arXiv quant-ph/98030551998-03quant-phgr-qc

Kochen-Specker定理的Topos观点：I.作为广义赋值的量子态

A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations

C. J. Isham, J. Butterfield

备注：: Clarification of situation for situation for operators with continuous spectra

摘要：

Kochen-Specker定理断言不可能以保持它们之间的函数关系的方式为量子量赋值。我们构造了一种新的类型的估值定义在所有运营商，并尊重一个适当的版本的功能组成原则。赋给命题的真值是（i）语境的;（ii）多值的，其中语境空间和每个语境的多值逻辑自然来自topos的预层理论。在我们的理论中的第一步是证明Kochen-Specker定理等价于这样的陈述，即定义在自伴算子范畴上的某个预层没有整体元素。然后，我们展示了如何使用从预层理论中得出的思想来定义量子理论中的广义赋值，其值是算子的筛子。特别地，我们展示了每个量子态如何导致这样一个广义估值。

英文摘要：

The Kochen-Specker theorem asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalized valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalized valuation.

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