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This may be done more precisely, taking spin and antiparticle states into account, using the ""Foldy-Wouthuysen Transformation"" (courtesy of [[http://www.physics.ucdavis.edu/~cheng/ Hsin-Chia Cheng]]; for a general discussion, see ""Costella & McKellar»""). For the present treatment, I will merely show the Lorentz force law for particles, ignoring spin.
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The term [[http://en.wikipedia.org/wiki/Gauge_symmetry gauge symmetry]] is something of a misnomer. It was introduced by [[http://en.wikipedia.org/wiki/Hermann_Weyl Herman Weyl]], as part of an attempt to extend the local scale invariance of general relativity to unification with electrodynamics. That attempt failed, but later Weyl, [[http://en.wikipedia.org/wiki/Vladimir_Fock Vladimir Fock]] and [[http://en.wikipedia.org/wiki/Fritz_London Fritz London]] adapted the idea and applied it to phase symmetry in quantum theory, and it is to phase symmetry that the term now applies. The relation to the of this phase symmetry to a corresponding symmetry in classical electrodynamics is shown in the section ""Gauge Invariance"".
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The term [[http://en.wikipedia.org/wiki/Gauge_symmetry gauge symmetry]] is something of a misnomer. It was introduced by [[http://en.wikipedia.org/wiki/Hermann_Weyl Herman Weyl]], as part of an attempt to extend the local scale invariance of general relativity to unification with electrodynamics. That attempt failed, but later Weyl, [[http://en.wikipedia.org/wiki/Vladimir_Fock Vladimir Fock]] and [[http://en.wikipedia.org/wiki/Fritz_London Fritz London]] adapted the idea and applied it to phase symmetry in quantum theory, and it is to phase symmetry that the term now applies. The relation to the of this phase symmetry to a corresponding symmetry in classical electrodynamics is shown ""Gauge Invariance"".
