Conserved Quantities Associated with Symmetry Transformations of Relativistic Free-Particle Equations of Motion

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Abstract
A general technique is presented for associating conservation laws with the symmetry transformations that leave invariant the relativistic equations of motion for a free particle. These transformations may be either continuous with the identity (such as infinitesimal transformations) or discontinuous (such as reflections). It is found that for each transformation there exist two classes of conservation laws. The number of separate laws within a class depends on the spin of the particle. The particular cases of the Dirac equation and Maxwell's equations are investigated in some detail. For the Dirac equation, conserved quantities involving discontinuous transformations and also matrix elements between particle and antiparticle states are obtained, in addition to the usual conservation laws. Application of the general method to Maxwell's equations yields not only the usual conserved quantities and Lipkin's ``zilch,'' but also twenty new gauge-independent conserved quantities and other additional integrals associated with discontinuous transformations.

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