[Paper Review] Towards a new quantization of Dirac's monopole

Alexander I. Nesterov, F. Aceves de la Cruz|ArXiv.org|Aug 31, 2002

Crystallography and Radiation Phenomena3 citations

TL;DR

This paper proposes a nonunitary representation of the rotation group SO(3) to describe a Dirac monopole system without enforcing Dirac's quantization rule. By modeling the charge-monopole system as a free anyon with spin degrees of freedom via a combined so(3) ⊕ so(2,1) algebra, the authors construct an infinite-dimensional, nonunitary representation where the magnetic charge μ need not satisfy the Dirac condition μ = integer/2, offering a new framework for nonassociative quantum mechanics with arbitrary monopole charge.

ABSTRACT

There are several mathematical and physical reasons why Dirac's quantization must hold. How far one can go without it remains an open problem. The present work outlines a few steps in this direction.

Motivation & Objective

• To explore the possibility of a self-consistent quantum mechanics for magnetic monopoles without Dirac's quantization rule.
• To investigate the physical implications of nonunitary representations of the rotation group in the context of the Dirac monopole problem.
• To construct a model where the charge-monopole system is described as a free anyon with spin, avoiding the standard quantization condition.
• To relate the monopole system to the noncompact group SO(2,1) and its infinite-dimensional representations, enabling arbitrary magnetic charge.

Proposed method

• Formalism based on the Hamiltonian of a charged particle in a magnetic monopole field, with the total angular momentum operator defined via a vector potential with Dirac string singularity.
• Use of the so(3) algebra for orbital angular momentum and so(2,1) algebra for spin degrees of freedom, combined via J_i = L_i + S_i to form a nonunitary representation of so(3).
• Construction of a highest-weight state |j,j⟩ = |l,l⟩ ⊗ |λ,0⟩, where j = l − λ, and application of lowering operators to generate the full nonunitary infinite-dimensional representation.
• Derivation of commutation relations for J_i operators showing closure under so(3) algebra despite nonunitarity, ensuring rotational invariance.
• Introduction of a symplectic structure on the unit sphere via the one-form θ = eA_i dx^i, leading to Poisson brackets {S_i, S_j} = ε_ijk S_k, identifying the electromagnetic angular momentum as a classical spin.
• Adoption of infinite-dimensional unitary representations of SU(1,1) for the so(2,1) generators, with S_0, S_+, S_- acting on states |λ,n⟩ with arbitrary λ and n ≥ 0.

Experimental results

Research questions

• RQ1 Can a consistent quantum mechanical description of the Dirac monopole be constructed without imposing Dirac's quantization rule?
• RQ2 What are the physical consequences of using nonunitary representations of SO(3) in the monopole problem?
• RQ3 How can the charge-monopole system be reformulated as a free anyon with spin degrees of freedom?
• RQ4 What is the role of the so(2,1) algebra in describing fractional or arbitrary spin in non-relativistic systems?
• RQ5 How does the nonunitary representation with m ≤ j instead of |m| ≤ j affect the spectrum and dynamics of the monopole system?

Key findings

• The model constructs a nonunitary, infinite-dimensional representation of SO(3) where the magnetic charge μ is not restricted to half-integers or integers, offering an alternative to Dirac's quantization.
• The total angular momentum operator J_i satisfies the standard so(3) commutation relations [J_i, J_j] = iε_ijk J_k, preserving rotational algebra despite nonunitarity.
• The representation is bounded above in m (with m ≤ j), differing from the standard finite-dimensional unitary case where |m| ≤ j.
• The allowed values satisfy μ² ≤ j(j+1) and j − μ ∈ ℤ, providing a new quantization condition independent of Dirac's rule.
• The system is reinterpreted as a free anyon with spin, where the spin is related to the parameter λ via s = 1 − λ, consistent with anyonic statistics in (2+1)D.
• The use of the symplectic form dθ = −(1/2S²)ε_ijk S_i dS_j ∧ dS_k on S² provides a geometric foundation for the spin-like behavior of the electromagnetic angular momentum.