从观测量和状态到希尔伯特空间再返回：2-范畴的附加
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，将其看作是物理可观测量的抽象集合，加上一个单位向量，人们通过限制获得代数上的一个状态。我们证明了Gelfand-Naimark-Segal（GNS）构造是这个限制的一个左伴随。为了正确地表述这个伴随，它必须被看作是一个弱自然变换，一个合适的2-范畴中的1-态射，而不是范畴之间的函子。弱自然性编码了GNS结构的功能性，而互操作的泛属性编码了GNS结构的特征性。数学定义和结果伴随着物理解释。
英文摘要：
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.
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