- 03-109 Tepper L. Gill and W. W. zachary
- Analytic Representation of Relativistic
Wave Equations I: The Dirac Case
Mar 12, 03
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Abstract. In this paper we construct an analytical separation (diagonalization) of the full (minimal coupling) Dirac equation into particle and antiparticle components. The diagonalization is analytic in that it is achieved without transforming the wave functions, as is done by the Foldy-Wouthuysen method, and reveals the nonlocal time behavior of the particle-antiparticle relationship. It is well known that the Foldy-Wouthuysen transformation leads to a diagonalization that is nonlocal in space. We interpret the zitterbewegung, and the result that a velocity measurement (of a Dirac particle) at any instant in time is +(-)c, as reflections of the fact that the Dirac equation makes a spatially extended particle appear as a point in the present by forcing it to oscillate between the past and future at speed c. This suggests that although the Dirac Hamiltonian and the square-root Hamiltonian, are mathematically, they are not physically, equivalent. Furthermore, we see that although the form of the Dirac equation serves to make space and time appear on an equal footing mathematically, they are still not on an equal footing from a physical point of view. It appears that the only way to justify a physical relationship between the Dirac and the square-root equations is via their relationship to the Klein-Gordon equation.
We then show explicitly that the Pauli equation is not valid for an analysis of the Dirac hydrogen atom problem in s-states (hyperfine splitting). We conclude that there are serious physical and mathematical problems with any attempt to show that the Dirac equation is insufficient to explain the full hydrogen spectrum.