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Maxwell's equations
Book chapter

Maxwell's equations

Robert Gilmore
Lie Groups, Physics, and Geometry, pp 259-283
17 Jan 2008
DOI: https://doi.org/10.1017/CBO9780511791390.016

Abstract

The electromagnetic field E(x, t), B(x, t) is determined by Maxwell's equations. These equations are linear in the space and time derivatives. In the momentum representation, obtained by taking a Fourier transform of the electric and magnetic fields, Maxwell's equations impose a set of four linear constraints on the six amplitudes E(k), B(k). Why? At a more fundamental level, the electromagnetic field is described by photons. For each photon momentum state there are only two degrees of freedom, the helicity (polarization) states, corresponding to an angular momentum 1 aligned either in or opposite to the direction of propagation. Thus, the classical description of the electromagnetic field is profligate, introducing six amplitudes for each k when in fact only two are independent. The remaining four degrees must be absent in any description of a physically allowed field. The equations that annihilate these four nonphysical linear combinations are the equations of Maxwell. We derive these equations, in the absence of sources, by comparing the transformation properties of the helicity and classical field states for each four-momentum.IntroductionThe electromagnetic field has been described in two different ways. Following the nineteenth century approach (pre quantum mechanics), a field is introduced having appropriate transformation properties. The price one pays is that not every field represents a physically allowed state: such fields must be annihilated by appropriate equations.

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Mathematics, Applied
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