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

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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>

::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.

[[File:Spinor_construction.png|frameless|]]

::The hook arrows denote the constructed embeddings, the double arrow gives the spin double cover of the rotations, and the downwards vertical arrows are the representations of the two groups acting on their respective vector spaces. The lack of an arrow between the two vector spaces indicates no direct relationship between vectors and spinors, only between their groups. In the cases of Euclidean and indefinite signatures of quadratic forms, a fixed orthonormal of <math> V </math> can be identified with the identity rotation such that all other bases/frame are related to it by rotations and thus identified with those rotations. Similarly for the spinors, there are "spin frames" which when choosing one to correspond to the identity, biject with the whole spin group. This theory of group representations on vector spaces and spinors in particular were first realized by mathematicians, in particular [https://www.google.com/books/edition/The_Theory_of_Spinors/f-_DAgAAQBAJ?hl=en&gbpv=1&printsec=frontcover Cartan] for spinors, in the study of abstract symmetries and their realizations on geometric objects. However, the next step was taken by physicists.

| desc = Use this book to learn about Clifford algebras and spinors directly, it covers the necessary prerequisite linear algebra and group theory

| desc = If following our main list [https://theportal.wiki/wiki/Read here], you will encounter Clifford algebras and spin representations here

| desc = Less general discussion of spin representations, but with focus on the low dimensional examples in quantum physics