Geometry of the Diric theory - In: A Symposium on the...

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In: A Symposium on the Mathematics of Physical Space-Time , Facultad de Quimica, Universidad Nacional Autonoma de Mexico, Mexico City, 67–96, (1981). GEOMETRY OF THE DIRAC THEORY David Hestenes ABSTRACT. The Dirac wave function is represented in a form where all its components have obvious geometrical and physical interpretations. Six components compose a Lorentz transformation determining the electron velocity are spin directions. This provides the basis for a rigorous connec- tion between relativistic rigid body dynamics and the time evolution of the wave function. The scattering matrix is given a new form as a spinor-valued operator rather than a complex function. The approach reveals a geomet- ric structure of the scattering matrix and simplifies scattering calculations. This claim is supported by an explicit calculation of the differential cross- section and polarization change in Coulomb scattering. Implications for the structure and interpretation of relativistic quantum theory are discussed. INTRODUCTION The Dirac equation is one of the most well-established equations of physics, having led to a great variety of detailed predictions which have been experimentally confirmed with high precision. Yet the relativistic quantum theory based on the Dirac equation has never been given a single complete and selfconsistent physical interpretation which all physicists find satisfactory. Moreover, it is generally agreed that the theory must be modified to account for the electron mass, but there is hardly agreement on how to go about it. This paper reviews and extends results from a line of research (Ref. [1–7]) aimed at clarifying the Dirac theory and simplifying its mathematical formulation. Of course, any such improvement in so useful a theory would be valuable in itself. But the ultimate goal is to achieve insight into the structure of the theory which identifies those features responsible for its amazing results, as well as features which might be modified to improve it. The central result of this research is a formulation of the Dirac spinor wave function which reveals the geometrical and physical interpretation of all its components. This makes it possible to relate the time evolution of the wave function to relativistic rigid body mechanics, thus giving insight into the dynamics and establishing a connection with classical theories of spinning bodies. A fairly detailed review of these results is contained in this paper. In addition, the general solution of the Bargmann-Michel-Telegdi equation for constant fields is obtained in simple form from a spinor formulation of the theory. Most of the new results in this paper arise from a reformulation of scattering theory in accord with the above ideas. A new spinor formulation of the S -matrix is obtained which combines the conventional spin scattering amplitudes into a meaningful unit. This makes it possible to relate the interpretation of the S -matrix to relativistic rigid body mechanics. Moreover, calculations are greatly simplified. Thus, the scattering cross section
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Geometry of the Diric theory - In: A Symposium on the...

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