Unitary Representations of the inhomogeneous Lorentz Group and their Significance in Quantum Physics

TL;DR

This paper reviews Wigner's and Mackey's theories on the inhomogeneous Lorentz group representations, highlighting their importance in describing quantum fields and establishing the spin-statistics connection.

Contribution

It provides a unified overview of the mathematical framework for inhomogeneous Lorentz group representations and their application to quantum field theory.

Findings

Unified description of free classical and quantum fields for arbitrary spin

Demonstration that locality implies the spin-statistics connection

Clarification of the mathematical structure underlying quantum field representations

Abstract

In honor of Minkowski's great contribution to Special Relativity, celebrated at this conference, we first review Wigner's theory of the projective irreducible representations of the inhomogeneous Lorentz group. We also sketch those parts of Mackey's mathematical theory on induced representations which are particularly useful for physicists. As an important application of the Wigner-Mackey theory, we shall describe in a unified manner free classical and quantum fields for arbitrary spin, and demonstrate that locality implies the normal spin-statistics connection.