Wigner Rotations and Precession of Polarization

Abstract

An easy and direct derivation of Thomas precession is obtained from infinitesimal Wigner rotations arising in unitary representations of the Poincare group. For spin > 1/2, multipole parameters are studied from this point of view. The canonical 3-component definition of polarization arising naturally in this context is compared with formalisms which start form a pseudo 4-vector and an antisymmetric tensor respectively. The full Thomas equations, including Larmor precession, is derived using time derivatives of finite Wigner rotations. Exact solutions, with arbitrary initial conditions, are presented for constant magnetic fields and for orthogonal constant electric and magnetic fields. For a class of plane wave external fields exact solutions are obtained for the Dirac equation generalized by the inclusion of anomalous magnetic moment (Pauli) and electric dipole moment terms. Using the front form of dynamics, well-adapted to this context and coinciding with proper time dynamics, expectation values are calculated. The polarization pseudo 4-vector thus obtained is shown to satisfy the BMT equation, which is equivalent to the Thomas equation. This shows that the validity of the classical precession equations is not necessarily restricted to slowly varying external fields. These solutions can also be of interest in the study of spin 1/2 particles in laser fields and in the study of electric dipole moments.