Stroop Theory - the possible unification of QM with GR through IT and CHO?

[Dcase]

The Reader is invited to help this thread become rigorous.

If any links should not fubction, inform me, I will try to correct them.

Currently this thread is an analogous interpretation of circumstantial evidence and incomplete information.

The writer has attempted to crudely use the techniques of Noble Economics Laureate John Harsanyi, a game theory mathematician rewarded along with John Nash and Reinhard Selten in 1994 “for their pioneering analysis of equilibria in the theory of non-cooperative games”.
http://nobelprize.org/nobel_prizes/economics/laureates/1994/press.html

More specifically Harsanyi was honored for “the analysis of games with incomplete information is due to you, and it has been of great importance for the economics of information”.
http://nobelprize.org/nobel_prizes/economics/laureates/1994/presentation-speech.html

The Outline

Stroop theory is a phrase coined by another.
Basically it is a combination of STRing an lOOP theories.
QM is quantum mechanics.
GR is general relativity.
IT is information theory.
CHO are complex harmonic oscillators.

The concept of this paper is that CHO are carriers of information that can tell time.
The information could either be utilized as is or be transformed into another or series of manifestations before being used.

CHO may come in many forms. Most often they are either loops or helices. Loops have various forms of circles, ellipses, probably hyperbolas and perhaps even parabolas. Helices have various forms of generalized helices or helicoids of various genus.

CHO may sometimes be simple harmonic oscillators [SHO].

The Invitation

Please don’t hesitate to correct any errors or make other suggestions, since the reader probably knows this material better than this writer.

The Background

The writer has an undergraduate degree in mathematics. The doctorate is in a field other than physics. Experience includes biophysiology, kinesiology and military ballistics.

The inspiration

This exercise has had many sources of inspiration. Only a few will be listed.

Perhaps the first was the two page Scientific American Profile on Richard Borcherds and the ‘Monstrous Moonshine is True’ by WW Gibbs in November 1998.
http://www.sciamdigital.com/index.cfm?fa=Products.ViewIssuePreview&ARTICLEID_CHAR=B62A3D21-20F3-4078-AE1B-6DD891DA0ED

Sometime between then and now, there was the realization that the Krebs citric acid cycle, most often represented as a circular loop with cusps, was more likely, actually either a helix or a helicoid.
http://www.sp.uconn.edu/~bi107vc/images/mol/krebs_cycle.gif

The most recent three items of inspiration would be:

1 - the 2004 John Baez paper ‘Quantum Quandaries: a Category-Theoretic Perspective’. In particular the first sentence of the abstract “General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two.”
http://arxiv.org/abs/quant-ph/0404040

2 - the development of twistor string theory by Edward Witten, Nathan Berkovits, Roger Penrose and with contributions from many others. Only two of these references are listed..
a - http://www.citebase.org/cgi-bin/fulltext?format=application/pdf&identifier=oai:arXiv.org:hep-th/0406051
b - http://users.ox.ac.uk/~tweb/00001/

3 - the proposed search for a fourth spatial dimension by Charles Keeton and Arlie Petters.
http://dukenews.duke.edu/2006/05/braneworld.html

 

[Dcase]

more on STROOP

The Interpretation

This writer suspects that John Baez is generally correct in this concept relating QM and GR.
This writer offers an alternative which is crudely based upon the Monstrous Moonshine proof of by Richard Borcherds in 1992.
The alternative is a complex-3D, geodesic-helical-string, time-D from the more general form of complex-(n-2)-D, string-D, time-D.

The writer suspects that one of the Monster subgroups such as a Mathieu group or one of the pariah groups may have some role depending upon the number of degrees of freedom.
The writer suspects that both the time and string D are also complex.
The string D is also thought to be of geodesic helical trajectory.
Loop and helical Modulo Mathematics play a role as complex or simple harmonic oscillators..
Transformations to e^ix are likely involved

There is an abstract use of entanglement concept.
http://plato.stanford.edu/entries/qt-entangle/

Conceptual Steps [1-3]:

1 - Let us perform a symbolic composition that is relatively equivalent to architectural and engineering blueprints / plans: Circle [o] + sinusoid [~] => Helix [o~].

Because the writing surface is 2D, let us clarify:
[SYMBOL font not avaolable, no access to LaTex]

a - Circle [o] will be represented by: (_) as a 1D object with 2D-conformation observed end-on;

Sinusoid [~] by: /\/\/\/\/ as a 1D object with 2D-conformation observed from the side or from above;

Helix [o~].by ... (_)/\/\/\/\/(_) ... as a 1D object [same gauge, period of the 2D conformation object] having 3D-conformation.

2 - Modular Mathematics:

The two, loop Modulo [2Pi]

la] 0Pi (_) 1Pi [modulo practice in accepted mathematics, ‘off-on’ in human-computer interface]

lb] 1Pi (_) 2Pi [clock-face arithmetic, ‘this or that‘ in alternative binary]

and the e^ix transformation:
lc] e^(i*0Pi) = +1 (_) e^(i*1Pi) = -1 [logarithmic binary]

...........+1 | | +1
.......... 0Pi | | 2Pi [= 0 Pi]
h] The helical Modulo [2Pi] ... ... (_)/\/\/\/\/(_) ...
tri-ave counterpart of musical octave ... 1Pi | | 3 Pi [= 1 Pi] or
... if reading triplets, restart ...... | 0-1-2 Pi sequence
if logarithmic binary ...... +1 | | -1

1 - when an object is at 0Pi in h it could simultaneously have a counterpart at la or lb or 1c, whichever binary [factor out Pi] is used.

2 - when an object is at 1Pi in h it could simultaneously have a counterpart at la or lb or 1c, whichever binary [factor out Pi] is used.

3 - when an object is at 2Pi in h it could simultaneously have a counterpart at la or lb or 1c, whichever binary [factor out Pi] is used.

Thus the helix and loop have the same period when in the same gauge and representing the same event: la 1b and lc as loop mechanics, h as helical mechanics.

 

[Dcase]

more on STROOP

Conceptual Steps [4]:

4 - There may be a simultaneous / reciprocal involvement of the eccentricities [1/R and R] of elliptical Riemannian curvatures and hyperbolic Gaussian curvatures.

For QM helical mechanics consider these references:

a - David Hestenes wrote the ‘The Kinematic Origin of Complex Wave Functions’ discussing Dirac and Schroedinger theories. He describes circular and helical Zitterbewegung and trajectory of the electron, relating them to the “complex phase factor in the complex function” yielding a physical origin for these statistical properties..
[Hestenes like many uses h-bar which does simplify numeric calculations. However h better identifies the eccentricity.]
http://modelingnts.la.asu.edu/pdf/Kinematic.pdf

b - Caspar Wessel basically proved the existence of the ‘imaginary unit” in 1797. This entity is likely more invisible than imaginary and not a simply a mathematical construct.
‘An Imaginary Tale: The Story of i [the square root of minus one]’ by Paul J Nahin (Hardcover - 24 August 1998)

c - 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).
He mentions electrical inductance and mechanical mass equivalence.
http://www.quantum-chemistry-history...onGabriel1.htm

d - Electrical engineers have used CP Steinmetz phasors about 25-30 years longer than physicists have used Schrodinger wave equations. Figure 3 of ‘Complex representation of Fourier series’ by C Langton. demonstrates that e^jwt plotted in three dimensions is a helix.
http://www.complextoreal.com/tfft2.htm

e - York University [Physics] reports Richard Feynman used phasor analysis to develop the 'sum over paths' method.
http://www.physics.yorku.ca/undergrad_programme/highsch/Feynm1.html

For GR:

a - Rotating objects, with an emphasis on those capable of generating magnetic fields, have elliptical Riemannian curvature [some are nearly circular] orbits. They tend to be nearly circular ellipses themselves as they rotate.
[No specific reference, Speculation interpreted from general reading]

b - Groups of objects [category] serving as satellites revolving around a primary object [moons around planets, planets around stars, stars around galactic cores] can be categorized as systems. Systems tend to have helical mechanics perpendicular to their axis of revolution.
If they left a wake, from the side or above their space-tine geodesic would be a helix, hence a complex-3D, geodesic-helical-D, time-D. Viewed end-on they tend to have the appearance of a logarithmic spiral. Taken in total the appearance is like a space-cyclone very much like a hurricane or tornado of the vortex of water draining from a bathtub.
[No specific reference, Speculation interpreted from general reading]

These are all pseudospheres or antispheres or tractrisoids - “Half the surface of revolution generated by a tractrix about its asymptote to form a tractroid“. Systems have hyperbolic Gaussian curvature.
http://mathworld.wolfram.com/Pseudosphere.html.

Thus there may be a nesting of alternating elliptic and hyperbolic surfaces from within the GR gauges. [Speculation]

 

[Dcase]

referesnces and examples used with STROOP

Other References and Examples

Ivars Peterson has a great article about helicoids in ’Surface Story’ [SN: 12/17/95] with links to the MSRI minimal surface library. Weber, Hoffman and Wolf proved that they are embedded.
http://www.sciencenews.org/articles/20051217/bob9.asp

Helix as a geodesic:
For the 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

The helix may also be a geodesic for hyperbolic surfaces.

This Japanese website [author: iittoo?] in English possibly has interesting illustrations of a psuedosphere or tractoid and how revolving systemss may be hyperbolic. The circle is a special type of both an ellipse and hyperbola. Rotating tractrix [matrix [loop?] pulled by a string] images are also interesting.
Figures 6, 6’ illustrate how the helix may be a geodesic for the moving tractoid.
Figure 16 is credited to Tore Nordstrand from Gallery of Curved Surfaces [French].
This may illustrate how a solar system or galaxy may retain the logarithmic spiral structure as they move through space-time using the apparent helical geodesic of this hyperbolic curve.
http://www1.kcn.ne.jp/~iittoo/us20_pseu.htm

Why Barred Spiral Galaxies have features in common with the logarithmic spiral?
Logarithmic Spiral - many images on the web, these two are the most interesting

a - Mice Problem “mice start at the corners of a regular n-gon [n=3->6] of unit side length, each heading towards its closest neighboring mouse in a counterclockwise direction at constant speed. The mice each trace out a logarithmic spiral, meet in the center of the polygon, and travel a distance ...”
http://mathworld.wolfram.com/MiceProblem.html

b - Hermann Riecke and Alex Roxin in their ‘Rotating Convection in an Anisotropic System’ features images remarkably like Barred Spiral Galaxies. Published in Phys. Rev. E 65 (2002) 046219 and on-line at
http://www.esam.northwestern.edu/~riecke/research/Modrot/research_klias.htm

Applied Twistor Theory?
The Lilja Precision Rifle Barrels, Inc has a Twist Machine!
“We feel very fortunate to have recently obtained a twist deviation inspection machine ... This device checks the twist rate of a rifle barrel and determines if there is any deviation to the actual rate.”
“The question that Manley [Oakley] solved was; how do you check for twist deviation? He reasoned that if two plastic washers were spaced about 3 inches apart and pushed through a barrel, that the first washer would speed up or slow down in its rotation in relation to the trailing washer, if the twist rate changed. He devised a method of holding these washers on a long, small diameter steel tube. A smaller diameter rod fit inside of the tube and was free to rotate. The first washer was fixed to the small rod and the trailing washer to the tube. So now, if there was any change in twist rate, the inner rod would rotate faster or slower than the outer tube. With the rod running completely through the tube, the differences in rotation between the rod and tube could be observed and measured on the other end of the tube.
Here is a view of the helical rifling guide that controls the twist rate on the "pull" end of the button/rod.”
http://www.riflebarrels.com/articles/barrel_making/twist_machine.htm

SOLARIA BINARIA by Alfred de Grazia and Earl R Milton demonstates the planetey orbits as loops with the sun stationary, but suggests that with the sun in motion planetary orbits become helices.
http://www.grazian-archive.com/quantavolution/QuantaHTML/vol_05/solaria-binaria_07.htm

A good reference for information theory is “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.

More Google images

Hilbert space.
The most intriguing for this writer are those similar to concentric circles or targets.
http://members.shaw.ca/ray_gardener/essays/favgame.htm

nCob images were not found, but are described it as similar to Hilb.

Logarithmic spirals
a - Galxies
http://antwrp.gsfc.nasa.gov/apod/ap950912.html
b - NASA Cosmicopia, The Heliosphere, The Sun's Magnetic Field, the Parker spiral?
http://helios.gsfc.nasa.gov/heliosph.html

end of STROOP (variant) introduction

 

[Kea]

Dcase

Thanks for starting the thread. Personally, my first reaction is as follows. In a purely mathematical sense, you refer to some very interesting ideas. However, the interpretation of Stroop theory that we have in mind actually involves even more difficult mathematics and...this is really the point...mathematics which some of us believe is essential to understanding the physics well enough to be able to make predictions. As an example, an investigation of planetary orbits in this context might begin by looking at some classic papers (such as one by Dirac) in General Relativity, because it isn't quite enough to see that the 'abstraction' of orbits to helical ideas is a good idea.

Your intuition about twistors is spot on. This is something we sometimes discuss here. Sorry to be brief for now. I hope this is a little helpful.

Cheers
Kea

 

[Careful]

***
1 - the 2004 John Baez paper ‘Quantum Quandaries: a Category-Theoretic Perspective’. In particular the first sentence of the abstract “General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two.”
http://arxiv.org/abs/quant-ph/0404040
****

Ok, I finally read this one since some people keep on mentioning it. I have some elementary questions/remarks which I am sure can be answered. First, the author seems to suggest that general relativity or any classical topological field theory incoorporates topology change; this is manifestly untrue since topology change involves lack of predictibility (singular fields) and any classical theory is deterministic (universes are globally hyperbolic). So, one can doubt wether nCob has any relevance for physics at all (and I certainly do know what Bryce deWitt would say about this). Assume for now that nCob is useful somehow, then the author wants to suggest that in a categorical sense it resembles more Hilb than Set. The reason for this would be the absence of a cartesian product structure as well as the lack of a natural duplication map : S -> [tex] S \times S [/tex]. Baez endows nCob with the disjoint union, that is to say that he allows for the existence of two disjoint (n-1) universes (I guess he considers these octopi as cuts in a flat background so that you can see them as particles) and disjoint n spacetimes (although in the paper it is sometimes explicitely stated that we deal with different universes - which makes no sense to me). Now, it is my humble guess that the main reason why he states that the disjoint union is not a cartesian product structure is because the nontrivial cobordisms aren't invertible, i.e. topology change cannot be undone. One can ask oneself whether this would be anything essential in a class or models which abandons classical physics to start with (and requires non elastic collisions !) since the invertibility property can be easily restored by allowing for the slightly more exotic possibility of lower dimensional branches and punctures in the ``unfolding'' - in this way one can reach any spatial topology one wants (one can perfectly give different amplitudes to these ``cobordisms''). The same comment applies to the duplication map (where one can figure out a more or less canonical procedure nevertheless). However, let's stay in nCob : admittedly, these are two distinctions with the cartesian product, the question however is wheter it comes any closer to Hilb. In particular what is the role of the superposition principle in nCob, there is no linear structure present - so what the heck is it supposed to mean ? Why would our universe obey unitarity so well if you give it up at the microscale - is there any mechanism to ``protect'' it ? Or does the author want to suggest that there is quantum information loss which is invisible at the scales on which we observe QM today? And what is the link between the lack of unitarity here and the projection rule in a ``classical'' basis in the copenhagen framework?

Careful

 

[marcus]

***
1 - the 2004 John Baez paper ‘Quantum Quandaries: a Category-Theoretic Perspective’. In particular the first sentence of the abstract “General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two.”
http://arxiv.org/abs/quant-ph/0404040
****

Ok, I finally read this one since some people keep on mentioning it. I have some elementary questions/remarks which I am sure can be answered. First, the author seems to suggest that general relativity or any classical topological field theory incorporates topology change;... So, one can doubt whether nCob has any relevance for physics at all..

Must a non-trivial cobordism necessarily involve topology change?
I was assuming it did not. Is there something I'm missing?

Does the author actually suggest that Gen Rel must involve topology change? I did not understand the author to be saying that at all. Classically it certainly does not.

 

[Careful]

**Must a non-trivial cobordism necessarily involve topology change?
I thought not. Is there something I'm missing?**

Sure, there has to be some section with different spatial topology. You perhaps imagine a cobordism from a circle to a circle which is a tube which splits, the left leg twists (the right one not) then they come together again and match the final circle. Clearly topology change is involved... (btw. pictures 4,5,7,8 are standard examples).

**
Does the author actually suggest that Gen Rel must involve topology change? I did not understand the author to be saying that at all. Classically it certainly does not. **

As a matter of information you can implement topology, signature change and so on in classical GR (for example: George Ellis has written about that in the context of a classical analogue of the Hartle Hawking no boundary proposal), but such constructs do not canonically follow from a well posed initial value problem. In the abstract, the author starts from saying ``that GR makes heavy use of nCob'' while it does actually no such thing at all - I think this is *very* suggestive to say the very least.

Careful

 

[marcus]

Must a non-trivial cobordism necessarily involve topology change?
I was assuming it did not. Is there something I'm missing?

I just learned something interesting about h-cobordisms from the Wiki
http://en.wikipedia.org/wiki/H-cobordism

"A cobordism W between M and N is an h-cobordism if the inclusion maps
M -> W
and
N -> W
are homotopy equivalences. The h-cobordism theorem states that if W is a compact smooth h-cobordism between M and N, and if in addition M and N are simply connected and of dimension > 4, then W is diffeomorphic to M × [0, 1] and M is diffeomorphic to N.
... An informal reason why manifolds of dimension 3 or 4 are unusually hard is that in lower dimensions there is no room for tangles to form, and in higher dimensions there is enough room to undo any tangles that do form...

If the manifolds M and N have dimension 4, then the h-cobordism theorem is still true for topological manifolds (proved by Michael Freedman) but is false for ... smooth manifolds of dimension 4 (as shown by Simon Donaldson)."

Baez is talking about differentiable manifolds----that is, the smooth case. That is what I am talking about also. For the h-cobordism theorem to be FALSE in dimension 4, there must be a counterexample of a cobordism W between M and N, where W is NOT diffeomorphic to M x [0,1]

that is what I mean by a trivial cobordism----one diffeomorphic to M x [0,1]

but somehow according to Michael Freedman's result. If one views W as a TOPOLOGICAL cobordism then the theorem is TRUE in dimension 4 (topologically speaking) and W in fact IS homeomorphic to M x [0,1].
And M is homeomorphic to N. So I would conclude that there is NOT TOPOLOGY CHANGE in going from M to N.

Yet the (smooth) cobordism is certainly non-trivial, by Donaldson.

So I conclude that in dimension 4 a non-trivial smooth cobordism does not necessarily involve topology change.

this contradicts what you said, so I must be missing something. What?

hmmmm, time for lunch. Be back later to read your answer.

 

[Careful]

**
this contradicts what you said, so I must be missing something.
***

Dear marcus,

I knew that you would come up with this and honestly you are looking for details which are entirely irrelevant.
First of all, Baez also applies these ideas in lower dimensions (that is where they were tested already), so there your comment does not apply at all and moreover, I am entirely convinced that none of these LQG people are interested in working with exotic differentiable structures (apart from a few singletons perhaps). Moreover, differentiable structure is entirely IRRELEVANT in LQG like approaches, which are only based upon *topological* invariants between spinnetworks in some topological embedding space (actually, by neglecting subtleties associated to smoothness, your hero Rovelli got rid of the annoying continuous dimension of the Hilbertspace in LQG). So this does not contradict at all what I said since you bring in facts (which I knew incidentally) which do not matter at all (so yes, you are missing something).

Please Marcus, don't play this silly game since all your comments are wrong/irrelevant until now - why don't you simply wait until someone who knows this stuff better gives a decent answer to ALL my questions/objections. In fact, you are simply arguing about some silly detail which does not matter while not adressing any of the issues I raised. Leave it, perhaps f-h or the author himself or someone else can anwer these things... I presume we still have the right to ask questions when reading a paper no?

Careful

 

[Hurkyl]

First, the author seems to suggest that general relativity or any classical topological field theory incoorporates topology change; this is manifestly untrue since topology change involves lack of predictibility (singular fields) and any classical theory is deterministic (universes are globally hyperbolic).

Why do you say GR doesn't incorporate topology change? How do you plan on dealing with the formation of a black hole? And why would topology change imply nondetermism?

the slightly more exotic possibility of lower dimensional branches and punctures in the ``unfolding'' - in this way one can reach any spatial topology one wants

Why do that? A cobordism in nCob is supposed to be interpreted as time evolution; it's the 4D manifold that spans the region between, and including, two spatial slices.

Now, it is my humble guess that the main reason why he states that the disjoint union is not a cartesian product structure is because the nontrivial cobordisms aren't invertible, i.e. topology change cannot be undone.

You can see the definition of "cartesian product" here. It has nothing to do with invertibility. (e.g. the category of Sets has cartesian products, and it has loads of noninvertible morphisms)

(I guess he considers these octopi as cuts in a flat background so that you can see them as particles) ... (and requires non elastic collisions !)

An ordinary TQFT isn't trying to model particles. Though, I understand that extended TQFT's do, and they do have conservation of momentum.

the question however is wheter it comes any closer to Hilb. In particular what is the role of the superposition principle in nCob, there is no linear structure present - so what the heck is it supposed to mean ?

Hilb doesn't have a linear structure either. It's the individual Hilbert spaces that have linear structure. And that's the whole point of the TQFT: it's a functor nCob --> Hilb that assigns a Hilbert space to each possible spatial topology, and a linear operator to each possible cobordism. (I think it can even be chosen to be unitary) In other words, a TQFT is simply a method of assigning linear structures to spatial topologies.

 

[marcus]

**Must a non-trivial cobordism necessarily involve topology change?
I thought not. Is there something I'm missing?**

Sure, there has to be some section with different spatial topology.

In other words you are saying that you were wrong about this detail.
That is all right. It seems like an interesting mathematical point, so I wanted to be sure.

There is no reason for you to act irritated. I am not attacking you personally---just want to get a math point straightened out.

And I gather then that you would retract what you said. Baez IS interested in the 4D case and so your objection (that nCob must involve topology change, so is not interesting physically) must be wrong. a nontriv cobordism does NOT necessarily involve top change.
So what he is talking about IS relevant to physics, contrary to what you were claiming. OK that seems simple enough.

Or have I made some careless mistake?

Please relax Careful and take a deep breath. We don't have to start being ad hominem. I am interested in some other things you said and have some more questions.

 

[marcus]

Why do you say GR doesn't incorporate topology change? How do you plan on dealing with the formation of a black hole? And why would topology change imply nondetermism?...

Interesting observation. The most recent paper of Ashtekar and Bojowald indicated that QG treatment of a black hole might lead to a bounce and a re-expansion of another region of spacetime.
It also stressed that the evolution thru the Planck regime (at the bounce) was DETERMINISTIC.

So one has the prospect of a deterministic fork-point, being allowed in QG even though you can't have one in Gen Rel. This is certainly a topology change!
So quantizing Gen Rel may make it allow topology change. Another reason that Baez may be on the right track with these cobordism pictures.

Just in case anyone is interested, here is the BH paper:
http://arxiv.org/abs/gr-qc/0509075
Quantum geometry and the Schwarzschild singularity
from the conclusions:
"...It suggests that the classical singularity does not represent a final frontier; the physical space-time does not end there. ...[Evolution]...remains regular during the transition through what was a classical singularity. Certain similarities between the Kantowski-Sachs model analyzed here and a cosmological model which has been studied in detail [10] suggest that there would be a quantum bounce to another large classical region. If this is borne out by detailed numerical calculations, one would conclude that quantum geometry in the Planck regime serves as a bridge between two large classical regions...."

 

[Careful]

**Why do you say GR doesn't incorporate topology change? How do you plan on dealing with the formation of a black hole? And why would topology change imply nondetermism? **

The formation of a black hole does not change the topological structure of spacetime, you just allow for a metric field which has a singularity behind the event horizon. The answer to your second remark is very simple, in an initial value problem the solution is only fixed on the globally hyperbolic neighborhood determined by the Cauchy hypersurface.


**
Why do that? A cobordism in nCob is supposed to be interpreted as time evolution; it's the 4D manifold that spans the region between, and including, two spatial slices. **

You are missing the point I try to make, in all discrete approaches to quantum gravity, there is no continuum, there are only networks (continuum topology is a coarse grained thing). It occurs to me as very unnatural to demand that no punctures in the continuum can be made; compare it with a soap bell which under extreme pressures can burst. If you take the soapbell in a continuum description, no dynamical law can make it break, the reason why it can is because it is made up of litte atoms in mainly empty space and extreme forces can force them apart. If you uphold the same ``atomic'' picture for spacetimes then punctures should be allowed. The fact that this ``destroys'' all differences with Set makes me very uneasy ...


**
You can see the definition of "cartesian product" here. It has nothing to do with invertibility. (e.g. the category of Sets has cartesian products, and it has loads of noninvertible morphisms) **

I was talking about WHY Baez claims that the disjoint union is *not* a cartesian product structure. It means that for two (n-1) hypersurfaces A and B there exist no ``cobordisms'' M_A and M_B such that for any cobordisms M_XA and M_XB there exists a unique cobordism M_XAB such that M_A M_XAB = M_XA and M_B M_XAB = M_XB. Now, this fails for ordinary cobordisms in general because for given M_A, M_B there might not even exist an M_XAB because M_A cannot undo some twists or something like that (this is the invertibility I was talking about) or because M_XAB is not unique. In case you allow punctures to be made in the interior all this is not a problem.

**
An ordinary TQFT isn't trying to model particles. Though, I understand that extended TQFT's do, and they do have conservation of momentum. **

These extended TQFT's build that in by hand in an analogue of the Feynman perturbation series for ordinary QFT.

**
Hilb doesn't have a linear structure either. It's the individual Hilbert spaces that have linear structure. And that's the whole point of the TQFT: it's a functor nCob --> Hilb that assigns a Hilbert space to each possible spatial topology, and a linear operator to each possible cobordism. (I think it can even be chosen to be unitary) In other words, a TQFT is simply a method of assigning linear structures to spatial topologies. **

Sigh, this is trivial of course and these toymodels have been developped for years, but what does it have to do with quantum gravity where you have to take a superposition of cobordisms itself ?! And of course I was referring to the individual linear structure of the *objects* in Hilb. (n-1) hypersurfaces in quantum gravity can be thought of as singular wavefuctions and should ultimately be superposed. The structures you are referring to are useful for defining a generalized quantum field theory on a say a causal set where it has been used by Markopoulou, Sahlmann and Hawkins I believe.

Did you really think I was missing these elementary points?

Careful

 

[selfAdjoint]

=quote=caredul]You are missing the point I try to make, in all discrete approaches to quantum gravity, there is no continuum, there are only networks (continuum topology is a coarse grained thing). It occurs to me as very unnatural to demand that no punctures in the continuum can be made; compare it with a soap bell which under extreme pressures can burst. If you take the soapbell in a continuum description, no dynamical law can make it break, the reason why it can is because it is made up of litte atoms in mainly empty space and extreme forces can force them apart. If you uphold the same ``atomic'' picture for spacetimes then punctures should be allowed. The fact that this ``destroys'' all differences with Set makes me very uneasy ... [/quote]

But what do these concerns have to do with "GR is all about cobordism"? Take, say, Wilson loops in classical GR and "cobord" them. No top change, meaningful math, no?

 

[Careful]

**In other words you are saying that you were wrong about this detail.
That is all right. It seems like an interesting mathematical point, so I wanted to be sure.

There is no reason for you to act irritated. I am not attacking you personally---just want to get a math point straightened out. **

No, it is not an interesting point for this kind of approaches and that is why I ommited it.


**
And I gather then that you would retract what you said. Baez IS interested in the 4D case and so your objection (that nCob must involve topology change, so is not interesting physically) must be wrong. a nontriv cobordism does NOT necessarily involve top change.
So what he is talking about IS relevant to physics, contrary to what you were claiming. OK that seems simple enough. **

Why do you always twist words of people ? I said : ``Baez has also applied it in the 3-D case where your comment is irrelevant''. I know he IS doing the 4-D case but also said that he is VERY LIKELY not interested in exotic diffentiable structures since the general spirit of this kind of work is to kick out differentiable structures ! So therefore I can conclude that his non trivial cobordisms DO change topology.

**
Or have I made some careless mistake?
**

LISTEN to people, you immediately jump in the air because I am not very enthousiastic (for good reasons I believe). And stop twisting words, it makes a good conversation very hard.

I have two main worries:
(a) any discrete theory of QG ought to be robust, Baez' construction depends in a very delicate way on properties of the continuum.
(b) Nowhere do I see the difficulties involving the superposition principle being adressed. The construction made indeed remembers one about the Markopoulou,Sahlmann and Hawkins work which is about QFT *on* causal sets, not about a quantum dynamics *of* causal sets.


Careful

 

[Careful]

=quote=caredul]You are missing the point I try to make, in all discrete approaches to quantum gravity, there is no continuum, there are only networks (continuum topology is a coarse grained thing). It occurs to me as very unnatural to demand that no punctures in the continuum can be made; compare it with a soap bell which under extreme pressures can burst. If you take the soapbell in a continuum description, no dynamical law can make it break, the reason why it can is because it is made up of litte atoms in mainly empty space and extreme forces can force them apart. If you uphold the same ``atomic'' picture for spacetimes then punctures should be allowed. The fact that this ``destroys'' all differences with Set makes me very uneasy ...

But what do these concerns have to do with "GR is all about cobordism"? Take, say, Wilson loops in classical GR and "cobord" them. No top change, meaningful math, no?[/QUOTE]

Sure, but in that case you do not take disjoint unions and you do not have a monoidal category either - but a simple category.

 

[Hurkyl]

Did you really think I was missing these elementary points?

I thought it odd, but since I'm not telepathic, I have to go by what you write and not what you actually know.

I was talking about WHY Baez claims that the disjoint union is *not* a cartesian product structure.

Well, the intuitive reason is that there simply aren't enough degrees of freedom in an n-dimensional manifold to encode all the ways to map into a pair of n-dimensional manifolds. (You really ought to have a 2n-dimensional manifold)

If you want a specific counterexample, the cleanest one comes from looking at the zero manifold. By considering the following diagram where both morphisms are the empty cobordism:

0 <--- 0 ---> 0

you can prove that 0x0 = 0, and the projections 0 <--- 0x0 ---> 0 have to be empty.

Then, when you consider any diagram:

0 <--- 0 ---> 0

where the two maps are different, you cannot possibly find a map 0 ---> 0x0 that let's the diagram commute.


If you really strongly don't like reasoning with empty spaces, then the same idea should work for any example; find a commuting diagram, add widgets to some of the diagonal maps, then prove you cannot make the new diagram commute.

These extended TQFT's build that in by hand in an analogue of the Feynman perturbation series for ordinary QFT.

I don't know that approach -- just the abstract nonsense approach. It turns out that an extended TQFT is nothing more than a functor of 2-categories -- from nCob2 to 2Hilb.

nCob2 is the 2-category whose objects are (n-2)-manifolds, 1-morphisms are (n-1)-cobordisms, and 2-morphisms are n-cobordisms with corners between the (n-1)-cobordisms.

2Hilb is the category of 2-Hilbert spaces. Some examples include Hilb, categories of group representations, and direct sums of such things.

 

[Careful]

***
Well, the intuitive reason is that there simply aren't enough degrees of freedom in an n-dimensional manifold to encode all the ways to map into a pair of n-dimensional manifolds. (You really ought to have a 2n-dimensional manifold)

If you want a specific counterexample, the cleanest one comes from looking at the zero manifold. By considering the following diagram where both morphisms are the empty cobordism:

0 <--- 0 ---> 0

you can prove that 0x0 = 0, and the projections 0 <--- 0x0 ---> 0 have to be empty.

Then, when you consider any diagram:

0 <--- 0 ---> 0

where the two maps are different, you cannot possibly find a map 0 ---> 0x0 that let's the diagram commute.


If you really strongly don't like reasoning with empty spaces, then the same idea should work for any example; find a commuting diagram, add widgets to some of the diagonal maps, then prove you cannot make the new diagram commute. ***

Well, this is what I said myself in one of the previous posts no? You constructed an example where you first show that both projections have to be the empty cobordism, and then that for another type of situation no suitable ``extension'' exists. I was not at all having trouble with this (it was clear to me how this worked in the beginning), I just mentioned that it seems rather artificial to me to rely upon such strong properties of the continuum in a discrete framework. As I said, once you allow for some punctures/branches to made, the above would fail (except for the empty case, cannot beat that one ). Ok, let me say it in another way, I think that reasonably speaking the octopi are meant to be particles (right ? , let's cut the ``but in ordinary TQFT this is not the case'' xxxxxxxx). In (old) string theory, the continuum is fundamental, strings are rigid and these objects are promoted to be the true degrees of freedom. In discrete approaches they are not, therefore (soapbell analogy) it occurs to me that such coarse grained picture (no punctures etc...) has to *emerge* rather than to be put in by hand - it has to be a result of the interactions. Another comment is that the claim was somehow made that a deeper insight is gained in such construction between gravity and QM. This, I found astonishing since the only result is the weak and unstable correspondence between two properties of nCob and Hilb (actually anyone who has played around with octopi once knows them - myself included) while no further light is shed on the superposition principle (which is the real troublemaker). Actually, as you said yourself, such abstraction reminds one of QFT in a more general background structure. Perhaps, people have gone further in this now and managed to make a stronger link with gravity. If so, let me know about it (but I doubt it)!

Careful

 

[Hurkyl]

Ok, let me say it in another way, I think that reasonably speaking the octopi are meant to be particles (right ? , let's cut the ``but in ordinary TQFT this is not the case'' xxxxxxxx).

I can't "cut it" -- this is exactly contrary to everything I've read about TQFTs.

The (n-1)-dimensional manifolds denote slices of space-time; they aren't particles. In fact, that would go against the whole spirit of a TQFT which is to study the aspects of a quantum theory that relate only to topology, right?

I can't say I've tried to imagine the significance of the Hilbert space that a TQFT assigns to an (n-1)-dimensional manifold M. Maybe that Hilbert space encodes information about particles living in M, but M itself is certainly not supposed to represent a particle!

while no further light is shed on the superposition principle

Right -- AFAIK it doesn't try to talk about anything like a superposition of space-times.

But a quantum theory of gravity that does talk about a superposition of these kinds of spatial slices will (presumably) still need to have the information described by the TQFT attached to each slice in the superposition.

Actually, as you said yourself, such abstraction reminds one of QFT in a more general background structure.

As I've hinted above, I think that is the (original) point of a TQFT.

Perhaps, people have gone further in this now and managed to make a stronger link with gravity.

Someone... I think Baez's students, published a paper recently, and its description really struck me as sounding very similar to the structure of an extended TQFT. But maybe the resemblence is superficial. *Shrug* I'm tired and don't remember the paper, but maybe marcus knows what I'm talking about and can give you the link.

 

[Careful]

**I can't "cut it" -- this is exactly contrary to everything I've read about TQFTs. **

Of course you can't cut them in TQFT's and neither does topology change happen in classical TFT's. Again, I was only wondering whether it is reasonable to expect that the *peculiar* features of a TQFT would have something deep to say about the connection between GR and QM.
Moreover, as I told a long time ago, topology change in *QG* has the nasty tendency to mess up renormalizability - only microscopic holes in a 1+1 toy model have been shown to behave ``reasonably'' well.


***
The (n-1)-dimensional manifolds denote slices of space-time; they aren't particles. In fact, that would go against the whole spirit of a TQFT which is to study the aspects of a quantum theory that relate only to topology, right? ***

In a topological field theory you can imagine the splitting and formation of the octopi to be topological defects in a flat background. These defects can be seen as particles, with momentum, mass and so on since flat connections can distinguish (flat) cones (with top removed) with different opening angles (which are related to the mass). A similar set of ideas in that direction concerns preons which are exclusive topological particle models.

**
But a quantum theory of gravity that does talk about a superposition of these kinds of spatial slices will (presumably) still need to have the information described by the TQFT attached to each slice in the superposition. **

Hmmm, I don't think so, vectors in the combined Hilbert spaces will have ``fuzzy particle as well as geometry information'', so the ones you have here are the distributional cases; the semiclassical approximation if you want to. Look, there are different ways of thinking about nature and Planck scale physics out there : TQFT, discrete, semiclassical, classical, lattice field theory approaches and so on. Now, I believe it should not really matter in the end what approach you take, which implies that in liaison towards the end result, a particular assumption should not really be of significant importance. Now, as said, the TQFT approach as advocated in **THIS** way, stands or falls with this basic assumption (of continuity) which excludes any discrete ideas towards QG (at least on the face of it). That is why it makes me feel uncomfortable...it is strange to say that there is a deep link between GR and QM (while the link is by far not even made !) which *excludes* genuine discrete approaches.

But again, explain me why you think these ideas will have any chance to obtain a deeper insight in the physical world, in particular why it could offer some different view upon the problem of quantum gravity.

Careful

 

[Dcase]

Anti-de Sitter Space relationship?

I found this interesting paper that employs a psuedosphere -
Quantum Fields in Anti-de Sitter Space and the Maldacena Conjecture
Nelson R. F. Braga Instituto de Física
http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332002000500010#fig

Figure 1, from this writer's perspective, may be viewed with the arrow pointing from z to delta to illustrate growth of a solar or galactic system from smaller to larger.

 

[Kea]

Dcase

I will reply here to your comments on the [itex]SO(4,1)[/itex] thread. Firstly, it is clear that biology is important in the big picture here. We do not deny this, but one cannot talk about everything at once. Moreover, the models that use [itex]SO(4,1)[/itex] are an interesting link between well-developed physics and the categorical ideas that we have in mind.

Re biology: I have now come across a few neuroscientists (sorry but I don't remember the names) who are interested in categorical reasoning. Their interest derives from a long history of thought about perception going back to the likes of C. S. Peirce. These ideas are far removed from present studies of de Sitter space physics. However, if one were to replace the Poincare group by the conformal group which one uses in studying massless equations in twistor theory, and also take covering spaces, then one would be more interested in the group [itex]SU(2,2)[/itex] anyway.

If you have been looking at Borcherd's proof and would like to explain some of that to us, that would be great!

 

[Dcase]

Hi Kea:

1 - I would be grateful if you would discuss Borcherd's proof on this or a new thread for the following reason.

The Klein's Quartic Curve by John Baez July 28, 2006 TWFMP [wk?] intrigues me - (7,3) hyperbolic plane tiled with regular heptagons [Don Hatch credit, with Mike Stay numbers] particularly the quotes:
a - “If we curl up this thing correctly, we get a 3-holed torus tiled by 56 triangles (or dually, by 24 heptagons).”
b - “You'll notice the heptagons are drawn in a fancy style: each one has been "barycentrically subdivided" into 14 little right triangles.” - after - ‘color version on Tony Smith's website”
http://math.ucr.edu/home/baez/klein.html

I suspect that a 3-holed torus is equivalent to a genus-3-torus for quote ‘a‘.

I noticed that for quote ‘b’ if the (7,3) hyperbolic plane can be re-written as [14,7,3] then this may be a way to partition the complex-24D of Borcherds.
Bosonic - Monster by Borcherds - 24+1+1 from 14+7+3 = n-2 when n=26?
Fermionic - F-theory by Vafa - 10+1+1 from 7+3 = n-2 when n=12?
5D space-time as 3+1+1 = n-2 when n=5?

2 - I have been reading papers from Institute for Mathematics and its Applications, U-MN in these areas -
a - n-Categories: Foundations and Applications June 7-18, 2004
which includes papers by Baez-3, Cheng, Leinster and others ‘concerns iterative structures that appear naturally in a wide variety of contexts“.
b - Mathematics of Materials and Macromolecules: Multiple Scales, Disorder, and Singularities September 1, 2004 - June 30, 2005
“... a program aimed at a synthesis of the problems at the interface between mathematics, materials science, condensed matter physics, and biology.”
c - Shape Spaces April 3-7, 2006
“Curves, shapes and images may undergo very large deformations while still remaining within a single semantic category, for instance an anatomical structure, human face or handwritten character. These transformations form an essential component of image understanding. A variety of spaces have been proposed as natural domains, especially finite-dimensional and infinite-dimensional manifolds and Lie groups (e.g., of diffeomorphisms). Abstract formulations have led to feasible, if intensive, computational algorithms for elastic matching based on geodesics in order to quantify differences between shapes. Possible topics also include hierarchical shape clustering, shape learning, shape synthesis and statistics on manifolds.”
d - Topology and its Applications Mississippi State University, Starkville July 10 - 28, 2006
“In a number of diverse areas, topological issues have begun to surface. In molecular biology, for example, the geometric features of the surface of a molecule have been shown to influence certain protein docking processes. Knot theory is becoming increasingly important in the study of DNA. Computer scientists encounter topological problems in attempts to reconstruct surfaces from sampled data. Topology in phase space can help overcome the inherent sensitivity in longtime simulations of dynamical systems ...
Week 1: Applications to Dynamical Systems
Week 2: Topological Approximation and Surface Reconstruction
Week 3: Applications to Molecular Biology”

3 - Now this information is overwhelming to me, but all of it appears headed toward a unified goal of gauge / scale unification.

 

[Kea]

I would be grateful if you would discuss Borcherd's proof...

Goodness, I'm not the person to ask. Why not try the maths subforums? But I have to say that I have been reading a little about monster moonshine for reasons related to the sort of stuff that you mention.

For the benefit of others: when one expands the [itex]J( \tau)[/itex] invariant of elliptic curve theory for the asymptotic case [itex]\tau \rightarrow i \infty[/itex] in terms of [itex]q = \textrm{exp}(2 \pi i \tau)[/itex] one gets

[tex]J (\tau) = \frac{1}{1728}(q^{-1} + 744 + 196884q + 21493760q^{2} + \cdots)[/tex]

and some guy noticed that these big numbers are related to a Verma module for the monster group. The blog NeverEndingBooks has an entry
http://www.neverendingbooks.org/index.php/borcherds-monster-papers/ with some links.

...if the (7,3) hyperbolic plane can be re-written as [14,7,3] then this may be a way to partition the complex-24D of Borcherds...

There is certainly a lot to think about here. At present I am actually thinking about a different sort of partitioning of complexes, but it has a similar flavour to this sort of stuff. Maybe I'll say something about it later.

Curves, shapes and images may undergo very large deformations while still remaining within a single semantic category...

A category theory expert on Shape Theory is T. Porter http://www.informatics.bangor.ac.uk/~tporter/ in Wales. This is a
really cool guy who works with the Brown of higher groupoid fame, the same Brown who had a lot to do with Grothendieck writing a very, very long mathematical letter in the early 1980s.

...this information is overwhelming to me, but all of it appears headed toward a unified goal...

Yes, it does.

 

[Dcase]

Hi Kea:

Thanks for your responses. I have already learned much from you.

1 - My reason for using the physics rather than math forums is to gain insight into the physics perspective - especially in regard to GR and QM - likely the most difficult unification due to the imaging limitations in viewing very small objects.
Since I have some knowledge of biophysiology, I think that I might be able understand such a perspective.
Information science may one day be a broad category encompassing the physical and social sciences.
Information theory may allow for a quicker pace of unification with multi-discipline collaboration. I think that physics may benefit from techniques employed in other scientific fields.

Two examples [that I understand minimally - and selected for scales in a, geodesics b]:
a - Richard D James (Department of Aerospace Engineering and Mechanics, University of Minnesota) has this paper “Deformable Thin Films: from Macroscale to Microscale and from Nanoscale to Microscale” that appears to attempt to unify these 3 scales with minimal surfaces and phase transitions. Various materials are used, but “Helical configurations are natural” at these scales for the proteins studied.
http://www.ima.umn.edu/talks/workshops/W11.18-20.04/james/james.pdf

b - Peter W Michor (Mathematics, Universität Wien) “Geometries on the space of planar shapes - geodesics and curvatures” discusses the Hamiltonian vector field, geodesic equation, momentum mapping and Bessel functions in Riemannian metrics.
http://www.ima.umn.edu/2005-2006/W4.3-7.06/activities/Michor-Peter/curves-hamiltonian-lec.pdf

2 - Thanks for the referral to Tim Porter's Home Page.
The (5,4)-torus knot on a torus reminds me of radial symmetry and “Geometric Whirlpools”.
I have not yet had time to do more than just scan the topics listed.
I was very impressed with the journal Editorial positions. The French journal “Cahiers de Topologie et Geometrie Differentielle categorique” appears to have a logarithmic spiral with radial symmetries on the cover.

3 - Thanks for the referral to NeverEndingBooks - main, related entries and recent posts.
I only had time to review this and “Symmetry and the Monster”. I then went to Amazon to read more about this Mark Ronan book.
http://www.neverendingbooks.org/index.php/borcherds-monster-papers/.

4 - More on the Monster:
a - The j-function relation to J(tau)
http://mathworld.wolfram.com/j-Function.html

b - “... coefficients of j turn out to be simple combinations of the degrees (traces on the identity) of representations of M ...” or Monster symmetry. [A form of radial symmetry?]
http://www.mathstat.concordia.ca/faculty/cummins/moonshine.html

c - Marcus has the thread ‘NCG predictions: Alain Connes in SciAm’.
I have posted to his thread referring to NASA Pitch, Yaw, and Roll Systems with a calculated eigen axis.
http://liftoff.msfc.nasa.gov/academy/rocket_sci/shuttle/attitude/pyr.html
I am beginning to speculate that this [with time implied]
i] may be a dynamic spatial dimensions of flight mechanics roll, yaw, pitch
ii] as opposed to static spatial dimensions of front <-> back, left <-> right, up <-> down.
iii] the eigen axis may be a geodesic string [between the initial and terminal points]
iv] hence 3+1+1 [?] ratger than 4+1

 

[Kea]

...hence 3+1+1 [?] rather than 4+1

One of my favourite articles about the relation between [itex]\hbar[/itex] and time is the one

Weinstein on quantum tori
http://arxiv.org/abs/hep-th/9309006

Anyway, I guess what you are trying to say is that we want 3+1, not 3+1+1, because the 1s get secretly identified.

One could take that to mean a number of radical things. For instance, maybe we should be looking at a whole family of [itex]q[/itex]-deformed [itex]SU(2)[/itex] (or rather [itex]SU(2) \times SU(2)[/itex]) algebras rather than a 4+1 model.

This idea is appealing because one can look at it in a higher dimensional (categorical) context and it starts to make sense. There is a number theoretic flavour to things. The category of representations looks like sheaves based on the [itex]p[/itex]-adic numbers [itex]\mathbb{Q}_{p}[/itex] for [itex]p[/itex] related to [itex]q[/itex], so we would be collecting together all the primes.

However, in a simple classical limit one doesn't want to throw the whole of time out the window, and this is where the de Sitter picture might come in handy, because it is really a 3+1 theory in the right sense.

 

[Kea]

Dcase

I will tell you my own pet theory (!) on why DNA has four bases, and you can tell me what you think. The maths is from this paper which is based on the work of Ghrist who has a homepage http://www.math.uiuc.edu/~ghrist/index.html . Better still, the book is here: http://cnls.lanl.gov/People/nbt/books/template_book.pdf

First up, one must not think of the billiard ball picture of the molecule.
Ghrist found that all knots and links can be embedded in a branched surface diagram with 4 bases (holes). The Lorenz attractor (butterfly) template is an example of such a diagram with 2 bases, but this isn't enough to do all knots. This means that knots are generated by making words in the 4 bases. Now one would need to think a bit about the base pairing, or rather the allowable words. Interestingly, I saw somewhere that experimental studies of simple knot types in DNA showed a preference for certain types. For instance, the achiral figure 8 knot was not prevalent. The basic diagram for this knot has both positive and negative crossings, so it wouldn't fit into the Lorenz template.

 

[Dcase]

Kea

I concede that you are correct with respect to my poor notation.
The 3+1+1 was my poor effort to use a ‘psuedo-Borcherds notation’ (complex-D, string-D, time-D).

I waded through the Weinstein paper ‘Classical Theta Functions and Quantum Tori’. He uses the terms wavefront sets’ and ‘frequency sets’ in his 4_Discussion on page 9.
Maybe David Hestenes [ASU] is correct “complex phase factors have a physical origin that has noting [sp] to do with probability per se” in ‘The Kinematic Origin of Complex Wave Functions’!
"http://modelingnts.la.asu.edu/pdf/Kinematic.pdf"
a - The concepts of roll, yaw, pitch as dynamic dimensions may explain the probability like physical origin of complex wavefront / frequency sets?
b - Hestenes also used h-bar in his paper. Personally I understand that this reduced Planck’s constant facilitates computation, but h-bar tends to obscure an important number ‘1/2Pi’ which may refer to eccentricity?

I reviewed Jacobi Theta Functions [128 equations] at MathWorld and linked topics.
http://mathworld.wolfram.com/JacobiThetaFunctions.html

The Hypergeometric Function lists a few “very special points” resulting in rational numbers, quadratic surd and other exact values [equations 18-24]
“Applying Euler's hypergeometric transformations to the Kummer solutions then gives all 24 possible forms which are solutions to the hypergeometric differential equation” [equations 63-84]. That number 24 [from the Monster] - strange occurrence?
http://mathworld.wolfram.com/HypergeometricFunction.html

Quintic Equation [31] has that number 1728 or [12^3] [(2^6)*( 3^3) ~ the first two factors of Mathieu group M12] associated with 1/60?
http://mathworld.wolfram.com/QuinticEquation.html

Is there any reason time numbers 24, 60 should relate to cubic inches in a cubic foot?

I also reviewed ‘Symplectic, Quaternionic, Fermionic’ by John Baez, September 7, 2000.
http://math.ucr.edu/home/baez/symplectic.html

I will read the “Knots and links in 3D Flows’ by Ghrist tomorrow.

I recall reading that there is evidence of a temperature phase transition that is suspected of resulting in couplet to triplet nucleic acid reading and will look for the reference.

RNA probably preceded DNA since U is requited for synthesis of both C and T. However ATP is needed to synthesize U and G while GTP is required for A which leads me to suspect that once upon a time orotic acid and hypoxanthine [IMP] or xanthine possibly played more important roles than simply as precursors as they do today.
From Michael W King, PhD-Biochemistry / IU School of Medicine discussion of nucleic acid metabolism [although he omits xanthine].
http://web.indstate.edu/thcme/mwking/nucleotide-metabolism.html

 

[Kea]

Personally I understand that this reduced Planck’s constant facilitates computation, but h-bar tends to obscure an important number ‘1/2Pi’ which may refer to eccentricity?

Yes, but physicists always drop factors of 2 and [itex]\pi[/itex] all over the place because they are usually easy to clean up at the end, so when discussing basic noncommutativity, for example, I just use [itex]\hbar[/itex] instead of [itex]h[/itex] because [itex]h[/itex] is just an ordinary letter, which could mean anything.

Is there any reason time numbers 24, 60 should relate to cubic inches in a cubic foot?

It is no accident that the ancients chose numbers that factored well to base their counting systems on. Pretty sensible really.

I also reviewed ‘Symplectic, Quaternionic, Fermionic’ by John Baez...http://math.ucr.edu/home/baez/symplectic.html

Good. I see that John explains how one should think of fermions as quaternionic and bosons as real Hilbert spaces. In particular, time reversal along with the usual [itex]SU(2)[/itex] gives a (1D) quaternionic space. This is a start to seeing how time generation might be associated to noncommutativity, and also how the antisymmetry of fermions is associated to noncommutativity.

RNA probably preceded DNA since U is requited for synthesis of both C and T. However ATP is needed to synthesize U and G while GTP is required for A which leads me to suspect that once upon a time orotic acid and hypoxanthine [IMP] or xanthine possibly played more important roles ...

Interesting, but I can see I'm going to have to tell you right now that my ignorance about genetics is horrifyingly vast. Anyway, even for RNA I think you'll find the Ghrist work interesting. There are already lots of papers on RNA using Feyman diagrams http://www.citebase.org/abstract?identifier=oai%3AarXiv.org%3Acond-mat%2F0106359&action=cite****s&cite****s=citedby
and I see that there are lots of books now on genetics and knots.

The large [itex]N[/itex] matrix models referred to in this paper are about putting lots of vertices on a Riemann surface and doing partition functions based on [itex]N \times N[/itex] random matrices. These models can actually describe Riemann surfaces fully, and are well studied in the context of String theory. To a category theorist, it is very nice to think of the (moduli of) surfaces in terms of ribbon diagrams.

The master of ribbon graphs is Mulase, who has a paper on quaternionic Feynman diagrams:

Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs
M. Mulase, A. Waldron
http://arxiv.org/abs/math-ph/0206011

which I am reading at the moment. This paper is I believe, er, of major importance to physics. Recall the use of twisted ribbons by S. Bilson-Thompson to characterise the particles of the standard model.

 

[Kea]

Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs
M. Mulase, A. Waldron
http://arxiv.org/abs/math-ph/0206011

Observe that the duality on page 20 explains the T-duality of String theory. Moreover, the three allowed values of [itex]\alpha[/itex], namely 1/2, 1 and 2, indicate the existence of three natural scales, one large, one small and one so-called self-dual point, which is the complex case.

 

[Dcase]

Kea

I became side-tracked with the discussion of attractors.

I reviewed my old ‘Chaos’ by J Gleick (paperback-1987).

On page 143 is a demonstration of how a torus may become distorted in cross [Poincare] section.

On p271 “Economists analyzing stock market data would try to find attractors of dimension 3.7 or 5.3”

The numbers 3.7 and 5.1 or 5.4 are prominent in the ‘The discovery of the [alpha]-helix and [beta]-sheet, the principal structural features of proteins’ by David Eisenberg in PNAS.
“Why did Pauling delay 3 years in publishing this finding that came to him in only a few hours? He gave the answer in his banquet address at the third symposium of the Protein Society in Seattle in 1989. He was uneasy that the diffraction pattern of -keratin shows as its principal meridional feature a strong reflection at 5.15-Å resolution, whereas the -helix repeat calculated from his models with Corey was at 5.4 Å. As he says in his fourth paper of the PNAS series with Corey: "The 5.15-Å arc seems on first consideration to rule out the alpha-helix, for which the c-axis period must be a multiple of the axis distance per turn..." But then came the paper in 1950 by Bragg, Kendrew, and Perutz enumerating potential protein helices. Pauling told his audience in 1989: "I knew that if they could come up with all of the wrong helices, they would soon come up with the one right one, so I felt the need to publish it."
“The origin of the discrepancy between the repeat of the -helix and the x-ray reflection of alpha-keratin was hit on a year later by Francis Crick (5), then a graduate student with Perutz, and also by Pauling. It is that keratin is a coiled-coil, with alpha-helices winding around each other. The wider excursion of the alpha-helix in the coiled-coil reduces its repeat distance to 5.1 Å. This knack of knowing which contradictory fact to ignore was one of Pauling's great abilities as a creative scientist.‘
http://www.pnas.org/cgi/content/full/100/20/11207

Is the helix itself an attractor?
Is chaos theory related to game theory?

I also looked at ‘attractor’ un MathWorld. The Rössler Attractor may relate to spiral galaxies? and perhaps to the 9_Conclussions p 30-32 of ‘Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs by M. Mulase, A. Waldron. in your 8-11 post.
http://mathworld.wolfram.com/RoesslerAttractor.html

In the ‘NCG predictions: Alain Connes in SciAm’ thread, I think that flight mechanics of animals or objects controlled by animals may be a subset of the NCG torus which may hold for any general n-body interaction.
I suspect this because of the evolution during or shortly after the Cambrian explosion of genus-3-torus semi-circular canals for balance and orientation based on angular momentum in chordates.
Phylum: Chordata
Subphylum: Vertebrata
Superclass: Pisces
Agnathans vs. Gnathostomes:
semicircular canals - agnathans have 1 or 2 - gnathostomes have 3
from ;BIO 342
Comparative Vertebrate Anatomy
Lecture Notes 1 - Chordate Origins & Phylogeny’
http://www.biology.eku.edu/ritchiso/342notes1.htm

‘Building a better semicircular canal: could we balance any better?‘ by Todd Squires (Caltech): “Every vertebrate organism uses fluid-filled semi-circular canals (SCC) to sense angular rotation -- and thus to balance, navigate, and hunt. Whereas the size of most organs typically scales with the size of the organism itself, the SCC are all about the same size--whether in lizards, mice, humans or whales. What is so special about these dimensions? We consider fluid flow in the canals and elastic deformations of a sensory membrane, and isolate physical and physiological constraints required for successful SCC function. We demonstrate that the `parameter space' open to evolution is almost completely constrained; furthermore, the most sensitive possible SCC has dimensions that are remarkably close to those common to all vertebrates”
http://flux.aps.org/meetings/YR03/DFD03/baps/abs/S240011.html

‘Electrical coupling between secondary hair cells in the statocyst of the squid Alloteuthis subulata’ by R Williamson: ‘The cephalopod angular acceleration receptor system has sensory response characteristics similar to those of the vertebrate semicircular canal system and, unusual for an invertebrate, contains secondary receptor hair cells. The experiments reported use intracellular recordings from pairs of hair cells to show that at least one subset of the hair cells is electrically coupled along the entire length of the crista section. The coupling can be reduced by application of heptanol or octanol. Intracellular injection of H+ ions into a hair cell reduces the coupling of cells on the opposite site of the injected hair cell but does not abolish it completely. It is proposed that the coupling is likely to result in an improvement in the signal-to-noise ratio of the receptor system, a reduction in overall frequency response, but an increase in the low frequency sensitivity.
http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=2541872&dopt=Abstract


BACK TO AP BIOLOGY HOME PAGE Chapter 40: Sense Organs by Jerry G. Johnson with the last line “4. Static equilibrium organs called statocysts are found in cnidaria, mollusks, and crustacea.”
http://www.sirinet.net/~jgjohnso/apbio40.html

Statocysts images can be found on Google, but otherwise I know very little about these SCC equivalent organs.

Sorry, but I still have not yet finished the Ghrist book.

 

[Dcase]

Kea

I became side-tracked with the discussion of attractors.

I reviewed my old ‘Chaos’ by J Gleick (paperback-1987).

On page 143 is a demonstration of how a torus may become distorted in cross [Poincare] section.

On p271 “Economists analyzing stock market data would try to find attractors of dimension 3.7 or 5.3”

The numbers 3.7 and 5.1 or 5.4 are prominent in the ‘The discovery of the [alpha]-helix and [beta]-sheet, the principal structural features of proteins’ by David Eisenberg in PNAS.
“Why did Pauling delay 3 years in publishing this finding that came to him in only a few hours? He gave the answer in his banquet address at the third symposium of the Protein Society in Seattle in 1989. He was uneasy that the diffraction pattern of -keratin shows as its principal meridional feature a strong reflection at 5.15-Å resolution, whereas the -helix repeat calculated from his models with Corey was at 5.4 Å. As he says in his fourth paper of the PNAS series with Corey: "The 5.15-Å arc seems on first consideration to rule out the alpha-helix, for which the c-axis period must be a multiple of the axis distance per turn..." But then came the paper in 1950 by Bragg, Kendrew, and Perutz enumerating potential protein helices. Pauling told his audience in 1989: "I knew that if they could come up with all of the wrong helices, they would soon come up with the one right one, so I felt the need to publish it."
“The origin of the discrepancy between the repeat of the -helix and the x-ray reflection of alpha-keratin was hit on a year later by Francis Crick (5), then a graduate student with Perutz, and also by Pauling. It is that keratin is a coiled-coil, with alpha-helices winding around each other. The wider excursion of the alpha-helix in the coiled-coil reduces its repeat distance to 5.1 Å. This knack of knowing which contradictory fact to ignore was one of Pauling's great abilities as a creative scientist.‘
http://www.pnas.org/cgi/content/full/100/20/11207

Is the helix itself an attractor?
Is chaos theory related to game theory?

I also looked at ‘attractor’ un MathWorld. The Rössler Attractor may relate to spiral galaxies? and perhaps to the 9_Conclussions p 30-32 of ‘Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs by M. Mulase, A. Waldron. in your 8-11 post.
http://mathworld.wolfram.com/RoesslerAttractor.html

In the ‘NCG predictions: Alain Connes in SciAm’ thread, I think that flight mechanics of animals or objects controlled by animals may be a subset of the NCG torus which may hold for any general n-body interaction.
I suspect this because of the evolution during or shortly after the Cambrian explosion of genus-3-torus semi-circular canals for balance and orientation based on angular momentum in chordates.
Phylum: Chordata
Subphylum: Vertebrata
Superclass: Pisces
Agnathans vs. Gnathostomes:
semicircular canals - agnathans have 1 or 2 - gnathostomes have 3
from ;BIO 342
Comparative Vertebrate Anatomy
Lecture Notes 1 - Chordate Origins & Phylogeny’
http://www.biology.eku.edu/ritchiso/342notes1.htm

‘Building a better semicircular canal: could we balance any better?‘ by Todd Squires (Caltech): “Every vertebrate organism uses fluid-filled semi-circular canals (SCC) to sense angular rotation -- and thus to balance, navigate, and hunt. Whereas the size of most organs typically scales with the size of the organism itself, the SCC are all about the same size--whether in lizards, mice, humans or whales. What is so special about these dimensions? We consider fluid flow in the canals and elastic deformations of a sensory membrane, and isolate physical and physiological constraints required for successful SCC function. We demonstrate that the `parameter space' open to evolution is almost completely constrained; furthermore, the most sensitive possible SCC has dimensions that are remarkably close to those common to all vertebrates”
http://flux.aps.org/meetings/YR03/DFD03/baps/abs/S240011.html

‘Electrical coupling between secondary hair cells in the statocyst of the squid Alloteuthis subulata’ by R Williamson: ‘The cephalopod angular acceleration receptor system has sensory response characteristics similar to those of the vertebrate semicircular canal system and, unusual for an invertebrate, contains secondary receptor hair cells. The experiments reported use intracellular recordings from pairs of hair cells to show that at least one subset of the hair cells is electrically coupled along the entire length of the crista section. The coupling can be reduced by application of heptanol or octanol. Intracellular injection of H+ ions into a hair cell reduces the coupling of cells on the opposite site of the injected hair cell but does not abolish it completely. It is proposed that the coupling is likely to result in an improvement in the signal-to-noise ratio of the receptor system, a reduction in overall frequency response, but an increase in the low frequency sensitivity.
http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=2541872&dopt=Abstract


BACK TO AP BIOLOGY HOME PAGE Chapter 40: Sense Organs by Jerry G. Johnson with the last line “4. Static equilibrium organs called statocysts are found in cnidaria, mollusks, and crustacea.”
http://www.sirinet.net/~jgjohnso/apbio40.html

Statocysts images can be found on Google, but otherwise I know very little about these SCC equivalent organs.

Sorry, but I still have not yet finished the Ghrist book.

 

[Kea]

Thanks for all the amazing genetics/physiology. I've never thought about the physiology side of it before. But I guess it makes sense that the evolution of balance is really a geometric question!

Are helices attractors? Well, that's what the Ghrist book suggests, I guess (please don't read it all at once - I haven't read it all!). However, I think it is a little premature to make any connection with chaos theory. Besides, the beauty of the ribbon template technology is that one can get numbers out from, essentially, the periodic orbits of the dynamical system. Everything comes from knot theory!

 

[Dcase]

... But I guess it makes sense that the evolution of balance is really a geometric question! ...

Great Point!
If one expands the concept of evolution beyond biophysiology, then physics evolution involves the existence and structure of space-time in apparently small and large [almost simultaneously by the eccentricities of hyperbolic and elliptic space through an E and 1/E relationship] then intermediate geometric forms such as nucleic acids eventually.

... I think it is a little premature to make any connection with chaos theory. Besides, the beauty of the ribbon template technology is that one can get numbers out from, essentially, the periodic orbits of the dynamical system. Everything comes from knot theory!

You are correct that this may be premature, but I am hopeful that when QM and GR are united so will nearly all branches of mathematics, including game theory.

I have been following the thread 'Lee Smolin's LQG may reproduce the standard model'.
We have been discussing attractors.
Attractors from my perspective look like
a - stringy loops or
b - looping strings.

The more I think about the Rössler Attractor, the more important it seems to become since it appears to be related to the logarithmic spiral thus utilizing Pi, e and i.
This spiral occurs in biology, weather [cyclones] and spiral galaxies.
Does it appear in QM?