QGQG Nottingham

Well, a week ago I got back from England, where I spent a week at the University of Nottingham at the conference “Quantum Gravity and Quantum Geometry 2008″, and a weekend visiting friends in London. London was enjoyable, though surprisingly expensive. It’s strange, when so many things are traded globally, that prices differ so much from place to place - the standard rule being to imagine that all prices in Pounds are actually in dollars, and they seem quite familiar. Clearly not everything is affected by trade, with restaurant meals among them. In any case, it was quite interesting to come come from London, Ontario to London, England, and walk around all the places whose names show up attached to completely dissimilar landmarks in the Canadian version.

As for the conference, it was a great experience. This was an outgrowth of the “LOOPS” series of conferences. The only one of those I’d been to previously was LOOPS ‘05 at the Albert Einstein Institute, in Germany. At that time the conference was a little more focused on some particular approaches to quantum gravity (though there was still a whole range of talks). This year, there seemed to have been some attempt to broaden the conference a little - one result being that there must have been about 200 people attending, with something on the order of 90 talks, most of them half-hour talks in the parallel sessions. As a result, I saw less than half of what was going on. However, there were some broad subject areas, such as loop quantum gravity, spin foam and combinatorial quantization, noncommutative geometry, quantum groups, as well as some less readily classifiable talks.

In one talk on the first day, Carlo Rovelli discussed the relation between the Loop Quantum Gravity and spin-foam approaches to a theory of 4D quantum gravity. In particular, he was talking about the fact that the two approaches agree with each other in 3D, but it’s not so clear they do in 4D - or at least, it’s not clear what the spin foam model is that does this in 4D. This is part of what’s behind the program to improve the Barrett-Crane spin foam model for 4D gravity. It has various technical problems as well, which various more technical talks got into in more detail later in the conference. Rovelli was describing work on the new models which agree with LQG. Various other people have done work on this, including (among others) Freidel (who talked about that in his own talk later) and Krasnov, and Engle, Pereira and Rovelli. Florian Conrady also talked about these new models later on. I know Igor Khavkine, just graduating here at Western, has also done some work on these.

Another talk based off the successes of these models was by Abhay Ashtekar, about Loop Quantum Cosmology - that is, applying loop QG methods to the universe as a whole - a quantum version of the Friedman-Robertson-Walker universe. What’s interesting about this is that they’re doing numerical and analytic simulations, and predicting something that otherwise has usually been added as a “what-if” afterthougoht. Namely, such a universe behaves a lot like classical FRW, except near the “big bang”, classically a singularity, where quantum geometric effects prevent that from happening. Continuing through the other side, one sees a collapsing universe - an overall “bounce” effect. An interesting prediction, if hard to check.

In any case, I was bombarded by a whole range of other talks on other points of view. Starting from the very first talk, by Vincent Rivasseau, there were several talks presenting noncommutative geometry, Alain Connes-style, as a setting for a quantum theory of gravity. There’s certainly an appeal to the idea of replacing measure-theoretic and topological information about spacetime with a quantum algebra of observables - just write the theory in quantum terms from the start, giving up the usual differential geometry for its noncommutative version. Rivasseau presented, among other things, the idea of QFT as weighted species, in the sense of Joyal’s combinatorial species. I thought this was great, since I looked at just that idea for the simplest QFT of all, the quantum harmonic oscillator.

(Speaking of which, I had some interesting conversations with Jamie Vicary in which I finally “got” part of what he did with his own paper about the oscillator - which is to show how “taking Fock space” for a quantum system is a monad, namely the monad associated with the “free commutative monoid” functor, and its adjoint.)

Shahn Majid, whom I knew as the author of some well-known books on quantum groups, also spoke about this C*-algebra approach to geometry, and quantum gravity. : begin with a space, like a manifold, or better yet a fibre bundle, which is where a lot of physics gets done, and look at the algebra of forms on it. It has nice properties (it’s a differential graded algebra, etc.), including being commutative. One can deform these to noncommutative algebras that are quite nice - “q-deformation” assumes the commutators between elements depend on some parameter q, so the old picture where q=0 is simply a special case.

So then one thing is to develop a deformed version of classical things from geometry and analysis - for example, the Fourier transform. Even in the big purple book on quantum groups, he outlined what this approach consists of: a criterion for a quantum theory of gravity, that it should be algebraically “self-dual”, under exchange of “position” and “momentum” variables. (That is, under a Fourier transform - being its own Fourier dual).

Well, speaking of quantum groups, I should mention Aaron Lauda’s talk on categorifying them - specifically, on categorifying “deformed classical Lie groups”, like (a q-deformed version of the universal enveloping algebra , which for is the algebra where the Lie bracket of is a genuine commutator). He described a graphical calculus - a particular kind of string diagram, with some relations on them - which is a categorification of the quantum group. In fact, as sometimes happens, it categorifies a specific presentation of the algebra in terms of some generators and relations.

An appealing thing about these string diagram methods and so forth is that it suggests why these algebraic gadgets - quantum groups, in this case - are good at encoding topological information about tangles, braids, knots, and so on. If diagrams that involve those shapes categorify (read “model the underlying structure of”) quantum groups, then it makes sense that quantum groups to give invariants for them.

Along similar lines, Joao Faria Martins talked about invariants for “welded virtual knots”, and for knotted surfaces from crossed modules (read “2-groups”, if you’re so inclined - they are equivalent). Martins also published a paper with Tim Porter about related work, which in turn builds on David Yetter’s, on a class of manifold invariants. Their paper talks about “extending the Dijkgraaf-Witten model to categorical groups” (Urs Schreiber, possibly among others, rephrased that to call it a “categorification of the Dijkgraaf-Witten model”. The DW model is the TQFT foundation for my own look at extending (read, “categorifying”) TQFT’s based on gauge theory using a group - (finite, for the DW model). These are categorifications in two different directions, though: one, from a gauge group to a gauge 2-group, the other from a TQFT - a functor - to a 2-functor given by a group. Probably for 4 dimensions and higher, the 2-group version or higher is the most interesting to study.

In fact, there was a fair bevy of talks relating to categorical methods in quantum geometry. For example, Jamie Vicary gave a talk introducing a “categorical framework for quantum algebra”, by means of non-threatening string diagrams. These can be used to show the axioms for a “-monoidal category”. Not incidentally to all this, he also shows that in finite dimensions, at least, a -algebra is “the same thing as” a -Frobenius algebra.

Benjamin Bahr gave another talk dealing with categorical issues - namely, how to get measures on certain groupoids, such as, indeed, the groupoid of connections on a manifold. In fact, he treated various cases under the same framework: flat and non-flat connections, on manifolds and on graphs - and others.

In all, I was pleasantly surprised by the mix of the physically and mathematically inclined points of view, and the trip itself was a lot of fun.

 

20th Century Space and 10th Century Space

A couple of posts ago, I mentioned Max Jammer’s book “Concepts of Space” as a nice genealogy of that concept, with one shortcoming from my point of view - namely, as the subtitle suggests, it’s a “History of Theories of Space in Physics”, and since physics tends to use concepts out of mathematics, it lags a bit - at least as regards fundamental concepts. Riemannian geometry predates Einstein’s use of it in General Relativity by fifty some years, for example. Heisenberg reinvented matrices and matrix multiplication (which eventually led to wholesale importation of group theory and representation theory into physics). More examples no doubt could be found (String Theory purports to be a counterexample, though opinions differ as to whether it is real physics, or “merely” important mathematics; until it starts interacting with experiments, I’m inclined to the latter, though of course contra Hardy, all important mathematics eventually becomes useful for something).

What I said was that it would be nice to see further investigation of concepts of space within mathematics, in particular Grothendieck’s and Connes’. Well, in a different context I was referred to this survey paper by Pierre Cartier from a few years back, “A Mad Day’s Work: From Grothendieck To Connes And Kontsevich, The Evolution Of Concepts Of Space And Symmetry”, which does at least some of that - it’s a fairly big-picture review that touches on the relationship between these new ideas of space. It follows that stream of the story of space up to the end of the 20th century or so.

There’s also a little historical/biographical note on Alexander Grothendieck - the historical context is nice to see (one of the appealing things about Jammer’s book). In this case, much of the interesting detail is more relevant if you find recent European political history interesting - but I do, so that’s okay. In fact, I think it’s helpful - maybe not mathematically, but in other ways - to understand the development of mathematical ideas in the context of history. This view seems to be better received the more ancient the history in question.

On the scientific end, Cartier tries to explain Grothendieck’s point of view of space - in particular what we now call  topos theory - and how it developed, as well as how it relates to Connes’.  Pleasantly enough, a key link between them turns out to be groupoids!  However, I’ll pass on commenting on that at the moment.

…

Instead, let me take a bit of a tangent and jump back to Jammer’s book.  I’ll tell you something from his chapter “Emancipation from Aristotelianism” which I found intriguing.  This would be an atomistic theory of space - an idea that’s now beginning to make something of a comeback, in the guise of some of the efforts toward a quantum theory of gravity (EDIT: but see comments below).  Loop quantum gravity, for example, deals with space in terms of observables, which happen to take the form of holonomies of connections around loops.  Some of these observables have interpretations in terms of lengths, areas, and volumes.  It’s a prediction of LQG that these measurements should have “quantized”, which is to say integer, values: states of LQG are “spin networks”, which is to say graphs with (quantized) labels on the edges, interpreted as areas (in a dual cell complex).  (Notice this is yet again another, different, view of space, different from Grothendieck’s or Connes’, but shares with Connes especially the idea of probing space in some empirical way.  Grothendieck “probes” space mainly via cohomology - how “empirical” that is depends on your point of view.)

The atomistic theory of space Jammer talks about is very different, but it does also come from trying to reconcile a discrete “quantum” theory of matter with a theory linking matter to space.  In particular, the medieval Muslim philosophical school known as al Kalam tried to reconcile the Koran and Islamic theology with Greek philosophy (most of the “Hellenistic” world conquered by Alexander the Great, not least Egypt, is inside Dar al Islam, which is why many important Greek texts came into Europe via Arabic translations).  Though they were, as Jammer says, “Emancipating” themselves from Aristotle, they did share some of his ideas about space.

For Aristotle, space meant “place” - the answer to the questions “where is it?” and “what is its shape and size?”. In particular, it was first and foremost an attribute of some substance.  All “where?” questions are about some THING.  The answer is defined in terms of other things: my cat is on the ground, under the tree, beside the house.  The “place” of an object was literally the inner shell of the containing body that held it (which was contained by some other body, and so on - there being no vacuum in Aristotle).  So my “place” is defined by (depending how you look at it) my skin, my clothes, or the walls of the room I’m in.  This is a relational view of space, though more hard-headed than, say, Leibniz’s.

The philosophers of the Kalam had a similar relational view of space, but they didn’t accept Aristotle’s view of “substances”, where each thing has its own essential identity, on which attributes are hung like hats.  Instead, they believed in atomism, following Democritus and Leucippus: bodies were made out of little indivisible nuggets called “atoms”.  Macroscopic things were composites of atoms, and their attributes resulted from how the atoms were put together.  Here’s Jammer’s description:

The atoms of the Kalam are indivisible particles, equal to each other and devoid of all extension.  Spatial magnitude can be attributed only to a combination of atoms forming a body.  Although a definite position (hayyiz) belongs to each individual atom, it does not occupy space (makan).  It is rather the set of these positions - one is almost tempted to say, the system of relations - that constitutes spatial extension….

In the Kalam, these rather complicated and surprisingly abstract ideas were deemed necessary in order to meet Aristotle’s objections against atomism on the ground that a spatial continuum cannot be constituted by, or resolved into, indivisibles nor can two points be continuous or contiguous with one another.

So like people who prefer a “background independent” quantum theory of gravity, they wanted to believe that space (geometry) derives from matter, and that matter is discrete, but space was commonly held to be continuous.  Also alike, they resolved the problem by discarding the assumption of continuous space, and, by consideration of motion, to discrete time.

There are some differences, though.  The most obvious is that the nodes of the graph in a spin network state don’t represent units of matter, or “atoms”.  For that matter, quantum field theory doesn’t really have “atoms” in the sense of indivisible units which don’t break apart or interact.  Everything interacts in QFT.  (In some sense, interactions are more fundamental units in QFT than “particles” are - particles only (sic!) serve to connect one interaction with another.)

Another key difference is how space relates to matter.  In Aristotle, and in the Kalam, space is defined directly by matter: two bits of matter “define” the space between them.  In General Relativity (the modern theory with the “relational” view of space), there’s still room for space as an actor in its own right, like Newton’s absolute space-as-independent-variable - in other words, room for a vacuum, which Aristotle categorically denied could even conceivably exist.  In GR, what matter determines is the curvature of space (more precisely the Einstein tensor of the curvature).

Well, so the differences are probably more informative than the similarities,

(Edit: To emphasize a key difference glossed over before…  It was coupling to quantum matter which suggested quantizing the picture of space.  Discreteness of the spectrum of various observables is a logically separate prediction in each case.  Either matter or space(time) could have had continuous spectrum for the relevant observables and still been quantized - discrete matter would have given discreteness for some observed quantities, but not area, length, and so on.  So in the modern setting, the link is much less direct.)

 but the fact that theories of related discreteness in matter, space, and time, have been around for a thousand years or more is intriguing.  The idea of empty space as an independent entity - in the modern form only about three hundred years old - appears to be the real novel part.  One of the nice intuitions in Carlo Rovelli’s book on Quantum Gravity, for me at least, was to say that, rather than there being a separate “space”, we have a theory of fields defined on other fields as background - one of which, the “gravitational field” has customarily been taken for “space”.  So spatial geometry is a field, and it has some propagating (through space!) degrees of freedom - the particle associated to this field is a graviton.  Nobody’s ever seen one, mind you - but supposing they exist makes many of things easier.

To re-state a previous point: I think this is a nice aspect of categorification for dealing with space.  Extending the “stuff/structure/properties” trichotomy to allow space to resemble both “stuff” and relations between stuff leaves room for both points of view.

…

I mention this because tomorrow I leave London (Ontario) for London (England), and thence to Nottingham, for the Quantum Gravity and Quantum Geometry Conference.  It’s been a while since I worked much on quantum gravity, per se, but this conference should be interesting because it seems to be a confluence of mathematically and physically inclined people, as the name suggests.  I read on the program, for example, that Jerzy Lewandowski is speaking on QFT in Quantum Curved Spacetime, and suddenly remember that, oh yes, I did a Masters thesis (viz) on QFT in curved (classical) spacetime… but that was back in the 20th century!

It’s been a while, and I only made a small start at it before, but that whole area of physics is quite pretty.  Anyway, it should be interesting, and there are a number of people I’m looking forward to talking to.

 

Categorification and Matter

First, the obligatory excuse found in most sporadic blogs: I haven’t taken the time to write anything here recently. I was busy for a while, between the trip to UC Davis to speak (giving a form of this talk) at the “Strings and Gravity” seminar there, and then catching up on teaching - the end of the term is coming up. There: now that’s out of the way.

Right now I want to say something a bit broader than I have been doing - somewhere between “intuitive justification” and “philosophy”. The motivation is that whenever I talk about ETQFT’s and how to see them as introducing matter into quantum gravity, there’s always some puzzlement about this “categorification” business. To people who think a lot about category theory, it may seem natural, but many of those interested in physical questions don’t fall in this category, and the whole idea of “categorifying” a theory seems like a weird, arbitrary imposition.

So talking to these different audiences has forced me to think about how to give an intuitive account of why this might be a good idea. Ideally this will not be so precise as to be incomprehensible, or so vague as to be useless. In reality, this will be at best a rough sketch of such a justification.

Stuff, Structure, and Properties

One aspect of the relationship which I wanted to comment on, one that almost seems like a pun, is the trichotomy which John Baez and Jim Dolan like to use in describing mathematical, um, widgets (I would use the more standard term “objects”, or maybe “structures”, but both of these words have technical meanings in the following) in categorical terms. This is the distinction between “stuff”, “structure”, and “properties”. (More details here and via subsequent links - some of which shows up in my first paper). Almost any usual mathematical widget can be broken down this way: (1) they consist of some “stuff”, often in the form of some sets; (2) the stuff is equipped with “structure”, often described by some functions; (3) the structure satisfies some “properties”, often expressed as equations.

For example: a group is (1) a set of elements, equipped with (2) a group operation (expressed as a function ), and a special identity element (picked out by a function from the one-element set, ), and an inverse for each element (given by an inverse function . These satisfy (3) the group axioms, which are some equations involving expressing some properties - associativity, the properties of and inverses.

In this case, the structure live inside the category of sets and functions - but similar things could be said in any other category. For instance, in the category of topological spaces and continuous functions, the same setup gives the definition of a topological group, likewise divided into “stuff” (objects, in this case topological spaces), “structure” (some morphisms), and “properties” (equations between morphisms).

Widgets which live in an -category of some kind have more of these layers - such a widget will be specified by one or more objects, equipped with specified morphisms and 2-morphisms, satisfying some equations. A monoidal category, for instance, is this kind of widget: it has a category worth of “elements”, equipped with a monoidal operation given as a functor, equipped in turn with specified 2-isomorphisms such as the “associator”, which satisfies some equations such as the Pentagon identity. There are now FOUR levels to specify. I think it was Jim Dolan who came up with the following way of extending the “stuff/structure/properties” terminology (his explanation).

The highest level - equations - always deserves the name “properties”, since they either hold, or don’t (at least, there’s a truth value associated to them - but let’s not worry about multiple-valued logics). By analogy, this suggests the data for our widget given by the -morphisms in the -category where it lives should be called “structure”. The -morphisms (which are the objects in a 1-category) should be called “stuff”.

For the , , and generally -morphisms, Jim introduces the prefix “eka”, as in “eka-stuff”, which follows Mendeleev’s nomenclature for elements predicted by his form of the periodic table of elements which were heavier than known ones. This nomenclature in turn comes from the Sanskrit “eka”, meaning “one” - the new elements were one level lower on the periodic table.

So specifying a widget in a 2-category involves “eka-stuff/stuff/structure/properties”. This is suggestive, in that it seems as if categorification - adding a new level - is like digging out a new sub-basement beneath a house. First “eka-stuff”, then “eka-eka-stuff”, and so on, to “eka^k-stuff”. Since, in many versions of -category, given two objects and , the totality of morphisms form an -category, this is somewhat correct: there is an -categorical structure describing each .

(The periodic-table analogy, I suppose, is meant to imply that the best-understood layer is the layer of equations - which describe properties. This opposes what is probably the more common intuition people have when first encountering higher categories, that we know what “objects” are, but find “higher morphisms” confusing. But when writing things concretely, it’s the highest-level morphisms which look most familiar, like functions.)

A key point here is that “stuff having structure satisfying properties” is a fairly intuitive framework for talking about things. Categorification gives us a more nuanced layering. It may seem odd to speak of “eka-stuff equipped with stuff equipped with structure satisfying properties” (even worse if you want to be consistent, and say “equipped with” instead of “satisfying”). But now the second layer - stuff, refers to 1-morphisms. Here is a layer which has some aspects we associate with “structure”: it describes relations between the eka-stuff (objects). On the other hand, it also has aspects we associate with “stuff” (it can be equipped with its own structure). When would one want something that is on the one hand something like a relational attribute between things (structure), and on the other hand something like an object in its own right (stuff).

One answer: to describe space. As a good Leibnizian, I prefer to think of space relationally: it describes how objects are situated in terms of structural relationships. On the other hand, General Relativity tells us that if we think about space, rather than spacetime, we need to describe it as having dynamics which satisfy some property. From this point of view, space is like material stuff that changes over time, according to some differential equation (classically, at least).

Matter = Stuff?

Now, part of the point of applying extended TQFT ideas to gravity is that the categorification introduces matter into the formerly empty background of topological gravity - in particular, the state of a bit matter is described by looking at the boundary conditions on a codimension-2 surface in spacetime (or codimension-1 surface in space) surrounding it. The “pun” I alluded to above is the idea that introducing matter amounts to introducing a new layer of “stuff”. Adding matter means adding “stuff”…

The pun isn’t quite dead on, however, because in the ETQFT setup, adding matter is actually adding “eka-stuff”: digging out a sub-basement on which the “stuff” of geometrized space and its dynamics can rest.

So how does the periodic table of stuff/structure/properties relate to an extended TQFT? To start with, consider the case of an ordinary TQFT in 2 dimensions. It’s well known that such TQFT’s correspond to commutative Frobenius algebras (though see e.g. this paper by Aaron Lauda and Hendryk Pfeiffer, where they explain this, and a generalization of it). That is, a TQFT defines an object with (1) Stuff: a vector space, equipped with (2) Structure: unit, counit, multiplication, and comultiplication maps, satisfying (3) Properties: a bunch of axioms, including the Frobenius relation, commutativity, and algebra axioms like associativity.

The key thing is that this correspondence comes from the fact that a 2D TQFT is a functor into from the category , which happens to be a symmetric monoidal category freely generated by one object (the circle), and some morphisms (corresponding to four cobordisms: the cap, cup, “pair of pants”, and “inverted pair of pants”), subject to just the topological relations making the circle with these maps into a “Frobenius object”. (Since the cobordisms are only defined up to diffeomorphism).

Then any actual “physical” setting will look like: a bunch of circles, say of them, connected to another bunch of circles, say of them, by some cobordism. We could call this a “string world sheet” (although not in the sense of string theory, exactly, since over there one typically has conformal structure on the cobordisms too, and talks about a CFT, not a TQFT, living on the sheet). In general, the cobordism will be an -punctured, genus- torus (with orientations that distinguish the inputs from the outputs). So if the dynamics of the “physical” world are described by a TQFT corresponding to Frobenius algebra , this topology will mean the space of states of the world is given by at the beginning and at the end (this is “stuff”). A state evolves through “time” by the morphism (”structure”) corresponding to the cobordism - a particular combination of multiplication and comultiplication maps for the

In a theory of gravity without matter, we can see three levels as well - “slices” of space with some geometric information, connected by spacetimes with geometric information, which satisfy some equations. In particular, the geometric information on spacetime has to satisfy Einstein’s equation, if we’re talking about the classical world, or some sort of Hamiltonian constraint in (some approaches to) quantum gravity. In any case, it must have some property to be admissible. So this suggests the classifications: “space geometry” - stuff; “spacetime geometry” - structure; “dynamical laws” - properties.

Categorification suggests adding to this list: “matter/boundary conditions” - eka-stuff. That is, the eka-stuff in a specific physical setting will be a “2-space of states” for matter as measured at a particular boundary. In a 3D ETQFT, for instance, the boundaries to space will be unions of circles (just as in a 2D TQFT), so this will be generated by a 2-space of states for a circle. The circle could be thought of as the boundary around a single excised particle, but in fact that only covers the irreducible 2-states: in general, it’s a boundary around some region containing a system. Space geometry relates such boundaries to each other: it is “stuff” relating the “eka-stuff”. That stuff (space geometry), in turn, can be equipped with structure - maps associated to a spacetime topology, which describe how it evolves in “time” (though a-priori there’s no special time direction - the “stuff” could equally well describe the world-sheet of the system boundary, and the structure describing how that evolution extends outward spatially).

It seems to me there’s a lot here, but to really say it properly would require being much more technically precise than I’m up to at the moment. So that’s about all I have to say about that.

 

Visiting PI - Colloquium by Robert Spekkens

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.

 

Visiting Perimeter Institute

Once again, I keep meaning to write some less math-heavy posts, if for no other reason than to keep in the habit of thinking up things to write in here. Now is a good occasion to do this, since I’m visiting at the Perimeter Institute in Waterloo to give a talk called “Extended Topological Quantum Field Theories and Quantum Gravity” at the quantum gravity seminar on Thursday (the 28th). This is basically an updated and refined version of the talk I gave for my thesis defense, in which I’ve tried to make more of the link to physics - in particular, to BF theory, and to 3D quantum gravity. This turns out to be hard to do in an hour-long talk and still cover things adequately. Still, I find it worthwhile to get the point of view of real physicists on these apparently physics-related ideas, after thinking about them as a mathematician for some time.

After I arrived, I had lunch with a bunch of the quantum gravity people here. The conversation ranged from hunting for jobs, through cultural differences between Europe, Canada, and the US (a standard conversation to be had anywhere in Canada at the drop of a hat), all the way over to “Why is spacetime 4-dimensional?” Lee Smolin put this last one to me when I was describing how categorification is related to considering higher co-dimensions of spacetime/space/surfaces in space. It’s a reasonable question, though not one I have any answer to. But when you cook up a theory - like this ETQFT stuff - which in principle works in any number of dimensions, and you want it to be physical, you’re left wondering “why so few dimensions?”

Okay - it’s not the main point of what I’m doing here, but it’s a nice light question to blog about, since I don’t pretend to have even a good guess at the answer.

It takes a certain mentality to think that 4 dimensions is astonishingly few - however, I have that mentality, as do many mathematicians. You can work with infinite-dimensional spaces in mathematics - why should “real”, “physical” space only have four? Actually, the segue into this had to do with the question of why all the Lie groups that turn up in physical gauge theories are so tiny - , , - rather than, say, , which describes rotations in a 745 (complex) dimensional space. Again: gauge theory makes just as much sense with big gauge groups as small ones - so what’s special about the low dimensions?

Well, I don’t know the answer - but it’s the kind of question mathematicians probably should be asked more often. We’re perfectly happy to deal with a 745 dimensional space and not worry about the fact that it’s non-physical. But if mathematics really underlies physics in any deep way, there should be some good mathematics in the answer.

There were some possibilities tossed around: what if the exceptional group really does turn out to be important in fundamental physics, and the real gauge group of the right physical theory has to lie inside it somewhere? Then there’s an upper bound on how many dimensions you can have - though, unfortunately, is 248-dimensional, so the upper bound is a bit high. (Mind you, the symmetries of 4D space is, in itself, a 10-dimensional group, so things are not quite as bad as they appear - but still worse than they should be). There’s also no obvious reason why should have such a special role.

A more physics-y answer is that in 5D and higher, you don’t get confinement - quarks and gluons just fly around like a dilute gas, and there would be no matter in the sense we know it. This is a great concise description of why we should be happy to live in a 4D spacetime. The objection to this is that it’s basically an appeal to the anthropic principle: “If space weren’t 4D, we wouldn’t be here to wonder why.” If you’ve read Lee Smolin’s most recent book, you’ll know he doesn’t care for appeals to the anthropic principle. Neither do I, for that matter. If you assume that every possible universe actually exists (which is at least metaphysically parsimonious - no need for two separate categories of “possible” and “actual”), the anthropic principle is undeniable. The problem is, it doesn’t predict very much until you work out enough about what universes are possible that you might as well just try to answer the question for its own sake. Still, maybe it’s just true that there are a huge number of actual universes, and some of them are no good for intelligent life. But that just means the question has no answer, so you might as well give up. It doesn’t take you anywhere. So suppose there’s a reason: what could it be?

In 3 and 4 dimensions, there are regular polyhedra - or, equivalently, discrete subgroups of the rotation group - that don’t correspond to the series which always exists. In 2D, there are infinitely many regular polygons, and in all dimensons, there are simplexes, cubes, and duals of cubes… but in 3 and 4D there are some extras, all of which boil down to the icosahedron, its dual, or things you can construct from it in 4D. Why this should make any difference, I have no idea.

And there are a couple of other special things in low dimensions, which are no more obviously relevant, but seem compelling to me, perhaps because I’m a mathematician…

In 4 dimensions, but no other dimensionality, there are “exotic” which are homeomorphic but not diffeomorphic to the usual . The heuristic explanation for why (which is as much as I really grasp) is that 4D is “big enough” for complicated twisty things to exist, but “too small” for there to always be room to untangle them - so only in 4D can “things be complicated”. Which is suggestive, but hardly a full answer.

4 dimensions is the only case where the classification of manifolds is not understood (now that the Poincaré conjecture has been settled - there were still some lingering doubts last I heard, but they seem to be evaporating day by day). in 2D, manifolds are basically just toruses with some genus; in 3D manifolds can be cut up into pieces each of which can be geometrized (a la Thurston). In 5D and higher, you can classify (in principle) manifolds by constructing them via surgeries. The reason this doesn’t work in 4D is that surgeries building new manifolds correspond to cobordisms between the input and output manifolds, and in 5 or more dimensions, cobordisms are rather trivial (actually, this only refers to cobordisms where the inclusions of the source and target manifolds are homotopy equivalences, which isn’t totally general).

This last bit seems the most intriguing to me, since I’ve been thinking about TQFT’s and ETQFT’s, which are field theories living on cobordisms. But that still doesn’t add up to an answer to the physical question. It would be nice to understand, for instance, whether the above fact means anything helpful in terms of the physics of such a theory.

Anyway, I’ll try to write up something about those theories from a physical point of view after I’ve had a chance to chit-chat about them with some physicists after my talk. It probably won’t answer this rather vague and (perhaps?) unanswerable question, but there seem to be some interesting things to say. Maybe before then (but after I’ve had a chance to give my talk, no doubt!) I’ll also give a little write-up of the colloquium talk by Robert Spekkens I attended today about foundations of quantum mechanics.

 

n-group Representation Theory - part 3: Rep(Poinc)

I’m going to be giving a talk on extended TQFT stuff and quantum gravity at Perimeter Institute next thursday, and then in mid-March I’ll be heading to UC Davis to give the same/similar talk for the String Theory and Quantum Gravity seminar being run by Derek Wise. So I have a bunch of things on my mind right now. However, before heading to Davis, I wanted to go back and look at some of the stuff Derek has done having to do with Cartan geometry, which I was following somewhat at the time, and blog about it a bit here. Before that, I’d like to wrap up this presentation of the talks I gave here about representation theory of the Poincaré 2-group, .

As a side note, thanks to Dan for pointing out these notes on representations of the (normal, uncategorified) Poincaré group, including some general comments on representations of semidirect products. It’s interesting to consider how this relates to the more general picture of 2-group representations - but I won’t do so here and now.

…

In Part 1 I talked about what representations 2-categories of 2-groups are like in general, and in Part 2 a fairly concrete description of . Here I’ll wrap up by summarizing the results of Crane and Sheppeard about what looks like concretely.

It has three parts: the objects are representations (also known as functors from as a 2-category with one object, into ); the morphisms are 1-intertwiners (a.k.a. natural transformations) between reps; and the 2-morphisms are 2-intertwiners (a.k.a. modifications) between 1-intertwiners.

1) Representations: A functor

will pick out some measurable space for the lone object of the 2-group - or rather, , the 2-vector space of all measurable fields of Hilbert spaces on . (This is a matter of taste since to know the one is to know the other.) Then for the morphisms and 2-morphisms of we get, respectively, 2-linear maps from to itself, and natural transformations between them.

The morphisms of are just the group in the crossed-module picture I described in Part 2. For the usual Poincaré 2-group, this is . For each such element, we’re supposed to get an invertible 2-linear map from to itself - that is, a measurable field of Hilbert spaces on (together with measures to do “matrix multiplication” with by direct integrals). This can only be invertible if the only Hilbert spaces which appear are 1-dimensional (since these maps compose by a “matrix multiplication” involving direct sums of tensor products of the components - and the discreteness of dimensions means that if any dimension is higher than 1, you’ll never get back the identity).

So any representation turns out to give what amounts to an action of on - the component is if and 0 otherwise. An irreducible representation gives an with a transitive action (otherwise, you can decompose it into orbits, each of which corresponds to a subrepresentation). Crane and Sheppeard classify several kinds of these, associated to various subgroups of , but an easy example would be a mass shell in Minkowski space - a sphere or hyperboloid (depending on ) that is the full orbit of some point under rotations and boosts (a “mass shell” because it gives all the possible momenta for a particle of a given mass, as seen by an observer in some inertial frame).

The 2-morphism part of gives a homomorphism from at each of these points. Now, one-by-one matrices of complex numbers are just complex numbers, so what we have here is a character of - at each point on . To be functorial, this has to be done in an equivariant way (so that acting on the point by affects the character by acting on by the same ).

2) 1-Intertwiners:

If representations and correspond to actions of on spaces and respectively, with characters , then what is a 1-intertwiner ? Remember from Part 1 that it’s a natural transformation: to the object of it assigns a specific 2-linear map

To each (object of it gives a transformation

This is a specified map which replaces the naturality square in the old definition of an intertwiner. It has to make a certain “pillow” diagram commute (Part 1).

Now, back in the posts on 2-Hilbert spaces, I explained that a 2-linear map is given by some field of Hilbert spaces on (a “matrix” of Hilbert spaces, though of course needn’t be finite), along with a family of measures on indexed by (which allow us to do integration when doing the sum in “matrix multiplication”). The transformations also can be written in components, so that

(Note this uses the two actions given by on - one forward, and one backward. This is the current form of what, in uncategorified representation theory, would be a naturality condition.)

What does this all amount to? One way to think of it is as a representation of itself! In particular, it’s a representation on the direct sum of all the Hilbert spaces which appear as components of . This is since the maps given by the have to satisfy a condition which says that composition is preserved (as long as you’re careful about indexing things):

To get a representation of the group, we can say that elements shuffle vector spaces over points in by the action of and then act within vector spaces by . So then has both intertwiner-like and representation-like properties.

The “intertwiner-ness” of has to do with how it interpolates between two actions on by turning them into an action on the product - but it also has some “representation-ness”, by giving this action of a (semidirect product) group on a big vector space.

3) 2-intertwiners

If a 1-intertwiner can be thought of as a representation of , it shouldn’t be too surprising that a 2-intertwiner between 1-intertwiners ends up being an intertwiner between the associated representations. If 1-intertwiners have some qualities of both reps and intertwiners, the 2-intertwiners are more single-minded.

In particular, a 2-intertwiner assigns to the only object of a 2-morphism in (that is, a field of linear maps between the vector spaces which are the components of ), which satisfies some “pillow” diagram. When we form the big rep. by taking a direct integral of all those spaces, the field of linear maps turns into one big linear map, and the diagram it satisfies just collapses into the condition that it be an intertwiner.

So the representation theory of this interesting 2-group looks a lot like the representation theory of the group of 2-morphisms. The extra structure involving actions on measurable spaces by would be mostly invisible if you just thought about irreducible reps of the group, since the space would be just a single point.

This phenomenon where a lower-order structure turns up in some form at the top level of morphisms of its categorified version has cropped up before in this blog - namely, when extended TQFT’s turn out to contain normal TQFT’s in individual components. In these examples, categorification is less a matter of building more floors “on top” of structures we already know, as “higher morphisms” suggests, but excavating additional floors of subbasement - interpreting what were objects as morphisms.

 

n-group Representation Theory - (part 2: the Poincaré 2-group)

It’s been a while since I wrote the last entry, on representation theory of n-groups, partly because I’ve been polishing up a draft of a paper on a different subject. Now that I have it at a plateau where other people are looking at it, I’ll carry on with a more or less concrete description of the situation of a 2-group. For higher values of , describing things concretely would get very elaborate quite quickly, but interesting things already happen for . In particular, the case that I gave the talk about, a while back, was mostly the Poincaré 2-group, since this is the one Crane, Sheppeard, and Yetter talk about, and probably the one most interesting to physicists.  It was first described by John Baez.

So what’s the Poincaré 2-group? To begin with, what’s a 2-group again?

I already said that a 2-group is a 2-category with only one object, and all morphisms and 2-morphisms invertible. That’s all very good for summing up the representation theory of as I described last time, but it’s sometimes more informative to describe the structure of concretely. A good tool for doing this is a crossed module. (A lot more on 2-groups can be found in Baez and Lauda’s HDA V, and there are some more references and information in this page by Ronald Brown, who’s done a lot to popularize crossed modules).

A crossed module has two layers, which correspond to the morphisms and 2-morphisms of . These can be represented as , where is the group of morphisms in , consists of the 2-morphisms ending at the identity of (a group under horizontal composition).

There has to be an action of on (morphisms can be composed “horizontally” with 2-morphisms), and a map (which picks out the source of the 2-morphism). The data have to fit together a certain way, which amounts to giving the axioms for a 2-category.

A handy way to remember the conditions is to realize that the action and the injection give ways for elements of to act on each other and for elements of to act on each other. These amount to doing first and then or vice versa, and both of these must amount to conjugation. That is:

and

Both of these are simplified in the case that maps everything in to the identity of - in this case, can be interpreted as the group of 2-automorphisms of the identity 1-morphism of the sole object of . In this case, by the Eckmann-Hilton argument (the clearest explanation of which that I know being the one in TWF Week 100) it turns out that has to be commutative, so the first condition is trivial since , and the second is trivial since it follows from commutativity. This simpler situation is known as an automorphic 2-group.

In any case, given a 2-group represented as a crossed module, automorphic or not, the collection of all morphisms can be seen as a group in itself - namely the semidirect product , which is to say with the multiplication . “What?” you may ask, or maybe “Why?”

Maybe a concrete example would help, since we’d like one anyway: the Poincaré 2-group, which is an automorphic 2-group. There are versions of various signatures , in which case , and .

The group , then, consists of metric-preserving transformations of Minkowski space with the metric of signature - rotations and boosts (if any). The (abelian) group consists of translations of this space - in fact, being a vector space, it’s just a copy of it. Between them, they cover the basic types of transformation. Thinking of the translations as having a “projection” down to the identity rotation/boost may seem a bit artificial, except insofar as translations “don’t rotate” anything. More obvious is that rotations or boosts act on translations: the same translation can look differently in rotated/boosted coordinate systems - that is, to different observers.

So where does the Poincaré group come in? It’s the group of all metric-preserving transformations of Minkowski space, and is built from these two types: but how?

Well, the vector space is the group of transformations of the identity Lorentz transformation , since the map is trivial. But suppose that there is another copy of over each point in . Then we have the set of points , but notice that to talk about this as a group, we’d want a way to act on an element of one copy of over by another over . The obvious way is to just treat the whole set as a product of groups, but this misses the fundamental relation between and , which is that can act on , just as morphisms can act on 2-morphisms by “whiskering with the identity”. (Via Google books, here is the description of this in MacLane’s Categories for the Working Mathematician).

Concretely, this is the fact that there is a sensible way for both parts of to affect the , so we can say (using additive notation for translations, since they’re abelian). The point is that the first rotation we do, , changes coordinates, and therefore the definition of the translation .

So that’s the construction of the Poincaré group from the Poincaré 2-group. What would be nice would be to have some clear description of some higher analog of Minkowski space where it makes sense to say the Poincaré 2-group acts as a 2-group. I don’t quite know how to set this up, but if anyone has thoughts, it would be interesting to hear them.

One reason is that, when describing representations of the 2-group, there’s an important role for spaces (or at least sets) with an action of the group - which raises questions like whether there’s a role for 2-spaces with 2-group actions in representation theory of higher -groups. Again - I don’t really know the answer to this. However, in Part 3 I’ll describe concretely how this works for 2-groups, and particularly the Poincaré 2-group.

 

2-Hilbert Spaces - pt 2

So I gave a little talk shortly before leaving London for Christmas. I had mostly written it up, but then I’ve been on the road for a while in Montreal, Ottawa, and Calgary, without consistent net access. However, now I have a moment to put this up.

The talk carried on from the previous one I described last post. It began to move in the direction of representation theory of 2-groups on 2-vector spaces and 2-Hilbert spaces, but didn’t get that far. This was partly because I had to finish describing what 2-linear maps and 2-maps look like for such spaces, and then because I had to explain about 2-groups and give some examples. I’ll say more about the representation-theory stuff in January. But here I’ll just summarize at least the rest of the description of the category , and also by describing 2-linear maps and so forth. Then I’ll comment a little more philosophically about what these are about.

So I explain how there’s a 2-vector space (in some suitable sense, not the KV sense) of measurable fields of Hilbert spaces on a space , analogous to the vector space of complex functions on a space. Also similarly, given a measure on , we get an inner product. Then there’s a (2-)Hilbert space where this inner product is always well-defined (as a complex scalar, or a genuine Hilbert space - which is the equivalent of a scalar at the next level up).

Well, then Crane and Yetter’s paper describes constructively how to get 2-linear maps (additive, linear functors) between such 2-vector spaces. They don’t as far as I can see, show that all functors arise this way, but it seems likely. The way is to say you get a functor from:

1) A measurable field of Hilbert spaces (this is similar to the linear maps between KV 2-vector spaces, which are like matrices of vector spaces)

2) A -indexed family of measures on - these give you the measures you need to do the “inner product” involved in “matrix multiplication” at each (note that this stuff is only well-defined up to sets of measure 0, as usual). So we have, on objects:

and a related expression for morphisms, using the identity on .

It’s probably worth pointing out that the measures on are used in the direct integral here, and so their only real role is to define the inner product on - the underlying vector space at each point in the new field would be the same no matter what these measures were (up to the fact that if the resulting inner product is degenerate, we need a quotient space where it’s not).

So this gives 2-linear maps, which are functors. Natural transformation between these functors come from the fact that is itself a category, and in fact a 2-vector space in the sense we’re using here ( is “enriched over itself”). So morphisms between these fields of Hilbert spaces basically amount to fields of bounded operators as usual. This is actually not quite right, because we need to account for the different measures: basically, you use a measure which is the geometric mean of those associated to source and target - check out Crane and Yetter’s paper if you want the details.

That finishes up a summary of how 2-Hilbert spaces work. The next thing I’ll be talking to our group about is how to use these for a categorified form of representation theory.

…

But first, what is the point of all this stuff? Not yet asking about representation theory in this setting - why is it interesting enough to bother? It’s worth thinking about what a categorification of a Hilbert space is supposed to be. In particular, let’s try locating them in the world of quantum mechanics.

A quantum system is usally portrayed as having states represented by vectors in a Hilbert space. The only things you can “do” to states involve applying operators to the whole space: project them into subspaces, “rotating” them by some unitary evolution operator, and so on. In a 2-Hilbert space, states, or “2-vectors” are objects in a category, which means there are not only these “macro” operations on the whole space, but also morphisms between any two states you pick. In fact, this is the source of the inner product on a 2-Hilbert space - there is a Hilbert space (in the usual sense) of morphisms between any two states, and in the world of 2-Hilbert spaces, this is the equivalent of a scalar.

In QM, the inner product is telling you an amplitude to observe a system in state if it was set up in state - this is saying something about “how related” is to . The categorifed picture saying this is just makes more explicit what kind of relationship this is.

Now, if you happen to pick the same vector to start and end with, considering , what this is saying is that there’s some bunch of “symmetry operations” on a state. (Taking just the invertible ones gives an actual symmetry group for a given state.) This is saying that “state 2-vectors” have some internal degrees of freedom. Their amplitudes give a measure of how many such degrees of freedom there are.

The fact that a 2-Hilbert space is described as an enriched category means that the usual picture of a quantum system returns when you look in individual components of a state 2-vector. In particular, the coefficients of a 2-state vector can be thought of as Hilbert spaces representing a system in that particular component. So, for instance, part of the big project I’m describing in these notes is to depict quantum gravity (at least in 3 dimensions) as an extended TQFT, which represents a physical system with these 2-Hilbert spaces. A 2-state vector here describes the situation on a boundary of space - matrix elements of a 2-linear map are Hilbert spaces of connections on a given manifold interpolating between chosen boundary states. Natural transformations between 2-linear maps are what give amplitudes for spacetimes joining such slices of space.

So what is a state 2-vector? All these properties should fit together into some nice scheme: classical configurations can exist in a “2-state” in some kind of superposition, where each configuration gets its own internal degrees of freedom. The inner product emerges naturally from this, considering morphisms between 2-states. Every morphism between 2-states has to respect the classical configurations, giving for each one a map between the internal spaces associated to it in the two 2-states. Is there a more elegant way to sum this up? Probably so, but at the moment I don’t quite see how to put it.

However: next time, I’ll carry on with some representation theory.

 

Musing, and a Tale of Two Kauffmans

Writing sizeable chunks of math blog takes longer than I expected. Here are a few non-intensive things that occurred to me.

…

While I was walking home from the UWO campus, I was reminded of the nature of Canada in late November: everything, from sky to plantlife to earth, is in shades of grey, brown, ochre, and the occasional desaturated greenish-whatever. Autumn leaves have pretty much stopped falling, and are on the ground turning greyish versions of whatever colours they were before. There are whole vistas of bare branches, dead underbrush, and so on.

Which seems dreary for a while, until you’re immersed in it, as I am on the particular route I walk home, along London, Ontario’s Thames River (not to be confused with the River Thames in London, England), which is lined with parks. Then, after a while, all the subtle differences in shading and texture start to jump out at you more and more, until brownish moss on a tree under overcast late-afternoon light is vibrant green, a patch of snow is glowing bluish white, the occasional flicker of sunset through the cloud cover is warm pumpkin-orange, that one particular bush’s leaves look startlingly red… and then you see something artificial, like someone’s nylon jacket, or a kid’s plastic play-structure, and their colours look implausibly oversaturated, like a badly photoshopped picture.

Which got me to thinking about fine distinctions that seem drab outside their context - the way these colours look at first. Or nitpicky, like having 30 different words for “cold” and the different qualities it can have, or recognizing 15 different types of snowflake from a distance. Coming back to Canada after several years in California, I noticed all this specialized knowledge I’d forgotten about, and seems terribly arcane outside its native habitat. It occurred to me that this is how mathematics probably seems to outsiders - like physicists, or statisticians… (I jest)

For instance: I often have the experience of using the term “categorification” in describing something I’m doing - often in scare-quotes, followed by some kind of explanation - only to have it echoed back as “categorization”, and wonder whether to risk pedantry and explain that they’re not the same thing at all. “Categorification, not to be confused with…”

…

On another note, I went looking for this paper by Carter, Kauffman and Saito, on a kind of invariant of 4-manifolds which generalizes 3D Dijkgraaf-Witten invariants, on the supposition that it would be closely related to some things I’ve been thinking about, from a diagrammatic point of view I’ve not paid much attention to in the last year or so. As I was looking through seach results, I noticed a paper from about 10 years ago by Kauffman and Smolin with an interesting sounding title, A Possible Solution to the Problem of Time in Quantum Cosmology. Since Lee Smolin has written on linking topological field theory and quantum gravity, I guessed it would also be interesting to look at. Only after reading the first few pages did I notice that the first listed author was not Louis Kauffman, who studies knot theory (and things tangent thereto), but Stuart Kauffman, who studies biocomplexity and complex systems.

I happen to be interested in the work of both Kauffmans - more immediately and professionally that of Louis, but I also read a couple of Stuart’s more accessible books, “At Home in the Universe”, and “Investigations” - and since the paper was short, I finished reading it. The basic premise is that the configuration space for 4D quantum gravity may not be constructible by any finite procedure (classifying spin networks, they say, might present a problem; doing path integrals over all 4-manifold topologies certainly does). So the “problem of time”, that there’s no role for time in describing dynamics in terms of paths through a configuration space, wouldn’t make sense - at least for a constructivist. (Or indeed a constructivist, though of course they shouldn’t be confused.) One thing that threw me off in noticing which Kauffman was involved was that part of this portion of the argument was about classifying knots.

That cleared itself up when they got to the part proposing a solution - that the total space of possible states isn’t a-priori given, but time re-enters the situation as the universe evolves, at each time step having some amplitude to move into each configuration in a (newly defined!) space called the adjacent possible. Having read Stuart K.’s books, this was when I realized my mistake - he describes this concept in “Investigations” in the context of a biosphere, or an economy, where a theorist also doesn’t have an explicit description of all possible future states given in advance.

It seems like this idea has a lot in common with type theory as a solution to Russell’s paradox: the collection of all sets isn’t a set, and so to get at it, sets are generated starting with nothing in successive stages. Whether this also doubles as a solution to the problem of time, I don’t know. In any case, it’s an interesting idea. It definitely would be a problem to have to do path integrals over a space of all topologies for 4-manifolds, when these can’t be classified, so some sort of suggestions are definitely a good thing here.

 

Morita Equivalence etc.

A recent talk in the noncommutative geometry seminar here, was by Farzad Fathizadeh. He was talking about a few ideas - the main part of the talk being about how to construct the Dixmier-Douady invariant, which is related to the question of whether or not you can put a spin structure on some manifold. It’s also related to a lof other things I want to figure out anyway for longstanding reasons. Indeed, Dixmier is one of the big early names behind the theory of fields of Hilbert spaces, which are used in Crane and Yetter’s “Measurable Categories”, which are a sort of infinite dimensional analog of the 2-vector spaces I’ve been talking about. (Actually, 2-Hilbert spaces, since that structure starts to look more important there).

Since I’ve started thinking about infinite dimensional 2-Hilbert spaces again I thought I’d check it out. It turned out to be somewhat related, but not very deeply. However, precisely because it’s related to things I’ve yet to figure out, I’m going to give a superficial gloss here, and later maybe try to say something more detailed. I should be giving a talk to our group soon about various aspects of 2-Hilbert spaces, so I’ll post more when I get to that.

…

The second part of the talk had a nice exposition of Morita equivalence, which was a notion that got a lot of use in the things people were talking about at Groupoidfest. I had heard about this concept before, but never quite got the hang of it until now, so here’s a quick little explanation for the record. There are two ways of describing Morita equivalence, and the content of Morita’s theorem is that the two definitions amount to the same thing.

One definition says that two algebras and are equivalent if the categories of modules over them, and are equivalent as categories. The other says that and are equivalent if there is an -bimodule, , and a -bimodule, with the properties that:

(as an -bimodule)

and

(as a -bimodule)

Where, if you’re unclear, an -bimodule is a set where the algebra acts on the left, and the algebra