Anybody who has taken an undergraduate course on quantum chemistry or quantum mechanics is familiar with the commutator on some level. In these courses, I was told how to use it to carry out some basic (but very important!) calculations, and was given a teaser of its physical significance in terms of incompatible observables. In short, I learned some very important facts about the commutator, but my understanding remained superficial until I started to read about the theory of Lie algebras and Lie groups.

In this post, I want to provide some geometric insight into the commutator. I want to build this intuition because I like to think geometrically – particularly when it comes to linear algebra and group theory – and because there is often a fundamental connection between physics and geometry. I will start by saying that Lie groups can have very interesting geometric structure, and that the rules of algebra provide insight into this structure. In the case of Lie groups and Lie algebras, this insight comes from the commutator.

The geometry of Lie groups and the commutator are tied together by a number of intermediate concepts, so before I go further down the rabbit hole, let’s step back and think about where geometric structure comes from. We already know that some vector spaces come endowed with a way to talk geometry that is, more or less, in line with our intuition for what is ‘normal’ as non-relativistic creatures living in Euclidean 3-space. In fact, any Hilbert space has enough structure for us to think about geometry in this way (modulo some extra dimensions). This all boils down to having a way to define lengths and angles between vectors – structure endowed by an inner product.

Now, suppose we take the vector space and continuously deform it into a smooth 2-dimensional surface (manifold) embedded in . Perhaps this deformation does nothing more than introduce a few hills and valleys. In any case, it destroys quite a bit of the structure that made life simple in ; and the resulting manifold is not a vector space. One casualty of this deformation is the inner product on . So how do we talk about the geometry in its absence? In this case, we can still talk about local geometry without getting into trouble by considering the plane tangent to wherever I happen to be standing on the manifold. This tangent plane is a perfectly legitimate vector space (isomorphic to ), and we can use the local structure of this tangent space to make sense of the local structure of the manifold.

We can get a feel for the local ‘shape’ the manifold by noting how it deviates from the tangent plane in each direction. In other words, we get insight from directional derivatives, and the fact that we can take directional derivatives is due to the local coordinates provided by the tangent space. From directional derivatives, we recover useful information about things like curvature – information that gives us geometric insight without any reference to the space in which the manifold is embedded.

This is not too hard to think about for a nice, well-behaved 2-d manifold embedded in 3-d space, where we can think about directional derivatives in the ‘normal’ calc II/III sense. But now I assert that we should be thinking about our favorite Lie group G as a manifold, and its Lie algebra as the tangent space of G at G‘s identity element. To help us digest this last statement, let’s pause to consider a simple example: the complex unit circle . This is a one-dimensional Lie group under regular multiplication. The identity of the group is ‘1’, and the tangent space at the identity is therefore . There are two points of interest here: first, the fact that the Lie group and Lie algebra are related through the exponential map really hits you in the face! Second: the algebra has linear structure, while the group is…curvy. The need not be true in general, but it illustrates that the Lie group can have a ‘shape’ to it that is a bit more interesting than that of a linear vector space.

Of course, the geometry of the complex unit circle is not terribly complicated (or interesting). To return to the concept of directional derivatives, we require a Lie group with a few more dimensions and a bit more algebraic structure, i.e. one for which the commutator does not vanish trivially. I think , the group of 3×3 proper rotation matrices, is a good place to start because it is simple (in both the literal and group theoretical sense) and serves as the foundation for the theory of angular momentum in 3-d space. We can think of as a manifold embedded in , and its Lie algebra, , is the space of 3×3 antisymmetric matrices endowed with the commutator.

Now we are finally ready to talk about directional derivatives. Choose two elements . Exponentiate the first and introduce a parameter so that . Suppose we allow to act on by conjugation. Then . Recall from linear algebra that conjugation is equivalent to a change of basis transformation – in this case, a rotation. And as we vary the parameter , the point moves along a trajectory in . The precise nature of this trajectory will, of course, depend on our choice of , but we can at least say that it is a closed loop (since we are conjugating by a rotation matrix). Since is a vector space, we can take derivatives along with respect to the parameter without any major issues. Now, in principle, we can take derivatives anywhere on our trajectory, but it turns out that the derivative evaluated at is of particular interest:

And there we have it – a formal definition of the commutator that really drives home the fact that it is a special sort of directional derivative – it evaluates how transforms in the ‘direction’ of under conjugation. This is a pretty neat idea! Let’s solidify it with an example, again from the algebra of rotations (this time from , which is very closely related to ). The angular momentum operators generate infinitesimal rotations. If this does not sound familiar, I will point you to my last post, which contains a simple example of how elements of a Lie algebra can generate infinitesimal transformations. Let’s consider the canonical commutation relationships for these operators: where is the Levi-Civita symbol. We know from above that tells us how changes in the ‘direction’ of . Think of as a unit vector oriented along the y-axis. Conjugation by (which represents a rotation by –t radians about the x-axis) rotates this vector clockwise in the yz plane as the parameter t increases. If we evaluate the derivative of this circular trajectory with respect to t at t=0, the result is a vector pointing in the negative z direction. In other words, we arrive at a familiar result: .

I think this example is very nice because it appeals to our geometric intuition. It is also easy to imagine how it could be generalized to other cases. For example: if we take it as a given that the Hamiltonian generates time translations, we have the tools to make sense of the fact that . But suppose we didn’t know the Hamiltonian generates time translations. How could we figure this out for ourselves from first principles? I hope to address this in the next post as part of a discussion of the transformation properties (symmetries) of operators compatible observables. Thanks for reading!

-B