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General transformation matrix for Dirac spinors and the calculation of spinorial amplitudes

### Abstract

A general transformation matrix *T*( *p*̃,*s*̃;*p*,*s*) is constructed which transforms a Dirac spinor ψ( *p*,*s*) into another Dirac spinor ψ( *p*̃,*s*̃) with arbitrarily given momenta and polarization states by exploiting the so‐called Stech operator as one of generators for those transformations. This transformation matrix is then used in a calculation to yield the spinorial matrix element *M*=ψ̄( *p*̃,*s*̃)Γψ( *p*,*s*) for any spin polarization state. The final expressions of these matrix elements show the explicit structure of spin dependence of the process described by these spinorial amplitudes. The kinematical limiting cases such as very low energy or high energy of the various matrix elements can also be easily displayed. Our method is superior to the existing one in the following points. Since we have a well‐defined transformation operator between two Dirac spinor states, we can evaluate the necessary phase factor of the matrix elements in an unambiguous way without introducing the coordinate system. This enables us to write down the Feynman amplitudes of complicated processes in any spin basis very easily in terms of previously calculated matrix elements of ψ̄Γψ which are building blocks of those Feynman amplitudes. In contrast, in the existing method, one has to figure out the suitable multiplication factor for each individual case. Furthermore, in the previous method one needs a coordinate system to evaluate phase factors, thus making the calculation more cumbersome. Also, the expression of the matrix element in the previous method is in the fractional form whose denominator may have a singularity structure (and not always an isolated one at that), while the spinorial amplitudes in our method take a much simpler form, free of coordinate systems in any spin basis. The usefulness of the results is illustrated on Compton scattering and on the elasticscattering of two identical massive leptons where the phase factor is important. It is also shown that the Stech operator as a polarization operator is simply related to the operator *K*=β(Σ⋅**L**+1)/2, which is often used in bound state problems.

© 1984 American Institute of Physics

Received 08 September 1982
Accepted 03 June 1983

/content/aip/journal/jmp/25/4/10.1063/1.526230

http://aip.metastore.ingenta.com/content/aip/journal/jmp/25/4/10.1063/1.526230

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/content/aip/journal/jmp/25/4/10.1063/1.526230

1984-04-01

2016-10-21

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