Clifford algebras are used for constructing spin groups, and are therefore of particular importance in the theory of quantum mechanics. An algebraist’s perspective on the many subgroups and subalgebras of Clifford algebras may suggest ways in which they might be applied more widely to describe the fundamental properties of matter. I do not claim to build a physical theory on top of the fundamental algebra, and my suggestions for possible physical interpretations are indicative only, and may not work. Nevertheless, both the existence of three generations of fermions and the symmetry-breaking of the weak interaction seem to emerge naturally from an extension of the Dirac algebra from complex numbers to quaternions.

Clifford 代数用于构建自旋群，因此在量子力学理论中特别重要。代数学家对 Clifford 代数的许多子群和子代数的看法可能会提出更广泛地应用它们来描述物质基本性质的方法。我并不声称要在基本代数之上建立物理理论，我对可能的物理解释的建议仅供参考，可能行不通。尽管如此，三代费米子的存在和弱相互作用的对称性破缺似乎都是从狄拉克代数从复数到四元数的扩展中自然出现的。