Polarized States on the Weyl Algebra

Abstract

A state σ on the Weyl algebra A(V,Ω) over a real symplectic vector space (V,Ω) is polarized when the vectors v for which the absolute value of σ on the corresponding Weyl generator δv is unity constitute a maximal Ω-integral additive subgroup of V ; such a state is necessarily pure, but has inseparable carrier space and is not regular. We determine fundamental properties of such states: in particular, we decide precisely when the GNS representations associated to a pair of polarized states are unitarily equivalent and decide precisely when a given symplectic automorphism is unitarily implemented in the GNS representation associated to a given polarized state.