Further uses of the quantum mechanical path integral in quantum field theory

Abstract

It has been shown how matrix elements of the form 〈x‖exp(−iHt)‖y〉 which arise when using operator regularization to do perturbative calculations in quantum field theory can be evaluated using the quantum mechanical path integral (QMPI). This technique has the advantage of eliminating loop momentum integrals and algebraically complicated vertices in gauge theories. A similar (but distinct) approach of Polyakov and Strassler has been applied to one loop processes with external vector particles and is related to the string based methods of Bern and Kosower. In this article, several features of the QMPI technique are examined. First, it is demonstrated how the path ordering in the QMPI can be handled by considering a model of three interacting scalar fields, each with a distinct mass. Next, it is shown how the QMPI can be used when the external wave function is not a plane wave field. The particular case of having an exponentially damped wave function is considered. Next, a discussion of the difference between the approach of Polyakov and Strassler and that employed here is given. Finally, it is demonstrated how the QMPI can be used to considerably simplify calculations in quantum gravity.