Urs tutorial on Connes spectral geometry

[marcus]

some of us will want to follow this
http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of.html

I always did think that "noncommutative geometry" was a little misleading and what it really had to do with was Gelfandry. Reformism in language is seldom needed and rarely works, but in this case I think Urs is right when he tries to make us call Connes stuff "SPECTRAL geometry"

Well this Urs thing was just posted today 6 September.
I hope it helps. I could use some help.

It is going to be continued.

 

[marcus]

Connes paper is the main thing people are trying to understand (at least in non-string QG, for the moment)

Here is an exchange at Woit's blog:

http://www.math.columbia.edu/~woit/wordpress/?p=455#comments

John Baez Says:
September 6th, 2006 at 3:43 am
Connes surely doesn’t get the right dimensionless constants in the Standard Model Lagrangian - if he did, folks at the Newton Institute would be drinking champagne and dancing naked in the streets. So, how does he manage to come so close yet not that far? And, what is his attitude towards these constants? Did his audience press him on this point?

[If you wish, you can skip the intervening and scroll down to Urs reply][/color]

steve Says:
September 6th, 2006 at 3:58 am
Peter,

I never thought I’d be defending string theory, but I think your remark above is too negative.

Lenny didn’t claim victory because there is a single robust prediction from the Landscape. He’s seems disturbed that most low-energy observables are unpredictable, even in an anthropic framework.

However, it does mean the Landscape is falsifiable — if Planck measures a positive curvature it will strongly disfavor the scenario.

Peter Shor Says:
September 6th, 2006 at 4:38 am
John,

It looked to me from his abstract that the dimensionless constants were parameters that Connes could put in his model, and the only actual predictions (so far) came at unification scale? I don’t understand any of this, so could somebody verify whether this is right?

Anyway, it seems to me that if the string theorists got this close (for whatever values of close he got), there clearly would be dancing in the streets.

A Says:
September 6th, 2006 at 6:23 am
Following your link, I tried to read the Susskind lectures, but they contain a few words, almost no equations, plenty of cartoons.
My child got interested.

amanda Says:
September 6th, 2006 at 7:57 am
“However, it does mean the Landscape is falsifiable — if Planck measures a positive curvature it will strongly disfavor the scenario”

It will disfavor the *particular* scenario that LS pushes, with Coleman-de Luccia instantons. Of course, LS has the bad habit of pretending that his way of doing things is the only way — cf his repeated declarations that black hole complementarity, which is nothing but the wildest of wild speculations, is a “law of nature”!

MathPhys Says:
September 6th, 2006 at 8:01 am
amanda,

What do you mean by “black hole complementarity” more precisely?

steve Says:
September 6th, 2006 at 8:11 am
Amanda,

I’m under the impression that if there are many metastable vacua (almost all with much larger vacuum energy than our own), then it is highly likely that our universe must have originated in a tunneling (bubble nucleation) event. If so, the curvature has to be negative. It so happens that Coleman-Deluccia worked out the bubble form, but I don’t see that Lenny is making a nontrivial assumption.

Am I missing something?

D R Lunsford Says:
September 6th, 2006 at 8:16 am
Has anyone improved on Brout’s paper as an exposition for physicists?

-drl

Peter Woit Says:
September 6th, 2006 at 9:23 am
steve,

I didn’t say he was claiming victory, just that he was using this to answer certain people who argue that this is not science. It’s good to hear that he’s disturbed by not having any low-energy predictions, but the problem with the Landscape is not just at low-energy, it doesn’t give predictions at any energy.

As for the single bit of info here, I remember a time when string theorists were going on about no-go theorems that showed that you couldn’t have string theory in deSitter space, i.e. with a positive cosmological constant. When a positive cosmological constant was found, they seem to have come up with a way of dealing with that problem. If the spatial curvature comes out positive, I’m sure they’ll come up with something.

urs Says:
September 6th, 2006 at 10:27 am
John Baez said

Connes surely doesn’t get the right dimensionless constants in the Standard Model Lagrangian - if he did, folks at the Newton Institute would be drinking champagne and dancing naked in the streets. So, how does he manage to come so close yet not that far? And, what is his attitude towards these constants? Did his audience press him on this point?

I don’t think the point of this quest for the spectral action of the standard model is to predict the standard model’s properties.

I think the main point is first of all to understand which spectral triple precisely is the one whose spectral geometry describes the standard model.

It would of course certainly be a nice side effect if some properties of the standard model were derivable this way, maybe in the sense that they might turn out to be forced to have a certain value to admit a spectral description at all.

So I think the point is that if you want to understand something deeply, you should first try to find its most elegant/powerful/compact description. And Connes rightly points out that encoding the entire standard model into a spectral triple does achieve such a description.

And we learn by that, for instance, that we observe a world of metric dimension 4 and KO-dimension 4+6 mod 8.

While not a prediction, that looks like a remarkable insight.

I am going to say more about that at the n-Café. So far there is an introductory entry.

 

[Haelfix]

This is rather beautiful stuff actually. Its begun to make some noise in my department and I suspect we'll be looking it over soon in theory division meetings.

Some of the formulation is somewhat adhoc and just repackaging in an elegant way, as well as some phenomonology put in by hand, but there are intriguing mathematical relationships and coincidences lingering in the background that deserve being studied.

 

[arivero]

Also in the workshop the rumour "someone told 2 mod 8 = 26 mod 8" is running at the dinner table. Also I am starting to think that the old model was not 4+0 but 4+8 dimensions.

 

[rtharbaugh1]

Hi all

Marcus

I give up. What is Gelfandry? I don't find it in the dictionaries.

Thanks.

Richard

 

[selfAdjoint]

Hi all

Marcus

I give up. What is Gelfandry? I don't find it in the dictionaries.

Thanks.

Richard

I don't know either, but I'll bet the word is based on the name Gel'fand, a Russian mathematician.

 

[marcus]

Richard, let's say Gelfandry is a kind of gallumphing gallimaufry in which instead of a continuum being described by usual coordinates it has an ALGEBRA consisting of all the smooth functions that can live on it.

and then when nobody is looking you actually throw away the manifold (the continuum, the spacetime) and the coordinatized patches, and all you have left is the algebra. my wife does things like that: she takes things to Good Will or the Salvation Army. things we've had around for years.

the former continuum is now only a memory, but everything significant is now ENCODED in the abstract algebra

it gets worse, selfAdjoint knows. ask him.

people doing Gelfandry can be recognized because they hiccup frequently---the hiccup sounds like they are saying "spec-trul-trippulz"

 

[CarlB]

Richard, let's say Gelfandry is a kind of gallumphing gallimaufry in which instead of a continuum being described by usual coordinates it has an ALGEBRA consisting of all the smooth functions that can live on it.

and then when nobody is looking you actually throw away the manifold (the continuum, the spacetime) and the coordinatized patches, and all you have left is the algebra.

It seems like this is what would be the easiest path to understanding if it happened that you picked the wrong manifold to begin with, but got the tangent space right.

Is that sort of like moving to a Wick rotated geometry, but without apologizing for not using the usual spacetime manifold?

Carl

 

[selfAdjoint]

Yeah, it has that triple that Urs is telling us about
1. The Hilbert space(s) of states
2. The operator that has to be honored by the functor (originally a "generalized heat kernel", now a "(generalized?) Dirac operator", and
3. That C* algebra.

So that's Gelfandry? In what sense? What part comes from Gel'fand and what from Connes et al?

Also all this heat kernal and Dirac operator talk raises a question in my mind; Has anybody invoked the Atiyah-Singer index theorem yet ? (topology and Dirac operators in bed together).

 

[marcus]

did you notice John Baez reference to CORYBANTIC DANCING at the Newton maths institute, Cambridge.

it was oblique, like well if such and such were really true then wouldn't they be having Corybantic dancing in the streets of Cambridge?

In fact, I do believe that morally the ARE engaged in such. these papers of Connes and Barrett are (I expect) the breakthrough of this year or so.

All the non-string Quantum Gravities from Laurent Freidel to Renate Loll to Carlo Rovelli will now be furiously thinking how they can assimilate and absorb this geometrical form of the Standard Model which seems to get so much right and even contain gravity. And legions of string thinkers will be likewise engaged.

(Please correct me if I am wrong---this is a mere guess.)

 

[marcus]

Yeah, it has that triple that Urs is telling us about
1. The Hilbert space(s) of states
2. The operator that has to be honored by the functor (originally a "generalized heat kernel", now a "(generalized?) Dirac operator", and
3. That C* algebra.

So that's Gelfandry? In what sense?

Non-sense. Let's call it something else.
Would anyone here like to say what the Dirac operator essential job is?
say what are the basic pieces of the spectal triple?
(A, H, D)

A the algebra, H the hilbert, D the dirac-operator

how they fit together?

 

[CarlB]

Would anyone here like to say what the Dirac operator essential job is?

Uh, I'll take a swing.

The Dirac operator defines the equations of motion for the particles. In the usual standard model, the Dirac operator defines how fermions move from place to place, it defines the propagator. The usual Dirac operator defines the activity of four particles that are related by a spacetime symmetry (they form a representation of the symmetry of spacetime), these could be the spin up and down electron, and the spin up and down positron. Or the four particles could be the spin up and down neutrino and the spin up and down antineutrino.

It's a hell of a coincidence that nature uses the Dirac equation over and over, for not just the neutrino and electron, but also for the various forms of quarks. And once you have the fermions, the gauge bosons appear by the gauge principle so they don't appear to be so fundamental.

In generalizing the Dirac operator, you expand the complexity of the operator so that you can use the same operator to simultaneously represent the propagators of all the elementary fermions. That is, just as a particle moving under the Dirac propagator could be spin up or down, or a particle or antiparticle, so a particle moving under the generalized Dirac operator should be able to be any (one) of the various elementary fermions in any of their various spin and charge states. By "(one)" I mean that the particle could also be a mixture of two elementary particles, just as we use the usual Dirac propagator. This kind of mixing is best illustrated in the mixing angles of the standard model.

But the propagators are only part of the standard model. If all you had was propagators, you'd have a theory with no interactions. To add interactions, you have to add stuff that mixes up the the elementary particles. That's where the "algebra" comes in. Now what you want is for the way the particles interact with each other to have something to do with how they are packed into the generalized Dirac operator.

Now I need to admit that I don't understand the Connes technique. It's written with too much high level mathematics, and when I see that sort of thing, my feeling is that it is too far away from the physics to be important. But what I've written here is my version of what I'd like to see them doing. One of the things that really agree with me are the splitting of the fermions into left and right handed parts that interact with a mechanism very similar to all the other (i.e. gauge) interactions. This is what unification should be about.

Carl

 

[marcus]

thanks Carl!, that's good, let's not bother to say thankyou in this thread, because if lots of people contribute we will be saying thankyou thankyou all the time

RICHARD did you get an idea from that?!

Here's what I found most helpful so far

!. read understandable chunks of A.C. landmarkblockbuster paper
http://arxiv.org/abs/hep-th/0608226

!. LISTEN to A.C. talk about these ideas----he's emotive and excited, he vividly remembers experiences along the way to the paper
http://www.Newton.cam.ac.uk/webseminars/pg+ws/2006/ncg/ncgw02/
http://www.Newton.cam.ac.uk/webseminars/pg+ws/2006/ncg/ncgw02/0904/connes/
click on "sound MP3" in left hand margin
you will get some sympathetic adrenalin

for me the most exciting part of the hour talk started 45% of the way thru, so you can drag the time-bar to that if you want to skip the history of earlier attempts and cut to the chase---but the whole thing could be worthwhle even though without slides.

!. read parts of Urs n-Café trilogy
http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of.html#more
http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_1.html
http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_2.html

Other people may have found other things helpful, like some of the links Urs gives and the papers that A.C. references.
Please say if you find something helpful and give a link.

==================
corybant. n. priest, votary or attendant of Cybele, ancient goddess of nature. corybantic, a. pertaining to wild and noisy rites performed by these; ...

Cybele came before and was more primitive. She was already in Anatolia and Greece before Greeks came and brought Olympian gods, a Neolithic NATURE deity. It was already traditional to have "wild noisy rites" for the Nature godess even before Ionians and Dorians immigrated there. So they probably just continued the customs they saw. I think so anyway.

Feynman probably saw Cybele smiling at him a few times and knew what a ***** and enigmatic flirt she is. Maybe he was a corybant or had a streak of that.

If the problems of quantum gravity and unification are ever solved, even a little bit, then ghosts of old physicists will dance naked in the Cambridge streets because it is the proper and honorable thing to do.
it pays respect.

http://en.wikipedia.org/wiki/Cybele
Neolithic
Anatolia was where Turkey is, before the Turks

 

[selfAdjoint]

Anatolia was where Turkey is, before the Turks

But after the classic Greeks; it was the Byzantine name for a province that took up most of the peninsula of Asia Minor, which in classic times was split up into a lot of little states, notably those constituting Ionia.

 

[rtharbaugh1]

Ok, it seems to me now that (AHD) is a system involving an algebra (a set of rules) a Hilbert space (a place in which the rules are in effect) and an operator, which I imagine to be a group of related functions which describe the possible paths (worldlines) of a set of objects.

I was wondering if the non-commutative part applies to only one, two, or all of the above? I think I see that non-commutative algebra involves complex numbers in which for example (BC)=(-CB). But algebra, or geometry, seems to me to be a kind of language with vocabulary and grammer designed to describe objects. The space H has to be defined in such a way that objects operating by D behave according to the rules in A.

Pity we are already throwing out the manifold, when I was just getting used to the idea of objects upon which the points transform so that if you do C and then B, you have to do B and then -C to return to the original condition.

Marcus, I will try to visit the sites you linked and download or copy if possible. I have the AC paper in hand and have read it through once and studied parts of it. I currently have only an hour a day online, or a little more before the ankle biters come into play video games. If no one else signs up for the terminal I can stay on it over my reserved time limit.

I am now free to move (responsibilities here are done) and may soon be able to spend more time studying online. Life is transition.

Maybe you or Carl or sA or others can decipher from this if I am on the right track or have taken a wrong turn...

as ever,

Richard

ps I am now printing out:
http://arxiv.org/PS_cache/q-alg/pdf/9503/9503002.pdf

which I found exploring in the links Marcus gave.

R

quick thought...while reading Urs...has anyone thought about supersymetric particles as candidates for dark matter?

R

 

[rtharbaugh1]

I didn't mean to kill this thread. Is my lack of math preparation just too abysmal?

"Poincare was motivated to develop topological methods due to the analytic complexity of the many body problem," -George D. Mostow, Yale, in the Encyclopedia Britannica.

"A distinctive feature of topological arguments in analysis is that they are qualitative and non-numerical." -ed. Encyclopedia Britannica

Well, EB was the only resource on topology I had available. I suppose things have changed since the above was written in the '80's. However I don't really see, from the evidence of the Connes paper (31 Aug. 06, the topic of this thread, linked in the first post) that danceing of any kind is in order. Where are the particle masses, charges, spins, composition in terms of quarks?

I am studying the Baez and Dolan '95 paper I linked above. A very useful (to me) exposition. Thanks,

R.

 

[marcus]

..don't really see, from the evidence of the Connes paper ...that dancing of any kind is in order...

Awww Richard, I thought dancing naked in the street was ALWAYS appropriate.


BTW you didnt stall the thread.

I was just singing in a Moz. Requiem 9/11 concert yesterday and we had rehearsals running up to it. It is corybantic in its own way altho the recommended concert attire is tux.

Urs TUTORIAL NUMBER 4 finally says what the finite internal space F is.

the thread isn't stalled, I am just stunned and confused by the historical Connes + Barrett work and don't know what to say. Other people probably have their own excuses----or they may be temporarily stunned too.

Arivero recommends listening to the Connes audio and he gives TIMES during the hour talk where specific things come up so you can drag the time-bar to them and listen just to those highlights if you want.

but if your computer does not do audio that is not to worry either.

 

[arivero]

but if your computer does not do audio that is not to worry either.

Er? Computer? It is a plain MP3. Just needs some space in the ipod clone

 

[marcus]

Er? Computer? It is a plain MP3. Just needs some space in the ipod clone

I yield to Arivero on this. My experience of these media is severely limited.

I just dragged the Barrett and Connes talks to desktop, so i would always have them even if Newton Institute gets hit by an asteroid.

this "finite space" F does not seem to be defined set theoretically but only symptomatically. It is something one imagines tacking onto the 4D manifold so that the algebra will become enlarged by a certain thing made up of the complex numbers, the quaternions, and the 3x3 complex matrices.

Anything you might say about this would help----correcting my perception, confirming, or just additional comment.

 

[rtharbaugh1]

Great. I've been wondering about that MP3 thingie. I don't have one and don't even vaguely know how it works. I am using a library terminal now only have an hour a day to play.

Dancing? Sure, I have nothing against any of the Muses.

Anyway, I am still wondering how the masses, charges, and quark compositions I am familiar with in the standard model of particles come out of Conne's paper. It seems to me that he only says the calculations are complicated. But, I am a neophyte in the math world.

Also, since the calculations are so complicated, shouldn't we still be hopeing for a more elegant theory?

Thanks,

R.

 

[arivero]

Anyway, I am still wondering how the masses, charges, and quark compositions I am familiar with in the standard model of particles come out of Conne's paper.
.

The point is that the representations of the standard model groups, which in the usual theory come from a quantisation consistency requeriment, here are the only ones possible because of the algebraic setting. The higgs comes induced from the noncommutativity of the space. And the masses of the fermions, well, there have the role of defining distances in the noncommutative space, but not restriction has been found. No Koide or similar

 

[rtharbaugh1]

Thanks, Arivero.

I am working slowly through Zee's QFT nutshell, but have only got to chapter two, and Zee has said he is going to put off the complications of fields in which mass changes with position until "much later."

It is an interesting idea, that mass should change with position. I suppose we are talking about an abstact space, not the 3-space 1-time in which physics is usually formulated. I still have been having some fun trying to imagine a physical space in which mass is not invarient. Inside a black hole maybe?

I don't remember the derivation, but I do recall concluding once that mass must be an inverse of space. So if an event horizon of a black hole (or other singularity?) is taken as a zero point, then the interior of the singularity might be thought of as a universe bud, infinite in its own right, where lengths are inverses of what we normally see, and masses vary with position.

I don't know the term Koide, but will go wiki it.

Thanks,

R.

no luck with Koide at Wiki. R

 

[marcus]

...
I don't know the term Koide, but will go wiki it.

Thanks,

R.

no luck with Koide at Wiki. R

Here are Yoshio Koide's papers on arxiv
http://arxiv.org/find/hep-ph/1/au:+Koide_Y/0/1/0/all/0/1

some silliness: change a letter and get Koine
which a few centuries it was the LINGUA FRANCA of the whole eastern Mediterranean world including Egypt, the Near East and part of Italy. I was going to say something about Connes and Koine, but realized it was off-topic and changed my mind

 

[rtharbaugh1]

Thanks, Marcus and Alejandro. I guess I'm in way over my head. That's the trouble with studying independently. I'll continue to watch what you are doing and try to understand what I can.

R.

 

[arivero]

Rt it is OK if you get first to run all across Zee book. At the end it goes GUT, which is not the same that Connes, but helps to understand. You could alternate it with reading from some older technical book, as Dyson 1051 (recently available in the ArXiV) or Sakurai so you will not be harassed by modern mathematical technology.

 

[rtharbaugh1]

Thanks, A.

Quiet day at the library and I havn't been bumped off yet. I'll go look for Dyson 1051 at arXiv.

R

 

[rtharbaugh1]

I found Dyson's book on arXiv, Dyson 1951, should anyone else lookfor it.

Actually I rather want to understand the modern math complications.

Thanks,

R

 

[arivero]

I found Dyson's book on arXiv, Dyson 1951, should anyone else lookfor it.

Actually I rather want to understand the modern math complications.

Well, this is the whole point of Dyson! It is the first set of lecture note done after the Feynman integral trick plus the SchwingerDyson equations. So it containst first-hand the "modern math complications" you are asking for. As I told, the idea is to use it as a second book to relax from the reading of an updated textbook (such as, for instance, Peskin his).

 

[rtharbaugh1]

Maybe I should start a different thread to discuss the Dyson book? Don't want to muddy this one.

R.

 

[arivero]

Ok, it seems to me now that (AHD) is a system involving an algebra (a set of rules) a Hilbert space (a place in which the rules are in effect) and an operator, which I imagine to be a group of related functions which describe the possible paths (worldlines) of a set of objects.

Really we have five objects nowadays: A H D g J. The two first ones are sort as you describe. I would not call D a set of related functions, but well, it describes the differential structure and then it can be told to describe the wordlines. At least, it can be used to calculate distances. Then we have some object describing the volume form, and this is g, and some object making sure we can select real structures out of complex ones, and this is J.

 

[rtharbaugh1]

Now I imagine that g is something like a simplex, only perhaps with a dimension along the possible worldlines included. Is this the idea?

I am going to listen to Connes as Marcus suggested in an earlier post, as I have now temporary access to a broadband bubble.

R.

Seems I can't understand the talk...maybe accent problem, or microphone, or my inability to recognise the concept vocabulary.

Oh well. Back to reading.

R

 

[mhill]

the idea is , if you drop commutativity of the derivative and similar then you define

the 'derivative' of f respect to 'x' as \frac{df}{dx}= [f, A]

but the question is what is 'A' operator ??

and for the 'integral' how can you define or justify this

\int f |D| = Res_{s=0} Tr( f |D|^{-s} )

and it would be so simple as this ? you would replace the 'normal' derivative and integral by taking a Trace or commutators ??