The Spin-Statistics Theorem in Arbitrary Dimensions

TL;DR

This paper explores the spin-statistics relationship across various dimensions, revealing dimension-dependent patterns for whether spinor and tensor fields behave as bosons or fermions.

Contribution

It extends the understanding of the spin-statistics theorem to arbitrary dimensions using a simple, unified method applicable beyond the familiar three-dimensional case.

Findings

Standard connection in dimensions 8n+3, 8n+4, 8n+5

Only bosons for spinors in dimensions 8n±1 and 8n

Spinors can be bosons or fermions in dimensions 4n+2

Abstract

We investigate the spin-statistics connection in arbitrary dimensions for hermitian spinor or tensor quantum fields with a rotationally invariant bilinear Lagrangian density. We use essentially the same simple method as for space dimension D = 3. We find the usual connection (tensors as bosons and spinors as fermions) for D = 8n + 3; 8n + 4; 8n + 5, but only bosons for spinors and tensors in dimensions 8n +/- 1 and 8n. In dimensions 4n + 2 the spinors may be chosen as bosons or fermions. The argument hinges on finding the identity representation of the rotation group either on the symmetric or the antisymmetric part of the square of the field representation.