Theoretical Atlas

A recent colloquium talk here at UWO caught my attention because it ties in quite directly to some of the things I’ve been talking about here. Alejandro Adem, from UBC (also the PIMS head-to-be) was talking about commuting n-tuples and spaces of homomorphisms. In particular, spaces of homomorphisms where is a discrete group and is a Lie group. If you take to be , then this is a space of -tuples of elements of which all commute (since is abelian).

In particular this turns up when you want to talk about the moduli space of flat -bundles on a manifold , which you do in the area of TQFT’s. Flat -bundles are determined by specifying holonomies in around any loop - the effect of doing transport around . If you take the discrete group , the fundamental group of , then this is an example of the kind of space Adem was talking about. In particular, speaking of commuting -tuples, that is the even more special case when is an -dimensional torus. However, it’s a tricky enough special case in its own right, as it turns out. Adem spent a fair amount of time on some of these.

In geometry, you’re perhaps more likely to be interested in the moduli space of flat bundles up to gauge equivalence - which amounts to saying that if you conjugate all your holonomies by , you have an equivalent bundle. The same thing happens with spaces - since acts on them by conjugation, you can take the quotient under this action. If you started with a finite group , the space was a manifold, but the quotient may not be. However, you do have a bundle , so that each point in the base space is a gauge equivalence class of connections, and the fibre over each point consists of all the gauge-equivalent connections in that class.

(Throughout the talk, I found myself trying to categorify things - in building an extended TQFT, rather than a TQFT, one uses the case where \Gamma = \pi_1(M)$). However, there you take a weak quotient, where instead of forcing gauge-equivalent objects to be equal, you just insert isomorphisms between them, getting a groupoid I’ll call . The bundle picture is related to but different from the groupoid picture. The groupoid is equivalent to its skeleton, where the objects are just the points in . The morphisms at object are the group - the points in the fibre over in the bundle are all stabilized by - it’s a coset space.

Also, when you include the morphisms, instead of looking at functions from this space into, say, , or - its cohomology - you tend to look at functors from the groupoid. The category of functors from it into is exactly the 2-vector space of states it gets in the extended TQFT picture I partially described back here and here. So this is a categorified version of a cohomology module - the non-categorified version being what a regular TQFT based on gauge group would assign to . I’m not sure quite how all the rest of the talk fits into this picture.)

First, though, he described some tools for dealing with such spaces. To start with, you use the classifying spaces and (where is a space whose fundamental group is and which has no other interesting homotopy groups). Since “taking the classifying space” is a functor, homomorphisms turn into continuous maps . (Even better is when for some Riemann surface (i.e. a torus of some genus ), then effectively is the classifying space: ). This correspondence may not be one-to-one, but the point is they tell us something about the shape of the moduli space we were interested in. Looking at homotopy classes of such , which form a space , we get information about the components of the moduli space - there’s a map

which we can try to understand. Alejandro Adem then went on to use this idea to look at spaces of commuting -tuples in a Lie group , namely . Since the image of generates an Abelian subgroup of , one basic result is that if every maximal such subgroup is path-connected, then so is - there’s just one component (since any tuple can be deformed into any other). This can be extended to groups “built from” Abelian subgroups (in various ways he left undefined for this talk).

The other important tool for looking at the geometry/topology of the moduli spaces which he spoke about was (Poincaré-)Alexander-Lefschetz duality, which provides information about the topology of one space embedded in another from the topology of its complement. In particular, it gives an isomorphism between the cohomology of a space and the of its complement, where is -dimensional. In particular, the spaces of commuting -tuples of elements of are subspaces of the manifold , which is much easier to understand.

So finally, among a number of other examples of how these tools come into play, the one Adem described that I was most interested in was the space , and particularly , the space of connections on a torus. The complement in is an open set in a manifold - hence it’s a manifold itself - and in fact it turns out to be equivalent to . You can get partway to seeing this by noting that the projection map turns into a bundle over - the projection never hits the centre of . This centre happens to be just two points, 1 and -1, leaving the base space homotopic to a sphere . The fibre over each point is , the whole group minus the centralizer of (i.e. everything which doesn’t commute with ). The centralizer of any point is just a circle, and the remaining set is homotopic to a circle itself.

So the complement of the moduli space, within , is homotopic to a bundle of circles over a 2-sphere. There are a few of these, and it takes a little more to find out that it happens to be the 3-sphere with the Hopf fibration, but that’s what it is. Then, to find out what the moduli space itself looks like, you have to use the Alexander-Lefschetz duality. Adem didn’t show all the details, so I’m not exactly sure how, but it seems that it turns out you have a space homotopic to the one-point union of three spaces:

Now, as I said before, this is telling us information about the objects of the groupoid (also known as the moduli stack of connections), and while the morphisms shouldn’t be too hard to work out in this case, it might be nice to have a more general picture. When I raised this, Rick Jardine suggested that looking at the maps in should help - the classifying spaces are simplicial sets, and so is the collection of maps between them, and the above is only talking about vertex information. There should be a way of looking at as an infinity-category - and in this case, it should be trivial above the level of morphisms. But I don’t quite know how this works yet.

One of the first things I did after arriving at PI on Wednesday (and having lunch) was to attend the colloquium talk which was being given by Robert Spekkens. It was called “Why the Quantum?”, but as he described it, the real point of the talk was to take a close look at the features of quantum physics that are commonly considered “weird” or “mysterious” and see what’s really innovative in the departure from classical physics. For the most part, “physics” here means “mechanics”, but he also touched on optics, theory of computation, and briefly on electromagnetism and gravity in a more speculative way.

The main message of his talk is that very few of the things about quantum physics which seem strange are really all that innovative. He showed this by describing a kind of classical theory that has many of them - interference, noncommuting observables, entanglement, “wavefunction collapse”, wave-particle duality, teleportation and a no-cloning theorem, superposition of states, and so forth. All of these, he told us, will show up in a model based on a classical mechanical system, where the “quantum” theory is a theory of probability distributions (or, equivalently, of the knowledge of observers about a classical system) subject to a restriction about what distributions are allowed.

The point is to start with some classical system: let’s say it’s a mechanical system of some moving particles. Then there’s a configuration space of all the possible (classical) configurations of the system - one point in this space for each configuration. Classical mechanics is then about defining a “flow” on this space, which tells you where a point will move over time (how the system will go from one configuration to another). Then Liouville mechanics is about probability distributions in this space: you might not know exactly which configuration the system is in, but you have a way of estimating the probabilities. Then you impose the restriction that the only allowed probability distributions are ones for which the products of the variances for conjugate variables are at least Planck’s constant. (Actually, I think Spekkens formulated this differently, but that’s about what it amounts to, as I understand it.) The result is equivalent to “Gaussian quantum mechanics” - one where probability distributions are all Gaussians.

This also puts limits on what the rule for evolving states can be: any rule for how individual states evolve over time also gives a result for how probability distributions evolve over time. (Picture a cloud of ink, with varying density, flowing along in moving water - knowing the flow lines tells you where the cloud goes.) If there are restrictions on what kind of probability distributions can be set up, these have to be preserved over time - otherwise, you could set up an allowed distribution, and then wait until it evolves into a disallowed one. In particular, for Gaussian quantum mechanics, he told us that systems with a quadratic Hamiltonian will satisfy this condition.

The important fact here is that this is a “realist” interpretation. It says the quantum mechanical uncertainty reflects that QM is a theory about your knowledge of the state of the system, which, however, really exists. Often in quantum mechanics, one defines a “wave function” as a function living on configuration space (complex-valued, not real-valued like a probability density, but a function nonetheless). However, it’s now pretty standard to think of this wave function as the “real” state of the system - the view that it represents a state of knowledge was popular for a while, but ran into various problems in the form of experiments that are hard to account for, such as Bell inequality violations. The point of the talk was to see just how many of the “strange” features of quantum mechanics are genuine problems for this view, and to show the answer is “not many”.

The features he claimed are really mysterious from this point of view are fairly few: Bell inequality violations, some no-go theorems for models of physics involving local hidden variables such as the Kochen-Specker Theorem, and a few others. So Spekkens’ suggestion was that this concept of quantum mechanics as a theory of probability with an “epistemic” restriction (i.e. limits on what’s knowable) might be salvaged if the underlying classical theory were non-local - and perhaps had some other odd features yet to be precisely delineated - to begin with. However, it might not have to be terribly strange apart from that, since quantum mechanical features like interference and superposition of states all show up in the restricted statistical picture.

The gist of his argument then seemed to be that to really straighten out some foundational issues in quantum physics, one approach would be: (a) come up with a well-founded justification for the assumption about restrictions on possible probability distributions, and (b) come up with at least one (and as few as possible) other principles to account for the remaining mysterious things - he also suggested they all seem to have something to do with “contextuality”. As I understand it, this last is the idea that an observable might have definite, but multiple, values - and that which values are seen depend on which groups of observables are measured together. I don’t know what, if anything, to make of that oddball-sounding idea.

However, he did argue that in some cases at least, the restriction can be justified by the observer effect: you have to look at a system using some apparatus, whose state you don’t know completely, and which interferes with the system in order to observe it (for instance, measuring the position of a particle by scattering it off another one, whose state is partly unknown, and imparts an unknown momentum).

My overall reaction to the talk is that it’s interesting to know that realist interpretations of quantum physics (where the “reality” is more or less classical, and quantum effects some kind of afterthought, or epistemic effect) aren’t as dead as they might have seemed. However, the view that says classical physics emerges as some kind of limiting case of quantum effects seems better developed, at least mathematically, than the reverse. As for his claim that we “understand” the classical picture “physically”, whereas it’s not so for the quantum picture - I personally can only agree that’s true for me, but I don’t entirely see what you can conclude from that.

The bottom line seems to be that there are still problems in epistemology. I suspected as much already - though I’m not sure if I “knew” it, whatever that means.