2024-03-04   08264

Clifford Algebra, Lorentz Transformations, and Spin Groups

Article 03910 introduced Clifford algebra, which is sometimes also called geometric algebra. This article uses Clifford algebra to construct the spin group, which is a double cover of the part of the Lorentz group that is generated by pairs of reflections. This is a prerequisite for the idea of a spinor field, which is an important ingredient in our current understanding of nature. The construction and basic topological properties of the spin group are explained for arbitrary signatures (p,q) with p+q≥ 2, including euclidean signatures (either p or q is 0), lorentzian signatures (either p or q is 1), and other signatures (p and q are both ≥ 2).

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