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arXiv:1810.06540 · Physical Review Letters

Small Magnetic Charges and Monopoles in Nonassociative Quantum Mechanics

Martin Bojowald A, Suddhasattwa Brahma, Umut Büyükçam, Jonathan Guglielmon, Martijn van Kuppeveld

Published Nov 15, 2018• 7 citations
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Abstract

Weak magnetic monopoles with a continuum of charges less than the minimum implied by Dirac's quantization condition may be possible in nonassociative quantum mechanics. If a weakly magnetically charged proton in a hydrogen atom perturbs the standard energy spectrum only slightly, magnetic charges could have escaped detection. Testing this hypothesis requires entirely new methods to compute energy spectra in nonassociative quantum mechanics. Such methods are presented here, and evaluated for upper bounds on the magnetic charge of elementary particles.

DOI
10.1103/physrevlett.121.201602

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Weak magnetic monopoles with a continuum of charges less than the minimum implied by Dirac's quantization condition may be possible in nonassociative quantum mechanics. If a weakly magnetically charged proton in a hydrogen atom perturbs the standard energy spectrum only slightly, magnetic charges could have escaped detection. Testing this hypothesis requires entirely new methods to compute energy spectra in nonassociative quantum mechanics. Such methods are presented here, and evaluated for upper bounds on the magnetic charge of elementary particles.
Formulations of magnetic monopoles in a Hilbert-space formulation of quantum mechanics require Dirac's quantization condition of magnetic charge, which implies a large value that can easily be ruled out for elementary particles by standard atomic spectroscopy. However, an algebraic formulation of nonassociative quantum mechanics is mathematically consistent with fractional magnetic charges of small values. Here, spectral properties in nonassociative quantum mechanics are derived and applied to the ground state of hydrogen with a magnetically charged nucleus. The resulting energy leads to new strong upper bounds for the magnetic charge of various elementary particles that can appear as the nucleus of hydrogenlike atoms, such as the muon or the antiproton.
The familiar concepts of state vectors and operators in quantum mechanics rely on associative products of observables. However, these notions do not apply to some exotic systems such as magnetic monopoles, which have long been known to lead to nonassociative algebras. Their quantum physics has remained obscure. This Letter presents the first derivation of potentially testable physical results in nonassociative quantum mechanics, based on effective potentials. They imply new effects which cannot be mimicked in usual quantum mechanics with standard magnetic fields.
The nonrelativistic quantum theory of a particle having both electric and magnetic charges moving in an arbitrary external electromagnetic field is presented. The theory is based on the hydrodynamic formulation of quantum mechanics. Dirac's quantization condition for the electric and magnetic charges is rederived as a consistency condition for the motion of the probability fluid. Neither the wave function nor the electromagnetic potential, which were the source of ambiguities in all other formulations, appears in our approach. Nevertheless, this theory has all the essential features of the standard quantum mechanics, including the superposition principle.
The standard quantum mechanical description of the motion of a charged particle in the field of a stationary magnetic monopole is notorious for the presence of unnatural singularities in the Hamiltonian operator originating in the vector potential A(r) used to describe the magnetic field of the monopole. In this paper, an elementary quantum mechanical formulation of the problem which involves only the physically observable field B(r) is presented. This is achieved by treating as a fundamental observable of the charged particle its kinetic momentum instead of the linear momentum p . An irreducible representation of the fundamental commutation relations involving the operators and is explicitly constructed. It is shown that the existence of an irreducible representation requires that Dirac’s charge quantization condition is satisfied. Also, it is demonstrated that, from the quantum mechanical perspective, the singularities (appearing when the vector potential is introduced) are in fact properties of coordinate representations of the fundamental commutation relations.
It is conjectured that Stark states of excited hydrogen atom posses magnetic charge for which the quantum mechanical operator is Gop=eℏ(σ⃗⋅A⃗){\cal G}_{op} = {e\over \hbar} (\vecσ\cdot\vec A)Gop​=ℏe​(σ⋅A) where A⃗\vec AA is the Runge-Lenz vector. The expectation value ggg of this operator for Stark states is found to be g=e(n1−n2) g = e(n_1-n_2)g=e(n1​−n2​) which obeys a Dirac-Saha type quantization formula egc=(n1−n2)α{eg\over c} = (n_1-n_2)αceg​=(n1​−n2​)α where ααα is the fine structure constant and n1n_1n1​ and n2n_2n2​ are parabolic quantum numbers. An experimental arrangement is outlined to test this conjecture.