SPIN-STATISTICS THEOREM IN PATH INTEGRAL FORMULATION

Abstract

We present a coherent proof of the spin-statistics theorem in path integral formulation. The local path integral measure and Lorentz-invariant local Lagrangian, when combined with the Green functions defined in terms of time ordered products, ensure causality regardless of statistics. The Feynman m-iε prescription ensures the positive energy condition regardless of statistics, and the abnormal spin-statistics relation for both the spin-0 scalar particles and spin-1/2 Dirac particles is excluded if one imposes the positive norm condition in conjunction with Schwinger's action principle. The minus commutation relation between one Bose and one Fermi field arises naturally in the path integral. The Feynman m-iε prescription also ensures a smooth continuation to Euclidean theory, for which the use of the Weyl anomaly is illustrated to exclude the abnormal statistics for the scalar and Dirac particles not only in four-dimensional theory but also in two-dimensional theory.