
#91
Jul2706, 07:49 AM

Emeritus
PF Gold
P: 8,147

The perennial question about categories is not "Can we do this theory in categories?" but "Can categories give us answers to these questions that we couldn't get without categories?" (Never mind "easier"; that's in the eye of the beholder. Some people get off on down and dirty hard analysis; look at Hardy; his hobby was simplifying awful complicated integrals). 



#92
Jul2706, 12:49 PM

Astronomy
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PF Gold
P: 22,808

it is a kind of "smart pills" (as in the expression "now you're taking smart pills") and using categorics seems to make some people frazzled or even wacky and some more creative. the result is not always good, but sometimes is. In this regard I am only interested in research say since 2002 because only lately did I see it impinge on physics (in ways that are explicit and make sense to my limited perception). Maybe all categorics was useless to physics before thatI don't know about that. but now I am beginning to see a correlation. the hidebound rejectionist attitude may be correlated with mediocrity and lack of inspiration. and some sense of "higher algebra" (whether categoric or some other) seems correlated to promising new physics ideas. I am waiting to seemy attitude is "by their fruits ye shall know them". We will just see if the people who come up with the necessary new ideas are the people who are taking smart pills, or the others. Probably trigonometry was not necessary. Hipparchus invented it around 140 BC roughly, and it was convenient but you PROBABLY COULD DO EVERYTHING just using geometry. nevertheless he made trig tables. Probably Cartesian graph paper was not necessary. You probably could do everything with elaborate geometric constructions and not using plotted formulas. Probably some hidebound rejectionists were scoffing this. But Descartes went ahead and promoted his coordinate methods. YOU CAN ALWAYS DO EVERYTHING THE OLD WAY. the question is whether the people using the new way appear to be more clever and do they invent the necessary things. and the question is do the people who reject the new way, do they seem mediocre and uninventive. Or is it different? I can only learn by watching the outcomes. 



#93
Jul2706, 01:16 PM

Astronomy
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PF Gold
P: 22,808

If they include a word or two of categorics then you win. If they have no explicit mention of categorics then I win. this is only if the paper comes out this year. if the problem proves unexpectedly intractible and they get stalled, bets are off. I wish someone would speculate what the Baratin Freidel 4D case will look like. I can see how they construct a flat Feynman spinfoam in 3D spacetime. It is just a PARTITION FUNCTION that somehow remembers that it is supposed to dwell in 3D even without a surrounding 3D spacetime to remind it. Like one of those shaperemembering pieces of metal, that go *boink* and flip back to their imprintment. Formally it is all seemingly straightforward, the trick is to get the right partition function. but spinfoams in 3D are regarded as somewhat rudimentary. maybe in the 4D case the partition function will be similar but just a bit gruesome. Is that all, do you think? Will everything look like the 3D case except messier? I think I could stand that, at least if I had a chocolate malted milkshake to steady my nerves. 



#94
Jul2706, 02:58 PM

P: 1,667

***
but now I am beginning to see a correlation. the hidebound rejectionist attitude may be correlated with mediocrity and lack of inspiration. and some sense of "higher algebra" (whether categoric or some other) seems correlated to promising new physics ideas. I am waiting to seemy attitude is "by their fruits ye shall know them". We will just see if the people who come up with the necessary new ideas are the people who are taking smart pills, or the others. *** So you say : (a) 99,8 percent of physicists is unimaginative and more mediocre than category theorists (b) you have to know category theory in order to be smarter Moreover, there is only a hidebound rejectionist attitude when a large community accepts the use of the subject under consideration. **** Probably Cartesian graph paper was not necessary. You probably could do everything with elaborate geometric constructions and not using plotted formulas. Probably some hidebound rejectionists were scoffing this. But Descartes went ahead and promoted his coordinate methods. YOU CAN ALWAYS DO EVERYTHING THE OLD WAY. the question is whether the people using the new way appear to be more clever and do they invent the necessary things. and the question is do the people who reject the new way, do they seem mediocre and uninventive. Or is it different? I can only learn by watching the outcomes *** This shows that you do not understand history. The method of Descartes was immediately recognized, just as special relativity, Maxwell theory and so on... . Instead of making erroneous political manifests you could contribute by explaining the useful, new insights for physics, as I asked you before. Careful 



#95
Jul2706, 06:04 PM

P: 859





#96
Jul2706, 11:50 PM

Astronomy
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PF Gold
P: 22,808

My point was that historically there were some holdouts to the method of Descartes. Indeed there were lots! Newton for example. Descartes explained his coordinates in 1637 (Discourse on Method, Geometry) and Newton's Principia appeared in 1687. Fifty years later. You can see him strictly avoiding Cartesian method. The example of Newton suggests that Cartesian coordinates WERE NOT FASHIONABLE at least in some circles even 50 years after exposition. Here http://members.tripod.com/~gravitee/booki2.htm you can see Newton using Euclidean method to discuss circular motion in a plane, where we would today normally use Cartesian coordinates. To make my point (the analogy with category theory) I only need to know that there were SOME holdouts Newton of the Principia Book I was hardly the sole holdout, Greek style plane geometry still has class (it is classic after all). What I am trying to say with this example, about categorics, is that one should not look for something that you CAN'T DO without the new method. There will often be some way to kludge around and make do, and that doesnt prove anything. It can even be a matter of taste. What one should look for is cases where someone GETS DIFFERENT IDEAS by solving the same problem by way of a different conceptual framework. If anyone wants to see more of Newton Principia http://members.tripod.com/~gravitee/toc.htm 



#97
Jul2806, 01:05 AM

P: 859

Yes, classic blooper. 



#98
Jul2806, 01:07 AM

Astronomy
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PF Gold
P: 22,808

Let me think about it. It probably wouldn't be a limerick. most likely a doggerel quatrain. But I'm the one who is going to win! Can you write justatouch disrespectful light verse? ================ I wrote this next when out of sorts, before I saw your post: I think we should just avoid or ignore complaining about category theory in this thread. People should use it if it gives them good ideas and inspires them to solve problems. And NOT use it if it DOESN'T. People who don't get any good from it should simply not bother. After a point, more talking to them will not help them. In some way it seems silly to argue about the Goods and Bads of some (to an extent optional) mathematical method or framework, with someone with a mindset unsuited to it. ============== From my viewpoint, Baez has already made abundantly clear to me as observer that it is a great source of new ways to look at things and that it is coming in to physics. Also Urs Schreiber is a bellweather in this respect. So I will be sure to keep my eye out for things happening with categorics and physics. I am also glad to see new stuff come out that does NOT use category theory. Whatever floats the researcher's boat. So I will do what I can to ignore arguing about the merit of categorics, or lack thereof, and hope I succeed. 



#99
Jul2806, 03:11 AM

P: 1,667

**I think you meant classic blooper. Hope you did anyway.
My point was that historically there were some holdouts to the method of Descartes. Indeed there were lots! Newton for example.** This example is not even a counterexample to what I said. The method of descartes was for sure accepted by more than 0,2 percent of scientists. By the way Marcus, for someone with a nonexpert opinion, you often refer to the notion of wrong/right mindset and to what is hopeful/sufficient evidence for something. Careful 



#100
Jul2806, 08:06 PM

P: 169

Such models aren't what I'm talking about. I'm talking about how the category of Hilbert spaces (Hilb) and the category of ndimensional cobordisms (nCob) are both monoidal categories with duals. The fact that Hilb has duals allows for quantum teleportation; the fact that nCob also has them is what allows you to straighten out a kink in a rope (in the case n = 1). This is simply a fact, not a "model"  and certainly not a model where quantum entangled particles are connected in some way, e.g. by an "invisible rope". Quantum entanglement arises from the fact that Hilb is noncartesian, unlike the category of sets. nCob is also noncartesian. (For a monoidal category to have duals, it must be noncartesian, but not vice versa. Or, in physics speak: we need entangled states to carry out quantum teleportation, but we also need more. All this is nicely explained in Bob Coecke's paper on Kindergarten quantum mechanics.) While these are just mathematical facts, they point the way towards models of quantum gravity, by showing us which class of mathematical structures combine the physically important features of general relativity and quantum mechanics. But, we need to take another step or two  and probably many more we haven't seen yet. For starters, nCob is better thought of as a monoidal ncategory with duals. This describes all the ways we can stick together small pieces of ndimensional spacetime; it captures the ndimensionality of spacetime in a way that a mere category can't do. This suggests trying to define "nHilb"  an ncategory of "nHilbert spaces"  and showing it's a monoidal ncategory with duals. I did this for 2Hilb a while ago, and it turns out to be quite interesting. In particular: just as Hilb gives rise to Feynman diagrams, 2Hilb gives rise to "spin foams"  a 2dimensional generalization of Feynman diagrams. If we went to nHilb for higher n, we'd get still higherdimensiaonal diagrams. I've never emphasized this aspect in my papers on spin foams, since I know most physicists don't like higher categories. But, I explain how it works in weeks 13 of the winter 2005 notes from my quantum gravity seminar. A lot of work has been done on spin foam models by now, but they're still mysterious. For example, we've all heard a lot about the BarrettCrane model, but it's still unclear why Simone Speziale and Dan Christensen are getting really good agreement with the graviton propagator based on calculations involving a single big 4simplex, refinements of Carlo Rovelli's original calculation. They made a lot of progress on this last week: Dan's supercomputer calculations match what Simone is getting analytically. But why should these calculations work at all  after all, if any model like this is right, you'd expect spacetime to be made of lots of small 4simplexes. Viqar Husain has some ideas.... And then there's the CraneSheppeard model. This explicitly uses infinitedimensional 2Hilbert spaces, namely representations of the Poincare 2group. But what does it mean, physically? Is it related to Baratin and Freidel's spin foam model for ordinary quantum field theory on Minkowski spacetime? I guessed it was... but my students Jeff Morton and Derek Wise have been doing a bunch of calculations with Baratin and Freidel, and they seem to be concluding that it's not. However, they found the CraneSheppeard model includes the BarrettCrane model in a certain sneaky way. And, perhaps the best part is: Freidel now understands 2Hilbert spaces and 2groups, and he wants to keep studying models based on them! 



#101
Jul2806, 08:38 PM

Astronomy
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PF Gold
P: 22,808

I'll fetch some related links, in case something turns up or anyone is interested. I saw some recent papers by Speziale and also by Christensen and friends. UWO must be a good place to do computational quantum gravity, which ought to become important.
http://arxiv.org/abs/grqc/0512004 http://arxiv.org/abs/grqc/0509080 http://arxiv.org/abs/grqc/0508088 ========== Here are some Speziale links 1. grqc/0606074 A semiclassical tetrahedron Carlo Rovelli, Simone Speziale 10 pages 2. grqc/0605123 Towards the graviton from spinfoams: higher order corrections in the 3d toy model Etera R. Livine, Simone Speziale, Joshua L. Willis 24 pages, many figures 3. grqc/0604044 Graviton propagator in loop quantum gravity Eugenio Bianchi, Leonardo Modesto, Carlo Rovelli, Simone Speziale 41 pages, 6 figures 4. grqc/0512102 Towards the graviton from spinfoams: the 3d toy model Simone Speziale 8 pages, 2 figures Journalref: JHEP 0605 (2006) 039 5. grqc/0508106 On the perturbative expansion of a quantum field theory around a topological sector Authors: Carlo Rovelli, Simone Speziale 7 pages 6. grqc/0508007 From 3geometry transition amplitudes to graviton states Authors: Federico Mattei, Carlo Rovelli (CPT), Simone Speziale, Massimo Testa 18 pages Journalref: Nucl.Phys. B739 (2006) 234253 Here is another interesting thing that turned up: Don't let me get in the way if someone wants to be reply to the general sense of JB's post, I am just assembling some detail to think about in that connection. 



#102
Jul2906, 01:50 AM

P: 102

Just an small sugestion.
In my opinion the programing analogue of cathegory theory would be UML (uniform modelling language). It is fine to plot diagrams and clarify flow of information. But you can do everything just implementing the apropiate class. Returngin to the maintopic, i have just made a first (and complete) reading of arXiv:grqc/0607014. To say it easy, I had readed the talks in other thread about desitter but the by far the part wich i understand less is the origin of the point lagrangian that they present in eq 3.1. I mean, it is basically the lagrangean of a classicla point particle carriying somehow information about so(3,1) álgebra or something similar? Wich sense makes that? I see that they later show that it describes a particles with "all that good behaviours" but even so i don´t see clear that lagrangian (yeah, sure it is my fault). And later, in chapter 5 when it makes a wilson loop with the exponential of that lagrangian, i simply don´t see the relation with the Feyman amplitudes. Maybe i need to read some previous papers? perhaps the ones aobut hidden quantum gravity in 3d Feyman diagrams? 



#103
Jul2906, 06:43 AM

P: 1,667

***
what I'm talking about. I'm talking about how the category of Hilbert spaces (Hilb) and the category of ndimensional cobordisms (nCob) are both monoidal categories with duals. The fact that Hilb has duals allows for quantum teleportation; the fact that nCob also has them is what allows you to straighten out a kink in a rope (in the case n = 1). *** Yes, and as I said, I acknowledge that  I merely was commenting on how serious we should take these analogies (which I deduced for myself a few years ago while thinking about topology change). *** This is simply a fact, not a "model"  and certainly not a model where quantum entangled particles are connected in some way, e.g. by an "invisible rope". Quantum entanglement arises from the fact that Hilb is noncartesian, unlike the category of sets. nCob is also noncartesian. ** Sure, but what is the spacetime interpretation you have in mind ?! I tried at least to offer some ways of looking at entanglement (or quantum teleportation) which would pave the road for such abstraction. I never claimed that your ``absolute truth'' was limited to those viewpoints, but said that it could be found by thinking in these ways (remember : abstraction of specifications of spin foam models). So, I have put myself in the weak position, not you; you have limited yourself to repeating the abstract results and their universality (as well as which of the latter pictures you do not have in mind) while I was interested in getting out the physics. I do not know what is worse, to keep on stressing the abstract results or the failure to recognize that the other party is begging for a specific way of looking at it (for example in terms of wormholes). *** While these are just mathematical facts, they point the way towards models of quantum gravity, by showing us which class of mathematical structures combine the physically important features of general relativity and quantum mechanics. *** I disagree here and I explained several times why  it is kind of silly you keep on repeating this without adressing these points  shows there was never an online discussion to begin with. I asked you how you would solve the problem of time while keeping measurement as it stands and still have a reasonable theory to end with (that is one which makes solid predictions). The reason why I ask you is that I came to the conclusion that doing so will require a profound change of the dynamics of measurement in QM (as some people in MIT try to realize). You could of course say : ``I limit myself to taking expectation values'' but then I would not see why one would not be pleased with merely imposing a sufficiently high UV cutoff (as well as a macroscopic nonlocality scale) in perturbative QG. In the beginning you said you are not doing physics anymore, so why not tell what problems you see in it ?? Careful 



#104
Jul2906, 08:06 PM

P: 859





#105
Jul3006, 06:29 AM

P: 1,667

**With you, there was never a discussion. Quite true.**
There has never been any discussion between category theorists and anyone here , since basically I am the only nonsilent ``opposition''. *** We are not keeping measurement as it stands, as I have said many times. Neither are we repeating in full detail here what has been said in many good papers, some of which I suggest you start reading. **** I am not asking for any detail, I want a rough physical picture ! Explain in words what the mechanism is you have in mind for dealing with entanglement (and please do not refer me to the aforementioned papers, since these do not contain any such thing  there is only abstraction) and other crucial issues ... ** Correct. ** So, please make a sketch of the mechanism *you* have in mind, then try to tell us why categories would come in handy here; not in the reverse order ! Roughly speaking, if you would not change measurement too drastically (no realism), then I guess you have to end up with something like an improved GRW, a scheme a la Penrose... (there are other possibilities too but those jump immediately to my mind). Anyway, something wich would make the total dynamics nonlinear. I think it would be good for many people here if you would start by doing that : put some of your cards on the table  let us talk about physics first and suggest then how your categories come out in a *deeper* way, just as this happened in the de Rahm theorem . Referring to papers is not the way to go, especially when no response to criticism comes out. Careful 



#106
Aug306, 09:21 PM

Astronomy
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PF Gold
P: 22,808

Probably a bunch of us have had a look at Baez paper QUANTUM QUANDARIES, which introduces the notion of a *category.
nCob and Hilb are both categories of this sort. today Robert Coecke posted a paper developing similar themes. It uses different terminology and a somewhat more restrictive definition. the notion of a "daggercompact" category I still have to find how to type a dagger. [tries various things] It looks like it is OPTION TEE! OK Coecke, I mean Okey Dokey. this paper will probably turn out to be used and cited some in the process of categories permeating physics through something Coecke calls CATEGORICAL SEMANTICS. So I had better post the abstract. BTW Coecke's reference [4] is Baez Quantum Quandaries. and his reference [30] is THE DISENCHANTMENT OF JOHN VON NEUMANN WITH HILBERT SPACES. von Neumann became dubious of Hilbert spaces and declared they were not where it's at. (that is, not where Quantum Mechanics is at) 



#107
Aug306, 09:32 PM

Astronomy
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PF Gold
P: 22,808

Here is the Coecke et al new paper:
http://arxiv.org/abs/quantph/0608035 Quantum measurements without sums Bob Coecke, Dusko Pavlovic 36 pages and 46 pictures; earlier version circulated since November 2005 with as title 'Quantum Measurements as Coalgebras''. Invited paper to appear in: The Mathematics of Quantum Computation and Technology; Chen, Kauffman and Lomonaco (eds.); Taylor and Francis sample exerpt from page 2 of the article Ever since John von Neumann denounced, back in 1935 [30], his own foundation of quantum mechanics in terms of Hilbert spaces, there has been an ongoing search for a highlevel, fully abstract formalism of quantum mechanics. With the emergence of quantum information technology, this quest became more important than ever. The lowlevel matrix manipulations in quantum informatics are akin to machine programming with bit strings from the early days of computing, which are of course inadequate. 1 ... ... A recent research thread, initiated by Abramsky and the first author [2], aims at recasting the quantum mechanical formalism in categorical terms. The upshot of categorical semantics is that it displays concepts in a compositional and typed framework. In the case of quantum mechanics, it uncovers the quantum informationflows [6] which are hidden in the usual formalism. Moreover, while the investigations of quantum structures have so far been predominantly academic, categorical semantics open an alley towards a practical, lowoverhead tool for the design and analysis of quantum informatic protocols, versatile enough to capture both quantitative and qualitative aspects of quantum information [2, 7, 10, 13, 31]. In fact, some otherwise complicated quantum informatic protocols become trivial exercises in this framework [8]. On the other hand, compared with the ordertheoretic framework for quantum mechanics in terms of Birkhoffvon Neumann’s quantum logic [29], this categorical setting comes with logical derivations, topologically embodied into something as simple as “yanking a rope” 2. Moreover, in terms of deductive machanism, it turns out to be some kind of “superlogic” as compared to the Birkhoffvon Neumann “nonlogic”. endquote Baez was talking about stretching out a piece of wet spaghetti. curious propositions in quantum theory, seeming paradoxes, become trivial exercises as Coecke says. Baez was trying to get that idea acrossbasically one of the reasons why one might see categorical semantics filter into physics. Reference [30] in the above exerpt is: "[30] Rédei, M. (1997) Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead). Studies in History and Philosophy of Modern Physics 27, 493–510. " Here is the abstract: "Sums play a prominent role in the formalisms of quantum mechanics, be it for mixing and superposing states, or for composing state spaces. Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics can be done without direct sums, expressed entirely in terms of the tensor product. The corresponding axioms define classical spaces as objects that allow copying and deleting data. Indeed, the information exchange between the quantum and the classical worlds is essentially determined by their distinct capabilities to copy and delete data. The sums turn out to be an implicit implementation of this capabilities. Realizing it through explicit axioms not only dispenses with the unnecessary structural baggage, but also allows a simple and intuitive graphical calculus. In categorytheoretic terms, classical data types are daggercompact Frobenius algebras, and quantum spectra underlying quantum measurements are EilenbergMoore coalgebras induced by these Frobenius algebras." 



#108
Aug306, 09:48 PM

Astronomy
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PF Gold
P: 22,808

the folklore (and I think this has been reliably confirmed by at least one scholar) is that in 1925 around the time he devised matrix mechanics version of QG Heisenberg did not know what a matrix was
and had never heard of Hilbert spaces. According to the JB history draft, page 5, Heisenberg came to show a formula to Max Born* who informed him that he had "reinvented matrix multiplication". Apparently the young physicists inventing QM at that time hadn't heard of Hilbert spaces. It was John von Neumann, a mathematician, who introduced them and showed them how to formulate QM with operators on a Hilbert space. However soon afterwards, von Neumann became disenchanted with the Hilbertspace formulation and wanted things to be done differently. BUT BY THEN IT WAS TOO LATE. The whole pack was already off like hounds after a fox. please correct any historical errors. *Max Born was Heisenberg's mentor at Göttingen, where he was a visiting student and later got a job. Heisenberg's thesis advisor was Sommerfeld, in Munich, and his thesis was in hydrodynamics. After it was accepted in 1923. he immediately returned to Göttingen and worked as Born's assistant.. http://www.aip.org/history/heisenberg/p06.htm It seems that although Max Born served as a mentor to the young Heisenberg, he did not supervise his PhD thesis. http://nobelprize.org/nobel_prizes/p.../bornbio.html ============= at first glance, it looks to me like what JB was calling a "starcategory", Robert Coecke would prefer to call a "daggercategory". I think someone with an ear for English will be apt to prefer "starcategory" to "daggercategory" for several reasons. Tthe phrase rings betterwith a better assortment of vowells. It has fewer syllables. The concept is all about things like adjoint of an operator A, something often written A*, the complex conjugate transpose of a matrix. Mathematicians frequently use the asterisk * for duals and adjoints and such. So if Coecke insists on the nomenclature "compact" then a sensible compromise would be "compact star category"instead of "daggercompact category"but we will just have to wait and see here is a picture of Bob Coecke (oxford computing lab) http://web.comlab.ox.ac.uk/oucl/people/bob.coecke.html he has an impressive list of publications since around 1999 http://arxiv.org/find/grp_physics/1/.../0/1/0/all/0/1 the coauthor Dusko Pavlovic is at Kestrel Institute in Palo Alto. It is the notforprofit institute connected with the software development company Kestrel Development. Both wings of Kestrel sound like interesting places to work. 


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