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August 29, 2008

New Structures for Physics I

Posted by John Baez

guest post by Bob Coecke

Following John’s great idea for having a public review here at the café of his and Mike Stay’s chapter for the New Structures for Physics volume(s) which I am editing, which happened here and here, John had the even better idea to have several chapters for these volumes reviewed here at the café.

I will submit them in pairs as guest posts. The ones we start with are two papers which, together with John and Mike’s, make up an ABC on category theory and monoidal categories and categorical logic in particular.

These chapters are:

The first one is the lecture notes of Abramsky’s category theory course here at Oxford, including exercises, intended for mathematicians and computer scientists. The other one has physicists, and in particular, quantum informaticians, as its target audience.

Any kind of feed-back is most welcome!

Posted at August 29, 2008 4:10 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1777

30 Comments & 1 Trackback

Re: New Structures for Physics I

Thie a typo-level criticism: the title on top of every page of “introduction to categories and categorical logic” is “Contents”. This was presumably set up in the table of contents, and never changed thereafter.

Posted by: Hendrik Boom on August 29, 2008 8:17 PM | Permalink | Reply to this

Re: New Structures for Physics I - more typoes

pg. 52 mid, ‘noindent’ (TeXnical mishap)

pg. 53 top, ex 37 ‘to show *the admissability* of the following:’

Posted by: Avery Andrews on September 2, 2008 1:14 AM | Permalink | Reply to this

Re: New Structures for Physics I

From Categories for the practicing physicist, p.3:

The framework of monoidal categories enables to model and axiomatise

Should be

The framework of monoidal categories enables modeling and axiomatising

Almost all non native English speakers get this wrong: the verb to enable either uses a gerund (-ing) to talk about the general case, or uses pronouns in combination with the infinitive (to x).

Posted by: Aaron on August 30, 2008 10:49 PM | Permalink | Reply to this

Re: New Structures for Physics I

Or “makes it possible to model etc “.

Posted by: Tim Silverman on August 31, 2008 1:26 PM | Permalink | Reply to this

Re: New Structures for Physics I

P. 13 of Bob and Eric

Also preorders play an important role e.g.:

• Also preorders play an important role e.g. they provide the only elegant and conclusive account on measuring quantum entanglement [49]. The relevant preorder is Muirheads majorization order [48].

Aside from the duplication, and the missing apostrophe in ‘Muirheads’, I don’t like ‘account on’ here and following sentences.

Posted by: David Corfield on September 1, 2008 4:23 PM | Permalink | Reply to this

Re: New Structures for Physics I

P. 69 in intro to cats: The rule for introducing linear implication has the wrong type for the variable ‘t’: it should be T, not U. The rule appears wrong twice on the page.

Posted by: Mike Stay on September 1, 2008 8:59 PM | Permalink | Reply to this

Re: New Structures for Physics I

On p. 73, there’s some sloppiness about the associator: the exchange rule has

ΓABΔ\Gamma \otimes A \otimes B \otimes \Delta

whereas in the table it uses an associator to push parentheses around.

Posted by: Mike Stay on September 2, 2008 8:21 PM | Permalink | Reply to this

Re: New Structures for Physics I

There are some issues with the strictness of the resulting symmetric monoidal closed category (SMCC) that aren’t clear to me.

1. Is the unit type necessarily one of the base types? If not, the monoidal unit is never mentioned anywhere, so it’s unclear how that part of the SMCC appears.

2. Is a product type necessary? Usually one forms pairs with the term

(1)P=λxyz.zxy P = \lambda x y z.z x y

so that given a pair PabPab, we can extract the first and second elements like this:

(2)Pab(λxy.x)=(λxy.x)ab=a P a b(\lambda x y.x) = (\lambda x y.x)a b = a
(3)Pab(λxy.y)=(λxy.y)ab=b P a b(\lambda x y.y) = (\lambda x y.y)a b = b

In linear lambda calculus, we don’t have the projections, but we do have

(4)Pab(λxyz.zxy)c=cab P a b (\lambda x y z.z x y)c = c a b
(5)Pab(λxyz.zyx)c=cba P a b (\lambda x y z.z y x)c = c b a

that applies cc to each part in order or in reverse order.

3. If the resulting SMCC is strict, then that would take care of the unit object, since tensoring with the unit would be an identity morphism. But then the use of the associator on page 73 doesn’t make sense; the rule

(6)Γ,A,BCΓ,ABC \frac{\Gamma, A, B \vdash C}{\Gamma, A \otimes B \vdash C}

should just be interpreted with the identity natural transformation.

Or is it trying to show how to form the morphism in the strictification of some non-strict category?

(I’d appreciate anyone’s response!)

Posted by: Mike Stay on September 23, 2008 10:12 PM | Permalink | Reply to this

Re: New Structures for Physics I

Mike,

You’re right that the unit II should be included as a base type, and the structural rules

IΓAΓ,IA\frac{}{\to I} \qquad \frac{\Gamma \to A}{\Gamma, I \to A}

ought to be included. You can’t just wish the unit away, because arrows like the double dual map

δ A:A(AI)I\delta_A: A \to (A \multimap I) \multimap I

are there and need to be accounted for.

My own preferred method for interpreting the sequent rules is to interpret contexts Γ\Gamma as objects in an ambient monoidal strictification containing the original smcc. I haven’t read through their paper, but it looks like maybe your points are well-taken and need to be fleshed out a bit in the paper.

Posted by: Todd Trimble on September 25, 2008 4:55 PM | Permalink | Reply to this

Re: New Structures for Physics I

My own preferred method for interpreting the sequent rules is to interpret contexts Γ as objects in an ambient monoidal strictification containing the original smcc.

Yes, thanks for that explanation! I’m going to take that tack in our paper. I also found this afternoon a very nice writeup of the whole correspondence in M. Hasegawa, Logical predicates for intuitionistic linear type theories.

Posted by: Mike Stay on September 26, 2008 12:14 AM | Permalink | Reply to this

Substantially New Structures for Physics

Is there something substantially new in these new structures, rather than just a new (and unfamiliar) language? In particular, do they have a chance to lead to new physics?

To illustrate my question, consider the concept of a phase space. Naively, this is an even-dimensional vector space with a Poisson bracket. Symplectic geometry is a nontrivial generalization of this concept, but it does not lead to new physics. In particular, we can not explain the photoelectric effect in classical physics by going to global symplectic geometry. To do that, we must perform quantization, but we can keep the naive concept of phase space. Symplectic geometry is of course still valuable in the quantum case, but it is progress in a direction orthogonal to quantization, and it does not help to understand the photoelectric effect.

Say that we want to quantize gravity. We know that QFT is naively incompatible with gravity. There could be some very fancy formalism in which QFT is compatible with gravity, but this is unlikely since the naive approach fails. In analogy with the photoelectric effect, one expects that substantially new physics is needed. Do categories have a chance to provide this, or is it progress in an orthogonal direction?

Phrased differently: substantially new physics typically comes with a deformation parameter:
* Special relativity has c.
* Quantum mechanics has hbar.
* New forces have their coupling constants.
* CFT has another c.
Do categories come with a deformation parameter?

Posted by: Thomas Larsson on September 2, 2008 7:12 AM | Permalink | Reply to this

Re: Substantially New Structures for Physics

Thomas wrote:

Is there something substantially new in these new structures, rather than just a new (and unfamiliar) language?

I hope Bob Coecke’s book has a little blurb on the back that answers this question for suspicious customers. It would be interesting to see his answer.

But here’s mine:

These structures are not really new. Category theory has been important in algebraic topology and algebraic geometry ever since its inception in 1945. Symmetric monoidal categories were introduced by Mac Lane in 1963. Braided monoidal categories were introduced by Joyal and Street in the late 1980s. It’s just taken a while for these structures to become part of the toolkit of the average mathematical physicist. But to boost sales of his book, I guess Bob decided that New Structures for Physics would be a more appealing title than Structures You Would Already Know About, Had You Been Paying Proper Attention.

Phrased differently: substantially new physics typically comes with a deformation parameter… Do categories come with a deformation parameter?

First, category theory isn’t a new theory of physics. It’s supposed to be a very general way of understanding lots of things, including physics.

Category theory is a lot like group theory in this respect. In fact, it’s a generalization of group theory. When group theory was introduced in quantum theory, there was a lot of grumbling from certain people — I think I’ve already told you how Slater, in his textbook on physical chemistry, proudly wrote that his treatment used no group theory. But eventually groups were accepted as useful tools in physics.

If you’d asked me do groups come with a deformation parameter?, I’d say “some don’t, some do”. As you probably know, there’s a whole theory of Inönü–Wigner contractions, which makes clear how the Galilean group deforms to the Poincaré group as 1/c1/c moves away from 00. This clarifies some aspects of special relativity — for example, how nonrelativistic quantum mechanics deforms to relativistic quantum mechanics.

Since you’re asking do categories come with a deformation parameter?, I’ll have to give the same answer. Some do, some don’t. There’s a whole theory of deformations of categories. And there are some very intriguing examples related to ‘qq-deformation’. But that’s not the main reason categories are important.

Still, let me say a bit about deformations of categories. For starters, any category of representations of a simple Lie group admits an essentially unique 1-parameter deformation, called a ‘qq-deformation’. People call these ‘categories of representations of quantum groups’. However, ‘quantum’ is probably not the right word for the physical meaning of this deformation! After all, even ordinary groups arise as symmetries of quantum systems.

So what does qq-deformation mean, physically? In 3d quantum gravity, the new deformation parameter is secretly the cosmological constant. It’s appealing to imagine that the cosmological constant points to deeply new physics just as the speed of light, Planck’s constant and the gravitational constant did. But, it’s too soon to say.

In particular, attempts to use qq-deformation to study 4d quantum gravity with nonzero cosmological constant show hints of promise — but so far I can’t tell if they really work.

In my attempts to understand qq-deformation more deeply, I was eventually pulled into a strange circle of ideas, which says that qq-deformation is especially nice when qq is a power of a prime number. I talked about this in week183, week184, week185, and subsequent Weeks. It’s incredibly cool math. But I have no idea how it might be related to the cosmological constant.

So: none of this is the main reason mathematical physicists should learn category theory. The main reason is that category theory lets us think more clearly, by giving names to what’s sitting right in front of our nose, letting us make analogies precise that would otherwise remain fuzzy, and giving us a huge arsenal of techniques to prove things that would otherwise be extremely difficult.

For more details, read Bob’s book.

Posted by: John Baez on September 2, 2008 11:58 PM | Permalink | Reply to this

Re: Substantially New Structures for Physics

Thanks for this John. And for the suggestion of a blurp. I am in the process to compose something to post here as an answer and will still do so. My idea was in fact that this could become more than a blurp, some kind of preface even, to which i would welcome contributions. This is how what I am drafting started:

“The above questions, or rather, stance, is a very typical one within the present physics community. Therefore it might be useful to provide a thorough response to it. I invite anyone to contribute and if the outcome is sufficiently substantial, could be included in the New Structures volumes as a multi-author preface. ”

Posted by: bob on September 3, 2008 6:43 PM | Permalink | Reply to this

Re: Substantially New Structures for Physics

The usual word is ‘blurb’. However, I really like the idea of a ‘blurp’.

More importantly, I think it’s great to explicitly address questions like “Is this stuff really new, or just unfamiliar language for talking about things I already know? What can I actually do with this stuff?”

It’s natural for people to raise questions like this — and it’s good to provide answers.

In our Rosetta Stone paper, Mike and I provide a list of references showing how people have used string diagram and category theory techniques to solve problems deep in the heart of mathematical physics. Some more discussion of these might be useful! I could have written a paper going into detail on these subjects… but I was more interested using this paper as an excuse to learn about new things: proof theory and lambda calculus.

Quoting:

In our rapid introduction to string diagrams, we have not had time to illustrate how these diagrams become a powerful tool for solving concrete problems. So, here are some starting points for further study:

  • Representations of Lie groups play a fundamental role in quantum physics, especially gauge field theory. Every Lie group has a compact symmetric monoidal category of finite-dimensional representations. In his book Group Theory, Cvitanovic develops detailed string diagram descriptions of these representation categories for the classical Lie groups SU(n)SU(n), SO(n)SO(n), SU(n)SU(n) and also the more exotic ‘exceptional’ Lie groups. His book also illustrates how this technology can be used to simplify difficult calculations in gauge field theory.
  • Quantum groups are a generalization of groups which show up in 2d and 3d physics. The big difference is that a quantum group has compact braided monoidal category of finite-dimensional representation. Kauffman’s Knots and Physics is an excellent introduction to how quantum groups show up in knot theory and physics; it is packed with string diagrams. For more details on quantum groups and braided monoidal categories, see the book by Kassel.
  • Kauffman and Lins have written a beautiful string diagram treatment of the category of representations of the simplest quantum group, SU q(2)SU_q(2). They also use it to construct some famous 3-manifold invariants associated to 3d and 4d topological quantum field theories: the Witten–Reshetikhin–Turaev, Turaev–Viro and Crane–Yetter invariants. In this example, string diagrams are often called ‘qq-deformed spin networks’ [see for example Smolin’s introduction ‘The future of spin networks’]. For generalizations to other quantum groups, see the more advanced texts by Turaev and by Bakalov and Kirillov. The key ingredient is a special class of compact braided monoidal categories called ‘modular tensor categories’.
  • Kock has written a nice introduction to 2d topological quantum field theories which uses diagrammatic methods to work with 2Cob2Cob.
  • Abramsky, Coecke and collaborators have developed string diagrams as a tool for understanding quantum computation. The easiest introduction is Coecke’s ‘Kindergarten quantum mechanics’.

The paper itself gives precise references to all these papers, and links to a lot of them.

Posted by: John Baez on September 3, 2008 7:37 PM | Permalink | Reply to this

Re: Substantially New Structures for Physics

As a well-meaning comment and not remotely a criticism, I’d say that an important point to get people interested in something is to present it as a work-in-progress that will lead to future interesting things, as that gives researchers the thought that learning about this stuff will lead to them producing new work. In that connection, talking too much about “clarity of thought” and “understanding” puts out the subliminal message that it’s primarily about dealing with things that people already know; I’d talk more about “enabling calculations” or “simplifying the development of blah”.

Mind you, I’m an (applied) computer scientist not a physicist and perhaps what qualifies as “attractive advertising” differs between fields.

Posted by: bane on September 6, 2008 3:26 AM | Permalink | Reply to this

Re: Substantially New Structures for Physics

Your comments about advertising probably apply to most physicists as well as computer scientists. So, Bob will do well to heed them.

Personally, what I enjoy most is understanding stuff. So when people glorify the manly, muscular virtue of ‘solving problems’, I start feeling rebellious and contrarian. I start wanting to say “Why be so proud of solving problems? If you understood stuff better, you wouldn’t have so many problems. It’s impressive how you can punch your way through brick walls… but isn’t it easier to use the door?”

But, I’m probably a bit weird.

Posted by: John Baez on September 6, 2008 10:25 PM | Permalink | Reply to this

Re: Substantially New Structures for Physics

The good thing is that improved understanding ultimately will lead (now close your ears for a second John) to `applications’. Following bane in his reference to computer science, a better understanding of comutational processes in terms of logics, calculi, and abstract semantics, has now led to the growing busyness of `program verification’, that is, a huge body tricks to make sure that a program truly obeys its specification. One does this by investgating that it `reasons in a logically correct manner’. If your program does well on the front of high-level information flows then it is very likely that it does so on all fronts. This is an essential development for maintaining the kind of progress in computer technology that we are used to, given the enormous complexity of the software we are using, and the fact that each piece of software is now connected to the rest of the world and needs to appropriately interact with it.

There is no way that these problems could be solved with the maths/logic available at the middle of the previous century. A whole body of maths/logic for concurrent behaviours had to be developed, and monoidal categories constitute the queen of that royal family. Since actual physical processes can be seen as complicated variants of the Boolean processes within a computer, involving far more variables, these structures are arguably the starting point of the formulation of any future physical theory.

Having said that, I must admit that I am also more the philosophy type guy John considers himself to be, who seeks understanding, …

Posted by: bob on September 7, 2008 2:24 PM | Permalink | Reply to this

Re: Substantially New Structures for Physics

I was actually making a much simpler point, which I’ll make in an even more vulgar way. The first thing I think about when deciding whether to read a new paper/book/etc – given that understanding a paper in depth takes time – is “is it likely reading this will lead to me producing new work that’s publishable?” The problem with work that strongly promotes itself only as “the way to properly understand X” is that it can give the impression that after reading it, I’ll understand X in as deep a way as the author but leaves me uncertain if it will it lead me to figure out new things (that I’ll hopefully be able to publish)? [That’s what I mean by “solving problems” although it could lead me to create more gadgets for understanding foundations.] Of course, if I didn’t have to keep producing papers to keep being employed then the issue would be less pressing and I’d be more interested in stuff that was more advertised around understanding stuff better.

I only mentioned this rather base viewpoint is because a lot of the people I know use the same priority rating for reading papers so it might be worth bearing in mind, particularly if the same mindset applies to physicists. (The one foolproof way to ensure no-one reads or cites a paper is to tackle a self-contained issue so completely that there’s clearly no more to be discovered in the area!)

Incidentally, when I studied applying these kind of models to programs at Bob’s august department over a decade ago it was very, very interesting stuff (which was clearly very much simpler stuff, eg, then-Professor Goguen’s course on OSA and program semantics using OBJ, than the much more sophisticated modern stuff). In the intervening years, despite occasionally trying to apply those techniques in my research, I haven’t actually used them “in anger”, let alone cited any of the papers. This isn’t intended to be pejorative – I think this stuff is great and really fascinating – but it’s not being used much yet (as far as I can see) outside of hardware verification and military systems.

Posted by: bane on September 24, 2008 8:00 AM | Permalink | Reply to this

Re: Substantially New Structures for Physics

bane wrote:
The one foolproof way to ensure no-one reads or cites a paper is to tackle a self-contained issue…

Reminds me of the referees response to my attempt to publish my thesis (Homotopy Associative H-spaces) that it was entirely self-contained and of no relevance to the rest of mathematics. We all know how prescient that opinion was.

Posted by: jim stasheff on September 25, 2008 1:45 PM | Permalink | Reply to this

Re: Substantially New Structures for Physics

Group theory is a good example for what categories are good for in physics.

Another good analogy is differential geometry:

all of physics can be - and has been for ages and still is in some corners – written up without ever mentioning manifolds, differential forms, exterior derivatives etc. All of these are tools to facilitate clear thinking.

Back then Maxwell wrote down his equations in components, filling a full page or so. Today we write half a dozen of symbols with the same information content. Does this have any gain, just repackaging information?

Posted by: Urs Schreiber on September 5, 2008 5:53 PM | Permalink | Reply to this

Re: Substantially New Structures for Physics

Group theory is indeed a compelling example, therefore we gave it a central place in our paper.

Even further back than Maxwell is of course the passage from Ptolemy’s epicycles, to the `new strucure of an ellipse’, essential for formulating Kepler’s axioms of planetary movement, and hence ultimately leading to Newtonean mechanics, giving rise to the constant of gravity.

Posted by: bob on September 7, 2008 2:31 PM | Permalink | Reply to this

Re: New Structures for Physics I

On page 10 of Coecke and Paquette’s paper it says:

“From this, we see that ‘groups’ provide an example of an abstract categorical structure. At the same time, all groups together, with structure preserving maps between them, constitute a concrete category. Still following?”

This can be tricky when you first meet it! I think the readers are more likely to follow if we don’t have the plural ‘groups’ attached to the singular ‘an abstract categorical structure’. I’d say this:

“From this, we see that any group is an example of an abstract categorical structure. At the same time, all groups together, with structure preserving maps between them, constitute a concrete category. Still following?”

(I’ll admit it: I also prefer to say something is an example than provides an example, unless I feel the reader is really starving for examples and I want to emphasize my intention to satisfy them by ‘providing’ one — as in “Here, have a cookie; dinnertime is coming soon!” But I try not to let the reader reach the point of starvation!)

Also: some readers may be nervous that you begin to talk about ‘abstract categorical structures’ without quite defining them. You say “abstract categorical structures arise by either endowing categories with more structure or by requiring them to satisfy certain properties.” But this is not quite a definition.

I suspect an “abstract categorical structure” is just a category that you take a certain special attitude towards — namely, you don’t bother equipping it with a forgetful functor to SetSet!

I’m reminded of the famous analyst Warren Ambrose, who explained to me why he didn’t go into physics. “In physics, they defined a ‘vector’ to be a ‘quantity with magnitude and direction’. But I didn’t know what a ‘quantity’ was, and I didn’t know what a ‘magnitude’ was, and I didn’t know what a ‘direction’ was. So I decided I couldn’t do physics.”

Posted by: John Baez on September 3, 2008 12:46 AM | Permalink | Reply to this

Re: New Structures for Physics I

On page 6 of Coecke and Paquette’s paper, they write:

Since the key structural data of a category is its composition, emphasis is given to the structure preserving maps rather than the structures themselves. Indeed, categorical structure neglects the structure of the objects themselves, which can be taken as a mere set of labels or types. Of course, for well-chosen notions of structure preservance, this ‘underlying’ structure is completely reflected within the compositional structure of the morphisms.

I know exactly what you mean. But as one of the professors here at UCR likes to say, “Trying telling that to the guys down at the gas station!”

To somebody just learning category theory, this passage is likely to be quite baffling. It might be enough to make them run away screaming.

One reason is that there’s no English word ‘preservance’. Well, sure, it appears on Google over 20,000 times, but if you look at the top hits you’ll see what I mean.

Another more important reason is that the word ‘structure’ appears over and over, with at least two different meanings (structure of a category, structure of an object in a category), neither of which has been defined — and both of which are rather tricky to define.

The phrase “Of course” may be the final death blow: it takes some expertise in category theory to see that of course, the structure of objects is completely known once you know the morphisms and how they compose. Indeed, getting this perspective usually takes quite a bit of struggle!

So, I think it’s better to make philosophical points like this after students have already understood them by examples.

For example, when I want to show kids that objects are ‘mere labels’, I point out that FdVectFdVect is equivalent to the category with objects 0,1,2,0,1,2,\dots and morphisms f:mnf : m \to n being n×mn \times m matrices. The once proud vector space 3\mathbb{R}^3 has been replaced by a mere label: 3. But everything works just as well! This can come as quite a shock.

But this shock can only occur after the student already understands ‘equivalence of categories’ — not just the definition, but how it works.

Posted by: John Baez on September 3, 2008 1:06 AM | Permalink | Reply to this

Re: New Structures for Physics I

Great comment, John. I have used another example to try to make students understand this point. I take a category whose objects are groups and whose morphisms are functions between the underlying sets of the objects. If you are within that category you cannot see that the objects might have some structure. In fact, within that category the objects do not really have any structure other than that detectable by using the morphisms. (It’s an observational viewpoint I suppose.) However this point usually takes some time to sink in.

Posted by: Tim Porter on September 3, 2008 11:57 AM | Permalink | Reply to this

Re: New Structures for Physics I

“… the word ‘structure’ appears over and over, with at least two different meanings (structure of a category, structure of an object in a category), neither of which has been defined – and both of which are rather tricky to define.”

Isn’t that (first time the student asks what you mean by structure) the time to explain Stuff, Structure, Properties?

Posted by: Jonathan Vos Post on September 3, 2008 4:32 PM | Permalink | Reply to this

Re: New Structures for Physics I

Firstly thanks for all comments/corrections so far, including those directly send to us by email.

We have to admit that we indeed sacrificed precise definitions on meta-concepts (structure, abstract vs concrete) with the excuse that physicists all the time rely on the intuition/experience of the reader cf John’s citation of Warren Ambrose.

But probably the word structure is indeed overused. The way we approached it was very much in the spirit of additional operations. The reason for relying as much as possible on structure rather than on properties is operationally motivated: we wanted the reader to think of structure as actual physical capabilities such as copying and deleting in the case of product structure, the ability to prepare Bell-states in the case of duals/compactness, etc.

The reason for emphasising the difference between abstract and concrete categories, a difference which again we introduced in a somewhat informal manner [but John is surely right about the fact that given that there are ways to formally define it we should have done it at a later stage in the text when the appropriate notions are available; which we will do; together with a pointer when we introduce it], is very much to distinguish between:

fully specified theory such as QM, SR, QFT, GR, …     concrete cat
—————————————————————————- = ——————
    family of theories with axiomatic freedom               abstract cat

I find it always so hard to talk to phycists about the latter; they just don’t seem to get their head around anything else than a fully specified model about life the universe and everything. Recently there has been lots of renewed activity on “variations on quantum theory” by the likes of Spekkens, Hardy, Barrett, Wilce, Barnum, Leifer, … which might help to get the message at least to the quantum information community. A slogan which I like to use in that context is: “categories as a canvas for quantum foundations”.

[Obviously this distinction between abstract and concrete is again not clear-cut. Computer scientists however precise a clear notion of abstract in terms of adjunctions, to pass from a fully specified programming language to a more abstract notion of computational process.]

The implicit manner in which we tried to make people used to the idea of abstract categorical structure is by emphasising that Dirac notation, which physicists use all the time, is nothing but abstract dagger symmetric monoidal structure. One of the writing strategies was to get to that point as quickly as possible, as well as to the teleportation example.

In particular, we assumed that our reader is NOT particularly motivated to learn category theory, but is just a bit curios what it is about, and also isn’t a great fan of abstract mathematics neither. Therefore we wanted to provide a big fat juicy worm and “hook him” [I guess this is not English] before he takes off.

There is also the to me very weird distinction physicists tend to make between maths and physics. When you talk differential equations, surely that’s physics. On the other hand, if any other mathematical structure than groups is involved [*symmetry* did make it into the physics camp] then they call it pure maths.

Posted by: bob on September 4, 2008 11:41 AM | Permalink | Reply to this

Re: New Structures for Physics I

Thanks for the post… I studied the course that Abramsky/Nikos’ paper is based on and it’s great (and no doubt useful!) to have a version in this prose format. Also I look forward to sending a link to other paper to some of my physicist friends and see what they think! It would be great for them to a) experience a new perspective and b) for them to understand the perspective used in the stuff I do (or rather, have done and will be doing.) I’ll let you know any comments they have.

Posted by: Martin Churchill on September 3, 2008 5:39 PM | Permalink | Reply to this
Read the post Reliability
Weblog: The n-Category Café
Excerpt: A response to an article on the unreliability of the mathematical literature.
Tracked: September 10, 2008 3:27 AM

Re: New Structures for Physics I

I recently presented (part of) the content of the tutorial Categories for the practising physicist to a large audience consisting mostly of physicists. Here are the slides. It seems to have gone well in with the audience, witnessed by the fact that not only theoreticians, but also several experimentalists asked me for a copy of the actual tutorial. It’s a funny thing which I already noticed before that experimentalists are more attracted by the categorical spirit. Francesco DeMartini, a renowned experimentalist from Rome, even opened is own talk stating that this new way of thinking should replace the reductionists/atomist philosophy that has ruled the previous century.

What went in particularly well was cooking as an example of monoidal categories. I have talked to some of these people before on categories of physical processes (e.g. these lecture notes), but it seems better to convey the idea with, initially, no reference to physics at all! Otherwise the physicists immediately start to think in terms of his favourite theory of physics and also wants to see all of its details.

The event itself was interesting since it had this wide range of people on the same stage, from experimentalists to (applied) category theoreticians, all listening to each other. In a way most of the critique to my presentation was coming from some of those with category theoretic background, in that I didn’t point at all to the use of limits etc, and they were somewhat surprised by the fact that one can get excited by category theory pure in terms of a modelling device. It was mentioned that this shouldn’t even be called category theory. I am a bit baffled by this attitude.

Posted by: bob on October 2, 2008 12:53 PM | Permalink | Reply to this

Re: New Structures for Physics I

Its a funny thing which I already noticed before that experimentalists are more attracted by the categorical spirit.

In my experience that’s because they don’t see Category Theory as something horribly technical that they might need to master, but appreciate it directly for its relational quality.

Posted by: Kea on October 3, 2008 12:13 AM | Permalink | Reply to this

Re: New Structures for Physics I

A version of ‘Categories for the Practising Physicist’ is now on the arXiv.

Posted by: David Corfield on May 22, 2009 12:05 PM | Permalink | Reply to this

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