SU(2) group ($SU(2)$)

The SU(2) group is the group of two-by-two unitary matrices of determinant one. It is a compact three-dimensional Lie group whose underlying manifold is a three-sphere.

For $\epsilon =\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, every two-by-two matrix satisfies $U^T\epsilon U=(\det U)\epsilon$. Thus ${\varphi }^T\epsilon \chi$ is exactly invariant under a common SU(2) group transformation. This is a bilinear contraction without complex conjugation. It vanishes for equal commuting spinors and is the alternating tensor defining the spin singlet state. Overall bosonic exchange symmetry must still be enforced if the spinors label identical particles.

An SU(2) matrix is a two-by-two complex unitary matrix with determinant one. It has the form $\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right)$ with $|a{|}^2+|b{|}^2=1$. Its action on the Riemann sphere is a rotational Möbius transformation; opposite matrices induce the same sphere rotation.