Evaluation of the Collision Matrix for Dirac Particles in an External Potential

Abstract

The collision matrix for a quantized Dirac field in an external electromagnetic potential is evaluated by a method which utilizes the close relation between the theory of positrons and the formal one-particle Dirac theory. The main mathematical tool employed is a theorem by Baker and Hausdorff on exponentials of noncommuting quantities. One obtains the collision matrix expressed as a product of exponentials, involving creation and destruction operators in a way suitable for the calculation of matrix elements in the occupation number representation. In the Appendix the evaluation of the collision matrix for a real scalar field in interaction with a given source distribution is carried out by means of a similar technique.