Picturing the Correspondences

We have an embarrassment of riches: so many correspondences that it is easy to get confused. This paragraph tries to fit things into a coherent framework, to make them easier to remember.

We have two fundamental concepts that are easy to visualize:

Rotations $\stackrel{\rm acts\ on}{\longrightarrow}$

          vectors in 3-space $\equiv$


angular velocities         

The rotation group acts on the space of vectors. For any representation of the rotation group and any representation of the vector space, we would like to have an intuitive grasp of the action. (Since the action preserves distances, one can also consider the action of the group just on the set of vectors of norm 1, i.e., on the sphere.)

Possible representations for the rotation group: $SO(3)$, $SU(2)$, quaternions of norm 1. Possible representations for the vector space: ${\bf R}^3$, $so(3)$, $su(2)$, and the space of traceless Hermitian matrices. For $SO(3)$ with ${\bf R}^3$, the action is matrix multiplication on the left: $v\mapsto Av$. For $SO(3)$ with $so(3)$, or $SU(2)$ with $su(2)$ or the traceless Hermitian matrices, the action is conjugation: $v\mapsto AvA^{-1} = AvA^*$.

The actions with $SO(3)$ are all faithful. The actions with $SU(2)$ are all two-to-one: $A$ and $-A$ determine the same action (i.e., rotation).

Let's look at the $SU(2)$ actions in more detail. The space of traceless Hermitian matrices consists of all matrices of the form $x\sigma_x+y\sigma_y+z\sigma_z$, $x,y,z\in{\bf R}$. This is in one-one correspondence with $su(2)$:

$$x\sigma_x+y\sigma_y+z\sigma_z \leftrightarrow i(x\sigma_x+y\sigma_y+z\sigma_z) = x{\bf i}+y{\bf j}+z{\bf k}$$

So if we understand one $SU(2)$ action, we understand the other. I'll use Pauli matrices from now on.

An arbitary element $A$ of $SU(2)$ looks like

$$A= \left[ \begin{array}{cc} a+id & c+ib \\ -c+ib & a-i... ...f 1}+bi\sigma_x+ci\sigma_y+di\sigma_z, \quad a^2+b^2+c^2+d^2=1$$

and we see that $A^*=a{\bf 1}-bi\sigma_x-ci\sigma_y-di\sigma_z$. So the result of acting with $A$ on $v$ can be computed simply by working out the product $(a{\bf 1}+bi\sigma_x+ci\sigma_y+di\sigma_z) (x\sigma_x+y\sigma_y+z\sigma_z) (a{\bf 1}-bi\sigma_x-ci\sigma_y-di\sigma_z)$. For this we need the multiplication table of the $\sigma$'s. This is simply:

$$\begin{array}{ccccccc} \sigma_x^2 &=& \sigma_y^2 &=& \sigma_... ...ma_z\sigma_x &=& -\sigma_x\sigma_y &=& i\sigma_y\\ \end{array}$$

The easiest way to check this is to work out the action of the $\sigma$'s on $(p,q)\in{\bf C}^2$:

$$\begin{array}{ccc} \sigma_x\left[\begin{array}{c}p\\ q\end{a... ...& \left[\begin{array}{c}p\\ -q\end{array}\right]\\ \end{array}$$

We have here non-commuting Hermitian matrices whose product is not Hermitian-- in fact is anti-Hermitian. Now, one has an analogy between matrices and complex numbers, under which ``Hermitian'' goes with ``real'', ``anti-Hermitian'' goes with ``purely imaginary'', and ``unitary'' goes with ``on the unit circle''. The $\sigma$ matrices provide a striking example of the analogy breaking down. Non-commutativity is the culprit-- for of course the product of commuting Hermitian matrices is Hermitian.

Example: what rotation does $i\sigma_z$ represent? Its action on the unit x vector, $\sigma_x$, is just $\sigma_x\mapsto i\sigma_z\sigma_x (-i)\sigma_z = i\sigma_y\sigma_z = -\sigma_x$. Similar calculations show that y goes to $-$y and z stays put, so we have a $180^\circ$ rotation about the z-axis. Similarly for $i\sigma_x$ and $i\sigma_y$.

The exponential map provides a mapping from $su(2)$ into $SU(2)$:

$$ib\sigma_x+ic\sigma_y+id\sigma_z\mapsto \exp(ib\sigma_x+ic\sigma_y+id\sigma_z)$$

(Warning: the exponential of a sum is not in general the product of the exponentials, because of non-commutativity.) For the rotation group (as we've seen) this says simply that an angular velocity determines a rotation-- e.g., by the prescription ``rotate at the given angular velocity for one time unit''. The basis of ``angular velocities'' in $su(2)$ is $(i\sigma_z,i\sigma_y,i\sigma_z)$. Let us consider rotations about the z-axis.

$$e^{ib\sigma_z}= \left[ \begin{array}{cc} e^{ib} & 0\\ ... ...ib} \end{array} \right] = \cos b\;{\bf 1}+ i\sin b\;\sigma_z$$

(since $i\sigma_z$ acts separately on each coordinate.) Perhaps the clearest way to exhibit the action of this rotation on $v=x\sigma_x+y\sigma_y+z\sigma_z$ is to work entirely in matrix form:

$$\left[ \begin{array}{cc} e^{ib} & 0\\ 0 & e^{-ib} \end... ...z & e^{2ib}(x-iy)\\ e^{-2ib}(x+iy) & -z \end{array} \right]$$

i.e., a rotation about the z-axis of $-2b$ radians. Exercise: the same sort of thing holds for $i\sigma_y$ and $i\sigma_z$.

So one can directly picture the action of $SU(2)$ on vectors in 3-space. The ``double angles'' $2b$, etc., stem from the two multiplications in the action: $v\mapsto AvA^*$. And the double angles in turn are the reason the map from $SU(2)$ to $SO(3)$ is two-to-one.

$SU(2)$ acts on ${\bf C}^2$ via left multiplication: $v\mapsto Av$, where $v$ is a column vector. Can one picture $v$ as some kind of geometric object in 3-space? Yes indeed! An object known as a spin vector embodies $v$ geometrically. But I won't get to them.

© 2001 Michael Weiss

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