Using the Quantum Mechanical Path Integral in Quantum Field Theory.

Matrix elements of the form {dollar}\\langle x\\vert\\exp({lcub}-{rcub}iHt)\\vert y\\rangle{dollar} arise when operator regularization, a symmetry-preserving regulating procedure that avoids explicit occurrence of divergences, is employed to do perturbative calculations in quantum field theory. In recent years, it is shown that this matrix element can be evaluated using a path integral encountered in single particle quantum mechanics. This technique has the advantage of eliminating loop momentum integrals and algebraically complicated vertices in gauge theories. We develop this technique to handle various problems. Our approach is similar to but distinct from the one of Polyakov and Strassler which is related to the string based method of Bern and Kosower. The formalism of the quantum mechanical path integral (QMPI) in quantum field theory is demonstrated by several explicit computations. The effect of having to path-order in the QMPI is illustrated by considering a theory with scalar fields, each having a distinct mass. Computations in which the background fields are not plane wave fields are considered. The two-point function {dollar}\\langle A\\sbsp{lcub}\\mu{rcub}{lcub}a{rcub}A\\sbsp{lcub}\\nu{rcub}{lcub}b{rcub}\\rangle{dollar} to both one- and two-loop order in the Chern-Simons field theory is evaluated. The generating functionals of the b-quark decay {dollar}b\\to s\\gamma{dollar} are also examined by applying QMPI.