Axial vectors, skew-symmetric tensors and the nature of the magnetic field

The direction assigned to the magnetic field today derives historically from magnetic navigation. The author argues that this is a null direction and that the true action of the magnetic field occurs in a plane perpendicular to the conventional direction. This has been recognized in physics since Ampère, but the momentum of tradition and the mathematical convenience of established conventions and notation appears to have prevented its widespread acceptance. 1. The mathematical structure of the magnetic field: vector or tensor? The direction of the magnetic field today remains that which William Gilbert (1544–1603) and magnetic navigation have assigned to it, and the field is still widely thought to possess a positive power of orienting a compass needle in this direction [1]. The discovery of electromagnetism in 1820, and the gradual shift of attention to the magnetic forces produced by, and acting on, electric currents, introduced certain ambiguities to the concept of the direction of the magnetic field. André Marie Ampère (1775–1836), in his seminal study of electrodynamics in 1823, found that a plane of electrodynamic action could be identified at any point in the neighbourhood of a closed current-bearing circuit. When a current element lies in this plane it experiences a force directed along the plane and perpendicular to the element, whatever the orientation of the latter. This plane was perpendicular to the conventional direction of the magnetic intensity. Ampère called it the ‘directive plane’ [2]. He also introduced a quantity perpendicular to this plane, which has the direction of the conventional intensity, and which he called the ‘directrix’. The magnetic intensity in the nineteenth century was generally represented by Ampere’s ‘directrix’ function, rather than by his ‘directive plane’. Mathematical investigations in the latter part of the 19th century began to reveal that the magnetic intensity has curious structural features. Hermann von Helmholtz (1821–1894) in 1858 drew attention to the close mathematical analogy between vortex motion and the magnetic intensity of an electric current [3]. Emil Wiechert (1861–1928) in 1899 may have been the first to recognize that Ampère’s directrix, or magnetic intensity vector, remains unaltered under a coordinate inversion and, therefore, has the mathematical properties of a rotation. He describes it as a rotation vector or ‘rotor’ [4]. Arnold Sommerfeld (1868–1951) in 1910 distinguished between the terms ‘polar’ vector and ‘axial’ vector in electromagnetism and states that the magnetic intensity is an axial vector [5]. 0143-0807/01/030193+11$30.00 © 2001 IOP Publishing Ltd Printed in the UK 193