The Cornerstone Of Spin Statistics Connection: The SU(2)$\times$ C $\times$ T Symmetry

TL;DR

This paper explores the fundamental reasons behind the spin statistics connection, showing that certain symmetries and spectral properties enforce this connection, with exceptions like the Schrödinger equation.

Contribution

It identifies symmetry conditions and spectral features that determine whether a free field theory obeys the spin statistics connection, clarifying the role of $C$, $T$, and SU(2) invariance.

Findings

Rotational ($SU(2)$) invariance combined with $C$ and $T$ symmetries enforces spin statistics connection.

The Schrödinger equation is a special case that does not obey the connection.

Particles with energy spectra involving branch points in the complex plane must obey the spin statistics connection.

Abstract

We investigate the intrinsic reason for spin statistics connection. It is found that if a free field theory is rotationally (SU(2)) invariant, and has time reversal ($T$) and charge conjugation ($C$) symmetries, it obeys the spin statistics connection, except for a special case. Shr{\"o}dinger equation belongs to this special case, and does not obey spin statistics connection. Further we show that if the energy spectrum of a particle $E(p)$ takes the form of a square root, namely contains branch points in the complex $p$ plane, the particle cannot belong to this special case, and must obey the spin statistics connection. This conclusion includes the relativistic particles as a particular example.