Dirac quantisation condition: a comprehensive review

Abstract

In most introductory courses on electrodynamics, one is taught the electric charge is quantised but no theoretical explanation related to this law of nature is offered. Such an explanation is postponed to graduate courses on electrodynamics, quantum mechanics and quantum field theory, where the famous Dirac quantisation condition is introduced, which states that a single magnetic monopole in the Universe would explain the electric charge quantisation. Even when this condition assumes the existence of a not-yet-detected magnetic monopole, it provides the most accepted explanation for the observed quantisation of the electric charge. However, the usual derivation of the Dirac quantisation condition involves the subtle concept of an "unobservable" semi-infinite magnetised line, the so-called "Dirac string," which may be difficult to grasp in a first view of the subject. The purpose of this review is to survey the concepts underlying the Dirac quantisation condition, in a way that may be accessible to advanced undergraduate and graduate students. Some of the discussed concepts are gauge invariance, singular potentials, single-valuedness of the wave function, undetectability of the Dirac string and quantisation of the electromagnetic angular momentum. Five quantum-mechanical and three semi-classical derivations of the Dirac quantisation condition are reviewed. In addition, a simple derivation of this condition involving heuristic and formal arguments is presented.

Massive Photon, Magnetic Charge and the Dirac Quantization Condition

“…In this work we have also included the magnetic field and current density source connected with the Dirac string -see equations ( 14) and (17). These fields and sources for the Dirac string are not generally emphasized, but recent works [6] [7] [8] have pointed out the subtle, but crucial importance of these quantities. In particular one may ask about the nature of the source of the effective string current in (17),…”

“…Usually, only the first, Coulombic term, gr r 2 is written down, but a careful analysis (see [6] [7] and also [8] for a recent, pedagogical and thorough exposition) shows the existence of the second, string term, which is required to make sure that the divergence of a curl is zero.…”

“…This takes some care due to the singularity along the ±z-axis. Following appendix D of [8] we convert the vector potential in (11) into cylindrical coordinates and regularize it via an infinitesimal ǫ to give…”

“…For this last term we have used e −mr δ(x)δ(y) → e −m|z| δ(x)δ(y) through the action of the two δ-functions. It is straightforward to see that (11) and ( 14) are exact solutions to the two magnetic equations (7) and (8) if the ±4πge −m|z| δ(x)δ(y)Θ(∓z)ẑ piece of the magnetic field is neglected.…”

Duality-invariant topological quantum phase of a moving dyon

“…and then (24) takes the form Ψ = e iδ Ψ 0 . Inserting the potentials A = Φ m φ/(2πρ ′ ) and C = −Φ e φ/(2πρ ′ ), and…”