From probabilities to mathematical structure of quantum mechanics

A large part of the mathematical formalism of quantum mechanics including the probability density, probability density current, uncertainty relations, wave function, momentum operator, kinetic energy, commutation relations and potentials is related to the probabilistic character of quantum measurements. Equations of motion of quantum mechanics can be obtained from the relativistic invariance of the space–time Fisher information. Some properties of potentials and existence of antiparticles are related to the CPT symmetry. Operators representing parallel or serial physical events in experimental setups can be represented by the sum or product of the corresponding operators.