Wigner rotations, Bargmann invariants and geometric phases

Abstract

The concept of the 'Wigner rotation', familiar from the composition law of (pure) Lorentz transformations, is described in the general setting of Lie group coset spaces and the properties of coset representatives. Examples of Abelian and non-Abelian Wigner rotations are given. The Lorentz group Wigner rotation, occurring in the coset space SL(2, R)/SO(2) ⋍ SO(2, 1)/SO(2), is shown to be an analytic continuation of a Wigner rotation present in the behaviour of particles with nonzero helicity under spatial rotations, belonging to the coset space SU(2)/U(1) ⋍ SO(3)/SO(2). The possibility of interpreting these two Wigner rotations as geometric phases is shown in detail. Essential background material on geometric phases, Bargmann invariants and null phase curves, all of which are needed for this purpose, is provided.