标题： 从可观测量和状态到希尔伯特空间再回到：二维范畴的伴随关系 显示英文标题

标题： From observables and states to Hilbert space and back: a 2-categorical adjunction

摘要： 给定一个C*-代数的表示，作为物理可观测量的抽象集合，再加上一个单位向量，通过限制可以得到代数上的一个状态。 我们证明Gelfand-Naimark-Segal（GNS）构造提供了此限制的左伴随。 为了正确表述这个伴随，必须将其视为弱自然变换，即在合适的2-范畴中的1-态射，而不是作为范畴之间的函子。 弱自然性编码了函子性，而伴随的普遍性质编码了GNS构造的特征属性。 数学定义和结果都伴随着物理解释。 显示英文摘要

摘要： Given a representation of a C*-algebra, thought of as an abstract collection of physical observables, together with a unit vector, one obtains a state on the algebra via restriction. We show that the Gelfand-Naimark-Segal (GNS) construction furnishes a left adjoint of this restriction. To properly formulate this adjoint, it must be viewed as a weak natural transformation, a 1-morphism in a suitable 2-category, rather than as a functor between categories. Weak naturality encodes the functoriality and the universal property of adjunctions encodes the characterizing features of the GNS construction. Mathematical definitions and results are accompanied by physical interpretations.