Physics, Topology, Logic and Computation:
Physics, Topology, Logic and Computation:
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发表时间：
2009-03
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通讯作者：
J. Baez;M. Stay
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作者：
J. Baez;M. Stay
在物理学中，费曼图被用来推理量子过程。到了20世纪80年代，人们发现这些图的背后是量子物理和拓扑学之间的一个强有力的类比。也就是说，线性算子的行为非常类似于“配边”：一个表示时空的流形，在两个表示空间的流形之间移动。这导致了拓扑量子场论和“量子拓扑学”的工作爆发。但这仅仅是个开始：类似的diag ram可以用来推理逻辑，它们代表证明，和计算，它们代表程序。随着人们对量子密码学和量子计算的兴趣的增加，很明显，在物理学、拓扑学、逻辑学和计算之间存在着广泛的类比网络。在本文中，我们使用“闭对称monoidal范畴”的概念使其中的一些类比精确化。我们假设没有范畴论，证明论或计算机科学的先验知识。
In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology. Namely, a linear operator behaves very much like a ‘cobordism’: a manifol d representing spacetime, going between two manifolds representing space. This led to a burst of work on topological quantum field theory and ‘quantum topology’. But this was just the beginning: similar diag rams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of ‘closed symmetric monoidal category’. We assume no prior knowledge of category theory, proof theory or computer science.