Spin-Statistics Theorem in Path Integral Formulation

TL;DR

This paper provides a rigorous proof of the spin-statistics theorem within the path integral framework, demonstrating how causality, positive energy, and positive norm conditions exclude abnormal statistics for scalar and Dirac particles.

Contribution

It offers a coherent proof of the spin-statistics theorem using path integral methods, clarifying the roles of measure, Lagrangian, and prescriptions in ensuring correct statistics.

Findings

Path integral formulation enforces causality and positive energy conditions.

Abnormal spin-statistics relations are excluded under positive norm and Schwinger's principle.

The approach applies to both 4D and 2D theories, confirming the theorem's robustness.

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 Green's functions defined in terms of time ordered products, ensure causality regardless of statistics. The Feynman's $m-i\epsilon$ prescription ensures the positive energy condition regardless of statistics, and the abnormal spin-statistics relation for both of 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 path integral. The Feynman's $m-i\epsilon$ 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 4-dimensional theory but also in 2-dimensional theory.