量子困境：范畴论视角
Quantum Quandaries: a Category-Theoretic Perspective
摘要：
广义相对论似乎与量子理论有很大的不同，但对量子引力的研究揭示了两者之间的深刻相似性。广义相对论大量使用nCob范畴，其对象是代表“空间”的（n-1）维流形，其态射是代表“时空”的n维协边。量子理论大量使用了希尔伯特范畴，它的对象是用来描述“状态”的希尔伯特空间，它的态射是用来描述“过程”的有界线性算子。此外，范畴nCob和Hilb彼此之间的相似程度远远超过它们中的任何一个与Set的相似程度，Set是一个对象是集合、态射是函数的范畴。特别地，Hilb和nCob都是具有非Carbohydronoidal结构的 *-范畴，但Set不是。我们说明了这是如何解释量子理论中许多著名的令人困惑的特征的：局域实在论的失败，量子信息的不可能复制，等等。我们认为，只有当我们试图把Hilb看作是集合而不是nCob的类似物时，这些特征才显得令人困惑，因此，当把量子理论看作是时空理论的一部分时，它才更有意义。
英文摘要：
General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose objects are (n-1)-dimensional manifolds representing "space" and whose morphisms are n-dimensional cobordisms representing "spacetime". Quantum theory makes heavy use of the category Hilb, whose objects are Hilbert spaces used to describe "states", and whose morphisms are bounded linear operators used to describe "processes". Moreover, the categories nCob and Hilb resemble each other far more than either resembles Set, the category whose objects are sets and whose morphisms are functions. In particular, both Hilb and nCob but not Set are *-categories with a noncartesian monoidal structure. We show how this accounts for many of the famously puzzling features of quantum theory: the failure of local realism, the impossibility of duplicating quantum information, and so on. We argue that these features only seem puzzling when we try to treat Hilb as analogous to Set rather than nCob, so that quantum theory will make more sense when regarded as part of a theory of spacetime.
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