# NCG predictions: Alain Connes in SciAm

1. Jul 27, 2006

2. Aug 7, 2006

### Dcase

Hi Marcus:

In reading the SCIAM article on Connes, there is a quote ‘"We now have to make a next step--we have to try to understand how space with fractional dimensions," which occurs in noncommutative geometry, "couples with gravitation," Connes asserts’ on page 2.

This seems to introduce fractal geometry or chaos or bifurcation theory into noncommutative geometry.

I have been noticing the peculiar repetition of the logarithmic spiral at many different gauges and scales [biology, weather, spiral galaxies] which reminds me of the Mandelbrot Set recurring initial image. Is there some relationship?
http://en.wikipedia.org/wiki/Mandelbrot_set

On page 1, I am surprised about the omission of “yaw” rotation by the author, although I agree that a different order of rotations will likely not commute.

This NASA site explains Pitch, Yaw, and Roll Systems
The orientation of the shuttle in space is defined as its attitude. This can be used to define such things as pointing the payload bay of the shuttle at the earth or orienting the nose of the shuttle to point at a celestial object, like the sun.
Orbiter attitudes are specified using values for Pitch, Yaw, and Roll. These represent a rotation of the shuttle about the Y, Z, and X axes, respectively, to the desired orientation. However, the shuttle doesn't actually perform each rotation separately. It calculates one axis, called the eigen axis, to rotate about to get to the correct orientation.
The attitude is be expressed as Pitch/Yaw/Roll or Roll/Pitch/Yaw, however the rotation is always performed in the order described in the previous paragraph.
The pictures below show the different rotations. [NOT shown in this post]

Since Penrose appears to have coined both the terms spinor and twistor, perhaps they might be related by considering one rotation as a spinoe, but a combination of three rotations as a twistor?

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3. Aug 8, 2006

### kneemo

Heck, maybe even a LQG/NCG mix as that presented in:

Intersecting Connes Noncommutative Geometry with Quantum Gravity
Authors: Johannes Aastrup, Jesper M. Grimstrup
http://arxiv.org/abs/hep-th/0601127" [Broken]

can be made testable. On page 17, the authors mention a possible construction using the group SO(3,1) and the difficulties they expect to arise. The group SO(3,1) doesn't seem to be a problem in BF theory. Perhaps a BF/LQG/NCG model is "beefy" enough to bring LQG to the level of testability.

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4. Aug 8, 2006

### marcus

you might also have mentioned a collaboration that appeared this year
http://arxiv.org/abs/hep-th/0601004
It has an "LQG/NCG" flavor because Laurent Freidel (whose current research is primarily with spinfoam QG and Feynman diagrams) co-authored with Shahn Majid.

as for testable predictions from the NCG direction, there is this statement by Majid on page 2 of
http://arxiv.org/abs/hep-th/0604130

"...Note that although (1) breaks usual Poincaré invariance, Special Relativity still holds as the quantum group ‘symmetry’. This is also the first noncommutative spacetime model with a genuine physical prediction[1], namely a variable speed of light (VSL). The NASA GLAST satellite to be launched in 2007 may among other things be able to test this prediction through a statistical analysis of gamma-ray bursts even in the worst case that we might expect..."

Both this NCG model and certain LQG/spinfoam models predict the same thing, namely a slight energydependence in the speed of very high energy photons
==============

It seems to me that a fair amount of theory research work is now on the line.
I am actually a bit nervous for some of the more promising lines of non-string QG. what will happen
to these efforts if the GLAST satellite does NOT see a slight advance in photon speed at very high gammaray energy?

Last edited: Aug 8, 2006
5. Aug 9, 2006

### Dcase

Hi Marcus:
.
I have been thinking about the NASA Pitch, Yaw, and Roll Systems with a calculated eigen axis.

I am beginning to speculate that this [with time implied]
i] may be 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 (?) rager thab 4+1

Last edited by a moderator: Apr 22, 2017
6. Aug 9, 2006

### Kea

Dcase

It is true that the NCG emergence of time is associated to the existence of a parameter for noncommutative spaces, but flight mechanics is not really a good way of thinking about it. The main example is the noncommutative torus, where the NC $C^{*}$-algebra generated by $U$ and $V$ has a quantum plane type relation depending on a parameter, which when irrational wraps a line about a torus without joining up with itself. Any introduction to NCG mentions this example.

Now you might prefer quaternions and hyperkahler geometry, but then the emergence of time is most well developed (meaning not very well) in the String theory use of this.

Last edited: Aug 9, 2006
7. Aug 9, 2006

Staff Emeritus
OT but interesting fact. Nicole Oresme actually discovered this fact about the irrationals on the torus in the fourteenth century. Considering two particles ("mobiles") moving on a circle at constant but independent velocities he wrote "If the velocities of the mobiles are incommensureable, they will never come in coinjunction at the same point of the circle twice, but there is no part of the circle so small that they will not come into conjunction within it". Considering the ratio of the velocities to be the parameter, the first assertion is that the track of its multiples never meets itself, and the second says that the point on it are dense in the diagonal of the torus.

Oresme was led to this theorem by his desire to debunk astrology. If the speeds of two planets are incommensureable, no astrological phase between them is ever exactly repeated in any degree of the zodiac, but any phase/degree relationship whatsoever, will be approximated as close as you like by some occurrance in history. So how can astrologers claim to predict?

8. Aug 9, 2006

### marcus

I recall visiting with Ralph Abraham one time in his office in Campbell Hall and his explaining this about the torus. He was trying to give me some intuition about "Dynamical Systems" and somehow we got onto this business of the orbit being dense (usually, because the slope would usually be irrational).

He didnt mention Nicole Oresme.

I found an English translation of a chapter by Oresme where he argues that the Earth might actually TURN and this would explain daily motion of the heavens.

He repeatedly uses the example of someone on a SHIP. For example to dispose of the counter argument that if the Earth rotated we would constantly experience a BIG EAST WIND. He explained that we might not, because the air could be being carried along with the rotation like the air in the cabin of a ship carried along with the ship's motion.

http://www.clas.ufl.edu/users/rhatch/HIS-SCI-STUDY-GUIDE/0040_nicoleOresme.html

1323-1382

9. Aug 9, 2006

Staff Emeritus
I believe the ship example was actually introduced by Jean Buridan, Oresmes' contemporary. Both of them wrote treatises on a "hypothetical turning sphere" (not the Earth of course, because the Church taught that it stood still, quoting the Bible). Oresme's was intended as a response to Buridan's. Buridan used his concept of impetus - in his usage a sort of informal precursor of momentum, which is conserved- to explain the lack of the Aristotelian wind. Oresme thought about this, (he was a mathematician, while Buridan was a sort of physicist) and came up with some counterexamples. If everything keeps rotating along with the surface, suppose you throw something up in the air. It will keep its same momentum, but the speed it has to travel on the bigger circle to stay above the same spot below will be greater, so it will fall back behind (to the East) of where it was thrown.

10. Aug 9, 2006

### Kea

Nice bit of history...but the NC torus is not really like a classical torus with a line drawn on it. The NC algebra has a lot more structure. Connes has lots of papers, slides etc on his website http://www.alainconnes.org/downloads.html, including descriptions of the Standard Model explaining how the Higgs sector arises from a noncommutative structure.

It is wonderful mathematics. The only question that remains is whether or not he is right about the physics of the Higgs mechanism.

11. Aug 9, 2006

### john baez

Cool! Where did you find out this? His second assertion is a special case of "[URL [Broken] theorem[/URL]. The general theorem concerns N particles moving around a circle at speeds that are linearly independent over the rational numbers; Oresme is considering the case N = 2.

http://en.wikipedia.org/wiki/Nicolas_Oresme" [Broken] (also known as Oresmus) is one of my heroes. Way back in the mid-1300's, he was the first to introduce graphs showing how a quantity changes as a function of time! This "spatializing" of the time variable is a crucial first step towards the concept of "spacetime". The Greeks, for example, never did this.

He called his idea the "http://www.maths.uwa.edu.au/~schultz/3M3/L10Oresme.html" [Broken]", since he was describing the time-dependence of "forms" such as temperature by drawing graphs where time corresponds to the horizontal direction. He did not introduce the concept of coordinate axes or coordinates for points in the plane - that's due to Descartes, much later.

Wow, that's almost a precursor of chaos theory - since he's using the existence of dense orbits to show a certain lack of predictability.

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12. Aug 9, 2006

Staff Emeritus
Yes, also the n=1 case of Tsebychav's theorem in Diophantine Approximation. He has a corollary that's even closer to the torus since it concerns an epicyle and its deferent, which have the obvious limitation in consequence. BTW Oresme also did the commentsurate (rational ratio) case. In that case there are just m places where conjunctions cann occer where m is the denominator of the ratio in lowest terms. More mud in th eye of the astrologer.

He does this in a treatise called "On the Commensurability and Incommensurability of the Heavenly Motions" or approximately that. A scholar named Grant, who was the go to guy on Oresme at the time, published a translation of it in an early issue of Archive for History of the Exact Sciences in about 1962 or 1963.

Oresme was my hero too. He proved the Merton College Mean Speed Rule by graphing the latitude form against the longtitude. The constantly increasing latitude made a right triangle with the latitude and the ordinate for the end of the time period (the beginning for this purpose could be assumed at zero). Then the distance made good was the area under the curve = area of triangle = (1/2)(height) X (base) = (1/2)(Change in V) X(Change in T) = (Average speed) X (Elasped Time).

He also constructed the exponential function - he almost thought of it in that much generality, using forms into forms - over the rationals. This was his "proportionum proportionibus". But using Euclid's (actually Eudoxos') theory of proportions, he was unable to extend his definition to the reals (incommensurable proportions). He recognized and stated this limitation. This is because the theory had no Completeness Axiom. There was no way to define a limit if the problem didn't give it to you in the data, like Archimedes' polygons inscribed in and circumscribing the circle. Oddly enough Oresme's statement equivalent to defining a dense set in the mobiles theorem didn't suggest to him a way out of his difficulty.

13. Aug 15, 2006

### Chronos

To quote an obscure roller coaster designer: "Jerk, I knew extra dimensions existed years ago." Perhaps we are descended from fish.

14. Aug 24, 2006

### Dcase

Spatio-Temporal Math History from an evolution perspective

Thanks for the history about Oresme - I was not aware of his work.
That this was nearly "a precursor of chaos theory" is particularly fascinating.

Genetic studies may elucidate the possible application of game theory through knots and / or chaos [categories of bifurcations?] as it relates to the trial and error of nucleic acid [NCG?] spatio-temporal evolution?

I searched the web for information on helix attractors.

This study examines Genetic Regulators - Could there be QM or GR application since the generalized helix is a geodesic?

http://www.inrialpes.fr/helix/people...tract-eng.html [Broken]
BacAttract - Theoretical and Experimental Analysis of Attractors of Genetic Regulatory Networks
Global Regulation of Transcription in Escherichia coli and Synechocystis PCC 6803
Coordinator : Hidde de Jong
Project funded in the framework of the Action Concertée Incitative IMPBio

"The study of genetic regulatory networks ... allowing the measurement of the spatio-temporal expression level of all genes in an organism under different conditions. In addition to high-throughput experimental methods, mathematical and computational approaches are indispensable for the analysis of genetic regulatory networks. A formalism based on piecewise-linear (PL) differential equations has been shown to be particularly adapted to the modeling of these networks."
"... first part ... study the attractors of the PL systems (equilibrium points and limit cycles), ... stability and their basin of attraction. The results of these mathematical studies will be used to develop efficient algorithms for the identification of attractors of a model of a given model."
"... second part ... study of the networks implied in the global regulation of transcription in the bacteria Escherichia coli et Synechocystis PCC 6803. We will develop the PL models of these networks by using data available in the literature, supplemented by plausible hypotheses. The predictions of the attractors, obtained by GNA [Genetic Network Analyzer] from these models, will be experimentally tested, by using measurements of the expression level of the genes, the concentration of certain proteins and metabolites, as well as the DNA topology."

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