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Prog. Theor. Phys. Vol. 41 No. 3 (1969) pp. 816-831

[ Full Text PDF : FREE ACCESS (835K) ]

Five Classes of Transformations of Dirac Spinors

— The Free-Particle Dirac Equation Is Brought to “p0-, “p1-, “p2-, “p3- and “m-Linear” Forms —

A. J. Bracken and H. A. Cohen

Department of Mathematical Physics, University of Adelaide, Adelaide, South Australia

(Received September 30, 1968)

Abstract:

The free-particle Dirac equation has two remarkable features: (1) It is linear in all four components of the energy-momentum pµ, and also in the mass m. (2) For its solutions there are five distinct simple modes of the invariant scalar product in the momentum representation.
In this paper, a theorem presented by Case is generalized and used to obtain five classes of transformations of the Dirac equation. Every transformation in a given class has two properties characteristic of the class: (1) The linearity in a corresponding one of the five quantities pµ, m is maintained in the transformed equation. (In this way “p0-, “p1-, “p2-, “p3- and “m-linear” forms of the Dirac equation are obtained.) (2) A corresponding mode of the invariant scalar product is presented. Thus all five classes consist of canonical transformations.
Included amongst the “p0-linear” forms are the Foldy-Wouthuysen-Tani equation, and the one commonly attributed to Cini and Touschek, together with equations appropriate to limiting situations other than the non-relativistic and extreme relativistic ones. The “canonical” form proposed by Chakrabarti is of the “m-linear” type. Belonging to all three of the “p1-, “p2- and “p3-linear” categories is a “p-linear” form of significance for large |p|.


URL : http://ptp.ipap.jp/link?PTP/41/816/
DOI : 10.1143/PTP.41.816

[ Full Text PDF : FREE ACCESS (835K) ] Citation:

References:

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