标题： 例外量子几何与粒子物理学 显示英文标题

标题： Exceptional quantum geometry and particle physics

摘要： 基于对夸克-轻子对称性的一种解释，即色群 $SU(3)$ 的单模性质以及三代的存在性，我们提出了一种论证方法，表明与实数维度为27的例外型约当代数（即欧几里得阿尔伯特代数）相对应的“有限量子空间”对于粒子理论中内部空间的描述是相关的。特别是，与例外代数的三个非对角八元数元素相对应的三重性与标准模型的三代相关联，而八元数作为四维复空间 $\mathbb C\oplus\mathbb C^3$ 的表示则与夸克-轻子对称性相关联（一个复空间对应轻子，三个复空间对应相应的夸克）。 更普遍地认为，用值在有限维欧几里得约当代数上的时空函数代数替代时空上的实函数代数是有意义的，该约当代数扮演着相应几乎经典量子时空上“实函数”的角色，在粒子物理学中有重要意义。 这引导我们研究约当代数的模理论，并发展了约当代数上的微分演算（即引入适当的微分形式概念）。我们给出了约当代数模上联络的相应定义。 显示英文摘要

摘要： Based on an interpretation of the quark-lepton symmetry in terms of the unimodularity of the color group $SU(3)$ and on the existence of 3 generations, we develop an argumentation suggesting that the "finite quantum space" corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation of the octonions as a complex 4-dimensional space $\mathbb C\oplus\mathbb C^3$ is associated to the quark-lepton symmetry, (one complex for the lepton and 3 for the corresponding quark). More generally it is is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of "the algebra of real functions" on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras, (i.e. to introduce the appropriate notion of differential forms). We formulate the corresponding definition of connections on Jordan modules.