Weakly projective representations, quaternions, and monopoles (Q2702382)

The authors propose a model describing quantum kinematics and dynamics of a charged particle in the field of a magnetic monopole which is realized in a Hilbert space of square-integrable quaternionic-valued functions on \(\mathbb R^3 \setminus \{\boldsymbol 0\}\). This model supports the view that there is a definite advantage to use a quaternionic Hilbert space formulation of quantum mechanics based on a generalized system of imprimitivity [\textit{A. Jadczyk}, Int. J. Theor. Phys. 14, 183-192 (1975)], since the quaternionic quantum mechanics leads to a less singular description of the dynamics. In fact, the generators of translations in the quaternionic Hilbert space \(\mathcal H_{\mathbb H}\) of the model have a simple explicit form as differential operators on a dense domain of differentiable functions, but this is not the case for their restriction to the complex Hilbert space defined by unitary anti-hermitian operator \(J\), a fixed imaginary unit \(\omega\), and the functions \(\psi \in \mathcal H_{\mathbb H}\) such that \(J\psi = \psi\omega\). The relation between generalized imprimitivity systems and weakly projective representations [\textit{S. L. Adler}, Quaternionic Quantum Mechanics and Quantum Fields, Oxford Univ. Press, New York (1995; Zbl 0885.00019)] is pointed out.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00049].