Probing Wigner rotations for any group

Narasimhaiengar Mukunda, P. K. Aravind, Richard M Simon

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.

Brian L. Silver

Jean-Pierre Amiet, Stefan Weigert

The Moyal formalism for a particle can be derived from the Moyal formalism for a spin. This is done by contracting the group of rotations to the oscillator group. A new derivation is given for the contraction of the spin Wigner-kernel to the Wigner kernel of a particle.

Jean-Pierre Amiet, Stefan Weigert

A general relation between the Moyal formalisms for a spin and a particle is established. Once the formalism has been set up for a spin, the phase-space description of a particle is obtained from contracting the group of rotations to the oscillator group. In this process, turn into a spin Wigner kernel turns into the Wigner kernel of a particle. In fact, only one out of ${2}^{2s}$ different possible kernels for a spin shows this behavior.

Alex Grossmann, P. Huguenin