Spans in Quantum Theory

October 1, 2007

Posted by John Baez

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Tomorrow I’m getting up at 5 am to catch an airplane, to give this talk the following day:

John Baez, Spans in quantum theory, lecture at Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, conference organized by Hans Halvorson, Princeton University, October 3, 2007.

This will be the first time I’ve been back to Princeton since my undergraduate friends all left sometime around 1984. It’ll be interesting to see how the place has grown. But, it will be strange staying at the Nassau Inn instead of the vegetarian hippie freak coop at 2 Dickinson Street.

It’s also strange that after all these years, I’m being invited by someone in the philosophy department, rather than math. But there’s a kind of poetic justice to it, since I did my senior thesis there under the supervision of John Burgess, in the philosophy department. I was a math major, and my thesis was on ‘Recursivity in Quantum Mechanics’, but I couldn’t get anyone in the math department to advise me on this project — I was young and dumb then, and didn’t realize you were supposed to let them pick you a project. Since my thesis used a lot of recursive function theory, and Burgess knows that stuff, I wound up working with him. And, he helped me a lot!

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I’ll eventually write a paper based on this talk. This will be the third of my ‘philosophy of quantum theory’ series. So, you might want to start with the previous two:

John Baez, Higher-dimensional algebra and Planck-scale physics, in Physics Meets Philosophy at the Planck Scale, eds. Craig Callender and Nick Huggett, Cambridge U. Press, 2001.
John Baez, Quantum quandaries: a category-theoretic perspective, in Structural Foundations of Quantum Gravity, eds. Steven French, Dean Rickles and Juha Saatsi, Oxford U. Press, 2006.

However, don’t be intimidated — you can also just dive in! Here’s the abstract:

Spans in Quantum Theory

Many features of quantum theory — quantum teleportation, violations of Bell’s inequality, the no-cloning theorem and so on — become less puzzling when we realize that quantum processes more closely resemble pieces of spacetime than functions between sets. In the language of category theory, the reason is that Set is a “cartesian” category, while the category of finite-dimensional Hilbert spaces, like a category of cobordisms describing pieces of spacetime, is “dagger compact”. Here we discuss a possible explanation for this curious fact. We recall the concept of a “span”, and show how categories of spans are a generalization of Heisenberg’s matrix mechanics. We explain how the category of Hilbert spaces and linear operators resembles a category of spans, and how cobordisms can also be seen as spans. Finally, we sketch a proof that whenever $C$ is a cartesian category with pullbacks, the category of spans in $C$ is dagger compact.

You’ll note that I’m calling ‘em ‘dagger compact’ categories instead of ‘symmetric monoidal categories with duals’. I’m hoping Bob Coecke will do the same in his talk, so we interfere constructively.

One sad thing is that both Chris Isham and Jeremy Butterfield will be unable to attend the conference, due to health reasons. So, Andreas Döring will have to hold up the topos side of things.

Hmm! I thought Princeton University had successfully crushed 2 Dickinson Street a while ago. But now I see they’re far from dead — in fact, they have a webpage!

Posted at October 1, 2007 11:16 PM UTC

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Re: Spans in Quantum Theory

> I’m hoping Bob Coecke will do the same in his talk,
> so we interfere constructively.

I will. Am already here in Princeton, talking most of the afternoon with Hans Halvorson about categories for physics, quantum logic, Doplicher-Roberts, …

Posted by: bob on October 2, 2007 1:25 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

John – I guess besides an interest in categories and physics we share the 2D coop in our past..

you might be very (pleasantly?) surprised by the inside of the coop –I visited recently and they had done a total remodel of the place, the kitchen is nice and beautiful rather than the homey but cramped and cockroach-infested place it was while I was there (in fact I think the new kitchen is where my bedroom used to be!)

Sadly I think they lost a lot of the old housebooks which contained so many amazing tales..

Is it true Queen Noor of Jordan was also a member? any other math alumni you know of?

Posted by: David Ben-Zvi on October 2, 2007 3:00 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

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David Ben-Zvi wrote:

John - I guess besides an interest in categories and physics we share the 2D coop in our past…

Wow! When were you there? I was at Princeton from ‘79 to ‘82, and I lived in 2D the last of those years.

Sadly I think they lost a lot of the old housebooks which contained so many amazing tales…

Including the cartoons I used to draw? Too bad. Or maybe it’s just as well…

Is it true Queen Noor of Jordan was also a member? any other math alumni you know of?

Queen Noor? I don’t see anything on the web about her and 2 Dickinson. According to Wikipedia she entered Princeton with its first co-educational freshman class, and got her degree in 1974. Was 2D operational back then?

It’s amazing to think Princeton went co-ed just 9 years before I went there. A couple years before I arrived, the slogan of 2D was “Cats, Co-Eds and Commies”.

I know two other mathematicians who lived in 2D.

One is my friend Steve Sawin, who works on topological quantum field theory and now teaches at Fairfield University. I met him when he was a postdoc at MIT and I was on leave from UCR teaching at Wellelsey.

Another is Judy Goldsmith, who now does computer science at the University of Kentucky. We graduated the same year. As you can see, she’s a typical staid 2D-dweller.

And now there’s you! I was sort of hoping I’d get some feedback from mentioning 2 Dickinson Street. I’m glad I did.

Posted by: John Baez on October 2, 2007 4:55 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

I wrote:

the slogan of 2D was “Cats, Co-Eds and Commies”

… but before anyone asks, I should explain that these “cats” were felines, and I had no inkling that someday I’d read “cats” and “2D” and instantly think “2-categories”.

(Cats were against the rules in university housing, which technically 2D was — but it was across the street from all the dorms, and a very independent-minded place, and some of the occupants were fond of those beasts, so they’d hide them whenever the inspectors came by.)

Posted by: John Baez on October 2, 2007 5:06 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

Do you mean to imply that you don’t read ‘co-ed’ and think it’s the dual of an ed?

(If I’m right in thinking that a co-ed is a woman student, it’s clear what an ed is.)

Posted by: Tom Leinster on October 2, 2007 5:24 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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And a commie is a dual of… me? (Bit phonetic that one.)

Posted by: someone who is a Cat(ster), an ed and a me on October 2, 2007 10:00 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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A commie is a pet name for a commuting diagram!

Posted by: Tim Silverman on October 3, 2007 9:15 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

I was in Princeton ‘90-‘94. I moved
into 2D “accidentally” in ‘93 - there was
a huge room available in the housing lottery
in some building I hadn’t heard of,
so my two roommates and I took it (apparently a mistake, they
were supposed to reserve 2D rooms for 2D members - especially since this room
was right adjacent to the kitchen!).

During the year we all fell in love with the place and its wacky liberal hippie spirit, something definitely in short supply in Princeton (I think I was sold during guided trips in the steam tunnels under campus by a guy who used to grow pot
there and who was usually writing songs on his guitar about the impending end of the world, and a group field trip to the Clinton inauguration), and joined my senior year (when it was decidedly less hippie but still a lot of fun).

Posted by: David Ben-Zvi on October 3, 2007 1:50 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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… ‘classical’ categories like Set to ‘quantum’ categories like nCob and Hilb.

We’ve had this discussion before, but it’s always struck me as a bit odd and confusing that you take Rel, the category of sets and relations, as ‘quantum’. (You don’t explicitly here, but you have elsewhere.) And likewise for nCob.

I can see there’s something special about categories with transposable matrix-like morphisms, but for me I’d want to reserve ‘quantum’ for when dealing with something stranger than $\{0, 1\}$ values. I’d prefer to use it when the rig of values has something of the oddness of $\mathbb{C}$ to it.

I suppose you can even see Set as matrix-morphismed. Thinking about kinds of morphism from $X$ to $Y$ in $\{0, 1\}$ -valued matrices, we have:

a) Unnormalised: Rel;
b) Row normalised: Set;
c) Column normalised: $Set^{op}$ ;
d) Row and column normalised: Permutations.

Only a) and d) are transposable.

Posted by: David Corfield on October 2, 2007 9:32 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

Rel is a degenerate model but still hosts some typical quantum features:
* superposition
* entanglement
* measurement and collapse
* conditional teleportation and other protocols
* …
What it can’t provide is incompatible observables. This is in fact also true for matrices in the semiring of positive reals, although it has many scalars, that is, endomorphisms of the tensor unit. On the other hand, now take the category of finite-dimensional Hilbert spaces and subject it to the following *physical* congruence: two morphisms relate if they are *equal up to a phase*. Then one again obtains a category with the positive reals as scalars, ut now with plenty incompatible observables. You can even relate morphisms when they are equal up to any non-zero scalar and still have many incompatible observables. (in the case the scalars, just like in Rel, are only 0 and 1)

Ross Duncan and I just finished a paper in which we propose axioms for having at least two mutually unbiased observables; these turn out to be a slightly tweaked version of the bialgebra actions.

Posted by: bob on October 2, 2007 3:44 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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David C. said

I’d want to reserve ‘quantum’ for when dealing with something stranger than {0,1} values. I’d prefer to use it when the rig of values has something of the oddness of $\mathbb{C}$ to it.

But isn’t John’s point precisely that you don’t need the oddness of $\mathbb{C}$ for quantum weirdness? Why should ‘quantum’ phenomena have anything to do with the complex numbers?

Posted by: Simon Willerton on October 2, 2007 10:52 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Note I didn’t say you needed $\mathbb{C}$ , but rather something of the oddness of $\mathbb{C}$ . Bob is pointing to what this oddness might be when he remarks that:

What it [Rel] can’t provide is incompatible observables.

I’d be as happy as everyone else here if we show we can do without $\mathbb{C}$ in quantum mechanics.

My point rather was that if you wander out of the Café and tell the first person you meet in the street that good old sets and relations are not ‘classical’ but rather ‘quantum’, they will likely give you a funny look.

Is there not a better nickname for what Rel, Hilb and nCob share?

Posted by: David Corfield on October 3, 2007 9:03 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

Is there not a better nickname for what Rel, Hilb and nCob share?

To me, it’s the idea of superposition . That’s traditionally thought of as a quantum-thingy, as people in the street would probably agree (Schrodinger’s cat). Rel and Hilb clearly have a superposition-aspect to them. nCob also has “superposition”, intuitively at least, because if you look at a cobordism from some imput manifold to some output manifold, and cut it up into time slices, then you get the idea that somehow the cobordism is “superposing” a whole lot of processes together.

So I’m kind of fond of the word “quantum” to describe these guys.

Posted by: Bruce Bartlett on October 3, 2007 2:41 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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David wrote:

Is there not a better nickname for what Rel, Hilb and $n$ Cob share?

Well, I encourage everyone to make up good nicknames, but I think the really good name for what they share is ‘being a dagger compact category’. It’s not evocative enough — but it’s precise.

Of course there are lots of other precise things one can say, too, and Bob Coecke has studied a lot of them. He’s worked out a kind of taxonomy of categories, based on various ways they can be ‘like quantum theory’ or not.

Bruce wrote:

To me, it’s the idea of superposition that’s traditionally thought of as a quantum-thingy, as people in the street would probably agree (Schrödinger’s cat). Rel and Hilb clearly have a superposition-aspect to them.

Yes, that’s right! They’re both enriched over $R$ -Mod for some rig $R$ , so we can add morphisms, and also multiply morphisms by scalars in $R$ . This is what superposition is all about.

(Note that the more familiar superposition of states is just a special case of superposition of processes, that is morphisms, since a state is basically just a morphism $f: I \to X$ where $I$ is the unit for the tensor product.)

Of course superposition is more shockingly ‘quantum’ when $R$ is ring, not just a rig. When $R$ has negatives, it’s possible for the superposition of processes to exhibit complete destructive interference. More ways to get from here to there can make it harder to get from here to there!

$n$ Cob also has “superposition”, intuitively at least, because if you look at a cobordism from some input manifold to some output manifold, and cut it up into time slices, then you get the idea that somehow the cobordism is “superposing” a whole lot of processes together.

That doesn’t seem like “superposition” of processes to me. It’s just “composition” — possibly a higher-dimensional form of composition, as in an $n$ -category. Superposition of morphisms is all about adding two morphisms from $X$ to $Y$ and getting a new morphism from $X$ to $Y$ . Composition is different.

Posted by: John Baez on October 7, 2007 3:22 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

“I was a math major, and my thesis was on ‘Recursivity in Quantum Mechanics’, but I couldn’t get anyone in the math department to advise me on this project — I was young and dumb then, and didn’t realize you were supposed to let them pick you a project.”

Are you being ironic here or serious? I’m asking because I’m working on a project of my own, and I don’t even have an advisor. I guess most people wouldn’t consider that a good idea, but do you think it is that hopeless?

Posted by: Squark on October 5, 2007 11:24 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

I picked my own problem for my doctoral thesis, and asked all my own questions. And now despite the fact that I find myself accidentally touching on all sorts of similarities with popular fields, I’m not actually working in one of them per se.

And so I still have to convince people that what I’m talking about is actually interesting and worthwhile. When I’m talking personally to people who work in the related areas, this is actually pretty easy. When I’m trying to make the case through my CV and research statement, not so much. It’s a harder row to hoe, but sometimes you gotta do what you gotta do.

Posted by: John Armstrong on October 6, 2007 12:17 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

By the way John, what is “Recursivity in Quantum Mechanics” all about? Sounds cool.

Posted by: Bruce Bartlett on October 6, 2007 1:34 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

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I’ll email you a copy of ‘Recursivity in Quantum Mechanics’. Someday later I’ll put it on my online collection of papers. In this paper I explained what it means for a one-parameter group of unitary operator

$U(t): L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$

to be computable, or in logicians’ jargon, ‘recursive’. I then showed that for a bunch of familiar quantum systems, like a finite collection of nonrelativistic quantum particles interacting by an inverse-square force law, the time evolution is computable in this sense.

In part, this was my reaction against Pour-El and Richards’ work showing that there’s a solution of the wave equation with smooth computable initial data for which the value of the solution fails to be computable at one point.

In part, it’s what was left of a mindblowingly visionary program after I realized I couldn’t actually accomplish most of it. Youthful dreams.

Posted by: John Baez on October 7, 2007 3:57 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Ok great - sounds interesting.

Posted by: Bruce Bartlett on October 7, 2007 10:09 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

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I was young and dumb then

One of the big bugs of this institution called life: when one really badly needs to know how things work, one is young and dumb and unexperienced. Later, when one doesn’t care anymore, one knows all the tricks.

I think: when you feel you can handle choosing a project on your own, than chances are you really can. But doing it may become really unenjoyable if there is not a minimum of intellectual support from the standard sources.

Posted by: Urs Schreiber on October 6, 2007 12:18 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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John wrote:

I was a math major, and my thesis was on ‘Recursivity in Quantum Mechanics’, but I couldn’t get anyone in the math department to advise me on this project — I was young and dumb then, and didn’t realize you were supposed to let them pick you a project.

Squark wrote:

Are you being ironic here or serious?

Serious. One of the main points of going to grad school is to completely absorb the expertise and style of a successful practitioner of the subject you’re studying — your advisor. It’s like buying an extra brain!

I’m asking because I’m working on a project of my own, and I don’t even have an advisor. I guess most people wouldn’t consider that a good idea, but do you think it is that hopeless?

It’s not hopeless, just radically suboptimal.

With any luck, you’ll have plenty of chances to work on your own projects: that’s what life as an academic is all about. Grad school is different: it’s the best chance you’ll have to work on something you couldn’t possibly invent yourself. If you do it right, you’ll expand beyond your own limits. You’ll add the skills, knowledge, and style of a successful older person to your cognitive repertoire. And, you’ll gain a powerful ally!

Have you looked at my advice page? I say quite a bit about choosing the right advisor. Choosing yourself violates most of that advice.

Posted by: John Baez on October 7, 2007 3:36 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Have you looked at my advice page? I say quite a bit about choosing the right advisor. Choosing yourself violates most of that advice.

But of course one must allow for exceptions. Bill Lawvere was a notable exception. Andre Joyal was another (is it true he never obtained a PhD?). In any event, it’s not clear that letting their advisers choose their topics for them would have been the right move, or that their graduate student careers were in any sense “radically suboptimal” as a result of their marching to their own drums.

Posted by: Todd Trimble on October 7, 2007 6:33 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

Todd, many of the greatest mathematicians never even went to graduate school in maths. The most famous examples are probably Euclid, Fermat, Newton and Ramanujan.

The great 20th century Russian functional analyst Israil Moiseyevich Gelfand never finished his secondary education! (However, he did later obtain a doctorate for his development of the theory of Banach algebras).

I don’t know anything about Squark, but Squark might consider doing what I do: make time to work on your own ideas outside of regular work or studies. Afterall, this is what Fermat did and he was no slouch !-)

Posted by: Charlie Stromeyer Jr on October 7, 2007 3:23 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

Yes, Charlie, thank you.

Getting back to the here and now, John’s advice is probably generally valid, but even in this academic-eat-academic world of today, there will always be people for whom it is best to for an adviser to just leave them be, listen, and learn from them. I’m sure John would agree with that: many of us have learned a ton from Jim Dolan, who at one point was John’s advisee.

Posted by: Todd Trimble on October 7, 2007 4:03 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

Todd, seriously, thanks for the tip about Jim Dolan. I am now learning about ‘decategorification’ and more in this paper he wrote with John Baez. Good stuff!

Posted by: Charlie Stromeyer Jr on October 7, 2007 6:16 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

Heh, you ain’t seen nuthin’ yet! Well, maybe you have, but it’s an awfully big world. :-)

As John has said over here, he and Jim are giving a seminar on geometric representation theory. Once the technical bugs with the videos are sorted out, you should be able to listen to Jim explaining some of his extraordinary ideas, and I can almost guarantee your mind will be blown. :-)

And by the way, John, that was a beautiful introduction!

Posted by: Todd Trimble on October 7, 2007 6:40 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

Todd wrote:

But of course one must allow for exceptions.

True. There are certain kinds of advice I feel duty-bound to give, even if it’s not suited to absolutely everyone. The few exceptions — the people who really shouldn’t follow it — will probably ignore it anyway. But I bet more people think they’re exceptions than actually are. Just like more people think they’re the next Einstein than actually are.

In particular, there are a lot of people who can make up and solve their own thesis problems, who would still greatly benefit from a good advisor. A good advisor helps in all sorts of ways that aren’t instantly obvious to the budding nerd.

For example, a good advisor helps you get to know the right people. And, they teach you things about the business and politics of academia that are very hard to pick up on your own — until it’s too late, that is.

Of course, there are more bad or mediocre advisors than good advisors.

Posted by: John Baez on October 8, 2007 1:37 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

As I once learned from reading something that Chris Hillman wrote, Albert Einstein did work on 7 or 8 different achievements each worthy of a Nobel prize. Even though I am not a physicist, I would say that expecting to be the next Einstein is very highly unrealistic.

That said, it would be cool if you physicists could find the next Einstein because so far no one can define string theory physics or loop quantum gravity rigorously enough.

Also, speaking of the Klein-Gordon equation, I saw this paper in the arxiv called “On the Klein-Gordon equation and hyperbolic pseudoanalytic function theory” which is cool for three reasons:

1) Hyperbolic numbers are cool.

2) The authors obtain infinite systems of solutions of the Klein-Gordon equation with potential (and anything infinite is cool).

3) You see about something new with solutions of the stationary Schrodinger equation just by reading the first paragraph.

global warming - your warm friend in any season

Posted by: Charlie Stromeyer Jr on October 8, 2007 2:14 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Todd asked if André Joyal ever obtained a PhD. I heard the following story from a senior category theorist, who I won’t name - not because he said anything malicious (he didn’t) but because the story has surely been passed from person to person and collected many inaccuracies along the way. I’ll probably add my own.

The story goes like this. Joyal didn’t acquire a PhD at the usual time, but got an academic job anyway. Then at some point much later in his life, the university at which he worked (UQAM, I guess) decreed that all of its faculty members must have PhDs. Panic! What to do? It was decided that he should be awarded a PhD for all his existing work - by his own institution or another, I forget which. Big hitters in category theory were called on to write letters of recommendation. Max Kelly was one of them. And according to legend, Kelly’s letter for Joyal ran as follows:

“A PhD is an apprenticeship in research. There is no need to serve an apprenticeship when you are a master.”

Posted by: Tom Leinster on October 10, 2007 2:57 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

Thanks for sharing that, Tom – great story. The quote attributed to Kelly certainly rings true: I can just hear him saying exactly that!

Posted by: Todd Trimble on October 10, 2007 6:24 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

This reminds me of G. E. Moore’s report on Wittgenstein submitting the Tractatus for a degree

In my opinion this is a work of genius; it is, in any case, up to the standards of a degree from Cambridge.

Posted by: David Corfield on October 10, 2007 8:52 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

Piling on with the recommendation quotes: Serge once wrote a letter for a student which read, in its entirety, “He’s okay.” One of the professors on the hiring committee that received this letter knew Howard Garland and called to ask him what the letter meant. Garland said without hesitation, “That means you should take him.”

Posted by: John Armstrong on October 10, 2007 1:42 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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In a better world I would have tried to post a summary and discussion of Jeffrey Morton’s long expected latest article Extended TQFT’s and Quantum Gravity by now.

With the world being as it is, days being way too short and all, this may have to wait a little while.

Posted by: Urs Schreiber on October 6, 2007 10:38 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Am browsing through Jeffrey’s article in an internet Café. Here are a couple of random comments:

a) on p. 5 it says:

It turns out that there is a close connection between the ideas of a theory having “no local degrees of freedom” in the discrete and continuum setting. In the continuum setting, this means that the theory is topological—the vector spaces and linear operators it assigns depending only on the isomorphism class of the manifold or cobordism. In the discrete setting of a triangulated manifold, it means that the theory is triangulation independent

For what it’s worth, this is a statement I would not agree with.

I do not think that the appearance of triangulations in TQFT is specific to “discrete” theories. FRS showed that Fukuma-Hosono-Kawai generalizes greatly, from finite group theory to all of rational 2D CFT (and is expected to generalize beyond).

I think that the definition of a TQFT in terms of triangulations is not a characteristic of the theory, but rather of its description. The question whether or not we see a triangulation appear in the definition of a quantum field theory (i.e. whether or not we see a state sum model) is precisely analogous to whether or not we see a triangulation in the computation of volume holonomy of higher gerbes: if you like you can express it in terms of local trivialization over a chosen local cover. But there are also other means.

In any case, triangulation independence is then just the obvious consistency requirement which ensures that indeed the triangulation is just a tool you use to describe a theory which exists independently of any such choices.

(Taking the risk of getting on everybody’s nerves, I mention that what I just said here is what I keep promoting as the cube. The description of Fukuma-Hosno-Kawai as the local data arising from local trivialization is here, the description of the FRS description of 2d RCFT from the same logic is Towards 2-functorial CFT. )

So I am saying:

- the description of TQFTs as local state sum models (triangulations and everything) is a choice of language/description. The state sum procedure is not an intrinsic property of a TQFT. Rather, it is analogous to describing global things in terms of descent data.

- there are “discrete” and non-discrete theories (all of 2-dimensional rational CFT! this does include the WZW model (non-finite group), for one) which have a state sum description.

Posted by: Urs Schreiber on October 6, 2007 12:40 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

There’s another weblog around that’s discussing this article. Now which one was it…

Oh yeah. Jeff Morton’s :D

Posted by: John Armstrong on October 6, 2007 3:13 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

Okay, let me see - I haven’t thought about this in a while, but I was wanting to come back to the relation between this and state sum models for higher dimensions, so it’s a good issue… Now, are you saying only that triangulation independence is not a complete description of what is implied by having no local degrees of freedom in a discrete background? What I was trying to suggest, maybe imprecisely, is that in a continuum setting, one can smoothly deform a manifold/cobordism into a new one; and in the discrete setting one can “deform” a triangulated space by retriangulating it through a series of moves. In each case, the condition I want will imply that the the states of the theory aren’t affected by this change. It sounds like you’re saying there should be an additional condition in the triangulated situation. What should it be?

I see what you mean by saying the independence is what’s necessary to get that the discrete theory can be used to define a theory on the continuous space - but all I was trying to suggest was what the no-locality condition says in that setting itself, not so much how the two are related.

Posted by: Jeffrey Morton on October 6, 2007 7:27 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Jeffrey wrote:

Now, are you saying […]

I guess I am mainly suggesting a slight shift in point of view, which, I find, is helpful for obtaining a nicely unified picture.

To maybe better see what I mean, consider the formula for the computation of surface holonomy for a gerbe with connection. This is the “classical” analogue of the quantum situation which we are looking at.

That formula tells us to choose a triangulation of the surface which we want to integrate over. Then we are asked to fatten the triangulation lines. Each (trivalent) vertex becomes a triangle. These triangles we are asked to label (even though nobody ever puts it that way – but it’s healthy to do so) by abelian Frobenius algebroids.

In addition to that, we label the fattened triangulation edges by some data and the remaining faces by some other data.

Now, consider the topological case: a flat gerbe connection.

In this case the above procedure collapses drastically: there is nothing to be assigned to the faces, and nothing to the edges. Only the assignments to the fattened vertices remain.

If you follow this pictorially, you see the faces shrink away, and the vertices grow fatter and fatter, until we arrive at the dual of the original triangulation.

Now composing everything in sight is completely analogous to performing the FHK state sum. Only difference is that we are restricting to the special case where the “algebra” is in fact an algebroid and the product restricted to be invertible.

My point is: this is not a coincidence. In both cases we are seeing a concrete case of the expression for expressing a locally trivialized 2-transport in terms of its transition data.

Once we start moving away from purely topological theories the situation becomes less degenerate and this statement would become more manifestly visible.

Posted by: Urs Schreiber on October 8, 2007 6:49 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

In this case the above procedure collapses drastically: there is nothing to be assigned to the faces, and nothing to the edges. Only the assignments to the fattened vertices remain.

Please explain.

Btw, one way of describing the associahedra and the cyclohedra is by fattening some but not all of the vertices, edges, faces, etc of the simplex.

Posted by: jim stasheff on October 31, 2007 1:25 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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In this case the above procedure collapses drastically: there is nothing to be assigned to the faces, and nothing to the edges. Only the assignments to the fattened vertices remain.

Please explain.

Let me say the same thing for one dimension lower, then it should become clear what I am talking about:

For a $U(1)$ -bundle with connection, the procedure for computing the holonomy around a circle in terms of local data is this, as you know:

a) choose a “triangulation” of the circle (a decomposition into segments)

b) integrate a 1-form over each segment (the local connection 1-form)

c) evaluate a 0-form on each vertex (the transition function)

d) add everything up.

In the case that the bundle is flat, we can assume that all the local 1-forms vanish. Hence the computation “collapses” to one where we assign nothing (i.e. the identity) to top-level simplices (edges, in this example) and only assign nontrivial information to lower dimensionanal simplices (here: to vertices).

A similar statement holds for all abelian $n$ -gerbes. There is a big formula which says that to compute the holonomy of an $n$ -gerbe with connection over a $(n+1)$ -dimensional surface using local data – i.e. using a Deligne cocycle – we should

a) first choose a triangulation of the surface, subordinate to an open cover of the basemanifold that it is embedded into

b) then assign integrals of local $(n+1)$ -forms over the $(n+1)$ -simplices of the triangulation, and integrals of $n$ -forms over the lower simplices, and so on.

c) then add everything up.

Again, in the special case that our $n$ -gerbe is flat, we may assume the top level forms to all vanhish. Then the above procedure simplifies, in that the assignment to the top-level simplices becomes trivial.

And all this has an anlogue for quantum field theory, and the computation of their correlators in terms of local data.

The prescription for defining a 2-dimensional QFT following Fukuma-Hosmo-Kawai is precisely that for computing the surface holonomy of a gerbe, as described above, for the special case that no data is assigned to top level simplices. So FHK gives a topological theory.

Topological $(n+1)$ -dimensional QFTs are the quantum analog of flat $n$ -gerbes with connection.

Please let me know if know if I managed to get this point across. If not, I’ll try to take the time and provice the formulas and diagrams.

Btw, one way of describing the associahedra and the cyclohedra is by fattening some but not all of the vertices, edges, faces, etc of the simplex.

Please explain!! :-)

Posted by: Urs Schreiber on October 31, 2007 6:05 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Please let me know if know if I managed to get this point across. If not, I’ll try to take the time and provide the formulas and diagrams.

The formula that I am talking about is originally due to Gawedzki and Reis (2.14), also to be found in A. Carey, S. Johnson & M. Murray (3.11) and many other places, like AschieriJurco .

Its diagrammatic version and nonabelian generalization is due to Urs Schreiber, badly displayed here and here and better described in an upcoming article with Konrad Waldorf.

The fact that 2d TFT constructions using Frobenius algebras is really just the same principle is usually not realized due to the fact that, due to being such a highly degenerate case, it is hard to recognize the Frobenius structure in the local data for a bundle gerbe. But it is there, as I tried to point out in Frobenius Algebroids with Invertible Products.

There I also presented a table, which was supposed to highlight the correspondence between

“classical transport”

in terms of local data, namely parallel transport, and

“quantum transport”

in terms of local data, namely state sum models

A more detailed description of this phenomenon is given in my Talks at “Higher Categories and their Applications”.

Posted by: Urs Schreiber on October 31, 2007 10:35 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

Is it time to wake up the baby yet? She’s been sleeping for a long time now :)

Posted by: Eric on November 1, 2007 5:24 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Is it time to wake up the baby yet? She’s been sleeping for a long time now :)

(Heh. Eric and I have our secret code.)

Eric, i know, sorry for that. You are not the only one wondering about my nurturing habits.

Did you notice, though, the recent step towards a wake-up call which I mentioned in the table at the end of this comment?

To speed the process up, we should challenge the hard-core category theorists here to help us with the following question:

Question:

Lie $n$ -algebras can be entirely encoded in a statement saying somehting like “the boundary of a boundary vanishes”. We know they integrate to smooth Kan-complexes. These in turn, being simplicial, certainly also involve something with boundaries of boundaries vanishing.

Is there a way to rephrase the definition of a Kan simplicial complex directly by saying something like:

a free graded abelian group $S = \oplus_n S_n$ equipped with an operation $D : S \to S$ of degree -1 which squares to 0 $D^2 = 0 \,.$

If it were just the simplicial structure underlying the Kan complex this would be a triviality. But I am asking for a setup $(S,D)$ such that $D^2 = 0$ actually captures not just the simplicial structure, but also the Kan conditions.

So it must be rather something like “the filler of a filler” is trivial. Or something like that. This is at least what the $D^2 = 0$ for Lie $n$ -algebras says, essentially.

One would think it would be straightforward to read off the desired operation I am looking for from staring sufficiently long at how it works for Lie $n$ -algebras. But the trouble is that when passing from n-groups to Lie n-algebras, some “quandling” is introduced. That makes it intransparent. At least to me.

Posted by: Urs Schreiber on November 1, 2007 12:42 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Did you notice, though, the recent step towards a wake-up call which I mentioned in the table at the end of this comment?

Hi Urs,

No, I missed that comment. Thanks for pointing it out. Maybe you were closer to waking the baby than I realized :)

Reading your comments here made me want to scold you though *tsk tsk* You’ve been talking about “triangles”, “simplices of a triangulation”, etc. If I learned anything from our months of (immensely fun) collaboration, it was that simplices are fundamentally sick. The barrier to progress for me all those years prior to you coming and solving all my problems was my insistence that triangulations and simplicial complexes should be the basic building blocks due to their cute algebraic/combinatorial properties. Something HAD to be right about them.

Then you came along and essentially proved that I couldn’t do what I wanted to do, i.e. build a fully self-consistent finitary version of differential geometry on a simplicial complex. BUT, you went further and demonstrated that we COULD build a finitary differential geometry on a “diamond complex”. This was obvious to you and I’m still not sure you appreciated the meaning (or potential meaning), but to me, it was a profound insight about the nature of space and time (or a pleasant/satisfying model thereof).

I’ve said over and over again how far you have progressed beyond anything I could possibly comprehend. Each time I say it, you’ve progressed by at least an order of magnitude beyond the last time, so I won’t even pretend to understand what you are working on these days, but in the spirit of our good old days, I can give you some speculative comments based on gut feelings.

I believe that in one form or another (that will likely only reveal itself after all the mysteries you’re experiencing have been resolved), you will find that essentially the same barriers I faced back before you paved the road to enlightenment regarding finitary differential geometry are the barriers to your progress now. The sooner you abandon triangulations and fall back on our old friend the “diamond”, the sooner you will get to where you’re going (wherever that is!).

Certainly, whether you’re working with triangulations or diamonations SHOULDN’T matter and once the details are worked out I have no doubt that will be so (in some mathematician’s sense of the word), but I highly suspect the secrets will be more easily revealed on a diamond.

The one thing that we never managed to prove (or I never managed to motivate you enough to prove) was my conjecture:

We know that any smooth manifold can be triangulated (I think! It’s been ages since I thought about this stuff). My conjecture was that any spacetime manifold can be “diamonated”.

If you were able to prove this statement, then I believe any statement you could make about diamonds (and n-curvatures thereon) would carry over in the continuum limit to a statement about any smooth spacetime manifold. At which point, I would say, “Who cares about the continuum?!” :)

Best regards,

Eric

Posted by: Eric on November 2, 2007 5:45 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Another random thought…

This “quandling” thing reminds me a LOT of a conversation we had with Connes about noncommutativity and time evolution. Not sure what to make of it, but maybe it means something to you? I tried and failed to relate the conversation to graph calculus, but maybe there is something there.

Posted by: Eric on November 2, 2007 6:11 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Connes about noncommutativity and time evolution

I had recently another conversation about that over on his blog (the latest after this came up last time here).

This clarified two important points which were not clear to me before that:

a) the claim is that in examples of quantum systems with type III vN factor algebras of observables, indeed only the outer part of the time evolution automorphism matters (which is remarkable, because for ordinary time evolution in finite DOF QM it is entirely inner, of course)

b) but at the same time there is at this moment no literature where this is actually described.

Posted by: Urs Schreiber on November 5, 2007 8:35 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

Note that Jeff has initiated discussion here and here.

Posted by: David Corfield on October 6, 2007 12:43 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Note that Jeff has initiated discussion here and here.

Thanks. I thought I’d collect a couple of remarks here (the limiting resource being time). Then I’d wrap this up into an $n$ -Café entry and send Jeffrey’s blog a trackback.

Posted by: Urs Schreiber on October 6, 2007 12:56 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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b) On bottom of p. 6 it says:

Double categories are too strict to be really natural for our purpose

Recently I got the impression that Ronnie Brown and his school are suggesting that we are all mislead by abandoning $n$ -fold categories (and $n$ -fold groupoids in particular). Compare for instance the remarks in the introduction of Kock on Higher Connections.

I am not sure I even know/understand what the claim really is. But the impression I got was that the idea was that in a suitable sense everything “one needs” are $n$ -fold categories.

Does anyone know more about this? What the claim/program really is, precisely? How far it carries? What it would imply?

I cannot make up my mind, since the question seems to be subtle: $n$ -fold categories are not just more restricted then weak $n$ -categories (in that they are taken to strict). They are at the same tie more flexible in another direction: they can have different $(k \lt n)$ -categories for all different $k$ -dimensional directions.

And finally: is there anything that would prevent us from inventing weak $n$ -fold categories?

Posted by: Urs Schreiber on October 6, 2007 12:54 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Urs wrote:

And finally: is there anything that would prevent us from inventing weak $n$ -fold categories?

Well, the main thing that would prevent me is this: Marco Grandis has already done it!

See his papers:

Marco Grandis, Higher cospans and weak cubical categories (Cospans in Algebraic Topology, I).
Marco Grandis, Collared cospans, cohomotopy and TQFT (Cospans in Algebraic Topology, II).
Marco Grandis, Cubical cospans and higher cobordisms (Cospans in Algebraic Topology, III).

You’ll see these papers were in part inspired by Jeff Morton’s work. Jeff in turn uses a sort of weak double category that was invented by Dominic Verity in his unpublished thesis.

Grandis doesn’t put his papers on the arXiv since the arXiv doesn’t have a policy statement guaranteeing open access.

Posted by: John Baez on October 8, 2007 3:18 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

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c) From p. 10 of the introduction on, states and propagation of them is discussed. I haven’t read on yet, but at this point it looks like what is discussed is Dijkgraaf-Witten theory with trivial twist. Is that right?

Given a manifold $B$ , the fundamental groupoid

$\mathrm{par} = \Pi_1(B)$

is the parameter “space” of the theory. This tells us how the $n$ -particle looks like which we are about to propagate. In DW theory we see a 3-particle propagating through $B G$ . Since we work on the level ofgroupoids, we use $\Sigma G$ for $B G$ (which Jefrrey’s denotes as $G$ ): the one-object groupoid obtained from the given gauge group.

So that’s target space

$\mathrm{tar} = \Sigma G$

$\mathrm{tar} = G$

in Jeffery’s (and John’s) notation.

A “field configuration” is a map from parameter space to target space

$\mathrm{par} \to \mathrm{tar} \,.$

Hence the “space” (groupoid) of field configurations (called configuration space) is

$\mathrm{conf} = [\mathrm{par},\mathrm{tar}] = [\Pi_1(B),\Sigma G] \,.$

This is, in the present case, the space of flat $G$ -bundles with connection on $B$ .

Next, in order to quantize the “charged $n$ -particle” (for Dijkgraaf-Witten and Chern.Simons: a 3-particle charged under a Chern-Simons 2-gerbe/3-bundle) we need to specify the “background gauge field”, which is an $n$ -bundle with connection on configuration space $\mathrm{conf}$ . The space of states assigned by the QFT to the parameter space is then the space of sections of that background field.

Jeffrey takes the space of sections to be

$[\mathrm{conf},\mathrm{Vect}] \,.$

A functor to vector spaces can be regarded as the section of a trivial line 2-bundle (as described for instance here.). So, I think, the discussion on the bottom of p. 10 amounts to saying that we couple our $n$ -particle to a trivial line 2-bundle with trivial connection.

Posted by: Urs Schreiber on October 6, 2007 1:21 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

Yes - this is definitely related to the Dijkgraaf-Witten model. The way you describe it here seems to be just about exactly how I think of it, except that, as you point out, I don’t describe it in terms of 2-bundles.

Posted by: Jeffrey Morton on October 6, 2007 7:07 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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except that, as you point out, I don’t describe it in terms of 2-bundles.

So you don’t incorporate the twist right? I mean, DW theory is parameterized by a group 3-cocycle. This you take to be the trivial one, it seems? (No problem with that, I am just trying to make sure that I understand.)

By the way, DW and CS theory actually involve a 3-bundle. A 3-bundle on the target space that the 3-particle (membrane) propagates on.

This target space is $B G$ . Sometimes, in this context, $B G$ is conceived as a smooth space and the 3-bundle on it as a Chern-Simons bundle 2-gerbe. That’s the point of view adopted in Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories (you probably know this, I am just saying it for the record and for completeness).

But we can think of other realizations of this background 3-bundle (this is a background structure even if and when you want to apply this to 3-dimensional quantum gravity). The best for the kind of approach that you follow is, I am convinced, one that expresses the CS 3-bundle entirely as a 3-transport.

One way to think of it is this: Consider $B G$ as the one-object groupoid $\Sigma G$ and then pass to the equivalent 2-groupoid

$\Sigma(\Omega G \to P G)$

on that 2-groupoid, I think, we canonically have a line 3-transport, namely a Gray 3-functor with values in endomorphisms of the 1-dimensional 3-vector space. Actually, it factors through the 1-object 1-morphism 3-category

$\Sigma \Sigma \mathrm{Vect}$

obtained by suspending the symmetric monoidal category $\mathrm{Vect}$ twice.

And there is one such 3-functor for every level $k$ : the 3-functor is nothing but the fiber-assigning-functor of the central String-extension

$\Sigma U(1) \to \mathrm{String}_k(G) \to (\Omega G \to P G) \,.$

I think that in order to introduce the twist in your setup, you need to consider such a 3-functor on the target, then pull it back to configuration space

$\mathrm{conf} = [\Pi_1(B),\Sigma G] \simeq [\Pi_1(B),\Sigma (\Omega G \to P G)]$

and then form the space of sections of this pulled back thing. That’s what will replace

$[[\Pi_1(B),\Sigma G],\mathrm{Vect}]$

in the general case, I think.

Posted by: Urs Schreiber on October 8, 2007 9:38 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

Jeffrey does not do the twist.

Posted by: John Baez on October 8, 2007 9:49 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Jeffrey does not do the twist.

Well, then, let’s twist again, like we did last summer (or was it winter?).

Actually, meanwhile we understand this better: we know how the universal $G$ -bundle on $B G$ is nicely encoded in the sequence

$G \to \mathrm{INN}(G) \to \Sigma G$

of groupoids, and know more about how the Chern-Simons line 3-bundle on $\Sigma G$ that we need is obtained as an obstruction from the failed lift of this to a $\mathrm{String}_k(G)$ -2-bundle.

In order to do not just DW but full CS theory this way we need to pass to the differential picture, expound on how the Lie algebroid scenario obtained this way fits into the AKSZ-picture such as to finally understand the necessary renormalization as in Kevin Costello’s Renormalisation and the Batalin-Vilkovisky formalism.

When this is done it should be a corollary to demonstrate that the resulting quantum CS 3-functor admits local trivialization which leads to the TFT description of WZW theory.

Posted by: Urs Schreiber on October 9, 2007 9:18 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

Well, then, let’s twist again, like we did last summer.

Lol, this is the second funniest thing ever posted at the n-cafe! (Dr Evil of course remains in numero uno)

Posted by: Bruce Bartlett on October 9, 2007 1:20 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Urs suggested

Well, then, let’s twist again, like we did last summer

Last summer? That post was sent in November! Maybe you’re turning into an Australian category theorist.

Posted by: Tim Silverman on October 9, 2007 9:43 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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Hi Jeffrey! Darn, I think your thesis came out on the archive while I was on vacation. There’s a lot of nice stuff in there, in particular I can learn a lot from the way you did the pull-push construction in Section 7. I think our basic viewpoints are very similar. Tomorrow I print it out and read in peace! Congrats for typing up a nice version .

I need to understand the cobordism-with-corners/double-category side of things. It’s a kind of strange development which I find confusing. In the world of TQFTs, when I see those corners/double-categories come-a-marchin’ I sometimes think to myself, “who ordered this?”. If they go on the left hand side, then what goes on the right?

As far as Dijkgraaf-Witten theory goes, I recently came across the following paper on the archive:

Mulase, Yu, Non-commutative matrix integrals and representation varieties of surface groups in a finite group .

Are you, or other cafe-patrons, aware of it? If I understand correctly, the ‘representation variety of a surface group in a finite group $G$ ’ is sort of a low-brow way of speaking about the moduli-stack, or groupoid, of $G$ bundles on the surface. The mapping class group of the surface acts on this space by pull-back - this is the part which interests me.

Anyhow, they have some cool formulas in that paper which I’d like to understand. Formula (0.3) says that

(1)

log \int_{\{x \in \mathbb{C}[G] | x = x^*\}} exp\left(-\frac{1}{2}\chi_{reg}(x^2)\right)exp\left(\sum_j \frac{t_j}{j} \chi_{\reg}(x^j)\right) d\mu(x) = \sum_{\Gamma connected ribbon graph} \frac{1}{|Aut_R \Gamma|} |G|^{\chi(S_\Gamma)-1}|Hom(\pi_1(S_\Gamma, G))| \prod_j t_j^{v_j(\Gamma)}

Well, it’s too long to fit on this page :-) But it’s a variant of the old TQFT formula for counting G-bundles on a surface. Except what’s interesting is that, normally, we’d be integrating over the moduli space of G-bundles. Here they are integrating over self-adjoint elements of the group ring , weighting each element with a Gaussian function. What’s going on?

Another Dijkgraaf-Witten related paper, which came out more recently, is

Kaufmann, Pham, The Drinfeld Double and twisting in stringy orbifold theory.

Posted by: Bruce Bartlett on October 9, 2007 12:39 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

So that’s target space

tar = Sigma G

or

tar = G

in Jeffery’s (and John’s) notation.

A “field configuration” is a map from parameter space to target space

par –> tar

Hence the “space” (groupoid) of field configurations (called configuration space) is

conf = [par, tar] = Pi_1(B), Sigma G ]

Oy veh! and it’s really BG !!

btw, [ , ] means? I’m used to homtopy classes of morphs/maps.

and a background gauge field as a bundle!!

as background - fine
but as a field???

btw, when i cut and paste, some symbols appear as garbage
and I have to delete them
how can I prevent this

Posted by: jim stasheff on October 31, 2007 1:35 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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$\mathrm{conf} = [\mathrm{par},\mathrm{tar}] = [\Pi_1(B), \Sigma G]$

Oy veh! and it’s really $B G$ !!

Well, as long as we agree to distinguish between groupoids and spaces, $\Sigma G$ is not the same as $B G$ , even though both map to each other under the identification of categories with their nerve.

$\Sigma G$ is the groupoid with a single object and one morphism per element of $G$ . Hence with $|\cdot|$ denoting the operation of taking the nerve and realizing it, we have $|\Sigma G| = B G \,.$

But one of the points of what I said above is: maybe its better not to pass to realizations of nerves.

btw, [ , ] means

Oh, I am sorry. That means (internal) Hom!

For $C$ and $D$ categories, I write

$[C,D]$

synonymously for

$\mathrm{Hom}_{\mathrm{Cat}}(C, D)$

hence for

$\mathrm{Funct}(C, D) \,.$

I’m used to homtopy classes of morphs/maps.

Yes, it’s related just like $\Sigma$ and $B$ are.

Homotopy classes of maps between spaces correspond to isomorphism classes of functors of the underlying categories.

So, to make that clear, what I called

$[\Pi_1(X), \Sigma G]$

is hence nothing but the category whose objects are flat $G$ -bundles with connection on $B$ , and whose morphisms are ismorphisms of these.

and a background gauge field as a bundle!!

Well, that’s not controversial, is it? A “background gauge field” for a particle is precisely a connection on a bundle.

as background - fine but as a field???

Right, that’s where it becomes interesting: for theories like Chern-Simons or Dijkgraaf-Witten, the bundles are the fields!

It’s really true. That’s one of the advantages of encoding $n$ -bundles with connection entirely in terms of their transport functors: this makes it manifest that and how they can be regarded as fields, namely as maps from some parameter space (paths in something) to some target (namely some suitable category of fibers, or, more commonly, the corresponding classifying space).

Well, you said that yourself, decades before I did!

Bonora, Cotta-Ramusino, Rinaldi, Stasheff, The evaluation map in field theory, sigma models and strings – II

you emphasize the point that it can be useful to regard gauge theories as sigma-models – with target the classifying space of the gauge group (p. 386-387).

That’s precisely what I am talking about here. Only that instead of passing to big spaces, I preferred to say everything in terms of small categories.

btw, when i cut and paste, some symbols appear as garbage and I have to delete them how can I prevent this

That happens when you try to cut and paste MathML output. Unfortunately, as far as I know, thre is no way to prevent this. The only workaround is to not copy-and-past formulas here. If you need to quote them, you will have to retype them. At least that’s the current state.

Posted by: Urs Schreiber on October 31, 2007 6:30 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

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For the moment I close with this comment:

Cool paper. Am enjoying it a lot. Looks like the perfect complementary discussion to what Bruce Bartlett is working on with Simon Willerton (which is vast extension of The Baby Version of FHT). It’s good that these things are finally being written up.

For political reasons it might eventually be helpful to cleanly separate the general discussion from applications and from speculations about quantum gravity. On the other hand, who cares about political considerations.

Posted by: Urs Schreiber on October 6, 2007 1:30 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

Dear John,

About recursivity in QM. This topic has been picked up by your academic brother Feng Ye. He developed a constructive theory of unbounded operators and investigated the applications to quantum physics. Ye also graduated with Bugress. I continued this work on unbounded operators in constructive mathematics here. My conclusion was that, once you see how to do it a recursive theory of unbounded operators is fairly natural.

I currently see this work as a stepping stone towards a development of quantum theory in a topos, with possible applications to algebraic quantum field theory.

Bas

Posted by: Bas Spitters on October 7, 2007 2:04 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

Dear John,

> In part, this was my reaction against
> Pour-El and Richards’ work showing that
> there’s a solution of the wave equation
> with smooth computable initial data for
> which the value of the solution fails to
> be computable at one point.

Did you ever see this paper by Weihrauch and Zhong? Their main conclusion seems to be that with a different topology the wave equation is computable. Finally they state: “Pour-El-Richards result probably does not help to design a wave computer which beats the Turing machine.” Note that Zhong is a student of Pour-El.

> In part, it’s what was left of a
> mindblowingly visionary program after I
> realized I couldn’t actually accomplish
> most of it. Youthful dreams.

What was the visionary program?

Bas

Posted by: Bas Spitters on October 7, 2007 8:24 PM | Permalink | Reply to this

Re: Spans in Quantum Theory

Bas Spitters wrote:

Did you ever see this paper by Weihrauch and Zhong?

No.

Their main conclusion seems to be that with a different topology the wave equation is computable.

Right. I knew that by the time I did my PhD thesis with Irving Segal. He made it clear that the “good” spaces of solutions for linear wave equations tend to be Hilbert spaces in which time evolution is unitary. And, these unitary time evolution groups tend to be computable.

For example, for the Klein–Gordon equation on Minkowski spacetime, we can use either the Hilbert space of finite-energy solutions or the Hilbert space with the Poincaré-invariant norm (which is what they use in quantum field theory). In either case, my methods show that time evolution is unitary and computable.

For the scalar wave equation on Minkowski space (i.e. the Klein–Gordon equation but with mass equal to zero), the space of finite-energy solutions is a bit awkward. However, the Hilbert space with the Poincaré–invariant norm is very nice — and again, time evolution is unitary and computable on this space.

I realized this after Segal had taught me a thing or two about quantum field theory and differential equations — sometime around 1984, I guess. After that, I decided that Pour–El and Richards’ result was sort of a red herring: cute, but not really relevant to physics.

John wrote:

In part, it’s what was left of a mindblowingly visionary program after I realized I couldn’t actually accomplish most of it. Youthful dreams.

What was the visionary program?

One great thing about being old is that you don’t have to answer questions like that. Some people enjoy revisiting the embarrassments of their youth, but I don’t.

Posted by: John Baez on October 8, 2007 2:10 AM | Permalink | Reply to this

Re: Spans in Quantum Theory

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The major advancements in physics have been unifications. Newton unified terrestial and celestial mechanics. James Clerk Maxwell unified electricity, magnetism, and optics. General relativity is a unification of special relativity and gravity. Quantum field theory is a unification of quantum mechanics and special relativity. Electroweak theory is a unification of electromagnetism and the weak force. The Standard Model is a unification of electromagnetism, the weak force, and the strong force. Grand unified theory is a much stronger unification of the electromagnetism, the weak force, and the strong force. Supersymmetry is a unification of bosons and fermions. String theory is a unification of all four forces: electromagnetism, the weak force, and the strong force. M-theory is a unification of the five string theories. Loop quantum gravity is a unification of general relativity and quantum mechanics.

At the same time, the major advances in physics have been generalizations of previous theories. Special relativity is a generalization of Newtonian mechanics in which we no longer assume v << c. Quantum mechanics is a generalization of Newtonian mechanics in which we no longer assume h is neglible. General relativity is a generalization of special relativity in which we no longer assume that spacetime is flat. Supersymmetry is a generalization in which we include anticommutators as well as commutators. String theory is a generalization of particle physics in which we no longer assume that p = 0. D-branes are a generalization of strings. M-theory is a generalization of string theory, in which we consider the entire moduli space, not just the five points that represent the five string theories. Elliptic cohomology is a generalization of K-theory.

In John Baez’s lectures on n-categories, he uses generalization to attempt to unify the categories relevant to general relativity and quantum mechanics. The category used in general relativity is nCob. The category used in quantum mechanics is Hilb. John Baez pointed out that both nCob and Hilb are examples of monoidal *-categories. Therefore, by generalizing them, you are able to unify them. I think there’s a deep insight in this observation that could eventually lead to unifying general relativity and quantum mechanics.

You can do this concept with anything. Let’s say you have two things, A and B. By pointing out that A and B are both examples of C, you can generalize and therefore unify both A and B. However, this only gives you insight if the result is not already obvious. This is only the case if A and B are of an intermediate level of similarity.

Let’s say A and B are very similar. In that case, C would also be similar to A and B, and there would be nothing surprising about that. Let’s say A is the series A = 1, 2, 3,… and B is the series B = 2, 4, 6,… Then C would just be the natural numbers multiplied by n, where with A, n = 1, and with B, n = 2. That is so obvious, that there is no insight to be gained from that observation.

On the other hand, if A are B are very different, C would have such little structure, there would be nothing surprising about that observation either. Let’s say A is the series A = 1, 2, 3,… and this time assume B is all the major baseball teams in the United States. In that case, the only thing they would have in common is that they are both Sets. Therefore, C would just be sets. Well, almost everything is a set, so the fact that two things are both sets is not surprising. There is no deep insight there.

Therefore, the process of unifying A and B by generalizing them to C only leads to deep insight if A and B are of an intermediate level of similarity. This is what you have in John Baez’s paper where A = nCob, B = Hilb, and C = monoidal *-categories. They are different enough that the result is not obvious, and they are different enough that C has enough structure to be meaningful.

What I’m wondering is that this process of unification and generalization has been so important to the progress of physics, is if there is someway we formalize this process in a more systematic manner. Is there some way to quantifyably measure the amount of similarity between A and B, and the amount of structure in the resulting C? Therefore, we could seek out to try to unify A’s and B’s within that window of similarity.

What do you think?

Jeffery Winkler

http://www.geocities.com/jefferywinkler

Posted by: Jeffery Winkler on October 9, 2007 8:36 PM | Permalink | Reply to this

The n-Category Café

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October 1, 2007