# Polarized States on the Weyl Algebra

### Paul Lee Robinson

University of Florida, Gainesville, USA

## 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 $δ_{v}$ 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.

## Cite this article

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

DOI 10.4171/PRIMS/39.3.1