Positive-type functions on groups and new inequalities in classical and quantum mechanics

Abstract

Out of any unitary representation of a group, positive-type functions on the group can be obtained. These functions allow one to construct positive semi-definite matrices that may be used to define new inequalities for higher moments of observables associated with classical probability distribution functions and density states of quantum systems. The inequalities stemming from the Heisenberg–Weyl group representations are considered in connection with Gaussian distributions. We obtain new inequalities for multi-variable Hermite polynomials.