# Least Action

Nontrivializing triviality..and vice versa.

## Group Theory II – SU(2)

In a previous post, I cursorily touched upon the standard orthogonal and unitary groups. It turns out that in quantum mechanics, the operators of principal interest are either Hermitian or unitary. So it is only natural that we should be interested in transformations from the Special Unitary Group $SU(N)$. In particular, we dwell on the $N = 2$ case.

First of all, $SU(2)$ has 3 independent parameters, something that is obvious if one demands that a matrix $U$,

$U = \left(\begin{array}{cc}U_{11}&U_{12}\\U_{21}&U_{22}\end{array}\right)$

be unitary, that is

$\left(\begin{array}{cc}U_{11}&U_{12}\\U_{21}&U_{22}\end{array}\right)\left(\begin{array}{cc}U_{11}^{*}&U_{21}^{*}\\U_{12}^{*}&U_{22}^{*}\end{array}\right) = \left(\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right)$

and have a determinant equal to $+1$. It follows that any such matrix can be written as

$U = \left(\begin{array}{cc}U_{11}&U_{12}\\-U_{12}^{*}&U_{11}^{*}\end{array}\right)$

in which there are three independent parameters. If $U$ is expanded about the identity, i.e.

$U = 1 + i\epsilon G$

then unitarity of $U$ demands that $G$ be Hermitian, and also traceless. For a finite transformation, this generalizes to

$U = \exp(iH) = \exp(i\alpha G)$

That is more generally, $det(U) = \exp(i\alpha\,tr(G))$ implies that $tr(G) = 0$ for $det(U) = 1$. So, $H$ is a traceless matrix.

We can write

$H = \sum_{k=1}^{3}\alpha_{k}G_{k} = \boldsymbol{\alpha}\cdot\boldsymbol{G}$

From nonrelativistic quantum mechanics, we know that the effect of a rotation through an angle $\theta$ on a spin-1/2 particle about an axis $\boldsymbol{\hat{n}}$ is given by the unitary matrix,

$U(\theta) = \exp(-i\theta\,\boldsymbol{\hat{n}}\cdot\boldsymbol{\sigma}/2)$

where $\boldsymbol{\sigma}$ is the Pauli spin matrix “vector” given by,

$\boldsymbol{\sigma} = \hat{x}\sigma_{x} + \hat{y}\sigma_{y} + \hat{z}\sigma_{z}$

Clearly, in this development, we can identify $G_j = \sigma_j/2$ for $j = 1, 2, 3$ as the generators of the group. The commutation relation for the generators is

$\left[\frac{\sigma_i}{2},\frac{\sigma_j}{2}\right] = i\epsilon_{ijk}\frac{\sigma_k}{2}$

This is called the fundamental representation of $SU(2)$. An $n$-dimensional representation of $SU(2)$ consists of $n \times n$ unitary matrices satisfying such commutation relations.