On the invariance groups of relativistic equations for the spinning particles interacting with external fields

Abstract

All relativistic free-particle motion equations, including the Dirac and Kemmer– Duffin–Petiau (KDP) ones, are invariant under the Poincare group P1,3. But such a group does not exhaust symmetry of the relativistic equations. It has been shown in [1] with help of non-Lie method, that any Poincare-invariant equation for a free particle with spin s ≥ 12 has additional invariance under SU2 ⊗ SU2 group. The same invariance group is possessed by Maxwell equations [2]. It has been shown in [3, 4], that the free equations of KDP (for s = 1) and of Rarita–Schwinger (for s = 32 ) have more extensive symmetry group than the group SU2 ⊗ SU2. It follows from the results of these papers, that any relativistic equation for a free particle of spin s ≥ 1 possesses SU3 symmetry. In this note, which is an extention of the paper [4], the invariance groups of the Dirac and KDP equations for the particles, interacting with an external field have been established. Theorem 1. The Dirac equation with the Pauli-type interaction