Some different ideas of “state”, part 2: ensembles, 2-states, and selection sectors

I just posted the slides for “Groupoidification and 2-Linearization”, the colloquium talk I gave at Dalhousie when I was up in Halifax last week. I also gave a seminar talk in which I described the quantum harmonic oscillator and extended TQFT as examples of these processes, which covered similar stuff to the examples in a talk I gave at Ottawa, as well as some more categorical details.

Now, in the previous post, I was talking about different notions of the “state” of a system – all of which are in some sense “dual to observables”, although exactly what sense depends on which notion you’re looking at. Each concept has its own particular “type” of thing which represents a state: an element-of-a-set, a function-on-a-set, a vector-in-(projective)-Hilbert-space, and a functional-on-operators. In light of the above slides, I wanted to continue with this little bestiary of ontologies for “states” and mention the versions suggested by groupoidification.

State as Generalized Stuff Type

This is what groupoidification introduces: the idea of a state in . As I said in the previous post, the key concepts behind this program are state, symmetry, and history. “State” is in some sense a logical primitive here – given a bunch of “pure” states for a system (in the harmonic oscillator, you use the nonnegative integers, representing n-photon energy states of the oscillator), and their local symmetries (the -particle state is acted on by the permutation group on elements), one defines a groupoid.

So at a first approximation, this is like the “element of a set” picture of state, except that I’m now taking a groupoid instead of a set. In a more general language, we might prefer to say we’re talking about a stack, which we can think of as a groupoid up to some kind of equivalence, specifically Morita equivalence. But in any case, the image is still that a state is an object in the groupoid, or point in the stack which is just generalizing an element of a set or point in configuration space.

However, what is an “element” of a set ? It’s a map into from the terminal element in , which is “the” one-element set – or, likewise, in , from the terminal groupoid, which has only one object and its identity morphism. However, this is a category where the arrows are set maps. When we introduce the idea of a “history “, we’re moving into a category where the arrows are spans, (which by abuse of notation sometimes gets called but more formally ). A span represents a set/groupoid/stack of histories, with source and target maps into the sets/groupoids/stacks of states of the system at the beginning and end of the process represented by .

Then we don’t have a terminal object anymore, but the same object is still around – only the morphisms in and out are different. Its new special property is that it’s a monoidal unit. So now a map from the monoidal unit is a span . Since the map on the left is unique, by definition of “terminal”, this really just given by the functor , the target map. This is a fibration over , called here for “phi”-bration, but this is appropriate, since it corresponds to what’s usually thought of as a wavefunction .

This correspondence is what groupoidification is all about – it has to do with taking the groupoid cardinality of fibres, where a “phi”bre of is the essential preimage of an object – everything whose image is isomorphic to . This gives an equivariant function on – really a function of isomorphism classes. (If we were being crude about the symmetries, it would be a function on the quotient space – which is often what you see in real mechanics, when configuration spaces are given by quotients by the action of some symmetry group).

In the case where is the groupoid of finite sets and bijections (sometimes called ), these fibrations are the “stuff types” of Baez and Dolan. This is a groupoid with something of a notion of “underlying set” – although a forgetful functor (giving “underlying sets” for objects in a category ) is really supposed to be faithful (so that -morphisms are determined by their underlying set map). In a fibration, we don’t necessarily have this. The special case corresponds to “structure types” (or combinatorial species), where is a groupoid of “structured sets”, with an underlying set functor (actually, species are usually described in terms of the reverse, fibre-selecting functor , where the image of a finite set consists of the set of all “$\Phi$-structured” sets (such as: “graphs on set “, or “trees on “, etc.) The fibres of a stuff type are sets equipped with “stuff”, which may have its own nontrivial morphisms (for example, we could have the groupoid of pairs of sets, and the “underlying” functor selects the first one).

Over a general groupoid, we have a similar picture, but instead of having an underlying finite set, we just have an “underlying -object”. These generalized stuff types are “states” for a system with a configuration groupoid, in . Notice that the notion of “state” here really depends on what the arrows in the category of states are – histories (i.e. spans), or just plain maps.

Intuitively, such a state is some kind of “ensemble”, in statistical or quantum jargon. It says the state of affairs is some jumble of many configurations (which we apparently should see as histories starting from the vacuous unit ), each of which has some “underlying” pure state (such as energy level, or what-have-you). The cardinality operation turns this into a linear combination of pure states by defining weights for each configuration in the ensemble collected in .

2-State as Representation

A linear combination of pure states is, as I said, an equivariant function on the objects of . It’s one way to “categorify” the view of a state as a vector in a Hilbert space, or map from (i.e. a point in the projective Hilbert space of lines in the Hilbert space ), which is really what’s defined by one of these ensembles.

The idea of 2-linearization is to categorify, not a specific state , but the concept of state. So it should be a 2-vector in a 2-Hilbert space associated to . The Hilbert space was some space of functions into $mathbb{C}$, which we categorify by taking instead of a base field, a base category, namely . A 2-Hilbert space will be a category of functors into – that is, the representation category of the groupoid .

(This is all fine for finite groupoids. In the inifinte case, there are some issues: it seems we really should be thinking of the 2-Hilbert space as category of representations of an algebra. In the finite case, the groupoid algebra is a finite dimensional C*-algebra – that is, just a direct sum (over iso. classes of objects) of matrix algebras, which are the group algebras for the automorphism groups at each object. In the infinite dimensional world, you probable should be looking at the representations of the von Neumann algebra completion of the C*-algebra you get from the groupoid. There are all sorts of analysis issues about measurability that lurk in this area, but they don’t really affect how you interpret “state” in this picture, so I’ll skip it.)

A “2-state”, or 2-vector in this Hilbert space, is a representation of the groupoid(-algebra) associated to the system. The “pure” states are irreducible representations – these generate all the others under the operations of the 2-Hilbert space (“sum”, “scalar product”, etc. in their 2-vector space forms). Now, an irreducible representation of a von Neumann algebra is called a “superselection sector” for a quantum system. It’s playing the role of a pure state here.

There’s an interesting connection here to the concept of state as a functional on a von Neumann algebra. As I described in the last post, the GNS representation associates a representation of the algebra to a state. In fact, the GNS representation is irreducible just when the state is a pure state. But this notion of a superselection sector makes it seem that the concept of 2-state has a place in its own right, not just by this correspondence.

So: if a quantum system is represented by an algebra of operators on a Hilbert space , that representation is a direct sum (or direct integral, as the case may be) of irreducible ones, which are “sectors” of the theory, in that any operator in can’t take a vector out of one of these “sectors”. Physicists often associate them with conserved quantities – though “superselection” sectors are a bit more thorough: a mere “selection sector” is a subspace where the projection onto it commutes with some subalgebra of observables which represent conserved quantities. A superselection sector can equivalently be defined as a subspace whose corresponding projection operator commutes with EVERYTHING in . In this case, it’s because we shouldn’t have thought of the representation as a single Hilbert space: it’s a 2-vector in – but as a direct integral of some Hilbert bundle that lives on the space of irreps. Those projections are just part of the definition of such a bundle. The fact that acts on this bundle fibre-wise is just a consequence of the fact that the total is a space of sections of the “2-state”. These correspond to “states” in usual sense in the physical interpretation.

Now, there are 2-linear maps that intermix these superselection sectors: the ETQFT picture gives nice examples. Such a map, for example, comes up when you think of two particles colliding (drawn in that world as the collision of two circles to form one circle). The superselection sectors for the particles are labelled by (in one special case) mass and spin – anyway, some conserved quantities. But these are, so to say, “rest mass” – so there are many possible outcomes of a collision, depending on the relative motion of the particles. So these 2-maps describe changes in the system (such as two particles becoming one) – but in a particular 2-Hilbert space, say for some groupoid describing the current system (or its algebra), a 2-state is a representation of the of the resulting system). A 2-state-vector is a particular representation. The algebra can naturally be seen as a subalgebra of the automorphisms of .

So anyway, without trying to package up the whole picture – here are two categorified takes on the notion of state, from two different points of view.

I haven’t, here, got to the business about Tomita flows coming from states in the von Neumann algebra sense: maybe that’s to come.

 

A Note on Quantum Groupoids

I’ve been looking over the last little bit at quantum groupoids, and how they can be used to deform the 2-linearization 2-functor (or into ) which I’ve discussed in here.

First a little motivation: that functor was part of the way I constructed extended TQFT’s. The inclusion realized cobordisms (with corners) in terms of spans of manifolds. Looking at fundamental groupoids using the 2-functor allows us to think about these in terms of the bicategory , and then applying gave 2-Hilbert spaces and 2-linear maps for manifolds and cobordisms (and then natural transformations for cobordisms with corners). Since I made the claim that, with gauge group – and a suitably infinitary version of , the extended TQFT gives a theory equivalent to the Ponzano-Regge model of quantum gravity, a reasonable question is: what about the Turaev-Viro model? The PR model is based on labelling edges of a triangulation with representations of , and the TV model, with representations of .

Now, the groupoids that show up in the above – groupoids of -connections on a manifold, modulo gauge transformations – are quite closely related to this. In particular, the groupoid of connections for a circle (the basic 1-dimensional manifold that the 3-dimensional theory builds from) is , the transformation groupoid produced from the action of on itself by conjugation. (That is: the objects are elements of , and the morphisms are all the conjugacy relations.) Applying gives the representation category of this, namely , so in particular, at the identity of , one has as a sub-2-vector space. (The “states” in the 2-Hilbert space for the circle in the ETQFT are labelled by “masses and spins” – the mass=0 case is what gives the representations of , and for nonzero mass, one has .)

More broadly: one can describe the state space of a gauge theory – or many other kinds of theory, in terms of transformation groupoids given by symmetries (gauge transformations, say) acting on states (connections, in that case). Is there a way of doing the same for systems whose symmetries are described by quantum groups? If so, then instead of getting 2-vector spaces which are representation categories of groupoids, we should get some which are representation categories of quantum groupoids.

This paper by Ping Xu describes quantum groupoids – or rather, quantum universal enveloping algebras. They’re described here as a “unification of quantum groups and star products” (star products being the partially-defined composition found in groupoids). This paper by Nikshych and Vainerman describes finite quantum groupoids and some applications – in particular, quantum transformation groupoids, which is the immediately relevant application.

First off, quantum groups: these are Hopf algebras, which in particular are bialgebras – they have both a product

and “coproduct”

.

This is because the point here is that we’re following the pattern in which spaces are replaced by algebras: in some simple examples, these are the algebras of functions on a space. The point of noncommutative geometry is that there’s a (contravariant) equivalence between the category of locally compact Hausdorff spaces and the category of commutative algebras, so generalizing to noncommutative algebras (and taking the opposite category) gives a generalization of “locally compact Hausdorff space”. Topological groups like Lie groups are group objects in this category of spaces – and quantum groups are group objects in . So in particular, the group operation shows up as the coproduct , and the inverse operation is the antipode

.

Of course there are also the unit

and co-unit

(where is the base field, say ). The co-unit is of course the “unit” map for the group object. These maps all satisfy some obvious relations.

Now what about quantum groupoids? These are “groupoid objects” – or rather, models of the theory of groupoids – in . We can’t quite say “groupoid objects”, since a groupoid internal to a category consists of two objects in . For example, a Lie groupoid is a groupoid in , the category of manifolds. It has a base manifold and a total manifold , and two maps , and so forth. The interpretation is that there is a set (or manifold, or what-have-you) of objects, and a set (etc.) of morphisms. There is a (partially-defined) composition operation allowing morphisms to be composed if the source of one is the target of the other, and so forth.

So (a slightly tweaked version of) the definition of a quantum groupoid given by Xu has it consisting of . These unpack in pretty natural ways: it helps to compare to both the definition of, say, a Lie groupoid, and a quantum group. is the “total algebra$ and the “base algebra”, and they correspond to the “noncommutative spaces” of morphisms and objects of a groupoid, respectively. Just as a group can be seen as a groupoid with just one object, a quantum group would be a quantum groupoid where the base algebra is just the base field .

But then, if is not , we need some nontrivial – the source and target maps respectively, which replace the unit map to . Notice they go from the base to the total algebra , not the other way around, because everything works as usual in . The other maps are likewise dual to those in the definition of a groupoid. The major difference is that we need the equivalent of a partially defined multiplication/composition and the dual “co-multiplication”/”co-composition” . This works because using and , we get left and right actions of the base on , which is thus an -bimodule, hence we can form the bimodule product , and thus:

and

The obvious analog of the unit we had for quantum groups is hidden in Xu’s definition (it seems like it should take the place of the requirement that be unital), but the co-unit

is the dual way of describing the “identity” function .

The antipode plays the role of the inverse map for morphisms in groupoids.

All these maps have to satisfy various identities which are implied by saying this is a model of the theory of groupoids – check out either of the above papers to see them all explicitly.

(A final observation about the definition: a groupoid is a category which has an inverse map from morphisms to morphisms. If we relax the assumption that we have an antipode , we end up with just the definition of a bialgebroid (having makes it a “Hopf” algebroid). So “bialgebroid” would seem to be the natural “quantum” version of the concept of a general category…)

So how might one construct such a “quantum action groupoid”? This is addressed (at least in the finite case) in the paper by Nikshych and Vainerman, in their section 2.6. This is generalizing the action groupoid arising from a group acting on a set. The set is replaced by an algebra (which must be separable, for them – the equivalent of a finite set – and thought of as a “quantum space”). The group is replaced by a quantum group (or, generally, Hopf algebra) . The equivalent of having action of the group on the set is that is a (right) -module.

Now, the action groupoid for a action on has for objects the elements of , and for morphisms, all relations , which we can write as morphisms , with source and target . The action quantum groupoid associated to the -module is the double crossed product , with multiplication, co-multiplication, etc. defined in fairly natural ways. (Note: those triangles should be semidirect products, but I can’t seem to make that symbol appear here.)

So finally, I seem to be claiming that a such a quantum groupoid, let’s call it is the right “classical” state space (if that’s not too blatant a contradiction in terminology) for a theory having quantum-group symmetry – at least in the categorified picture. No doubt in many cases there is additional structure, capturing the equivalent of, say, symplectic structure, that should also be included (such things certainly can be found in NCG, but I’m still absorbing how exactly).

Then the 2-vector space for the quantized version of such a theory is the category , and a “2-state” just an object in here – a representation of .

One thing that’s not quite clear to me just now is how this relates to the usual idea of “state” in NCG – a state for a “quantum space” (which is an algebra) being a linear functional on that algebra. Not necessarily a character (i.e. a homomorphism into ), mind you – that would be a 1-dimensional representation, but just a functional.

 

Recent Talk – Enxin Wu on Algebra Deformations

I’m going up to Ottawa for a few days, in part to talk about spans and groupoids (basically, some cross section of the material in these posts here) at a conference put on by the Ottawa U math department, primarily for grad students and postdocs in the general vicinity. This is nice – gives me a chance to visit my parents and friends there (the fraction of my life I lived in Ottawa is now creeping down toward a mere third, but it probably has as strong a claim to “home” as anywhere). May is also one of the most tolerable months to be there. One of the grad students in our department is also going. Enxin Wu recently decided to start working with Dan Christensen too, so probably in future we’ll have various things to talk about. Last week, he gave a seminar talk on algebra deformation that was a long version of the one he’ll be giving in Ottawa.

Enxin is one of those guys who seems to really understand – it’s tempting to say grok- algebra, which I always find impressive. I’m a predominantly visual thinker, and the kind of symbolic computations common in algebra always seem a little mysterious to me at first until I can find a picture, or at least practice them a lot. Lie groups, for instance, make some sense to me – you can picture rotation groups, or at least keep a geometric picture of a manifold in mind. Lie algebras, being infinitesimal versions of Lie groups, are also not so hard to visualize. General associative algebras? Harder.

The talk was about associative algebras, to give some background on deformation, but the things whose deformations Enxin has been thinking about are -algebras (see this brief intro, for instance), an “invention” of Stasheff. The talk was about deformation of these algebras – the kind of deformation that pertains to deformation quantization. This has been studied by Kontsevich. Deformation quantization has to do with replacing things valued in some algebra by new things, valued in the bigger algebra of formal power series in with coefficients in , so that the original structure you started with is just the constant part that appears when you set . (The term “quantization” applies when you consider algebras of functions on a manifold, with a Poisson bracket – in other words, algebras of observables of a physical system).

Some of the main results have to do with the Hochschild cohomology for some complex associated to the algebra you start with, and the fact that this cohomology classifies obstructions to the deformation. I expected to get lost in a maze of notation – and there certainly is a lot – but as it turns out, I had some mental pictures to attach to these things, because related things came up a few years ago in the quantum gravity seminar at UCR (week 8 on that page especially), which provides a few pictures that helped a lot. Diagrammatic notation makes algebra a lot more comprehensible to me.

So let’s get more specific.

The point is to replace a multiplication operator with a power series whose coefficients are “multiplication” operators. That is, a deformation of an associative algebra (where is the multiplication for ) is , where the new multiplication is defined (by linearity) by its action on elements of , which works like this:

for some operators . Then there are a bunch of conditions on the that are needed to make associative. There’s one condition for each power of , since the coefficients in the associator should be zero:

The condition just says that is associative – so it’s the from the original algebra, which you get back when .

Then given an algebra , you can create the deformation category of whose objects are its deformations. The morphisms are continuous algebra homomorphisms that get along with the multiplication operations. It turns out that since formal power series with nonzero term are invertible (a consequence of the Lagrange theorem) this is actually a groupoid. Then the question is to classify the isomorphism classes of deformations – that is, . One can easily imagine that there might be no nontrivial deformations of some algebra – that is, every one is isomorphic to the deformation where all the are trivial except . So when does this happen? More generally, how can one classify the deformations up to isomorphism?

The answer has to do with Hochschild cohomology, which is related to a complex you can make from . Taking , the space of -ary multilinear operations on , you build this complex:

where the differential maps are defined by an alternating sum:

(Intuitively: there are too many arguments, so you start with the extra one on the left, push it into the middle as a “lump under the rug” where two arguments are combined, and push the lump all the way to the right. To ensure that , you do this with alternating signs. This kind of algebraic manipulation is the kind of thing I can do, and clearly works, but I don’t exactly grok.)

Then you take the Hochschild cohomology groups in the standard cohomology way: . A cohomology class in one of these groups is a class of multilinear maps from copies of to (up to a factor which is of something). As usual with cohomology, they describe obstructions to something – to exactness. Exactness, in this setting, would mean that has no interesting deformations at the level.

What does “level” mean here? Well, for example, at level 2 we’re talking about maps , such as the multiplication map. In fact, we have for an associative algebra – you can check that is twice the associator , which is zero. So is a cochain. Is it a coboundary? Sure – it’s . So is in the trivial class in . The point then is that it turns out that if this is the only class – if – then there are no interesting deformations of the multiplication of in the sense described above. The groupoid $\mathcal{D}$ has just one object. (One thing that occurs to me is that this makes it a group – which group is something Enxin didn’t discuss. My algebra instincts aren’t quite up to answering that off the top of my head.) For example, if (as an algebra over ), there are no nontrivial deformations: .

What do the other levels mean? Really, this is where you’d want to look at the generalization from associative algebras to -algebras. Whereas for an associative algebra , the associator $a(x,y,z) = x(yz) – (xy)z$ is zero, in general an -algebra will have an associator map (that is, in the complex above), which might not be zero, but which is .

This is the beginning of a story relating -algebras to weak -categories: a bicategory, for example, has an associator for composition of morphisms. In a bicategory, you expect the associator to satisfy a certain identity – the Pentagon identity – but in general you’d just ask for a “pentagonator” (something in ), and so on (this is where those seminar notes above help me think in pictures, by the way). An -algebra is a vector space equipped with maps at all these levels – described by Stasheff’s associahedra – satisfying some relations. The general story of deformation relates the Hochschild cohomology groups at different levels to deformations of -algebras. Enxin didn’t go into this in his talk, but he did say a little something about the next level:

An infinitesimal deformation of is a deformation not in , but in the quotient . This only needs two maps, . The third Hochschild cohomology measures obstructions to extending an infinitesimal deformation to a full deformation in – if , then any infinitesimal deformation can be extended to a full deformation.

All in all, I thought the talk was interesting – it tied in much more closely to things I already knew about TQFTs and higher categories than I’d expected. I’ll be really impressed if he can condense it into a 25-minute version…