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 non-associative quantum mechanics is mathematically consistent with fractional magnetic charges of small values. Here, spectral properties in non-associative quantum mechanics are derived, 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 hydrogen-like atoms, such as the muon or the antiproton.
Highlights
Eigenvalues and eigenstates can be defined and derived completely algebraically, without using a Hilbert-space representation of observables as operators
Physical examples can be found mainly in situations in which fractional magnetic charges may be present that do not obey Dirac’s quantization condition [1], which can be defined at the level of a nonassociative algebra of observables even though no Hilbert-space representation exists [2,3,4,5]
Using 1⁄2px; r 1⁄4 −i∂d r=∂x 1⁄4 −ixr−1. This result proves the virial theorem for standard quantum mechanics and for nonassociative systems in the presence of magnetic monopoles: While some associativity is applied in computing the commutator in (25), none of the brackets (2) appear that would be modified for nonzero magnetic charge
Summary
Eigenvalues and eigenstates can be defined and derived completely algebraically, without using a Hilbert-space representation of observables as operators. Since the assumption of an associative product would imply the Jacobi idenity for the commutator, magnetic monopoles are seen to require nonassociative algebras of quantum observables [2,3,4,5]. Algebraic derivations that do not make use of specific representations are usually more challenging than standard quantum mechanics, in particular if associativity cannot be assumed. As a consequence, such systems remain incompletely understood, and it remains to be seen whether they can be viable. The present paper presents details of the latter derivation as well as a discussion of new methods that may be useful for further applications
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