spinor

From Mathematics wiki

A spinor is a generalization of a vector that is more sensitive to the geometry of the rotation group. Spinors were discovered by Élie Cartan^[1] in 1913.

1 Introduction
2 Two-component spinors
3 Types of spinors
4 See also
5 References

Introduction

When looking at the transformation laws of vectors under the rotation group $SO(3)\,$ , one discovers that a vector transforms under the direct product of $SU(2)\,$ with itself. To see this, associate to the vector with components $r^i\,$ the $2\times2\,$ matrix

$r = r^i \sigma_i\,$ (summation implicit)

where $\sigma_i\,$ are the three Pauli matrices. Explicitly,

$r= \left( \begin{matrix} r^3 & r^1-i r^2 \\ r^1+i r^2 & -r^3 \end{matrix}\right)$ .

Then $\det r = -\Vert r\Vert^2\,$ and $r\,$ is Hermitian. A linear transformation

$r^i \to r^{\prime i} = O^i_j r^j\,$

can be realized by a linear transformation on $r\,$ that preserves the determinant as well as the Hermiticity of $r\,$ . The most general is

$r \to r' = U r U^{\dagger}\,$ ,

where $U \in SU(2)\,$ . I.e., $r_{\alpha\beta} \to r'_{\alpha\beta} = U_{\alpha\gamma} U^{*}_{\beta\delta} r_{\gamma\delta}\,$ . This is not the simplest transformation law that can be obtained. We therefore define a spinor $\psi\,$ which transforms as

$\psi \to \psi' = U \psi\,$ .

The transformation law $r \to r' = U r U^{\dagger}\,$ is also the transformation law obeyed by the outer product of two spinors, namely $\psi \chi^{\dagger} \to U \psi \chi^{\dagger} U^{\dagger}\,$ . We have therefore learned that vectors transform under the direct product of the spinor representation $SU(2)\,$ with its conjugate. For this reason spinors are sometimes referred to as the square roots of vectors.