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 δ_v_ 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.
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Paul Lee Robinson, Polarized States on the Weyl Algebra. Publ. Res. Inst. Math. Sci. 39 (2003), no. 3, pp. 415–434DOI 10.4171/PRIMS/39.3.1