What are C*-algebras good for?

John Baez

March 1, 2000

I'd like to say a bit about the importance of C*-algebras in physics. I'll carefully avoid any sort of mathematical details and focus on the basic physical ideas. Everything will be nonrigorous, handwavy, and vague. I won't even define what a C*-algebra is! I just want to get you interested. For details, try some of the references at the end.

In quantum mechanics we often start by taking classical observables and writing down some formulas which say that actually these observables don't commute. The famous example is of course

pq - qp = iℏ

but there are many others.

When we do this, what we're really doing is defining an algebra - physicists would usually call it an "algebra of observables". C*-algebras are a way of making this precise. They were invented by Irving Segal in 1947. However, his work was based on that of others, especially von Neumann's papers with Murray on the foundations of quantum mechanics, and also the ideas of Gelfand and Naimark.

Now, observables aren't much use without states. One way to get ahold of states is to take your algebra of observables and represent it as an algebra of operators on a Hilbert space. Then unit vectors in your Hilbert space represent states.

However, the same algebra of observables can have different representations as operators on a Hilbert space. In the example mentioned above, Heisenberg figured out how to represent p and q as infinite-dimensional matrices, while Schrodinger figure out how to represent them as differential operators. In this particular case the two representations are "equivalent", so there's no serious conflict between Heisenberg's matrix mechanics and Schrodinger's wave mechanics: they are just two different viewpoints on the same theory.

In 1931, von Neumann proved a famous theorem - now called the Stone-von Neumann theorem - that explains why we don't have to worry about different representations in this particular case. But later, people found examples where the same algebra of observables can have many fundamentally different representations!

The simplest example is the quantum field theory of a noninteracting massive spin-zero particle. We have observables analogous to the p's and q's in this theory, so we get a C*-algebra of observables built from these p's and q's. This algebra does not depend on the mass of the particle. However, its representation as operators on Hilbert space depends in a really important way on the particle's mass. Different masses correspond to inequivalent representations!

What does this mean in practical terms? Well, if you take the representation corresponding to a particle of mass m, and try to write down the operator for the Hamiltonian of a particle of mass m', you'll get a meaningless divergent integral unless m' = m.

A similar problem occurs if we try to treat the vacuum state for the theory with mass m' as a unit vector in the Hilbert space for the particle of mass m. You can write down a formula that's supposed to give the answer, but again, it contains an integral which diverges unless m' = m.

In fact, many of the infinities in quantum field theory can be understood this way: they come from assuming that all representations of your algebra of observables are equivalent. To avoid these infinities, it helps to stop making this false assumption. It's not a panacea, but it's a necessary start.

A great example is the case of spontaneous symmetry breaking. Certain quantum field theories allow many different vacuum states. Each different vacuum gives a different representation of the C*-algebra of observables as operators on a Hilbert space. Crudely speaking, each vacuum state lives in a different Hilbert space: to get from one to the other would require the creation or annihilation of infinitely many particles, so you can't think of them as living in the same Hilbert space.

Since quantum field theory is closely related to statistical mechanics, it should come as no surprise that all these phenomena have analogues in statistical mechanics as well. For this reason C*-algebras are also useful in statistical mechanics. Actually there's an interesting feedback loop here: ideas from statistical mechanics have also had an important effect on the theory of C*-algebras. The most famous example is the work of Kubo, Martin and Schwinger on thermal equilibrium, which led to something called Tomita-Takesaki theory. But I said I wouldn't get too mathematical, so I can't talk about this!

I've already hinted that there's a close relation between "vacuum states" and representations of C*-algebras. In fact there's a theorem called the GNS construction which makes this precise. But physically, what's going on here? Well, it turns out that to define the concept of "vacuum" in a quantum field theory we need more than the C*-algebra of observables: we need to know the particular representation. This becomes most dramatic in the case of quantum field theory on curved spacetime - a warmup for full-fledged quantum gravity. It turns out that in this setting, it's a lot harder to get observers to agree on what counts as the vacuum than it was in flat spacetime. The most dramatic example is the Hawking radiation produced by a black hole. You may have heard pop explanations of this in terms of virtual particles, but if you dig into the math, you'll find that it's really a bit more subtle than that. Crudely speaking, it's caused by the fact that in curved spacetime, different observers can have different notions of what counts as the vacuum! And to really understand this, C*-algebras are very handy.

Well, this article is so lacking in detail that I'm getting sort of sick of writing it - the really fun part, to me, is how the mathematics of C*-algebras makes the vague verbiage above utterly precise and clear! So I'll stop here. If you want some details, try these books:

For books that talk about the math but not the physics, try these (in rough order of increasing difficulty):

© 2000 John Baez