AI 中文总结
探讨量子理论物理合理数学模型的基本要素，提出需具备跃迁概率等三个特征，介绍源于原子JBW因子及E6作用的相关模型，最后指出关于该问题及相应数学结构分类的一些开放问题。
AI 中文摘要
量子理论数学形式的物理基础仍是个不确定的谜团。本文假定物理合理的数学模型只需三个基本特征。一是量子理论特有的跃迁概率，另外两个是对连续可逆动力学过程存在并在基础空间上可迁作用这一假设的变体。一类具有这些特征的数学模型源于原子JBW因子，在有限维时包含原子冯·诺依曼因子且等同于约旦矩阵代数。还有一个已知模型，例外李群E6在其上可迁作用。有趣的是，E6有时被视为粒子物理内部对称性的候选者，但此时量子理论许多常见特征会丢失。本文最后提出了一些关于此问题及具有这三个特征的数学结构分类的开放问题。
英文摘要
What are the minimal requirements for a physically reasonable mathematical model for quantum theory or a potential extension? In addressing this question, rather than relying on the generalized probabilistic theories or other usual approaches, we propose a reset based on the following fundamental features: transition probabilities, which are typical of quantum theory, and continuous reversible dynamical processes (e.g., represented by a Lie group), which play the same crucial role in classical as well as in quantum physics. In doing so, we get back to the very elementary transition probability space framework originally introduced by Bogdan Mielnik in 1968 and combine it with the continuous reversibility condition. The primary class of valid spaces arises from the irreducible atomic JBW algebras, which encompass the irreducible atomic von Neumann algebras and the Jordan matrix algebras. The von Neumann algebras represent the model of common quantum theory. Beyond these, there is a non-standard valid space, where the exceptional Lie group $E_6$ acts transitively. While $E_6$ has been proposed as a candidate for internal symmetries in particle physics, we highlight that this model deviates from standard quantum theory in critical ways. Specifically, it fails to guarantee the existence of post-measurement states in all situations. Though the above postulates are powerful, they still allow for some physically meaningless models. A further feature of quantum theory (the correspondence between the pure states and the minimal projections) leads us to an additional requirement that rules out these models. However, a complete classification of the valid models remains one of the open issues, pointed out in the paper, which is intended for the mathematical and physical experts, inspiring them to tackle these problems.
Commentsmajor enhancements in v02, + 6 pages, now 15 pages; comments and contributions very welcome
