Isham new quantization

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Do you know what a category is?

  • I can quote the definition from memory

    Votes: 7 87.5%
  • Hmm I heard of them.

    Votes: 0 0.0%
  • No.

    Votes: 1 12.5%
  • Sort of philosophical concept?

    Votes: 0 0.0%

  • Total voters
    8
  • #1
arivero
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Readers of gr-qc could have lost the preprint from Isham at quant-ph this January, Quantising on a category
http://arXiv.org/abs/quant-ph/0401175

I guess it is related to his series A New Approach to Quantising Space-Time ( gr-qc/0303060 gr-qc/0304077 gr-qc/0306064 ) which is so abstruse that nobody has quoted it.

Mutters about using Topos theory to describe quantization functors has being going around for a time, as well as other collateral links: Doplicher took some attemps to use category theory in a practical way, and even Moerdijk (coauthor of one textbook on sheaf theory) did some feints to Connes noncommutativity, in the context of foliations, of course. On the mathematical side, I have heard nothing about using topoi for foundational purposes since the homonimous australian book (in the seventies? fron Goldblatt?).
 
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  • #2
marcus
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Alejandro,

John Baez was talking about deriving quantum theory from
category theory----it was on SPR only a week or so ago I think.
Do you have a link to what he says or was referring to and is
it the same as what Isham is doing. I hope it is a lot more
accessible than Isham.

Is there actually some prospect that people will eventually understand why quantum theories are constructed in the customary
fashion----it seems that for many people the answer still is
"because it is customary".
 
  • #3
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a category is a collection of objects along morphisms between the objects such that the morphisms satisfy associativity (with the appropriate domains and codomains) and for every object there is an identity morphism.

it is possible to forget the objects and work only with the morphisms since identity morphisms are in one to one correspondence with objects.

a morphism is called an isomorphism if there is a left and right inverse (with appropriate domains and codomains).

a functor is a linear map between categories, it takes objects to objects and morphisms between objects to morphisms between the images of objects.
 
  • #4
marcus
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Categories make sense

here is a Stanford U link to supplement what Lethe says

http://plato.stanford.edu/entries/category-theory/


If quantum theory and the procedures customarily used to quantize
classical theories
could be made sensible in a Category Theory context
that would be great
the whole thing would become more clear and make better sense
IMHO
 
  • #5
arivero
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Thanks Marcus, I missed it, it is at
http://math.ucr.edu/home/baez/week200.html
and, just as Isham's, it has not got any followups :-(

I'd add two references to the current list from Baez. Both have been mentioned from time from time in spr and I alluded to them above.

Goldblatt. Topoi: The Categorial Analysis of Logic - is about using Topoi as a new foundation for mathematics. Note the following comment from Baez time ago at spr:

The problem is, there's a lot more to category theory than working within a single given category: there are functors between categories, and natural transformations between those! It's just like group theory: you'd be insane to talk about groups but not homomorphisms. Goldblatt is weak on this stuff, and especially on the all-important concept of adjoint functors. For this reason, serious toposophers disdain his text. So you gotta dig deeper... but it's a good place to start.

S. MacLane & I. Moerdijk Sheaves in Geometry and Logi. It is the practical (well, sort of-) side of the theory.

Note that one of the popular uses of the theory (Moerdijk, Renault...) is to classify arbitrary foliations, and that the same kind of things can be done by using the Connes's C* algebras of groupoids. So some surprising links could happen.
 
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  • #6
selfAdjoint
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Relating categories to another topic we've been discussing, here's a http://arxiv.org/PS_cache/math/pdf/0402/0402150.pdf [Broken].
 
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  • #7
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Arivero,

do you know Prof. Van Oystaeyen (Univ. Antwerp)? , he is in topos and together with Alain Connes in noncommutatives.
 
  • #8
marcus
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Originally posted by selfAdjoint
categorization of the Gelfand-Naimark-Segal construction.
Whoah! could be way cool!

http://arxiv.org/math/0402150 [Broken]

new. and from UC Riverside Baez hometown

(Mozart echo: variations on a theme of Haydn)
 
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  • #9
arivero
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Originally posted by pelastration
do you know Prof. Van Oystaeyen (Univ. Antwerp)? , he is in topos and together with Alain Connes in noncommutatives.
First news I have of it! Can you comment on it, or at least quote some reference?
 
  • #10
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Originally posted by arivero
First news I have of it! Can you comment on it, or at least quote some reference?
He gives NCG at Antwerp University (http://www.esf.org/esf_article.php?activity=1&article=19&domain=1&page=791 [Broken])
Works also with Connes: http://ercom.mis.mpg.de/cgi-bin/ercom_conferences.pl [Broken]

He told me this week that he made some breakthroughs in topos.
 
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  • #11
arivero
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beyond groupoid

I have taken a first contact with Isham papers. He goes beyond Connes cuantization of groupoids because he uses a new concept, sort of "arrow field" over the objects of category. Interesting; it is a bold decision, because it breaks the symmetry between domain and range, giving preference to one of them.

It is the same that deciding between the backward or the forward derivative. BTW, I am becoming interested on osmotic derivatives...
 
  • #12
arivero
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week 202 too

Amazingly, week 202 of Baez,
http://math.ucr.edu/home/baez/week202.html
besides continuing on the relationship between categories and quantums, does a feint to Galois Theory, which we invoked last week in out threads about dimensional analysis and space dimensionality. Even more amazingly, week 201 does a complete attack upon it.

(Sorry the fencing lingo, I am just back from a tournment this Saturday ;-)
 
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  • #13
jeff
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hi arivero,

I was wondering if you'd mind discussing the appeal this sort of math-centric research has for you in relation to physics and how you choose what to work on?
 
  • #14
marcus
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there was an enlightening discussion on SPR some months ago
about why categories
maybe we can find a link back to this lively discussion
for anyone who is curious

the question is to Alejandro, but if I may be
allowed to add my own opinion,
I suppose there was a time when
Platonic Solids and Conic Sections
seemed like pure mathematics and very "mathematical"
as opposed to "physical" (a shaky distinction at best!)
But then in 1618 Kepler explained the solar system
in terms of them.

And he happened to be right about the Conic Sections
and wrong about the Platonic Solids, and so it goes.

And there was a time, no doubt, when Group Theory
seemed very "mathematical" as opposed to "physical".

Baez had some crisp remarks about this IIRC which might
even be good to recall and copy here.

At certain historical periods in the growth of physics,
it turns out that the creative physicists (who actually
took care of business and made the necessary advances) were
in fact mathematicians

IIRC Baez mentioned Newton, Laplace, Maxwell maybe others

cant remember the details but it was a fascinating discussion
on SPR and I came away with the impression that
the artificial division was a sterile and sterilizing distinction.

that is, having physical intuition (a gut feel for real physics) and a sense of mathematical beauty are probably so intimately connected that they are the same thing at root

maybe we should start a new thread discussing the connection of mathematics and physics, and the interest of today's physicists in Category Theory.

A new thread, because this thread is what Alejandro started about "Isham's new quantization"

and that is an interesting thing I'd like to hear more about
and not get sidetracked parsing general issues like what is math and what is phys.
 
  • #15
jeff
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Originally posted by marcus
...maybe we should start a new thread discussing the connection of mathematics and physics...
...or a new forum. It's difficult to gain helpful perspectives on the significance of the highly speculative and mathematically abstruse ideas residing on the margins of research without a firm grounding in the basics of mainstream physics including the requisite math. Anyway, you guys do tend to seriously misjudge those papers you're always endorsing with so much enthusiasm.

Originally posted by marcus
...the interest of today's physicists in Category Theory...
...is virtually nill and rightly so.


From the mary tyler moore show:

Ted: Hey Lou, I'm teaching myself poker so I can play with you guys.

Lou: Oh, and what have you learned?

Ted: Well, four of a kind beats three of a kind, and three of a kind beats two of a kind...uhm, Lou?

Lou: Yes Ted.

Ted: What's a kind?
 
  • #16
arivero
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Jeff, note the poll in the thread. It implies at least that the inhabitants of this subforums are aware of these abstruse mathematics.

As four your first question, I would say that I endorse such mathematics only if I see a way of contact to physics. Because of that, for instance, I have been personally more interested on groupoids that in general categories.

A clear geometric link gives a lot of help, too.

On the other hand, topoi theory has the intrinsic attractive of foundations.
 

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