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September 10, 2008

New Structures for Physics II

Posted by John Baez

guest post by Bob Coecke

This is the continuation of John’s idea for having a public review here at the café of some chapters for the New Structures for Physics volume(s) which I am editing.

So far we’ve been discussing Baez and Stay part I, Baez and Stay part II on categories, topology, logic and computation, Abramsky and Tzevelekos’ tutorial lecture notes on categorical logic, and Coecke and Paquette’s categories for the practicing physicist who is not so interested in category theory.

We continue the theme ‘physics meets computation’ with:

I am very much looking forward to your comments on these ones!

Posted at September 10, 2008 4:19 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1790

6 Comments & 1 Trackback

Re: New Structures for Physics II

I might have missed the point where this project of public reviews gained its momentum, but wouldn’t it be helpful to actually start each entry with a (ever so brief) review of the articles being linked to?

Unless I am missing somehting, currently these entries just say: “hey, did you know that there exist four articles with the following titles?”

The standard reaction to that is: no, I didn’t, and I am too busy to read them unless you give me some good reason why I should be interested. Such as a brief summary of the coolest parts to whet my appetite.

What do you think?

Posted by: Urs Schreiber on September 10, 2008 9:10 PM | Permalink | Reply to this

Re: New Structures for Physics II

Right. Given that these papers are tutorials the titles sounded fairly self-explanatory to me, overlooking the fact that not everyone here is familiar with computer science semantics and quantum computational models. I must say so far this idea of public reviews has proven to be very successful: besides the comments posted here at the cafe there are also those directly emailed to us, which sometimes have been really extensive.

Having said that, here are the blurbs:

Domains, an order-theoretic structure [hence deserving its place here at the cafe], were Dana Scott’s brainchild in his search for a denotational semantics for the Lambda calculus, that is, a calculus of functions which also allows for recursion. Key results of Domain theory include fixed point theorems. One can think of domains as a novel way of doing topology. In particular, they provide topology with an operational interpretation, and also with a corresponding logic, as exposed by Samson Abramsky’s generalisation of Stone duality to Domains. Steve Vickers’ book is the place to read about this.

But it was realised that Domains also provide a powerful framework to study analysis, with work by Keye Martin, Abbas Edalat and Martin Escardo as the most notable examples. Martin’s notion of ‘measurement’ is key to these developments. This aspect of Domains is very present in Martin’s tutorial.

More recently, Keye Martin has been exploring other possible applications of Domain theory, in (quantum) information theory, thermodynamics, and most notably, space-time structure. The joint chapter with Prakash Panangaden provides details on this. A punchline of that chapter is that Dana Scott’s notations for Domain theory and Penrose’s axiomatisation of space-time structure perfectly coincide, and structurally they are basically talking about the same thing. At the time their respective theories were developed they were both at the Mathematical Institute here in oxford - no coincidence it seems.

Here in Oxford, Domain theory is a course on the curriculum for computer scientists and mathematicians, although now that Samson Abramsky and I both have special fellowships which prohibit us from teaching –isn’t that sad– it has been harder to maintain the course.

Here’s the wikipedia entry on domain theory .

Anyonic quantum quantum computing, or topological quantum computing as it also known, is a model of quantum computation, in which unitary gates are encoded in terms of low-dimensional topological properties of particles. More than any model of quantum computing it requires a categorical description, and this is not even controversial anymore. If a quantum computer of this kind would ever been build then this would be a ***killer application*** for category theory.

Here’s the wikipedia entry on topological quantum computing .

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

Re: New Structures for Physics II

Thanks!

If a quantum computer of this kind would ever been build then this would be a killer application for category theory.

Yes, even an >applied application.

More than any model of quantum computing it requires a categorical description, and this is not even controversial anymore.

On the more non-applied side of things it is slowly becoming non-controversial that topological QFT even requires higher categorical descriptions.

What about higher categorical language in quantum computing? Is it beginning to be used? Is it conceivable that it might even be eventually become uncontroversial that it is required here?

Posted by: Urs Schreiber on September 11, 2008 1:47 PM | Permalink | Reply to this

Re: New Structures for Physics II

What about higher categorical language in quantum computing? Is it beginning to be used?

Not that I’ve seen.

I’m trying to get a paper published illustrating the way that

1) topological quantum algorithms are representations of the category of tangles.
2) we can approach such representations by categorifying knot invariants, extending the result to tangles, and decategorifying

and using coloring numbers as a motivating example.

Quantum topology people say it’s interesting to see it worked out. Quantum computer engineers even like it. But the referee is evidently a knot theorist and keeps insisting that despite the innovations in the approach it’s not new if there’s nothing new once you decategorify. Viewpoint is worth nothing.

Posted by: John Armstrong on September 11, 2008 3:20 PM | Permalink | Reply to this

Re: New Structures for Physics II

if there’s nothing new once you decategorify. Viewpoint is worth nothing.

The higher categorical description of QFTs is supposed to reproduce the standard description on cobordisms without boundary and but capture more information in the more general case. This thus typically leads to a refinement of topological invariants.

The standard example, as you know, 2d TFT on closed cobordisms being the same as a commutative frobenius algebra, while the extended 2-categorical version is a commutative Frobenius algeba together with extra stuff and property.

can you obtain from your aproach new structures of this kind? Some that maybe your referee did not know of before?

Posted by: Urs Schreiber on September 11, 2008 4:09 PM | Permalink | Reply to this

Re: New Structures for Physics II

can you obtain from your aproach new structures of this kind?

I’m sure one can do so with other more complicated invariants. The paper is for the proceedings of a conference centered on quantum information theory, and is intended as exposition – a fully worked-out example – to present this categorical viewpoint to a more general Q-Comp audience. Unfortunately, there’s conflicting reports as to whether that’s appropriate for a conference proceedings (though the organizers thought it was appropriate for a conference!).

Posted by: John Armstrong on September 11, 2008 4:44 PM | Permalink | Reply to this
Read the post New Structures for Physics III
Weblog: The n-Category Café
Excerpt: Mike Stay and I have finished what we hope is the final version of our paper for Bob Coecke's book on New Structures for Physics. Peter Selinger's paper for this book is also done.
Tracked: February 27, 2009 5:04 AM

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