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Lie Groups, Physics, and Geometry by Robert Gilmore

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15

Maxwell’s equations

 

 

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 ...

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