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20: Sir Roger Penrose - Plotting the Twist of Einstein’s Legacy: Difference between revisions
20: Sir Roger Penrose - Plotting the Twist of Einstein’s Legacy (edit)
Revision as of 20:55, 24 July 2023
, 24 July 2023→Spinors
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:::the adjoint action, conjugation, by an invertible element <math> \phi </math> of the Clifford algebra: <math> Ad_{\phi}(x)=\phi x \phi^{-1} </math> | :::the adjoint action, conjugation, by an invertible element <math> \phi </math> of the Clifford algebra: <math> Ad_{\phi}(x)=\phi x \phi^{-1} </math> | ||
:;3) | :;3) | ||
::Spinors | ::Spinors require more algebra to construct in general, such as understanding the representations of Clifford algebras as algebras of matrices. In the simplest case, one can choose an orthonormal basis of <math>V: \{e_1,\cdots,e_n\} </math> and correspond these vectors to n <math> 2^k\times 2^k </math> matrices <math> with k=\lfloor n/2\rfloor </math> such that they obey the same relations as in the Clifford algebra: <math> \gamma^{\mu}\gamma^{\nu}+\gamma^{nu}\gamma^{mu}= -2\eta^{\mu\nu}\mathbb{I}_{k\times k} </math> where <math> \mathbb{I}_{k\times k} </math> is the <math> k\times k </math> identity and </math> \eta^{\mu\nu} </math> is the matrix of dot products of the orthonormal basis. | ||
== Notes == | == Notes == | ||