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blefloch / tables-for-supersymmetry

Compact document containing many tables about supersymmetry and related hep-th topics.
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Clifford algebra clock #45

Closed blefloch closed 8 years ago

blefloch commented 8 years ago

The Clifford algebra clock is invariant under (p-q) => 2-(p-q) because Cliff(p,q) tensor M_2(R) is Cliff(q+2,p). This is seen as follows. Starting with the matrices gamma_a in Cliff(p,q) consider the 2x2 matrices gamma_a tensor i sigma_2, in other words the block matrix [(0, gamma_a); (-gamma_a,0)]. These matrices anticommute and square to the opposite of the gamma_a, thus realize Cliff(q,p). To conclude, note that the 2 extra matrices sigma_3 and sigma_1, in other words [(1,0); (0,-1)] and [(0,1), (1,0)] anticommute with the others and square to +1.

blefloch commented 8 years ago

Probably not worth including the explanation on the sheet, but still useful to know.