Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2004-06-28
Physics
High Energy Physics
High Energy Physics - Theory
28 pages, Latex, no figures, 7 problems, download associated Mathematica notebook at http://ox.wit.edu/~shurtleffr/; Version 2
Scientific paper
In the quantum theory of fields one writes the relativistic field operator as a linear combination of annihilation operators, with invariant coefficient functions. The annihilation operators transform as physical, massive, single particle states with a unitary representation of the Poincare group, while the relativistic field operator transforms with a nonunitary spin 1/2 representation of the homogeneous Lorentz group. The Lorentz group represents translations trivially, i.e. as multipliction by unity. Here the nonunitary representation is provided with translation matrices, so that the unitary and the nonunitary representations represent the same group, the Poincare group. Translation matrix invariance is shown to give the free particle Dirac equation, without invoking parity. The coefficient functions for a given momentum determine a current. These currents turn out to be, within a constant factor, the electromagnetic vector potential of the free particle source moving with that momentum. Thus it is shown that the Dirac and Maxwell equations can be related to the inclusion of translation matrices in the transformations of field operators.
Shurtleff Richard
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