对称性下作为统计理论的量子理论
Quantum theory as a statistical theory under symmetry
摘要：
统计学和量子理论都用概率来处理预测问题。我们将证明这两个领域之间可以建立联系。这同时也将提出一种新的、不那么形式化的方式来看待基本的量子理论。一个全参数空间$Φ$，配备了一组$G$变换，给出了一些量子系统的心理图像，以这样一种方式，只有某些组件，总参数$ϕ$的函数可以估计。选择一个实验/问题$a$，并从中得到一个参数空间$Λ^{a}$，也许在一些与群结构兼容的模型简化之后。本文的基本统计结构在自然的假设下导出了量子力学的基本公理，从而意味着对这个传统上非常正式的理论的一种新的统计解释。通过玻恩公式引入概率，并从一般的、合理的假设，本质上是对称的假设中证明该公式。该理论是由一个简单的宏观例子，并通过自旋1/2粒子的例子来说明。作为最后一个例子，我们展示了在对称性下相关宏观实验之间的推理联系。
英文摘要：
Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic quantum theory. A total parameter space $Φ$, equipped with a group $G$ of transformations, gives the mental image of some quantum system, in such a way that only certain components, functions of the total parameter $ϕ$ can be estimated. Choose an experiment/ question $a$, and get from this a parameter space $Λ^{a}$, perhaps after some model reduction compatible with the group structure. The essentially statistical construction of this paper leads under natural assumptions to the basic axioms of quantum mechanics, and thus implies a new statistical interpretation of this traditionally very formal theory. The probabilities are introduced via Born's formula, and this formula is proved from general, reasonable assumptions, essentially symmetry assumptions. The theory is illustrated by a simple macroscopic example, and by the example of a spin 1/2 particle. As a last example we show a connection to inference between related macroscropic experiments under symmetry.
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