Geometric and algebraic aspects of spectrality in order unit spaces: a comparison

Strongly symmetric spectral convex bodies are Jordan algebra state spaces

We show that the strongly symmetric spectral convex compact sets are precisely the normalized state spaces of finite-dimensional simple Euclidean Jordan algebras and the simplices. Spectrality is the property that every state has a convex decomposition into perfectly distinguishable pure states; strong symmetry is transitivity, for each integer N, of the affine automorphism group of the state space on lists of N perfectly distinguishable pure states. Additional assumptions combine with this theorem to give simple characterizations of finite-dimensional complex quantum state space. Important as…

Spectral resolutions in effect algebras

Effect algebras were introduced as an abstract algebraic model for Hilbert space effects representing quantum mechanical measurements. We study additional structures on an effect algebra $E$ that enable us to define spectrality and spectral resolutions for elements of $E$ akin to those of self-adjoint operators. These structures, called compression bases, are special families of maps on $E$, analogous to the set of compressions on operator algebras, order unit spaces or unital abelian groups. Elements of a compression base are in one-to-one correspondence with certain elements of $E$, called p…

量子物理中的对偶代数对和多项式李代数：基础和几何方面

Dual Algebraic Pairs and Polynomial Lie Algebras in Quantum Physics: Foundations and Geometric Aspects

本文讨论了对偶代数对$({\cal G}_1,{\cal G}_2)$在量子多体物理中的应用。它们出现在哈密顿量$H$具有不变性群$G_i$的模型中。然后可以取${\cal G}_1 = G_i$，而另一个对偶伙伴${\cal G}_2= g^D$是由$G_i$不变量生成的，具有李代数结构并描述了模型的动态对称性;因此多项式李代数$\hat g = g^D$出现在具有本质上非线性哈密顿量的模型中。这种方法导致模型运动学和动力学的几何化。

Aspects of Algebraic Quantum Theory: a Tribute to Hans Primas

This paper outlines the common ground between the motivations lying behind Hans Primas' algebraic approach to quantum phenomena and those lying behind David Bohm's approach which led to his notion of implicate/explicate order. This connection has been made possible by the recent application of orthogonal Clifford algebraic techniques to the de Broglie-Bohm approach for relativistic systems with spin.

量子不可逆性的代数学问题

Algebraic aspects of quantum indiscernibility

拟集合论被提出作为一种数学背景来研究不可区分对象的集合。在介绍了这个理论的概要之后，我们定义了一个代数，它具有正交完备正交模格的大部分标准性质，正交完备正交模格是希尔伯特空间的闭子空间的格。我们把这样得到的数学结构称为$\mathfrak{I}$ -格。在初步讨论了这种结构的某些方面之后，我们指出了相应逻辑公理化的下一个问题，即以$\mathfrak{I}$ -格作为其代数模型的逻辑。我们认为，“量子力学的逻辑”将不是经典逻辑（及其布尔代数）的直觉，与从一开始就考虑不可解释性的想法是一致的，也就是说，作为一个原始概念。换句话说，不可分割性似乎“直接”导致$\mathfrak{I}$ -格。在第一部分中，我们提出了主要的动机和一个“经典”的情况，反映了我们专注于论文的最后一部分。本文是我们第一次研究的代数结构内的拟集理论的不相容性。

超越张量积结构的纠缠：量子不可分性的代数方面

Entanglement beyond tensor product structure: algebraic aspects of quantum non - separability

量子不可分性的代数方法被应用到两个量子比特的情况。它是基于分区的代数的可观的独立的子代数和张量积结构的希尔伯特空间是不利用。即使在这种简单的情况下，这种一般公式也有一些优点。利用代数形式，我们可以明确地表明纠缠概念与系统中测量的可观测量的相关性，并描述可分离和不可分离的纯态。作为纯态不可分性的普遍度量，我们建议采用所谓的全关联。这个量取决于状态以及代数划分。其数值由相应相关矩阵的范数给出