arXiv quant-ph/02050532002-05-10quant-phgr-qchep-th

一种基于Borelian正规数的实在确定性量子理论

A Realistic Deterministic Quantum Theory Using Borelian-Normal Numbers

T. N. Palmer

备注：: 38 pages, 1 figure. Proc.Roy.Soc.Lond. Submitted

摘要：

确定性的量子理论的元素被开发，重新制定和扩展标准量子理论。所提出的理论是“现实的”，在这个意义上，一个一般的M能级量子态是由一个单一的实数r表示。令人惊讶的是，这个实数被证明包含与标准希尔伯特空间状态相同的概率信息，加上确定测量结果的额外信息。实现这一点的一个关键概念是Borelian（数论）正规性。复数在标准量子理论中的重要作用被包含在一组自相似置换算子对一个基M的Borelian-normal r的基M展开的位数和位置的作用中;这些置换算子被证明具有复杂的结构，并保持基本实数的正规性不变。由这些排列产生的实数集不仅定义了标准量子理论的希尔伯特空间，而且还定义了可以客观确定量子测量结果的样本空间。动态实数态约简是由确定性数论算子精确描述的，这些算子降低了r的正态度;从正态度可以推导出测量概率的标准量子理论迹规则。所有的基本困难的标准量子理论的描述提出的理论。结果表明，所提出的理论的实数状态正是它的beables。

英文摘要：

The elements of a deterministic quantum theory are developed, which reformulates and extends standard quantum theory. The proposed theory is `realistic' in the sense that in it, a general M-level quantum state is represented by a single real number r. Surprising as it may seem, this real number is shown to contain the same probabilistic information as the standard Hilbert-space state, plus additional information from which measurement outcome is determined. A crucial concept in achieving this is that of Borelian (number-theoretic) normality. The essential role of complex numbers in standard quantum theory is subsumed by the action of a set of self-similar permutation operators on the digits and places of the base-M expansion of a base-M Borelian-normal r; these permutation operators are shown to have complex structure and leave invariant the normality of the underlying real number. The set of real numbers generated by these permutations defines not only the Hilbert space of standard quantum theory, but also, in addition, the sample space from which quantum measurement outcomes can be objectively determined. Dynamical real-number state reduction is precisely described by deterministic number-theoretic operators that reduce the degree of normality of r; from the degree of normality one can infer the standard quantum-theoretic trace rule for measurement probability. All the foundational difficulties of standard quantum theory are described in terms of the proposed theory. It is shown that the real-number states of the proposed theory are precisely its beables.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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