Proof of Spin-Statistics Theorem in Quantum Mechanics of Identical Particles

TL;DR

This paper provides a nonrelativistic proof of the spin-statistics theorem in quantum mechanics, linking particle spin to their commutation relations through field operators and permutation symmetry.

Contribution

It introduces a nonrelativistic proof of the spin-statistics theorem using coordinate space field operators and permutation symmetry, offering an alternative to relativistic approaches.

Findings

Integral-spin particles obey commutation relations.

Half-integral-spin particles obey anticommutation relations.

The proof connects rotational properties to particle statistics.

Abstract

A nonrelativistic proof of the spin-statistics theorem is given in terms of the field operators satisfying commutation and anticommutation relations, which are introduced here in the coordinate space as a means to build the permutation symmetry into the brackets of identical particles. An eigenvalue problem of a $\pi$-rotation for a product of two annihilation operators is combined with an analysis on its rotational property to prove the connection that the field operators for integral-spin and half-integral-spin particles obey the commutation and anticommutation relations, respectively.