Polarized States on the Weyl Algebra

Abstract

A state $\sigma$ on the Weyl algebra $A(V,\Omega )$ over a real symplectic vector space $(V,\Omega )$ is polarized when the vectors $v$ for which the absolute value of $\sigma$ on the corresponding Weyl generator ${\delta }_v$ is unity constitute a maximal $\Omega$-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.