Section 6.1 Spinors
We begin with a canonical example, namely the isomorphism between \(\su(2)\) and \(\so(3)\text{.}\)
Recall that \(\so(3)\) consists of \(3\times3\) real, antisymmetric matrices. Such matrices act on \(3\)-component (real) column vectors, which are usually interpreted as vectors in \(\RR^3\text{.}\) Similarly, \(\su(2)\) consists of \(2\times2\) complex, antihermitian, traceless matrices, acting on \(2\)-component (complex) column vectors. Now, however, these column vectors are usually interpreted as spinors.
Let's consider these Lie algebras in more detail. Recall that the standard basis of \(\so(3)\) consists of the infinitesimal rotations \(r_m\text{,}\) and the standard basis of \(\su(2)\) consists of the matrices \(s_m=-i\sigma_m\text{.}\) Recall further that we can identify \(\RR^3\) with the three-dimensional vector space spanned by the Pauli matrices \(\sigma_m\text{,}\) and that \(M\in\SU(2)\) rotates these vectors via \(X\longmapsto MXM^\dagger\text{;}\) this identification is at the heart of the relationship between \(\SU(2)\) and \(\SO(3)\text{.}\) But the infinitesimal version of this action is, using the product rule, precisely
where the minus sign arises from the antihermitian nature of \(\SU(2)\text{.}\) Thus, both the elements \(s_m\) of the Lie algebra \(\su(2)\) and the vectors \(\sigma_m\) on which they act can be thought of as \(2\times2\) complex matrices. Furthermore, the Lie algebra elements are just the product of two Pauli matrices, since for instance \(\sigma_x\sigma_y=i\sigma_z=-s_z\text{.}\) What other products are there? The square of any Pauli matrix is the identity matrix, and the product of all three is \(i\) times the identity matrix.
All orthogonal groups \(\SO(p,q)\) admit such double covers, denoted \(\Spin(p,q)\text{.}\) 1 The corresponding matrices typically have different sizes, acting on vectors and spinors, respectively, although the Lie algebras \(\spin(p,q)\) and \(\so(p,q)\) are isomorphic. However, the case \(\spin(3)\cong\so(3)\cong\su(2)\) is exceptional in that almost none of the spin groups are related to unitary groups.
Furthermore, the algebraic structure discussed above generalizes to all of the orthogonal groups. The vectors can themselves be interpreted as matrices, whose products yield the elements of the Lie algebra, together with the identity matrix. Higher-order products also exist. The resulting algebra, generated by the original vectors (as matrices), is called a Clifford algebra, as discussed further in Section 6.2.