A Quantum Mechanical Evolution Equation for Mixed States from Symmetry and Kinematics

The idea of this contribution goes back to an article published in Symmetry in Science [1]. We proposed there a quantization method for a system S localized on a smooth Riemannian manifold (M,g) and presented preliminary results for the kinematics which were developed and formulated systematically and rigorously in [2,3], with applications in [4]. This approach, the Quantum Sorel Kinematics, is based geometrically on a representation of a pair (B (M), Vect (M)), or equivalently S (M) = (C ∞ (M, ℝ), Vect (M)), on some Hilbert space H,with B (M) as the Borel field and Vect(M) as the Lie algebra of smooth vector fields on M,and C ∞ (M,ℝ) as the space of smooth functions. For M = ℝ3 the results of the quantization were derived independently in connection with a representation of a certain subgroup of the diffeomorphism group Diff(M) of M in [5]. The pair is a purely kinematical quantity. Sorel sets are generalized positions and vector fields are generalized momenta. To describe a dynamical stuation, S must be furnished with a time dependence. A conventional method to do this is to write an evolution equation or a class of evolution equations. The choice in classical mechanics for point particles is the class of second order (or Newtonian) equations. A construction of a quantum analogue of this class has to be based on the quantization of the kinematics, i.e. on the unitarily inequivalent Quantum Borel Kinematics.