Stroop Theory (lives in category land)

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selfAdjoint

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Marcus and Hurkyl, have you really got into understanding of what 2-categories are and do? It seems to me that undestanding this behavior will enable you to envision the geometric/topological situation better. Just what exactly is a 2-morphism and what are its axioms (note use of zig zag axioms in physics, cited by Baez in the talk).
 

selfAdjoint

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arivero said:
A good one. The points are vector spaces on the field F, and a matrix NxM has F^N as targen and F^M as source, if acting on a "column" vector, and the inverse asignment if acting on a "row" vector, this double possibility exemplifying the concept of duality between categories.
Well a practical "duality" in mathmatics is the algebraic versus topological thinking. Algebras and their behavior are good for some developments and commutative diagrams are good for others. Note that the whole subject of this thread can be described as a riff on Feynmann diagrams!
 

marcus

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selfAdjoint said:
Marcus and Hurkyl, have you really got into understanding of what 2-categories are and do? ...
Hurkyl is the point man on this one. I was thinking this morning of posting simply "hurkyl I owe you one" or some such message, but decided to leave it unsaid

thanks arivero also!

so far AFAICS Baez has NOT given a very complete set of hints, or it is hard to see how they fit together. any attempt to guess how it fits involves risk of making mistakes. my feeling is that it is better for learning to take that risk and go ahead and guess. Probably Baez will eventually explain it better in some TWF. For now, thanks to H. for conjecturing how it might go.
 
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Hurkyl

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I have a good idea about what they are. The timing of this thread is interesting because I encountered "Grothendieck's dream" last week, and have been talking to an algebraic topologist friend, so I can understand how n-groupoids relate to homotopy n-types. :smile: I can't really go beyond n=2 yet... but then again the experts can't go all that much further, so I don't feel bad. :wink:


I sort of feel that, on the topological side, the 2-category is entirely transparent. I can picture loops, cobordisms of loops, and "cobordisms" of cobordisms. (And now I can picture them even better after marcus's #23) We have this clean algebraic structure, and realizing it's a 2-category just lets you do bookkeeping. I can't picture n-holed torii, cobordisms of n-holed torii, and cobordisms of cobordisms, but knowing that it's a 2-category doesn't help. :wink:

I had been making some headway into figuring out 2-Hilb, but I've forgotten all of it. :blushing: Though I did mention earlier that I realized that there's a subcategory whose objects are simply groups... or more accurately, categories of unitary group representations.

So some eTQFTs are something like a choice of group to represent (exponentiated) momenta!




Thinking some more about the 3-D case, our 1-morphisms are cobordisms of collections circles... so they're 2-manifolds that have boundaries. We can always sew disks onto those circles to turn it into a compact, oriented 2-manifold. The compact, oriented 2-manifolds are, of course, the n-holed torii. (Including the sphere as a 0-holed torus)

And once we've done that, I think we can homeomorphically move those disks anywhere on our 2-manifold that we please... so, up to homeomorphism (!), a 1-morphism is nothing more than a choice of the global (spatial) topology for our universe.

If we always stick to the same genus, I can picture the 2-morphisms. If it's a sphere, I can puncture it then flatten it out to the plane... otherwise, I can take one of those plane models where we identify edges and stuff, but at the moment I'm having trouble imagining things where the number of holes change. :frown:
 
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arivero

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selfAdjoint said:
Well a practical "duality" in mathmatics is the algebraic versus topological thinking.
This is not just reversing arrows, but also different structures, related then with "functors". A functor is covariant or contravariant depending of when it reverses arrows.

The main example here is of course topological spaces versus boolean algebras.
 
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Observations with image examples:

In some ways planetary orbital loops are like quantum loops as R is to 1/R.
http://www.grazian-archive.com/quantavolut...figs/sb_f14.jpg

The Earth rotates and displaces as the sun moves somewhat like a cannon ball fired through a rifled barrel.
http://modelingnts.la.asu.edu/html/UGC.html [Broken]

David Hestenes discusses the importance of the helix in 'The Kinematic Origin of Complex Wave Functions'.
http://modelingnts.la.asu.edu/html/UGC.html [Broken]

Thus in some ways planetary helical displacements are like quantum helical strings as R is to 1/R.

Speculative questions:

1 - Is it possible that various loop theories can be integrated into helical string theories?

2 - Conversely, can one find loop theories that are derivatives of string theories?

3 - Could loop and / helical string theories be applied to both GR gauges and the QM gauges?

4 - Did Smolin give up too soon in his attempt to relate quantum loops to strings (via the helix)?

5 - Is twistor string theory a subset of the Monstrous Moonshine proved by Borcherds?

6 - Should the string-D and time-D of the Monstrous Moonshine be complex as are the other 24-D due to the j-function (via e^ir2Pi)?

Note - #6 would accommodate helical strings as complex harmonic oscillators and complex time (consistent with Hawking imaginary time and perhaps 2T physics of Bars.
 
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marcus

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Hello Dcase,

I see you posted something about your helix ideas at Motl blog and mentioned this thread in passing.

Your ideas (about helixes) are distinctive, but, as I assume you realize, are not closely related to the topic of this thread.

In this thread I mainly wanted to talk about Baez paper
Quantum Quandaries; a Category-Theoretic Perspective
http://arxiv.org/quant-ph/0404040 [Broken]

I am afraid the thread title "stroop theory" is rather fanciful

However category theory does give a loose flexible framework where one can see analogies between, not only the cobordisms of Gen Rel and the Feynman diagrams of quantum physics, but also perhaps in a very general way between string worldsheets and spinfoams.

As you surely recognize, these constitute no more than ANALOGIES and not any firm mathematical connection

my advice would be:

1. make a separate thread for your helix theories, because they don't belong in this thread discussing quant-ph/0404040

2. read quant-ph/0404040 ("Quantum Quandaries") and see what you can make of it

3. discuss that John Baez paper with us in this thread, if you would care to. further comment "Quantum Quandaries" by you or anyone would be very welcome (as long as focused on that) because it would help to continue discussion of the paper

This is re the Doug post of 29 June 2006 8:52 PM at Motl blog on the "Barton Zwiebach letter" thread
 
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An alternative perspective may be helpful

Hi Marcus:

1 - Please tell me if this is acceptable for continuing discussion in this thread.

Baez does have 34 references for this paper. I definitely agree with his abstract statement that there is a QM / GR “analogy“! I can visualize planetary orbital loops as analogous to quantum loops and twistor [helical] strings as analogous to planetary orbital trajectories.

Although Penrose, Witten, Hestenes and Freedman emphasize the helix, I suspect that even they may underestimate the true significance of this entity.
http://library.ictp.it/FP-DB/docs/1993/Dirac-1993-Lecture-Freedman.pdf

I suspect that the loop and helix are harmonic oscillators capable of telling time and transmitting information in music, electricity, QM, probably GR and in nucleic acids as in information science. They are likely complex because of usually being correlated with a charge, often ionic. They are important in solar magnetic reconnection [GD Holman], geodynamics [GA Glatzmaier] and observed at the galactic core magnetic fields [M Morris].

My goal is for someone smarter than I to take up where Smolin left off and be able to unify loops with [helical] strings!.

2 - I was somewhat suspicious that your thread was meant to be ‘fanciful’, but this term ’stroop’ that you coined, expresses my interest in how QM may relate to GR and perhaps all intervening gauges. I truly suspect that loops [circles, ellipses and possibly hyperbolas] may have such a relationship to helical strings.

I think that Euler with his identity circle proved this for that specific case [circle].
Electrical engineers have used phasors for 25-30 years more than physicists have used Schrodinger’s wave equation - see Figure 3 of Complex representation of Fourier series – e^jwt plotted in three dimensions is a helix.
http://www.complextoreal.com/tfft2.htm [Broken]

This can be composed from 2D representations:
Let circle [o] +{or x?] sinusoid [~] => helix [o~] when interpreted as a 2D architectural diagram.
In Figure 18.1 of Zwiebach's "A First Course in String Theory", the left most sinusoidal curve is helix from my perspective may be a representation of a complex-D24, string-D, time-D like Borcherds Monstrous Moonshine. [I remain unclear why the string and time D are also not complex.]

Gabriel Kron, an electrical engineer for General Electric, had an interesting paper: 'Electric Circuit Models of the Schrödinger Equation', Phys. Rev. 67, 39-43 (1945)
http://www.quantum-chemistry-history.com/K...ronGabriel1.htm [Broken]

I am uncertain if this is true for Riemannian or Gaussian curvature.

I have a great deal of respect for the astronomer Fed Hoyle who coined the term ‘Big-bang’ in a somewhat ‘fanciful’ manner. This term became significant in the literature.
I am hoping that ‘stroop’ may similarly be applied one day since I suspect that loops are related to helical strings.

3 - I quite agree that I am being more analogous [remember the Baez abstract] than rigorous, but must disagree that that there is no firm mathematical connection. I have been trying to make this clear by using words such as speculate rather than conjecture or from my perspective. As much as I respect Hilbert, even his 10th problem was disproved.

I have been absent from pure mathematics for a long time.
I do commonly use biometrics.
Most often I attempt to integrate from this polymorphisms set [history symptoms, physical signs, laboratory data] with the goal to establish a differential diagnosis and formulate a treatment plan.
This process is somewhat analogous to QM, but the probabilities in medicine are very ill-defined for decision analysis. There are attempts underway to improve this process. But the current state of affairs makes for a more analogous than truly rigorous process.
[The Mayo School of Graduate Medical Education (MSGME) HSR 5850 Medical Decision Making.]
http://www.mayo.edu/msgme/crtp-curriculum.html [Broken]

I have been trying to become more rigorous by attempting to update my mathematical abilities utilizing the internet and various texts. I may be misinterpreting some of what I read.
The web:
a - David Hestenes has an excellent site on geometric calculus and algebras. I particularly found this useful in attempting to understand Grassmann, Clifford and Lie Algebras and the kinematics of complex wave functions.
http://modelingnts.la.asu.edu/
b - John Baez has an excellent site on mathematics ‘This Week's Finds in Mathematical Physics‘. I am particularly interested in weeks 233 and 234, but there is some nugget I think I can learn from nearly each week - such as the addendum to week 73 on biological chiralty.
http://math.ucr.edu/home/baez/TWF.html
c - MathWorld is a great reference site.
http://mathworld.wolfram.com/
d - I have used many other sites but not as frequently as these three.
Texts:
I am reading or have read some texts of the Scientific American Book Club by Nahin, Moar, Livio, Slatner and Seife.
Recently I completed ‘The Limits of Mathematics’ by GJ Chaitin with his interesting ideas on the need for experimental mathematics and his work on incompleteness and definable but not computable probability.

Carlo Rovelli sparked my helical interest in ‘Loop Quantum Gravity’.
http://relativity.livingreviews.org/Articles/lrr-1998-1/index.html [Broken]

Specifically consider the section 6.10 ‘Unfreezing the frozen time formalism: the covariant form of loop quantum gravity’. On this page - Figure 3: The elementary vertex - reminds me of vertex algebra - while Figure 4: A term of second order - reminds me of a twisted cylinder.
http://relativity.livingreviews.org/Articles/lrr-1998-1/index.html [Broken]

Once there is a cylinder, consider Generalized Helix: “The geodesics on a general cylinder generated by lines parallel to a line l with which the tangent makes a constant angle.” Squirrels use such a geodesic when climbing the trunk of a tree. Ballistics and celestial mechanics also appear to use a helix in their trajectories.
http://mathworld.wolfram.com/GeneralizedHelix.html

4 - I should add that Lubos did think my perspective a “joke”, but was kind enough to provide two arXiv papers from 2001 to demonstrate that others have had a similar perception.
a - ‘D(NA)-Branes’ by Simeon Hellerman, John McGreevy, Stanford
http://arxiv.org/abs/hep-th/0104010
b - ‘Super D-Helix’ by : Jin-Ho Cho, Phillial Oh, SU-Korea
http://arxiv.org/abs/hep-th/0105095

Awaiting your critique.
 
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marcus

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Dcase said:
Hi Marcus:

1 - Please tell me if this is acceptable for continuing discussion in this thread.
Thanks for asking. No it is not acceptable AFAICS. Everything you have posted in this thread appears quite off topic.

If you will kindly read the first 30 posts on the thread, you will see that we have been discussing some category theory ideas that Baez introduced us to.

the most concise outline of the topic is in the lecture
Higher-Dimensional Algebra: a Language for Quantum Spacetime
I guess I could abbreviate it as HDA:QS

an easy partial introduction was given in quant-ph/0404040

Awaiting your critique.
Your ideas, which you lay out in your posts, may have some interest, but I CANNOT COMMENT ON THEM IN THIS CONTEXT because it would just crowd this thread with off-topic stuff. If you want people to critique your ideas, why not start a thread?
 
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marcus

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Unless there is some objection, I'd like to continue with the topics we were discussing in the first 30 posts of the thread which are the categorical ideas in
HDA:QS (with a partial and gentle introduction in quant-ph/0404040).
http://math.ucr.edu/home/baez/quantum_spacetime/

Here is a thumbnail summary of what HDA:QS is talking about.

there are 23 slides, with an important break between 14 and 15, which is where he starts talking about TWOCATEGORIES.

Slides 1-14 are about MONOIDAL (one)CATEGORIES with DUALS for objects.

this is the type of category I was earlier calling "tensor star" as a kind of nickname----in the spirit of the introductory essay 0404040 "Quantum Quandaries".

It seems too imposing to call these things "monoidal categories with duals for objects."

The point that John Baez makes in slides 1-14 is that
both Hilb and nCob are this kind of category, and Set isn't, and this is a HINT

And then in slides 15-23 he follows out what he thinks are the consequences of taking this hint seriously----and he talks about TWOCATEGORIES.

Notice that this is a persuasive and suggestive (not strictly logical) development of ideas. he is taking us into his intuition. So there is an intuitive step at slide #15.

To me, that means we should try to deeply assimilate the intuitive content of what he says in 1-14-----what it suggests to him that Hilb and nCob are both a special kind of category unlike the category of Sets (which is conventionally the basis of mathematics).

So I will devote a post to that.
 
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marcus

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more about
http://math.ucr.edu/home/baez/quantum_spacetime/

Hilb is the core category of Quantum Mechanics
nCob is the core category of Gen Rel

JB is saying that these two categories are LIKE each other in a distinctive way that SUGGESTS they might each be different facets of the same thing

quantum mechanics might be seen to arise from the structure of spacetime, if we only understood the latter better----yeah sure, I know quite a few people have speculated about that but what he is describing is a PARTICULAR TAKE on the idea that QM can be better understood if we see it as part of the spacetime machine.

there are links to Baez HDA:QS webpage in this thread and he gives the definitions of Hilb and nCob, but we can remark in passing that Hilb is the category of hilbertspaces with linear maps from one space to the other as the morphisms.

And nCob is where orientable (n-1)-dimensional manifolds are the objects and the morphisms are n-dimensional manifolds JOINING them.

Usual Gen Rel is about 4Cob, which is where the objects are (orientable) 3D spaces and the morphisms are 4D spacetimes connecting them. So a 4D spacetime is something that morphs you smoothly from one version of 3D space to another.

======================

I look at HDA:QS as having a goal of PERSUADING the listener to be interested in twocategories (and higher groups, higher gauge theory, essetially in doing geometry with higher algebra)
and the message of slides #1-14 is one of comfortableness and RESOLUTION OF PUZZLES

This gets the listener ready to venture into the higher algebra realm of twocategories, because he sees that just a LITTLE ordinary category theory can help a lot to resolve puzzles.

So we need to study slides #1-14 carefully.

They say that Hilb is very analogous to nCob, and if you look at some things that puzzled you about QM they will turn out to have ANALOGS in spacetime, i.e. in nCob, that are TOTALLY OBVIOUS. So what may seem peculiar and paradoxical in Quantum Mechanics becomes VERY INTUITIVE if you go over to the analogous thing in Spacetime.

things like the "clone-taboo" and "teleportation" turn into obvious stuff with wet spaghetti when you look at them in nCob.

This, IMO, makes slides 1-14 worth the price of admission EVEN IF YOU DON'T BUY TWOCATEGORIES.

But remember that the speaker is also hoping that we will get interested in twocategories as well, so I should devote a post to slides #15-23 also.

===================
these sets of slides could also be helpful
http://math.ucr.edu/home/baez/barrett/

they are for three talks given at Knoxville 29 April-1 May at a workshop on Geometric Topology
(the Barrett Lectures)

sometimes they give more pictures, and spell things out in more detail, than was possible in the
single Perimeter talk given 31 May-----what I am abbreviating "HDA:QS" for Higher-Dimensional Algebra: a Language for Quantum Spacetime.
 
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Hi Marcus:

I must first ask for some clarifications then I will honor your request and not post further on your thread.
I am trying to be constructive and diplomatic but I will understand if you find some of these comments not to your satisfaction.

Did you actually read the first sentence of this Baez paper abstract?
"General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two."

I am agreeing with Baez although my perspective and language is different and imprecise.
I read 'Higher-Dimensional Algebra: A Language for Quantum Spacetime'. The last unlabeled figure of two intersecting tubes [worldsheets] has beside it a series of intersecting and non-intersecting loops - but a diagram is missing for the geodesic helical string interaction of the two tubes. Since this may be a geodesic in either Riemannian or Gauassian curvature, such a diagram would likely yield a "noncartesian tensor [perhaps only vector] product, given by the disjoint union of manifolds".
http://math.ucr.edu/home/baez/quantum_spacetime/

For the author to use an analogy for QM and GR and for me not to be able to use analogy for comments on this thread seems - at best hypocritical and disingenuous - at worst dogma and not even wrong.

Hurkyl mentioned in an earlier post [06-03-2006 08:00 PM] difficulty with 1D point particles but would be happy if they were strings. I am attempting to offer a way that they may be helical strings.

My language for this topic is currently inexact. With time my language will improve. I had collegiate experience with German, French and Russian not to mention numerous programming languages - but for now my doctorate language is within another field. I do not mean to imply that I am another Helmholtz, physician and physicist [I hope I am at least half that good], but even though the mathematics is difficult it is not impossible and less difficult than the language of medicine.

I reviewed the 14 slides of ‘Higher Gauge Theory’. I can put this to use in medicine - things, processes and processes between processes - whish is basically the morphism of DNA instructions into amino acid proteins - and more over do so with some of the simpler elements of game theory - which is basically the interaction of these proteins into a relatively cooperative multi-organ system.
In the subsequent slides I visualize a complex rather than real space-time.
I do get a little confused between 2-morphisms and polymorphisms which are more common in my field. The crossed modules are somewhat difficult. I would probably have used virtual rather than 'vanishing fake' curvature. I am probably misunderstanding gerbes by visualizing trajectories. Difficulty certainly increase as the slide progress.
http://math.ucr.edu/home/baez/barrett/

C Keeton, Rutgers and A Petters, Duke hope to search for the braneworld universe contains an extra fourth dimension of space for a total of five dimensions - but an alternative is complex-3D, helical-string-D and time-D which may be consistent with one of the Mathieu subgroups discussed by Baez in week 234.
http://www.space.com/scienceastronomy/060626_mystery_monday.html

In reviewing the guidelines, I do not thank that I am overly speculative, since helicity is discussed by many prominent physicists and this is the part of my discussion that appeared to be most provocative to you.

If your problem is with my inexact terminology, how can I learn without interacting and receiving critical feedback.

I will continue to use the term "stroop" in other settings, since it is so descriptive, but will not credit you unless you specifically ask me to do so.
 

marcus

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Dcase said:
I must first ask for some clarifications then I will honor your request and not post further on your thread.
Good! you may get some useful response if you start a thread.
As I explained, I can not respond to your ideas in this thread (it would be O.T.)
Farewell, and good luck discussing your ideas in some appropriate context!
 

Kea

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Dcase

It is great to have someone here who (a) knows something about DNA and (b) can see the relevance of category theoretic thinking...but Marcus is right: you need to start a separate thread. Many of the threads here are about similar topics. We are simply asking you to differentiate between helical ideas and basic Stroop theory as discussed in this thread.

Anybody can start a thread here.

Cheers
Kea :smile:
 

Kea

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question for Dcase

P.S. Are you the American biostatistics professor?
 
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response to Kea

Hi Kea:

I do not think that I am the person to whom you refer.

Please feel free to comment on my thread.
I could really use assistance to improve rigor since my ideas may rely too much analogy.
This is probably because I know a little about biophysiology and kinesiology and less about particle physics and mechanics.
I do have military ballistics experience.
 
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Hi Marcus:

I’m posting as a courtesy in respond to Kea.

I have posted a separate thread as you suggested.

I think that the applied mathematicians in electrical engineering have proved that same gauge and period loops and helices are equivalent.
I have not been able to locate such a proof.

I suspect that Penrose spinors are related to Penrose twistors.

Please feel free to comment. I could really use assistance to improve rigor.

The more math I read, the more I realize that medical decision theory is closer to mathematical game theory, especially that of Harsanyi.

I am going to ask Lubos for assistance. I really appreciate that he provided evidence that I was not alone in left field, even though he did not agree with my ideas.
 

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