state. It also elucidates why the standard classical field angular momentum definition fails to capture this in two dimensions, due to non-vanishing boundary terms.

Comments
5 pages, 2 figures

Related papers

Rotational symmetry of classical orbits, arbitrary quantization of angular momentum and the role of the gauge field in two-dimensional space

辛俊丽, 梁九卿

We study quantum–classical correspondence in terms of the coherent wave functions of a charged particle in two- dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.

中国物理B：英文版•Jan 1, 2012•0 citations

Rotational symmetry of classical orbits, arbitrary quantization of angular momentum and the role of the gauge field in two-dimensional space

Jun-Li Xin, Jiu-Qing Liang

We study quantum—classical correspondence in terms of the coherent wave functions of a charged particle in two-dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than ħ is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.

arXiv:1010.4352•Chinese Physics B•Apr 1, 2012•3 citations

Remarks on the Theory of Angular Momenta

O. Chavoya-Aceves

A rigorous application of the correspondence rules shows that the operator of the angular momentum of a quantum particle---corresponding to the classical magnitude $\mathbf{l}= m \mathbf{r} \wedge \mathbf{v}$---is given by $\mathbf{\hat{l}}=\mathbf{r}\wedge(-i\hbar\mathbf{\nabla} -\frac{e}{c}\mathbf{A})$ in the presence of an electromagnetic field. Thus, despite the general opinion on the corresponding rules of quantization, the eigenvalues of the angular momentum depend on the configuration of the electromagnetic field. The usual rules of commutation $[{\hat{l}}_i,{\hat{l}}_j]=i\hbarε_{ijk}{\hat{l}}_k$, that are at the foundation of the calculus of angular momentum and of the theory of \emph{spin}---and Bohm's example of the EPR argument---are not valid in the presence of an electromagnetic field. The expected value of the operator $\mathbf{\hat{l}}=-i\hbar\mathbf{r}\wedge\mathbf{\nabla}$ is not gauge invariant, it depends on the calibration of the electrodynamic potentials.

arXiv:quant-ph/0305049•May 9, 2003•2 citations

Angular momentum in two dimensions

D. K. Sunko

It is shown by direct computation that angular momentum cannot be reduced to two dimensions by assuming lack of dependence on the z coordinate. Formally, this is because the independence of z breaks rotational invariance. As a physical example, it is shown that a gauge- and rotation-invariant description of planar electrons in a perpendicular magnetic field must have a three-dimensional form, even though all the terms refer to planar coordinates alone. The physical concepts underlying common formal manipulations with the quantum angular momentum operator are treated at a level directly useful to students encountering them thoroughly for the first time. The letter may also serve as a contextual introduction to advanced quantization issues, usually studied at graduate level.

European Journal of Physics•Apr 19, 2010•1 citation

Quantum Mechanics of Fractional-Spin Particles

Frank A. Wilczek

Composites formed from charged particles and vortices in (2+1)-dimensional models, or flux tubes in three-dimensional models, can have any (fractional) angular momentum. The statistics of these objects, like their spin, interpolates continuously between the usual boson and fermion cases. How this works for two-particle quantum mechanics is discussed here.

Physical Review Letters•Oct 4, 1982•1,693 citations

Canonical group quantization, rotation generators, and quantum indistinguishability

Carlos Benavides, Andrés Fernando Reyes-Lega

Using the method of canonical group quantization, we construct the angular momentum operators associated to configuration spaces with the topology of (i) a sphere and (ii) a projective plane. In the first case, the angular momentum operators derived this way are the quantum version of Poincaré's vector, i.e., the physically correct angular momentum operators for an electron coupled to the field of a magnetic monopole. In the second case, the operators one obtains represent the angular momentum operators of a system of two indistinguishable spin zero quantum particles in three spatial dimensions. The relevance of the proposed formalism for the progress in our understanding of the spin–statistics connection in nonrelativistic quantum mechanics is discussed. Introduction The connection between the spin of quantum particles and the statistics they obey is a remarkable example of a simply stated physical fact without the recognition of which many physical phenomena (ranging from the stability of matter and the electronic configuration of atoms to Bose–Einstein condensation and superconductivity) would not have an explanation. Nevertheless, the simplicity of the assertion “integer spin particles obey Bose statistics, and half-integer spin particles obey Fermi statistics” stands in bold contrast to its intricate physical origin. Indeed, Pauli's proof of the spin–statistics theorem [Pau40] (improving on earlier work by Fierz [Fie39]), showed that the spin–statistics connection was deeply rooted in relativistic quantum field theory. The path to a rigorous proof of this theorem (from the mathematical point of view) was a long one and involved the efforts of many people (see, e.g., the book by Duck and Sudarshan [DS98b]).

arXiv:0806.2449•Cambridge University Press eBooks•Apr 29, 2010•2 citations
The fractional angular momentum of a two-dimensional charge-flux composite is a well-established phenomenon usually derived from a semiclassical Hamiltonian. However, when the composite is treated as an isolated system in free two-dimensional space, the fundamental rotational and reflection symmetries of the O(2)O(2)O(2) group demand that its total angular momentum is strictly quantized. We address this conceptual discrepancy by applying a full quantum electrodynamic (QED) approach combined with Noether's theorem. We demonstrate that the interaction between the charge and flux, mediated by the vacuum electromagnetic field, generates an intrinsic interaction angular momentum composed of both field momentum and hidden relativistic momentum. This gauge-invariant interaction angular momentum exactly compensates for the fractional part of the kinetic angular momentum. Consequently, the net angular momentum of the composite strictly follows the integer or half-integer quantization rule. Our formalism clarifies that the conventional fractional spin corresponds to the expectation value of the kinetic angular momentum within the perturbed QED ground state. It also elucidates why the standard classical field angular momentum definition fails to capture this in two dimensions, due to non-vanishing boundary terms.
We study quantum–classical correspondence in terms of the coherent wave functions of a charged particle in two- dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.
We study quantum—classical correspondence in terms of the coherent wave functions of a charged particle in two-dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than ħ is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.
A rigorous application of the correspondence rules shows that the operator of the angular momentum of a quantum particle---corresponding to the classical magnitude l=mr∧v\mathbf{l}= m \mathbf{r} \wedge \mathbf{v}l=mr∧v---is given by l^=r∧(−iℏ∇−ecA)\mathbf{\hat{l}}=\mathbf{r}\wedge(-i\hbar\mathbf{\nabla} -\frac{e}{c}\mathbf{A})l^=r∧(−iℏ∇−ce​A) in the presence of an electromagnetic field. Thus, despite the general opinion on the corresponding rules of quantization, the eigenvalues of the angular momentum depend on the configuration of the electromagnetic field. The usual rules of commutation [l^i,l^j]=iℏεijkl^k[{\hat{l}}_i,{\hat{l}}_j]=i\hbarε_{ijk}{\hat{l}}_k[l^i​,l^j​]=iℏεijk​l^k​, that are at the foundation of the calculus of angular momentum and of the theory of \emph{spin}---and Bohm's example of the EPR argument---are not valid in the presence of an electromagnetic field. The expected value of the operator l^=−iℏr∧∇\mathbf{\hat{l}}=-i\hbar\mathbf{r}\wedge\mathbf{\nabla}l^=−iℏr∧∇ is not gauge invariant, it depends on the calibration of the electrodynamic potentials.
It is shown by direct computation that angular momentum cannot be reduced to two dimensions by assuming lack of dependence on the z coordinate. Formally, this is because the independence of z breaks rotational invariance. As a physical example, it is shown that a gauge- and rotation-invariant description of planar electrons in a perpendicular magnetic field must have a three-dimensional form, even though all the terms refer to planar coordinates alone. The physical concepts underlying common formal manipulations with the quantum angular momentum operator are treated at a level directly useful to students encountering them thoroughly for the first time. The letter may also serve as a contextual introduction to advanced quantization issues, usually studied at graduate level.
Using the method of canonical group quantization, we construct the angular momentum operators associated to configuration spaces with the topology of (i) a sphere and (ii) a projective plane. In the first case, the angular momentum operators derived this way are the quantum version of Poincaré's vector, i.e., the physically correct angular momentum operators for an electron coupled to the field of a magnetic monopole. In the second case, the operators one obtains represent the angular momentum operators of a system of two indistinguishable spin zero quantum particles in three spatial dimensions. The relevance of the proposed formalism for the progress in our understanding of the spin–statistics connection in nonrelativistic quantum mechanics is discussed. Introduction The connection between the spin of quantum particles and the statistics they obey is a remarkable example of a simply stated physical fact without the recognition of which many physical phenomena (ranging from the stability of matter and the electronic configuration of atoms to Bose–Einstein condensation and superconductivity) would not have an explanation. Nevertheless, the simplicity of the assertion “integer spin particles obey Bose statistics, and half-integer spin particles obey Fermi statistics” stands in bold contrast to its intricate physical origin. Indeed, Pauli's proof of the spin–statistics theorem [Pau40] (improving on earlier work by Fierz [Fie39]), showed that the spin–statistics connection was deeply rooted in relativistic quantum field theory. The path to a rigorous proof of this theorem (from the mathematical point of view) was a long one and involved the efforts of many people (see, e.g., the book by Duck and Sudarshan [DS98b]).