20: Sir Roger Penrose - Plotting the Twist of Einstein’s Legacy: Difference between revisions

:::the adjoint action, conjugation, by an invertible element <math> \phi </math> of the Clifford algebra: <math> Ad_{\phi}(x)=\phi x \phi^{-1} </math>

::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 with <math> 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. The diagonal of <math> \eta^{\mu\nu} </math> can contain negative elements as in the case of the Minkowski norm in 4d spacetime, as opposed to the Euclidean dot product where it also has the form of the identity. Then the spinors are complex vectors in <math> \Delta=\mathbb{C}^{2^k} </math>, however an important counterexample where spinors don't have a purely complex structure is the Majorana representation which is conjectured to be values of neutrino wavefunctions.