退相干泛函在历史量子理论中的表示理论
The representation theory of decoherence functionals in history quantum theories
摘要：
本文第一部分回顾了量子理论发展的历史概况。历史量子理论提供了一个概念和数学框架，以制定量子理论没有一个全球定义的哈密顿时间演化和引入时空事件的概念到量子理论。在数学水平上，历史量子理论的特征在于代表时空事件的历史空间和代表历史方法中量子力学态的退相干泛函空间。本文的第二部分详细研究了某些物理合理的历史空间的退相干泛函空间的结构。标准哈密顿量子理论的时间重构建议考虑这样一种情况，即历史空间由（i）某个希尔伯特空间上的投影算子格，或者--稍微更一般一些--（ii）某个冯诺依曼代数中的投影算子集给出。在情形（i）中，我们确定了退相干泛函可以分别由底层Hilbert空间的张量积上的迹类算子、有界算子或迹类算子族表示的条件。此外，我们将讨论自然产生的表示的退相干泛函作为sesquilinear形式。本文最后讨论的后果的一般公理框架的历史理论的结果。
英文摘要：
In the first part of this paper the general perspective of history quantum theories is reviewed. History quantum theories provide a conceptual and mathematical framework for formulating quantum theories without a globally defined Hamiltonian time evolution and for introducing the concept of space time event into quantum theory. On a mathematical level a history quantum theory is characterized by the space of histories, which represent the space time events, and by the space of decoherence functionals which represent the quantum mechanical states in the history approach. The second part of this paper is devoted to the study of the structure of the space of decoherence functionals for some physically reasonable spaces of histories in some detail. The temporal reformulation of standard Hamiltonian quantum theories suggests to consider the case that the space of histories is given by (i) the lattice of projection operators on some Hilbert space or -- slightly more general -- (ii) the set of projection operators in some von Neumann algebra. In the case (i) the conditions are identified under which decoherence functionals can be represented by, respectively, trace class operators, bounded operators or families of trace class operators on the tensor product of the underlying Hilbert space by itself. Moreover we shall discuss the naturally arising representations of decoherence functionals as sesquilinear forms. The paper ends with a discussion of the consequences of the results for the general axiomatic framework of history theories.
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