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arXiv:2011.07596 · Physical review. D/Physical review. D.

Ground state of nonassociative hydrogen and upper bounds on the magnetic charge of elementary particles

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

Published Nov 10, 2021• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.104.105009

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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.
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.
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.
We discuss a certain generalization of the Hilbert space of states in noncommutative quantum mechanics that, as we show, introduces magnetic monopoles into the theory. Such generalization arises very naturally in the considered model, but can be easily reproduced in ordinary quantum mechanics as well. This approach offers a different viewpoint on the Dirac quantization condition and other important relations for magnetic monopoles. We focus mostly on the kinematic structure of the theory, but investigate also a dynamical problem (with the Coulomb potential).
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 noted that the elementary electric charge is equal to e , from which fact the well-known principle of quantization of the electric charge definitely follows, namely, the electric charge is quantized and a quantum is the charge of the electron e (or positron). Or any change in the charge is equal to an integer number of electrons (or positrons). The formally identical transformation of the principle of quantization of the electric charge allows formulating the principle of quantization of the magnetic flux, namely, the quantity inverse to the magnetic flux is quantized and a quantum is the inverse of the quantum of the F. London magnetic flux. Or any change in the reciprocal of the magnetic flux is equal to an integer number of quantities inverse to the quantum of the F. London magnetic flux. The reciprocal of the magnetic flux quantum is equal to the sum of two quantities inverse to the quantum of the F. London magnetic flux. The validity of the principle of quantization of the magnetic flux with respect to the hydrogen atom is illustrated by Theorem 1: quantization of the energy of the hydrogen atom is a consequence of the principle of quantization of the magnetic flux. Theorem 2 is proved: the magnetic flux of a hydrogen atom in the ground state is equal to the quantum of the F. London magnetic flux. Theorem 3 is proved: the quantum of the magnetic flux is not the minimum possible for a nonzero magnetic flux. It is established that the quantum of the magnetic flux is not a quantum in the sense of a portion (like the F. London quantum). A quantum is the inverse of the quantum of the F. London magnetic flux. It is established that the magnitude of the magnetic flux quantum is not the minimum possible for a nonzero magnetic flux. It is established that the magnetic flux of a hydrogen atom in the ground state is equal to the quantum of the F. London magnetic flux. A discrete set of energies of the hydrogen atom is noted to be a consequence of the solution of the Schrödinger equation, which, in turn, is phenomenological. The reverse discourse used in the proof of Theorem 1 can show that the Schrödinger equation is a consequence of the principle of quantization of the magnetic flux.