On the gauge invariance of Wigner–Dunkl quantum mechanics in the presence of a constant magnetic field

In this paper we consider a charged Wigner–Dunkl quantum system in the presence of a constant magnetic field. It is shown that this system obeys gauge invariance if minimally coupled to a vector potential following the Dunkl-Maxwell relations. A family of vector potentials, which generate the constant magnetic field, is constructed explicitly. The gauge invariance of the Wigner–Dunkl quantum system is established with a gauge transformation exhibiting a deformed unitarity. For vector potentials following the standard Maxwell relations it is not possible to establish gauge invariance for the Wigner–Dunkl quantum system.