Minimal length scale in quantum mechanics from a deformation quantization perspective

Abstract We develop a deformation quantization approach to quantum mechanics that exhibits a nonzero minimal uncertainty in position through the modification of commutation relations for the operators of position and momentum. Deformation quantization is a method of quantizing a classical Hamiltonian system by suitably deforming the Poisson algebra of the system. The developed theory of deformation quantization is non-formal. An appropriate integral formula for the star-product is introduced, along with a suitable space of functions on which the star-product is well defined. A C* -algebra of observables and a space of states are constructed. Moreover, an operator representation in momentum space is presented. Finally, an example of states of maximal localization is given in terms of generalized Wigner functions.