We have two fundamental concepts that are easy to visualize:
vectors in 3-space |
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: , , quaternions of norm 1. Possible representations for the vector space: , , , and the space of traceless Hermitian matrices. For with , the action is matrix multiplication on the left: . For with , or with or the traceless Hermitian matrices, the action is conjugation: .
The actions with are all faithful. The actions with are all two-to-one: and determine the same action (i.e., rotation).
Let's look at the actions in more detail. The space of traceless
Hermitian matrices consists of all matrices of the form
,
. This is in one-one
correspondence with :
An arbitary element of looks like
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 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 represent? Its action on the unit x vector, , is just . Similar calculations show that y goes to y and z stays put, so we have a rotation about the z-axis. Similarly for and .
The exponential map provides a mapping from into :
So one can directly picture the action of on vectors in 3-space. The ``double angles'' , etc., stem from the two multiplications in the action: . And the double angles in turn are the reason the map from to is two-to-one.
acts on via left multiplication: , where is a column vector. Can one picture as some kind of geometric object in 3-space? Yes indeed! An object known as a spin vector embodies geometrically. But I won't get to them.
© 2001 Michael Weiss