Unitary Representations of the Inhomogeneous Lorentz Group and Their Significance in Quantum Physics

Abstract

Minkowski’s great discovery of the spacetime structure behind Einstein’s special theory of relativity (GlossaryTerm

SR

) had an enormous impact on much of twentieth-century physics. (For a historical account of Minkowski’s Raum und Zeit lecture and Poincaré’s pioneering contribution, we refer to [1] and Chap. 2.) The symmetry requirement of physical theories with respect to the automorphism group of Minkowski spacetime – the inhomogeneous Lorentz or Poincaré group – is particularly constraining in the domain of relativistic quantum theory and led to profound insights. Among the most outstanding early contributions are Wigner’s great papers on relativistic invariance [2]. His description of the (projective) irreducible representations of the inhomogeneous Lorentz group, that classified single particle states in terms of mass and spin, has later been taken up on the mathematical side by George Mackey, who developed Wigner’s ideas into a powerful theory with a variety of important applications [3] [4] [5]. Mackey‘s theory of induced representations has become an important part of representation theory for locally compact groups. For certain classes it provides a full description of all irreducible unitary representations.

We find it rather astonishing that this important classical subject is not treated anymore in most modern textbooks on quantum field theory.

I shall begin with general remarks on symmetries in quantum theory, and then repeat Wigner’s heuristic analysis of the unitary representations of the homogeneous Lorentz group (more precisely, of the universal covering group of the one-component of that group). This will lead us to those parts of Mackey’s theory of induced representations which are particularly useful for physicists. In the final section, we shall describe free classical and quantum fields for arbitrary spin, and show that locality implies the normal spin–statistics connection. We shall see that the theory of free fields is a straightforward application of Wigner’s representations of the inhomogeneous Lorentz group. (Since the quantum theory for massless fields poses delicate problems – as is well known for spin 1 – we treat only the massive case.)