A first blog post on noncommutative rings

In this post, I’d like to record a few basic definitions and results regarding noncommutative rings. This is a subject clearly of great importance and generality, but I haven’t had much exposure to it, and I’m trying to fix that. I am working mostly from Lam’s A first course in noncommutative rings.

A less biased definition of a group

Here’s what seems like a silly question: what’s the universal group? That is, what’s the universal example of a set together with maps

satisfying the identities

1. ,
2. ,
3. ?

A moment’s reflection shows that there isn’t such a group; the existence of the groups , where is an arbitrary set, shows that there exist groups of arbitrarily large cardinality, so no particular group can be universal.

Estimating roots

In lieu of a real blog post, which will have to wait for at least another two weeks, let me offer an estimation exercise: bound, as best you can, the unique positive real root of the polynomial

.

The intermediate value theorem shows that , which was the subject of a recent math.SE question that provided the inspiration for this question. I provide a stronger lower bound on using elementary inequalities and entirely by hand in an answer to the linked question, although I don’t try to improve the upper bound.

What should I do next semester?

Once again, apologies for the lack of updates. In my defense, I am taking almost entirely graduation requirements so that I can graduate from MIT this semester, and then I plan on taking a gap semester in the spring. I have some incomplete plans for next semester, but I thought I’d throw out the following question anyway: what should I do with all of that time?

My current plans involve going through my backlog of books and papers I haven’t had time to read and writing posts about them, but I’m sure there are plenty of other ways I could mathematically enrich my life before graduate school and I’d be very interested to hear your suggestions.

Morality

Apologies for the lack of updates; I’ve been attempting to apply to graduate school. In the meantime, I want to link to a fantastic paper I just heard about by Eugenia Cheng on moral truth in mathematics. In private (or for me, on MathOverflow), mathematicians often say things like “well, morally, this should be true because…” and Cheng extensively discusses what this could mean and why it’s important.

I’m glad I finally have a word for this. I’ve cared about moral truth more than proof for awhile now, and that’s a major reason I’ve been trying to teach myself physics: even if it isn’t a good source of proofs, it seems like a great source of moral truths.

Constructing Poisson algebras

(Commutative) Poisson algebras are clearly very interesting, so it would be nice to have ways of constructing examples. We know that is a Poisson algebra with bracket uniquely defined by ; this describes a classical particle in one dimension, and is the classical limit of a quantum particle in one dimension (essentially the Weyl algebra).

More generally, if are Poisson algebras, then the tensor product can be given a Poisson bracket given by extending

linearly. At least when are unital, this Poisson algebra is the universal Poisson algebra with Poisson maps from such that the images of elements of Poisson-commute with the images of elements of . In particular, it follows that is a Poisson algebra with the bracket

.

This describes a classical particle in dimensions, or different classical particles in one dimension, and it is the classical limit of a quantum particle in dimensions, or different quantum particles in one dimension.

Today we’ll discuss the question of how one might go about constructing Poisson brackets more generally.

Poisson algebras and the classical limit

In the previous post we described the Heisenberg picture of quantum mechanics, which can be phrased quite generally as follows: given a noncommutative algebra (the algebra of observables of some quantum system) and a Hamiltonian , we obtain a derivation , which is (up to some scalar multiple) the infinitesimal generator of time evolution. This is a natural and general way to start with an algebra and an energy function and get a notion of time evolution which automatically satisfies conservation of energy.

However, if is commutative, all commutators are trivial, and yet classical mechanics somehow takes a Hamiltonian and produces a notion of time evolution. How does that work? It turns out that for algebras of observables of a classical system, we can think of as the classical limit of a family of noncommutative algebras. While is commutative, the noncommutativity of the family equips with the extra structure of a Poisson bracket, and it is this Poisson bracket which allows us to describe time evolution.

Today we’ll describe one way to formalize the notion of taking the classical limit using the deformation theory of algebras. We’ll see how Poisson brackets pop out along the way, as well as the relevance of the lower Hochschild cohomology groups.

Update

I put up a post over at the StackOverflow blog describing a little of what I’ve been up to this summer.

Curiously enough, the Zipf distribution which shows up in that post is the same as the zeta distribution that shows up when trying to motivate the definition of the Riemann zeta function. I’m sure there is a conceptual explanation of this connection somewhere, probably coming from statistical mechanics, but I don’t know it. I suppose the approximate scale invariance of the zeta distribution is relevant to its appearance in many real-life statistics, as described in Terence Tao’s blog post on the subject here.

The Heisenberg picture of quantum mechanics

In an earlier post we introduced the Schrödinger picture of quantum mechanics, which can be summarized as follows: the state of a quantum system is described by a unit vector in some Hilbert space (up to multiplication by a constant), and time evolution is given by

where is a self-adjoint operator on called the Hamiltonian. Observables are given by other self-adjoint operators , and at least in the case when has discrete spectrum measurement can be described as follows: if is a unit eigenvector of with eigenvalue , then takes the value upon measurement with probability ; moreover, the state vector is projected onto .

The Heisenberg picture is an alternate way of understanding time evolution which de-emphasizes the role of the state vector. Instead of transforming the state vector, we transform observables, and this point of view allows us to talk about time evolution (independent of measurement) without mentioning state vectors at all: we can work entirely with the algebra of bounded operators. This point of view is attractive because, among other things, once we isolate what properties we need this algebra to have we can abstract them to a more general setting such as that of von Neumann algebras.

In order to get a feel for the kind of observables people actually care about, we won’t study a finite toy model in this post: instead we’ll work through some classical (!) one-dimensional examples.

The representation theory of SU(2)

Today we will give four proofs of the classification of the (finite-dimensional complex continuous) irreducible representations of (which you’ll recall we assumed way back in this previous post). As a first step, it turns out that the finite-dimensional representation theory of compact groups looks a lot like the finite-dimensional representation theory of finite groups, and this will be a major boon to three of the proofs. The last proof will instead proceed by classifying irreducible representations of the Lie algebra .

At the end of the post we’ll briefly describe what we can conclude from all this about electrons orbiting a hydrogen atom.