Introduction

These notes attempt to develop some intuition about Lie groups, Lie algebras, spin in quantum mechanics, and a network of related ideas. The level is rather elementary-- linear algebra, a little topology, a little physics. I don't see any point in copying proofs or formal definitions that can be had from a shelf full of standard texts. I focus on a couple of concrete examples, at the expense of precision, generality, and elegance. See the first paragraph on Lie groups to get the flavor of my ``definitions''. I state many facts without proof. Verification may involve anything from routine calculation to a deep theorem. Phrases like ``Fact:'' or ``it turns out that'' give warning that an assertion is not meant to be obvious.

A quote from the Russian mathematician V. I. Arnol'd:

It is almost impossible for me to read contemporary mathematicians who, instead of saying ``Petya washed his hands,'' write simply: ``There is a $t_1<0$ such that the image of $t_1$ under the natural mapping $t_1 \mapsto {\rm Petya}(t_1)$ belongs to the set of dirty hands, and a $t_2$, $t_1<t_2 \leq 0$, such that the image of $t_2$ under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence.''

A taste of things to come: consider the spin of an electron. One would like to visualize the electron as a little spinning ball. This is not right, yet not totally wrong. A spinning ball spins about an axis, and the angular velocity vector points along this axis. You can imagine changing the axis by rotating the space containing the ball. ¹ Analogously, the quantum spin state of an electron has an associated axis, which can be changed by rotating the ambient space.

The classical concepts of rotation and angular velocity are associated with $SO(3)$, the group of rotations in 3-space. $SO(3)$ is an example of a Lie group. Another Lie group, $SU(2)$, plays a key role in the theory of electron spin. Now $SO(3)$ and $SU(2)$ are not isomorphic, but they are ``locally isomorphic'', meaning that as long as we consider only small rotations, we can't detect any difference. However, a rotation of $360^\circ$ corresponds to a element of $SU(2)$ that is not the identity. Technically, $SU(2)$ is a double cover of $SO(3)$.

Associated with every Lie group is something called its Lie algebra. The Lie algebra is a vector space, but it has additional structure: a binary operation called the Lie bracket. For the rotation group, the elements of the corresponding Lie algebra can be thought of as angular velocities. Indeed, angular velocities are usually pictured as vectors in elementary physics (right hand rule of thumb). The Lie bracket for this example turns out to be the familiar cross-product from vector algebra. (Unfortunately, I won't get round to discussing the Lie bracket.)

The Lie algebras of $SO(3)$ and $SU(2)$ are isomorphic. This is the chief technical justification for the ``electron = spinning ball'' analogy. The non-isomorphism of $SU(2)$ and $SO(3)$ has subtle consequences. I can't resist mentioning them, though these notes contain few further details. Electrons are fermions, a term in quantum mechanics which implies (among other things) that the Pauli exclusion principle applies to them. Photons on the other hand are bosons, and do not obey the exclusion principle. This is intimately related to the difference between the groups $SU(2)$ and $SO(3)$. Electrons have spin $\frac{1}{2}$, and photons have spin 1. In general, particles with half-odd-integer spin are fermions, and particles with integer spin are bosons.

The deeper study of the electron involves the Dirac equation, which arose out of Dirac's attempt to marry special relativity and quantum mechanics. The relevant Lie group here is the group of all proper Lorentz transformations.

© 2001 Michael Weiss

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