Talk:Spinors in three dimensions

Ring theory

From the point of view of ring theory, a spinor is an element of an ideal (ring theory). The ring in this article is M(2,C) and any non-zero element m with determinant m = 0, has linearly dependent rows or columns. Once one of the pair is known, and their proportionality, the second is known. Physicists call this the factorization property, or column times row representation of m. The principle ideal <m> is a real line of null vectors in M(2,C). See

P.R. Holland (1983) "Tensor conditions for algebraic spinors", Journal of Physics A 16: 2363

“A spinor may be defined as an element of a minimal ideal in a Clifford algebra”.

P. Budinich & L. Dabrowski (1985) "On Factorization of Algebraic Spinors", Letters in Mathematical Physics 10:7 to 11

"spinors are elements of a minimal left ideal" — "Holland found factorization property (column times row)".

These articles are cited for the recognition of ideal theory, not for their opinions on history of mathematics. — Rgdboer (talk) 23:32, 12 September 2017 (UTC)

Questions

• In the § "Formulation"

1. the matrix $X\ =\left({\begin{matrix}x_{3}&x_{1}-ix_{2}\\x_{1}+ix_{2}&-x_{3}\end{matrix}}\right)$ is said to be Hermitian (line 1) where the $x_{i}$ are real or complex numbers (line 2). But if $X$ is Hermitian then the $x_{i}$ must be real.
2. Cartan said (p.42) "A spinor is thus a sort of 'directed' or 'polarised' isotropic vector". It seems that spinors are only define in relation with isotropic vectors. The article says that "Provisionally, a spinor is a column vector". It seems now to be defined in a general way, independently of any vector. Is that a new definition (citation needed)? Moreover, what means "Provisionally"?

• In the § "Reality"
  

  The above considerations apply equally well whether the original euclidean space under consideration is real or complex. When the space is real, however, spinors possess some additional structure which in turn facilitates a complete description of the representation of the rotation group. Suppose, for simplicity, that the inner product on 3-space has positive-definite signature:$\left|x\right|^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$

  "The above considerations" concern isotropic vectors. But "When the space is real", the above inner product have no isotropic vector.

I think that all this needs some clarifications.--KharanteDeux (talk) 13:02, 13 June 2021 (UTC)

Agree with question about "Reality"

With regard to the § "Reality" I came to the talk page to see what people are saying about it. As said above, "When the space is real" there are no isotropic vectors. The construction suggested in this section falls apart. It needs clarification. --ArtKalb (talk) 09:43, 29 August 2021 (UTC)