Foundations of Quantum Mechanics: On Rotations by $4\pi$ for Half-Integral Spin Particles

Abstract

Rotations in Quantum Mechanics are a very well-known subject. When one is faced with rotations related to the SO(3) group, for instance, all the underlying operators are well-known and built from their classical counterparts. However, when it comes to represent rotations related to the SU(2) group, it is always argued that there is no classical counterpart from which the expressions for the quantum mechanical operators can be built. The approach is always done using matrix representation (and not by solving some Schrödinger equation). In the way of this aproach, one assumes that the operators related to SU(2) are pseudovectors (or antisymmetric rank two tensors) and end up with the weird result that only rotations by multiples of $4\pi$ would bring the system to its original situation. In Olavo (Physica A 262:181–196, 1999) we have shown that any half-integral spin system can be represented in the Schrödinger picture by solving a Schrödinger equation with well defined expressions for the SU(2) differential operators. It turned out that two of these operators are symmetric tensors, not pseudovectors or antisymmetric tensors. In this paper we use these results to dismiss with the $4\pi$ conundrum.