February 4, 2009

The Cocktail Party Version

Posted by John Baez

guest post by Jeffrey Morton

In this guest post, I thought I would step back and comment about big picture of the motivation behind what I’ve been talking about on my own blog. I recently gave a talk at the University of Ottawa, which tries to give some of the mathematical/physical context. It describes both “degroupoidification” and “2-linearization” as maps from spans of groupoids into (a) vector spaces, and (b) 2-vector spaces. I will soon write a post setting out the new thing in case (b) that I was hung up on for a while until I learned some more representation theory. However, in this venue I can step even further back than that.

Over the Xmas/New Year break, I was travelling about “The Corridor” (the densely populated part of Canada: London, where I live, is toward one end, and I visited Montreal, Ottawa, Toronto, Kitchener, and some of the areas in between, to see family and friends). Between catching up with friends — who, naturally, like to know what I’m up to — and the New Year impulse to summarize, and the fact that I’m applying for jobs these days, I’ve had occasion to think through the answer to the question “What do you work on?” on a few different levels. So what I thought I’d do here is give the “Cocktail Party Version” of what it is I’m working on (a less technical version of my research statement, with some philosophical asides, I guess).

In The Middle

The first thing I usually have to tell people is that what I work on lives in the middle — somewhere between mathematics and physics. Having said that, I have to clear up the fact that I’m a mathematician, rather than a physicist. I approach questions with a mathematician’s point of view — I’m interested in making concepts precise, proving facts about them rigorously, and so on. But I do find it helps to motivate this activity to suppose that the concepts in question apply to the real world — by which I mean, the physical world.

(That’s a contentious position in itself, obviously. Platonists, Cartesian dualists, and people who believe in the supernatural generally don’t accept it, for example. For most purposes it doesn’t matter, but my choice about what to work on is definitely influenced by the view that mathematical concepts don’t exist independently of human thought, but the physical world does, and the concepts we use today have been selected — unconsciously sometimes, but for the most part, I think, on purpose — for their use in describing it. This is how I account for the supposedly unreasonable effectiveness of mathematics — not really any more surprising than the remarkable effectiveness of car engines at turning gasoline into motion, or that steel girders and concrete can miraculously hold up a building. You can be surprised that anything at all might work, but it’s less amazing that the thing selected for the job does it well.)

Physics

The physical world, however, is just full of interesting things one could study, even as a mathematician. Biology is a popular subject these days, which is being brought into mathematics departments in various ways. This involves theoretical study of non-equilibrium thermodynamics, the dynamics of networks (of chemical reactions, for example), and no doubt a lot of other things I know nothing about. It also involves a lot of detailed modelling and computer simulation. There’s a lot of profound mathematical engagement with the physical world here, and I think this stuff is great, but it’s not what I work on. My taste in research questions is a lot more foundational. These days, the physical side of the questions I’m thinking about has more to do with foundations of quantum mechanics (in the guise of 2-Hilbert spaces), and questions related to quantum gravity.

Now, recently, I’ve more or less come around to the opinion that these are related: that part of the difficulty of finding a good theory accommodating quantum mechanics and general relativity comes from not having a proper understanding of the foundations of quantum mechanics itself. It’s constantly surprising that there are still controversies, even, over whether QM should be understood as an ontological theory describing what the world is like, or an epistemological theory describing the dynamics of the information about the world known to some observer. (Incidentally — I’m assuming here that the cocktail party in question is one where you can use the word “ontological” in polite company. I’m told there are other kinds.)

Furthermore, some of the most intractable problems surrounding quantum gravity involve foundational questions. Since the language of quantum mechanics deals with the interactions between a system and an observer, applying it to the entire universe (quantum cosmology) is problematic. Then there’s the problem of time: quantum mechanics (and field theory), both old-fashioned and relativistic, assume a pre-existing notion of time (either a coordinate, or at least a fixed background geometry), when calculating how systems (including fields) evolve. But if the field in question is the gravitational field, then the right notion of time will depend on which solution you’re looking at.

Category Theory

So having said the above, I then have to account for why it is that I think category theory has anything to say to these fundamental issues. This being the cocktail party version, this has to begin with an explanation of what category theory is, which is probably the hardest part. Not so much because the concept of a category is hard, but because as a concept, it’s fairly abstract. The odd thing is, individual categories themselves are in some ways more concrete than the “decategorified” nubbins we often deal with. For example, finite sets and set maps are quite concrete: here are four sheep, and here four rocks, and here is a way of matching sheep with rocks. Contrast that with the abstract concept of the pure number “four” — an element in the set of cardinalities of finite sets, which gets addition and multiplication (abstractly defined operations) from the very concrete concepts of union and product (set of pairs) of sets. Part of the point of categorification is to restore our attention to things which are “more real” in this way, by giving them names.

One philosophical point about categories is that they treat objects and morphisms (which, for cocktail party purposes, I would describe as “relations between objects”) as equally real. Since I’ve already used the word, I’ll say this is an ontological commitment (at least in some domain — here’s an issue where computer science offers some nicely structured terminology) to the existence of relations as real. It might be surprising to hear someone say that relations between things are just as “real” as things themselves — or worse, more real, albeit less tangible. Most of us are used to thinking of relations as some kind of derivative statement about real things. On the other hand, relations (between subject and object, system and observer) are what we have actual empirical evidence for. So maybe this shouldn’t be such a surprising stance.

Now, there are different ways category theory can enter into this discussion. Just to name one: the causal structure of a spacetime (a history) is a category — in particular, a poset (though we might want to refine that into a timelike-path category — or a double category where the morphisms are timelike and spacelike paths). Another way category theory may come in is as the setting for representation theory, which comes up in what I’ve been looking at. Here, there is some category representing a specific physical system — for example, a groupoid which represents the pure states of a system and their symmetries. Then we want to describe that system in a more universal way — for example, studying it by looking at maps (functors) from that category into one like Hilb, which isn’t tied to the specific system. The underlying point here is to represent something physical in terms of the sort of symbolic/abstract structures which we can deal with mathematically. Then there’s a category of such representations, whose morphisms (intertwiners in some suitably general sense) are ways of “changing coordinates” which get along with what’s important about the system.

The Point

So by “The Point”, I mean: how this all addresses questions in quantum mechanics and gravity, which I previously implied it did (or could). Let me summarize it by describing what happens in the 3D quantum gravity toy model developed in my thesis. There, the two levels (object and morphism) give us two concepts of “state”: a state in a 2-Hilbert space is an object in a category. Then there’s a “2-state” (which is actually more like the usual QM concept of a state): this is a vector in a Hilbert space, which happens to be a component in a 2-linear map between 2-vector spaces. In particular, a “state” specifies the geometry of space (albeit, in 3D, it does this by specifying boundary conditions only). A “2-state” describes a state of a quantum field theory which lives on that background.

Here is a Big Picture conjecture (which I can in no way back up at the moment, and reserve the right to second-guess): the division between “state and 2-state” as I just outlined it should turn out to resolve the above questions about the “problem of time”, and other philosophical puzzles of quantum gravity. This distinction is most naturally understood via categorification.

(Maybe. It appears to work that way in 3D. In the real world, gravity isn’t topological — though it has a limit that is.)

Posted at February 4, 2009 1:03 AM UTC

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Re: The Cocktail Party Version

Thanks for cross-posting this from my blog, John!

I originally wrote it to sum up what I had come up with, trying to describe as non-technically as I could the relation between category theory and quantum gravity. I wasn’t sure whether it would translate out of that context - I’m pleased you thought it did. If my terribly vague conjecture seems off base to anyone, I’d also be pleased to hear about why.

Posted by: Jeffrey Morton on February 4, 2009 4:49 AM | Permalink | Reply to this

Re: The Cocktail Party Version

Thanks for letting us post your article. It hits smack dab in the middle of our interests here: math, physics and philosophy. Someone reading your papers might not notice the philosophical interests underlying them — and that’s probably a good thing. But here at this café, we can get into that side of things.

I like the idea of ‘state versus 2-state’. If I understand this idea correctly, it’s really all about first specifying a Hilbert space as a hom-space inside a 2-Hilbert space, and then picking a vector in that 2-Hilbert space. At the cocktail party level of precision you describe this as ‘first specifying a geometry of space, and then a state of a quantum field theory on this background’. I’d instead describe it as ‘first specifying boundary conditions, and then a state of a quantum field theory with these boundary conditions’. Am I mixed up?

I’m not sure how much this will help with the problem of time when we go beyond 3d quantum gravity. But regardless, it seems like a sensible thing to study. Right now, I’d really enjoy seeing the idea worked out very precisely for 3d quantum gravity. You’ve already done it for untwisted Dijkgraaf–Witten models, but replacing the finite gauge group by something like $SU(2)$ will make the geometry and physics a lot more vivid — we’ll get particles with masses and spins showing up! It will also give us an excuse to dabble in more fun math, like coherent sheaves and infinite-dimensional 2-Hilbert spaces.

The whole program has already been worked out to some extent in topological open-closed string theory, but not using a bicategory of cobordisms — it would be nice to translate existing work into that language.

It would also be nice to tackle some 2d conformal field theories that aren’t topological! We need to go beyond topological theories to see real physics, and this might be the easiest place to start.

Hmm, but now I’m talking like a math phys wonk instead of a typical party-goer. Maybe I should be saying stuff like “So what’s your ontology? I’m a Gemini, so of course I lean toward Platonism.”

Posted by: John Baez on February 4, 2009 4:56 PM | Permalink | Reply to this

Re: The Cocktail Party Version

Uh - ObCocktailParty: I’m a Wood Rabbit, Aries with Scorpio Rising, so naturally I don’t believe in astrology (though I hear it works even if you don’t believe in it) and lean toward quasi-empiricism. (End ObCocktailParty).

I think you got the state/2-state idea, though I’d correct “vector in a 2-Hilbert space” by removing the “2”. (Although I notice there’s an unfortunate orientation mismatch in the notation… to be consistent with “morphism/2-morphism”, probably states and 2-states should be named the other way around.)

You’re right in saying that in extended TQFT’s, choosing a basis 2-vector is about specifying boundary conditions rather than geometry - a general 2-vector gives some (direct) sum of such choices. That’s the general picture for extended TQFT - it’s only in 3D (or the $G \rightarrow 0$ limit in 4D) that these are really the same for gravity.

It does seem that when there are local degrees of freedom for gravity, the hom-Hilbert spaces that appear as components of the 2-linear maps associated to spacelike slices will have some decomposition, indexed by particular geometries - finer than the decomposition of a big Hilbert space into a 2-linear map (i.e. indexing by boundary conditions). So while your description is the state-of-the-art as I understand it, I’m wondering if it’s possible to capture the finer decomposition in some way at the 2-state level (obviously not in a theory of cobordism representations, where the objects are just boundaries, but in some other way). The motivation is pretty much as in the post.

On the subject of the open-closed string theories, I’m hoping to make the link with that clear in the final paper writing up what’s in my thesis. Basically, since that’s a 2D version of what I was looking at in 3D, it’s a bit less well-motivated to use a 2-functor. The only objects are points - which have only one connection on them, and therefore the only 2-Hilbert space that ever appears is just Hilb itself (one copy for each corner, anyway), and so the 2-linear maps that appear are most naturally just Hilbert spaces.

In any case, I agree that CFT is a good place to look next. I’ve had that question several times at talks. Adding conformal structure to the cobordisms means, among other things, that the monoidal structure is more complicated - getting into fusion products and such - but it definitely seems like it should be possible to produce something like an “Extended CFT” as a monoidal 2-functor.

Another generalization I might point out is the following: we expect the extended TQFT associated to SU(2), once it’s rigorously defined (in particular, using infinite dimensional 2-Hilbert spaces, which I’ve been thinking about a bunch recently), is expected to reproduce the Ponzano-Regge model. Bruce Bartlett asked a while back about Turaev-Viro model, partly because my factorization using that $\Lambda$ 2-functor was bothering him. I think he’s right, in that doing this would seem to involve quantum groupoids (see, e.g. here). Of course, there is no such thing as a quantum groupoid - just its category of representations, which destroys that factorization. On the other hand, thinking about the resulting 2-Hilbert spaces in terms of such a factorization still might be helpful…

Posted by: Jeffrey Morton on February 5, 2009 5:04 AM | Permalink | Reply to this

Re: The Cocktail Party Version

Jeffrey wrote:

Another generalization I might point out is the following: we expect the extended TQFT associated to SU(2), once it’s rigorously defined (in particular, using infinite dimensional 2-Hilbert spaces, which I’ve been thinking about a bunch recently), is expected to reproduce the Ponzano-Regge model.

By ‘rigorously defined’, do you mean finite?

Posted by: Jamie Vicary on February 17, 2009 9:58 AM | Permalink | Reply to this

Re: The Cocktail Party Version

In your work, is there any notion of the evolution of a 2-state?

In perturbation theory, we have a free Hamiltonian, like a quadratic potential for a QHO, plus a perturbation. We can write the perturbation in terms of creation and annihilation operators. If we had two QHOs, for instance, we could talk about a system perturbed by $(a^\star_1 a_2 + a^\star_2 a_1)$. Here, photons in each QHO can be emitted and absorbed by the other. The evolution of this system is a sum over diagrams, where each diagram consists of free evolutions interspersed with “kicks” of the form above.

The objects of a symmetric monoidal category are finite tensor products of some base objects. The object $X^{\otimes n}$ is rather like having $n$ photons in the QHO labelled $X$. So just as we had a Hilbert space whose states encoded how many photons of each kind we had, in a 2-QHO we have a 2-Hilbert space whose 2-states encode how many Hilbert spaces of each kind we have.

A morphism $f:X\to Y$ is like a perturbation $(a^\star_Y a_X)$. But it’s annihilating and creating a Hilbert space, not a photon. I imagine that the evolution of the 2-state would be a sum over diagrams, where we have some kind of evolution from the “free 2-Hamiltonian”–one in which you have n base objects but no morphisms between them–punctuated by “kicks”, which are morphisms. A diagram shows one possible composition of morphisms in the symmetric monoidal category.

But it’s unclear to me even what the free evolution would be like.

Posted by: Mike Stay on February 6, 2009 8:41 PM | Permalink | Reply to this

Re: The Cocktail Party Version

It might be surprising to hear someone say that relations between things are just as “real” as things themselves — or worse, more real, albeit less tangible. Most of us are used to thinking of relations as some kind of derivative statement about real things. On the other hand, relations (between subject and object, system and observer) are what we have actual empirical evidence for. So maybe this shouldn’t be such a surprising stance.

So, what do you think of relational quantum mechanics?

Posted by: Blake Stacey on February 4, 2009 2:36 PM | Permalink | Reply to this

Re: The Cocktail Party Version

I liked the relational interpretation as soon as I read Rovelli’s discussion of the EPR paradox in that interpretation, which fit very nicely with my own intuition, namely, that there is no paradox because the fact of correlation only exists for people who can observe it. More broadly, the motivating intuitions - basically, take QM seriously without unnecessary extras like “hidden variables” or “macroscopic systems” - seem very natural.

I’m not sure if anyone has explicitly described it in categorical language, but it seems pretty natural. There’s some category whose objects are subsystems of the world, and where the hom-set between any two objects is the Hilbert space describing the correlations between them, where “states” live.

(I reiterate, this terminology is kind of unfortunate - rays in this Hilbert space probably should be called 2-states, so that objects in the category can be called 1-states, but this is in conflict with established use).

My bias is to say that a good way to say this is to say relational QM amounts to working in a particular 2-Hilbert space, describing the universe. Though this is rather different from the cobordism-world I’m more familiar with. since we’re looking at a category where objects are subsystems, not submanifolds with some codimension… Though it’s a bit more complicated since we have to think about more than just containment relations, but also causal structure since the systems have to interact. Would this be like a causal site?

(Okay, as long as I’ve gone ahead and said that, I may as well say that if anyone thinks the phrase “working in a particular 2-Hilbert space” sounds like “working in a particular topos”, I generally agree, but I kind of don’t want to get into that now.)

Posted by: Jeffrey Morton on February 5, 2009 5:28 AM | Permalink | Reply to this

Re: The Cocktail Party Version

This quote caught my eye also and I think the reason is that it’s a bit imprecise. Empirical evidence usually means perceived by the senses and that means talking about a perceived object not one’s relationship to the object. Category theory or Process Philosophy seem too abstract to be deemed empirical evidence, without denying the usefulness of conceptualizing in terms of relationships between objects.

http://plato.stanford.edu/entries/quantum-field-theory/
“Since mathematical reasoning dominated the heuristics of QFT,
its interpretation is open in most areas which go beyond the
immediate empirical predictions. Philosophical analysis might
help to clarify its semantics. QFT taken seriously in its
metaphysical implications seems to give a picture of the world
which is at variance with central classical conceptions like
particles and fields and even with some features of quantum
mechanics (QM).”

SH: I don’t think J. Morton’s description of empirical relationship fits into the category of QFT’s empirically enabled predictions. If quarks and fields are irreducible with a correlation between them, does the term relationship describe that correlation? Formal math/logical systems do not generate empirical evidence, which is acquired by experiment.

Posted by: Stephen Harris on February 5, 2009 6:10 AM | Permalink | Reply to this

Re: The Cocktail Party Version

That’s true. It was a highly imprecise statement. I meant it to indicate that every bit of empirical evidence is, necessarily, a datum about a relation (between subject and object). You say that empirical evidence is about observing objects, but in the majority of my experiences, there has also been a subject involved. I’m not sure if that clarifies what I was trying to say. I’m also not sure if there’s a substantive disagreement here or just a semantic one.

Also, I didn’t intend to imply that mathematics “generates” evidence, but if I did, I retract the implication.

I’d prefer (not being a Platonist) to say that math summarizes evidence - which is what my comment about its “unreasonable effectiveness” was about. Mathematics has been developed over several millennia of experience with the real world, and is intimately connected with it as a result of probably several tens of millions of person-years of mental labour. So I’m not sure what this “formal math” might be. We often use concepts without knowing where they came from, but they generally came from looking at reality.

Posted by: Jeffrey Morton on February 5, 2009 7:02 AM | Permalink | Reply to this

Re: The Cocktail Party Version

Jeffrey wrote:

One philosophical point about categories is that they treat objects and morphisms (which, for cocktail party purposes, I would describe as “relations between objects”) as equally real. Since I’ve already used the word, I’ll say this is an ontological commitment (at least in some domain — here’s an issue where computer science offers some nicely structured terminology) to the existence of relations as real. It might be surprising to hear someone say that relations between things are just as “real” as things themselves — or worse, more real, albeit less tangible. Most of us are used to thinking of relations as some kind of derivative statement about real things. On the other hand, relations (between subject and object, system and observer) are what we have actual empirical evidence for. So maybe this shouldn’t be
such a surprising stance.

Even after your response to my post, I still didn’t understand this paragraph. I should have been more specific: what was real as compared to “real” what was being compared that was “more real”? So I wasn’t going to post. But then I went to your blog and found JB’s comment:

This sentence could use a bit of copy-editing:

What it mean is, the claim that relations between things are just as “real” as things themselves, albeit maybe less tangible might be a sort of surprising if you’re used to thinking of relations as some kind of derivative statement about real things.

Jeffrey Morton wrote:

I’ve broken that clunky sentence into two. On reflection, I also made it a little bolder, hinting that morphisms are more “real”, or at least more fundamental, than objects (you can describe a category entirely in terms of its morphisms, but not its objects, after all).

Now with this explanation I was able to reread the main paragraph quoted initially and understand what you meant. The reason I didn’t grasp your ideas from what you wrote is that we don’t share some underlying philosophical assumptions such as comparing using the idea of realness.

Posted by: Stephen Harris on February 6, 2009 5:52 AM | Permalink | Reply to this

Interval in UK means Intemision in USA; Re: The Cocktail Party Version

Musicontologically, are Intervals are Real as Notes?

In music theory, as simplified at Wikipedia, “the term interval describes the relationship between the pitches of two notes.”

Intervals may be described as:

* vertical (or harmonic) if the two notes sound simultaneously
* linear (or melodic), if the notes sound successively.[1]

Interval class is a system of labelling intervals when the order of the notes is left unspecified, therefore describing an interval in terms of the shortest distance possible between its two pitch classes.[2]

Note that “The term ‘interval’ can also be generalized to other elements of music besides pitch. David Lewin’s Generalized Musical Intervals and Transformations uses interval as a generic measure of distance in order to show musical transformations which can change, for instance, one rhythm into another, or one formal structure into another” [8].

The recent breakthrough research on Orifolds in Music Theory cry out for Categorification, or n-Categorification. NOte that John Horton Conway is coauthoring an Orbifolds and Symmetry book, although I think that he gets by mathematically with genius, rather than n-Categories.

[1] Lindley, Mark/Campbell, Murray/Greated, Clive. “Interval”, Grove Music Online, ed. L. Macy (accessed 27 February 2007), grovemusic.com (subscription access).

[2] Roeder, John. “Interval Class”, Grove Music Online, ed. L. Macy (accessed 27 February 2007), grovemusic.com (subscription access).

[8] Lewin, David (1987). Generalized Musical Intervals and Transformations,[clarification needed “p. ?”]. New Haven: Yale University Press. Reprinted Oxford University Press, 2007. ISBN 978-0-19-531713-8

Posted by: Jonathan Vos Post on February 6, 2009 5:41 PM | Permalink | Reply to this

Re: The Cocktail Party Version

I think I see the source of the confusion, which appears to be the word “real”. I was using it loosely since this was, after all, a “Cocktail Party Version”. On the whole, when trying to be rigorous, I prefer to avoid this concept entirely, since I’m not sure what, if anything, it refers to. But this viewpoint hasn’t really sunk into my habitual ways of using language.

I’m glad you did choose to reply - it’s always useful to understand the source of confusions that come up.

Posted by: Jeffrey Morton on February 6, 2009 7:19 PM | Permalink | Reply to this

Re: The Cocktail Party Version

In fact, I notice on closer inspection that this version of my original post doesn’t include one of the links in the original, namely the one to the Wikipedia article on ontologies in computer science. Specifically, I linked to the notion of domain ontologies, mentioning that this gives a nice terminology.

The CS notion of an “Ontology” is as a kind of data structure or other formal representation of the basic concepts in some domain of discourse, and their relationships. We could also use the word “theory” for “domain of discourse”: a physical theory involves making certain ontological commitments. That is, identifying what kinds of things the theory makes statements about, and including them in the ontology of the theory.

This is part of what I think is interesting about quantum theories, whether QM, QFT, or what-have-you, even before you get to the (also interesting) notions of Hilbert spaces, expectation values, von Neumann algebras. Namely, one of the very first ontological commitments of a quantum theory is to respect the fact that it is trying to describe empirical data, and empirical data is always relational. Thus, what the theory actually predicts are expectation values for observables, say as an inner product between a “state” and a “costate” (which describes an observation). So, for example, in one formalism, these features of the ontology are written as “bras” and “kets” - and what’s meaningful, empirically (and here we get into the connection between the ontology and the epistemology of the theory), is a combination of a bra with a ket.

Indeed, both bras and kets can be represented as morphisms in a categorical formulation (for example, as described by Abramsky and Coecke). On the other hand, the inner product itself can be interpreted (in a categorified setting) as a hom-set (i.e. collection of morphisms) in its own right. So a category-theoretic formulation of a quantum theory seems to make plenty of ontological commitments to morphisms, but not many to objects.

Which makes sense, in that you can describe the theory of categories without making any ontological commitments to objects.

I guess that’s a more precise way of expressing the thought behind that clunky sentence.

Posted by: Jeffrey Morton on February 6, 2009 7:56 PM | Permalink | Reply to this

Re: The Cocktail Party Version

Jeffrey wrote:

In fact, I notice on closer inspection that this version of my original post doesn’t include one of the links in the original, namely the one to the Wikipedia article on ontologies in computer science.

Fixed. I’d inserted the links by hand, and missed this one.

Posted by: John Baez on February 10, 2009 6:19 AM | Permalink | Reply to this

Re: The Cocktail Party Version

This is how I account for the supposedly unreasonable effectiveness of mathematics – not really any more surprising than the remarkable effectiveness of car engines at turning gasoline into motion, or that steel girders and concrete can miraculously hold up a building. You can be surprised that anything at all might work, but it’s less amazing that the thing selected for the job does it well.)

Yes, if there’s some surprise to be had, it’s that there are humanly understandable means of capturing aspects of the universe, when there was no direct advantage for us as a species in capturing those aspects.

Something I’d like to understand better is why Gauss already in the early nineteenth century is $q$-deforming mathematics (even using q as his symbol).

Is it that the space of ‘rich’ mathematics is very restricted, so no surprise that humans should stumble over parts which the universe uses, when the universe has little choice?

Posted by: David Corfield on February 5, 2009 8:56 AM | Permalink | Reply to this

Re: The Cocktail Party Version

David Corfield wrote:

Yes, if there’s some surprise to be had, it’s that there are humanly understandable means of capturing aspects of the universe, when there was no direct advantage for us as a species in capturing those aspects.

I’m on the side of the unsurprised, even more than this: I suspect many animals have found it hugely beneficial as a species to have implicit knowledge of some basic inferential machinery. We didn’t get to formalise this stuff until we had language, and parts of it are still a struggle rather than intuitive, but even young babies will stare in amazement at many basic contradictions of the implicit rules that we use to reason about physical objects, and those rules are where logic and mathematics come from.

Posted by: Greg Egan on February 5, 2009 10:30 AM | Permalink | Reply to this

Re: The Cocktail Party Version

I suppose the question is why should the parts of the mathematics for quantum mechanics, whose understanding doesn’t seem to be something to be selected for, be so close to that which is useful to count and measure middle-sized objects that we stumbled upon it before we needed it for physics.

John finds it odd:

One eerie thing about these modified versions of calculus is that people discovered them before quantum mechanics

and

Pretty much anything you can do with calculus, you can do with the q-calculus. There are q-integrals, q-trigonometric functions, q-exponentials, and so on. If you try books like this:

2) George E. Andrews, Richard Askey, Ranjan Roy, Special Functions, Cambridge U. Press, Cambridge, 1999.

you’ll see there are even q-analogues of all the special functions you know and love - Bessel functions, hypergeometric functions and so on. And like I said, the really weird thing is that people invented them before their relation to quantum mechanics was understood.

Posted by: David Corfield on February 5, 2009 11:00 AM | Permalink | Reply to this

Re: The Cocktail Party Version

One eerie thing about these modified versions of calculus is that people discovered them before quantum mechanics

I find that plenty of people are studying plenty of structures with great enthusiasm whose whose true origin and meaning is clearly unknown. I find this eerie, too, but maybe in a different sense: it’s not so hard to just fiddle around with structures and study axiom systems. What is harder is finding out where these naturally live.

Posted by: Urs Schreiber on February 5, 2009 12:19 PM | Permalink | Reply to this

Re: The Cocktail Party Version

David, John’s TWF Week 183 that you linked to is fascinating, but I’d love to know why:

Gauss […] wrote about a q-analogue of the binomial formula and other things.

Was he just messing about, or what? Does anyone know what the attraction was at the time?

Posted by: Greg Egan on February 5, 2009 11:33 AM | Permalink | Reply to this

Re: The Cocktail Party Version

Well, q is the successor to p, and p is a prime. So q is a power of a prime. Some people were interested in studying stuff over finite fields. The quantum 6j symbols and such work out just as nicely (or even more so) over finite fields. The representation theory is a symmetric monoidal category. So there are not interesting manifold invariants that come about that way.

Ramanujin proved a whole host of these quantum identities in the context of hypergeometric series. Clearly, his motivation would have been number theoretic.

I guess the question is some other contexts for xy = q yx from a Gaussian viewpoint.

Meanwhile, when I asked the woman at the cocktail party, “What’s your sign?”, she replied, “Taurus, what’s yours, negative?”

Posted by: Scott Carter on February 5, 2009 9:51 PM | Permalink | Reply to this

Re: The Cocktail Party Version

Greg wrote, of Gauss’ work on $q$-deformed special functions:

Was he just messing about, or what? Does anyone know what the attraction was at the time?

I still don’t know. I somehow doubt Gauss was ‘just messing about’, but a brief attempt to track down the history has led to just one small fact: q-deformed binomial coefficients are also known as ‘Gaussian coefficients’. Did Gauss invent these? If so, why did he do it?

Scott wrote:

Well, $q$ is the successor to $p$, and $p$ is a prime. So $q$ is a power of a prime. Some people were interested in studying stuff over finite fields.

But was one of these people Gauss, or not? The $q$-deformed version of the binomial coefficient ‘$n$ choose $k$’ counts the number of $k$-dimensional subspaces of an $n$-dimensional vector space over the field with $q$ elements. But did Gauss know this? And is this why those coefficients are called Gaussian coefficients?

Posted by: John Baez on February 10, 2009 6:55 AM | Permalink | Reply to this

Re: The Cocktail Party Version

If I recall correctly from a conversation with Henry Cohn a long time ago, Gauss used some sort of q-analogues to prove the sign of the quadratic gauss sum. I’ll try to dig up more if I remember, I think I still have Gauss’s paper printed out somewhere…

Posted by: Noah Snyder on February 10, 2009 7:17 AM | Permalink | Reply to this

Re: The Cocktail Party Version

Thanks! That would be very interesting.

Posted by: John Baez on February 10, 2009 9:11 PM | Permalink | Reply to this

Re: The Cocktail Party Version

Here’s an MAA award winning expository paper by Henry Cohn which in footnote 1 attributes the “Gaussian binomial” moniker to Gauss’s investigation of the sign of the quadratic Gauss sum. He cites a paper of Gauss’s, “Summatio quarumdam serierum singularium”, which is in latin and doesn’t seem to be on the internet.

However, Section 2 of this Bulletin paper by Berndt and Evans appears to recap Gauss’s proof using q-binomial coefficients.

Posted by: Noah Snyder on February 11, 2009 8:01 PM | Permalink | Reply to this

Re: The Cocktail Party Version

I wonder if Gauss was interested in the interplay between “continuum” and “discete” theories? For example, the Gaussian distribution is the continuum limit of a discrete random walk (involving binomial coefficients). “Calculus” on a lattice often gives rise to q-deformations.

Just a thought from left field…

Posted by: Eric on February 10, 2009 3:22 PM | Permalink | Reply to this

Gauss Sums; Re: The Cocktail Party Version

What did Gauss know and when did he know it?

This committee has subpoenaed Gauss to ask this, and some related questions. Members should notice that he failed to appear, per that subpoena. Now, we must presume him innocent until proven guilty, but…

Meantime, I have distributed copies of the book:
Gauss and Jacobi sums
B.C. Berndt, R.J Evans, K.S. Williams, 1998

and the papers:
Nonlinear Bayesian estimation using Gaussian sum approximations
D. Alspach, H. Sorenson - Automatic Control, IEEE Transactions on, 1972

O.D. Mbodj - Finite Fields and Their Applications, Elsevier, 1998

We now call Bruce C. Berndt and Ronald J. Evans as witnesses.

Their prior testimony:

The Determination of Gauss Sums, AMERICAN MATHEMATICAL SOCIETY, 1981.

Posted by: Jonathan Vos Post on February 11, 2009 5:13 PM | Permalink | Reply to this

Gauss’s diary; Berndt and Evans; Re: Gauss Sums; Re: The Cocktail Party Version

The codebreakers at the NSA provide this analysis of an intercepted message on the topic in question.

On August 30, 1805, Gauss wrote in his diary [63, pp. 37, 57], “Demonstrate theorematis venustissimi supra 1801 Mai commemorati, quam per 4 annos et ultra omni contentione quaesiveramus, tandem perfecimus.” (At length we achieved a demonstration of the very elegant theorem mentioned before in May, 1801, which we had sought for more than four years with all efforts.)

Gauss’s proof, which is elementary, was published in 1811 [64], [66, pp. 9-45,
155-158].

It may be remarked that eigenvector decompositions for the finite Fourier transform (Schur’s matrix) have been given by McClellan and Parks [188] and by Morton [189]. Regarding §2.3, note that in Eichler’s book [184, p. 137], a reciprocity formula for Gauss sums attached to quadratic forms is proved
from the transformation formula for the theta-function. In connection with the paragraph preceding (10.1), note that Joris [186] has shown how the functional equation for Dirichlet L-series can be used to evaluate imprimitive Gauss sums in terms of primitive ones. Regarding the two paragraphs following (10.1), note that conductors of Gauss sums as Hecke characters have been investigated by Schmidt [190]. To the list of papers near the end of §10 giving interesting generalizations of Gauss sums, one should add the papers of O’Meara [187] and Jacobowitz [185], which use Gauss sums over lattices to classify local integral quadratic (respectively, hermitian) forms.
Also, Cariitz [183] has evaluated a character sum which generalizes a cubic
Gauss sum over GF(2^n).

Using an estimate for generalized Kloosterman sums due to De Ligne
[40, p. 219], one can easily show that
SUM (over Chi)_ G(Chi)^k = O(p^(k+1)/2)
as p tends to oo, where k is an arbitrary, fixed natural number and where the sum is over all characters Chi(mod p). In particular, it follows that the arguments of the Gauss sums G(x) are asymptotically uniformly distributed as p tends to oo [191].

Posted by: Jonathan Vos Post on February 11, 2009 5:29 PM | Permalink | Reply to this

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