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

1. Jul 5, 2006

### 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 osillators [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.c...LEID_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 [Broken]
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

Last edited by a moderator: May 2, 2017
2. Jul 5, 2006

### 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.

3. Jul 5, 2006

### 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 [Broken]

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 [Broken]

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 [Broken]

e - York University [Physics] reports Richard Feynman used phasor analysis to develop the 'sum over paths' method.

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]

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4. Jul 5, 2006

### 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 [Broken]

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.

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

Last edited by a moderator: May 2, 2017
5. Jul 5, 2006

### 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

Last edited: Jul 5, 2006
6. Jul 6, 2006

### 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 -> $$S \times S$$. 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

Last edited: Jul 6, 2006
7. Jul 6, 2006

### marcus

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.

Last edited: Jul 6, 2006
8. Jul 6, 2006

### 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

Last edited: Jul 6, 2006
9. Jul 6, 2006

### marcus

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.

Last edited: Jul 6, 2006
10. Jul 6, 2006

### 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). :rofl: 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

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11. Jul 6, 2006

### Hurkyl

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

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 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)

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

Hilb doesn't have a linear structure either. :tongue: 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.

12. Jul 6, 2006

### marcus

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.

13. Jul 6, 2006

### marcus

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...."

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14. Jul 6, 2006

### Careful

**Why do you say GR doesn't incorporate topology change? How do you plan on dealing with the formation of a black hole? :tongue: 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. :tongue: 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. :tongue: 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? :grumpy:

Careful

Last edited: Jul 6, 2006
15. Jul 6, 2006

Staff Emeritus
=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?

16. Jul 6, 2006

### 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

17. Jul 6, 2006

### Careful

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.

18. Jul 6, 2006

### Hurkyl

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

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 lets 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.

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.

Last edited: Jul 6, 2006
19. Jul 7, 2006

### 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 lets 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 :rofl: ). Ok, let me say it in another way, I think that reasonably speaking the octopi are meant to be particles (right ? , lets 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

Last edited: Jul 7, 2006
20. Jul 7, 2006

### Hurkyl

Staff Emeritus
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!

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.

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

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.