The Dirac Equation- Old and New

Peter Rastall

Kamal Barley, Andreas L. Ruffing, Sergeĭ K. Suslov

We review Bohr's atomic model and its extension by Sommerfeld from a mathematical perspective of wave mechanics.The derivation of quantization rules and energy levels is revisited using semiclassical methods.Sommerfeld-type integrals are evaluated by elementary techniques, and connections with the Schr odinger and Dirac equations are established.Historical developments and key transitions from classical to quantum theory are discussed to clarify the structure and significance of the old quantum mechanics.

Steven Kenneth Kauffmann

A carefully motivated symmetric variant of the Poisson bracket in ordinary (not Grassmann) phase space variables is shown to satisfy identities which are in algebraic correspondence with the anticommutation postulates for quantized Fermion systems. "Symplecticity" in terms of this symmetric Poisson bracket implies generalized Hamilton's equations that can only be of Schroedinger type (e.g., the Dirac equation but not the Klein-Gordon or Maxwell equations). This restriction also excludes the old "four-Fermion" theory of beta decay.

Miroslav Pardy

The equivalent system of equations corresponding to the Dirac equation is derived and the WKB approximation of this system is found. Similarly, the WKB approximation for the equivalent system of equation corresponding to the squared Dirac equation is found and it is proved that the Lorentz equation and the Bargmann-Michel-Telegdi iquations follow from the new Dirac-Pardy system. The new tensor equation with sigma matrix is derived for the verification by adequate laboratories.