Grimstrup merger of Loop with Noncommutative Geometry

[marcus]

Grimstrup combination of Loop with Noncommutative Geometry

This is something I wish we would talk more about. Some posters here like Arivero know a lot about Alain Connes Noncommutative Geometry and they could discuss the basics some. It would be good if more of us grasped the basic concepts of "spectral triple" and "dirac operator". It would help me personally since I pick up understanding from other people, perhaps we all do.

This is a breaking QG news story, from at least two directions. One direction it is coming from is that the next "Loops" conference---following Loops 05 and Loops 07---is actually going to be called QG-squared 2008 that is "Quantum GEOMETRY and Quantum Gravity 2008"
and the scientific committee helping to put together the conference has Noncommutative Geometry (NCG) people. the leadership wants to get QG people and QG people together.

The story is breaking from another direction, not as visible but I think equally interesting and important. This is the ongoing work of Hans Aastrup and Jasper Grimstrup, to combine LQG with NCG
in the course of this they make some improvements in LQG---dispensing with the foliation of spacetime for example.
(maybe you will not think it is an improvement but it does seem to make LQG conceptually simpler)

PLEASE ANYBODY IF YOU HAVE GOOD LINKS TO NCG STUFF, especially tutorial explanations, PLEASE SHARE.

All I know of at the moment, besides Connes et al writings, is Urs Schreiber notes at the "Web Cafe" he has with John Baez and another friend.
http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of.html
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

There is also the paper by John Barrett
http://arxiv.org/abs/hep-th/0608221
A Lorentzian version of the non-commutative geometry of the standard model of particle physics

there is also a NCG blog
but AFAIK mostly to find NCG sources you just look for Connes articles.

Anyway Hans and Jasper are combining LQG with NCG and the first question to ask is WHAT IS THE PURPOSE OF THIS. So I will make a separate post about it.

 

[marcus]

Motivation for INTERSECTING LQG with NCG

To find out about the direction of Aastrup and Grimstrup's work you just need to look at their two papers

http://arxiv.org/abs/hep-th/0503246
Spectral triples of holonomy loops

http://arxiv.org/abs/hep-th/0601127
Intersecting Connes Noncommutative Geometry with Quantum Gravity

One good thing about their writing is they are really clear about the motivation. In fact their writing is on the whole unusually simple and understandable, given the level they are working on.

Actually they aren't talking about a MERGER (Loop and Noncommutative remain two distinct fields) but about an OVERLAP OR INTERSECTION. A place where they work together to achieve a common goal.

The motivation is that each approach has what the other lacks.
On the one hand, LQG has a quantum state of the geometry of the universe, but it does not have the generations of the particles

On the other hand, NCG can tell you all the particles of the standard model and even predict some masses that Standard does NOT predict! This is a truly great achievement. But NCG is purely classical---nobody has ever been able to quantize it.

Loop is a quantum geometry of space and time (to which generic matter can be added).
Noncommutative gives an (unquantized) geometric catalog of all the particles.

If I am mistaken here, or oversimplifying, I hope someone will set this right. But if this is essentially correct the motivation for a merger is crystal clear.

=============
BTW for simplicity, to have just one name as a handle for the Aastrup Grimstrup collaboration, I have referred to Grimstrup as the lead author. This is because he was the one who presented their preliminary results in a talk at Loops 05. The video of that talk is online at the Albert Einstein Institute website.
http://loops05.aei.mpg.de/

A lot of the videos of Loops 05 talks are available online. BTW a remarkable thing that you see is how young the Loop community is. Most of the talks are by postdocs! This is my impression from what I have seen. On the whole I think it is a good thing.
If you go to the AEI site and can't find some talk you want to watch, please ask and I will help navigate, but the site is fairly obvious to navigate, so unless there is some trouble I won't bother with individual links.

 

[marcus]

the idea of the Dirac operator

in physics there are two very common types of operator---a linear operator on a Hilbert, and a differential operator.

a differential operator is a simple animal made up by combining some partial derivatives.
some algebraic combination of slopes and slants in different directions
and the Laplacian is an example. the heat equation and the wave equation----ooze and ripple---stuff described by letting differential operators scrutinize the present shape of things and tell you how it will look later on.
"this heap will spread and subside, Captain!" "thank you Mr. Operator, carry on"

at some point in the history of humanity Paul Dirac desired to take the square root of a differential operator. I believe it was the Laplacian. He wanted a new operator that didnt give you the Laplacian answer right away but you had to iterate it---apply whatever differentiation recipe TWICE IN A ROW and then you got the Laplacian. That was probably the original "Dirac operator".

there is also the other kind---the linear operator on a Hilbert. One can think of the hilbert as a big vector space and the operator as a MATRIX which mooshes the vectorspace around or rotates or squashes or whatever. You apply the matrix to a vector and you get another vector. WELL oft-times one can take the SQUARE-ROOT OF A MATRIX.
so that if you iterate the new matrix, and apply it twice in a row, you get the same result as once with the old matrix.

In a quantized theory, the state of the world or the experiment is a vector in a Hilbert, and everything carries over. Corresponding to the classical Laplacian differential operator there is a Laplace linear operator on the Hilbert. Corresponding to the Dirac square root differential operator there is a Dirac linear operator on the Hilbert which is the square root of the Laplacian.
They sometimes relax restrictions and make exceptions but that is a rough idea for starters. Here's Wikipedia
http://en.wikipedia.org/wiki/Dirac_operator

Jeez I wish Wikipedia would get on the ball and do one for SPECTRAL TRIPLE. That would really be a help.
For that we have Urs Schreiber's notes. Maybe someone else can give link to some others.
The Spectral Triple is the GOAL of Hans and Jasper's work! They almost have it, and they will know they are done and have arrived at the goal when they have a spectral triple which expresses Loop's quantum state of the geometry of the universe and also expresses Noncommutative's crystal algebraic mandala of the particles.

Here's a sample of the Noncommutative Geometry blog:
http://noncommutativegeometry.blogspot.com/2007_08_01_archive.html

 

[jal]

Are these two papers included in what you mean?
http://www.math.uni-hamburg.de/home/schreiber/ad.pdf
On 2D QFT - from Arrows to Disks
Schreiber
December 22, 2006
----------
http://math.ucr.edu/home/baez/fields/ptld.pdf
Parallel Transport in Low Dimensions
Urs Schreiber
with
John Baez
Jim Stasheff
Konrad Waldorf
January 5, 2007
--------

 

[marcus]

thanks for offering help, Jal.
those papers do not seem essential to the matter at hand, so I would not mention or discuss them here. But they are recent work of Schreiber and of Baez in the same general n-category and quantum mechanics field that Schreiber's NCG notes are in.
So there is certain a relationship!
The question is, is the relationship enough to justify diverting mental resources into understanding those papers when there is a lot of work to do just going directly at it.

People will differ as to what they think is the best way. Kea might go via n-categories. But Aastrup Grimstrup don't go heavy into any category biz. They focus on the goal---constructing the Hilbert (with its Dirac op defined on it)----these are more familiar everyday maths. You could have explained it to John von Neumann in the 1930s, I guess.

A spectral triple is essentially made of three things (H, D, A) and all three are good oldfashion mid-20th century things
H is a hilbertspace (quantumstates of both geometry and matter)
D a dirac-op which will be the key to the dynamical evolution because it is "like" having a laplacian that tells how the wavefunctions evolve except that it is the squareroot of a laplacian. A laplacian operator has been a STANDARD EVOLUTION MACHINE for hundreds of years, so this dirac-op is in a sense a familiar domestic animal
A is an ALGEBRA OF OTHER OPERATORS ON THE HILBERT

this too is familiar and unexotic. a Hilbert is just a vector space, not unlike ordinary R³ the 3D euclidean space. the cool thing is that it represents quantum states of the world. it is an infinite dimensional sister of R³
and WHENEVER you have a vectorspace you can have an ALGEBRA of ops defined on it, like simple stuff, like rotations for example. A bunch of matrices basically, and you can multiply two matrices together and get a matrix----you can do algebra with matrices.
So the bunch of matrices is CALLED and algebra, because you can do algebra with it just like you can with ordinary numbers.

so the Aastrup Grimstrup goal that they are working towards is a very plain everyday unexotic thing-----simply a triple of familiar objects (H, D, A) each already in use for on the order of a hundred years.

the question is how can this simple thing possibly be the goal? how can this triple of objects REPRESENT SO MUCH INFORMATION that it holds a world in it, with living and dying stars, and galaxies and clusters of galaxies. How can it hold a quantum state which is in some sense the history of the universe?

I am asking myself this.
We are listening to the opera by Bellini called PURITANI performed by Joan Sutherland and Luciano Pavarotti and Luciano is singing. I will stop for a moment.

we must understand how a spectral triple can represent a geometry of space and time.

 

[eendavid]

In Rovelli's book there is mention of the 'noncommutativity of the geometry' (last section of 6.7), where is argued that the area operators of intersecting surfaces do not commute. To me, this seems to imply that NCG is necessary for LQG (rather than a promising approach). Am I missing something?

 

[francesca]

A long time ago I asked about NCG to a friend of mine who worked on it, and I was suddenly alerted not to confuse NCG whit what I already know about NC algebra, such as the QCD one...
I'm sorry, I hadn't already gone deeper in the question, but I've thought that it would be usefull to point out it... I'm glad if some one would explain it better ;-)

 

[ensabah6]

A long time ago I asked about NCG to a friend of mine who worked on it, and I was suddenly alerted not to confuse NCG whit what I already know about NC algebra, such as the QCD one...
I'm sorry, I hadn't already gone deeper in the question, but I've thought that it would be usefull to point out it... I'm glad if some one would explain it better ;-)

So how much of the 17 unknown parameters of the SM is explained by NCG?