POSITIVE ELEMENTS IN THE ALGEBRA OF THE QUANTUM MOMENT PROBLEM

• 成果类型: Article
• 署名作者: JORGENSEN, PET; POWERS, RT
• 署名单位: University of Pennsylvania
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/BF01366901
• 发表日期: 1991
• 页码: 131-139
• 关键词: operators

摘要

Let U denote the extended Weyl algebra, U0 subset-of U, the Weyl algebra. It is well known that every element of U of the form A = SIGMA-B(k)* B(k) is positive. We prove that the converse implication also holds: Every positive element A in U has a quadratic sum factorization for some finite set of elements (B(k)) in U. The corresponding result is not true for the subalgebra U0. We identify states on U0 which do not extend to states on U. It follows from a result of Powers (and Arveson) that such states on U0 cannot be completely positive. Our theorem is based on a certain regularity property for the representations which are generated by states on U, and this property is not in general shared by representations generated by states defined only on the subalgebra U0.