what is the importance of non-commutative geometry, especially in quantum mechanics? In all the approximations used in quantum mechanics we assumed that interaction between electrons -electrons,noyau-electons.... Are cummutative!!!!
Dear Dr. Naceur Selmane,
Your question is very broad, however, there are some important aspects to your question. So, the description of the theory of called non-commutative geometry rests on two essential points:
1. The existence of many natural spaces for which the classical set-theoretic tools of analysis, such as measure theory, topology, calculus, and metric ideas lose their pertinence, but which correspond very naturally to a non-commutative algebra.
2. The extension of the classical tools, such as to measure theory, topology, differential calculus, and Riemannian geometry, to the noncommutative situation. This extension involves, of course, an algebraic reformulation of the above tools,
but passing from the commutative to the non-commutative case is never straightforward. On the one hand, completely new phenomena arise in the non-commutative case, such as the existence of canonical time evolution for a non-commutative measure space.
As for its importance, it is used in many aspects, including the following:
1. It was used in the interplay of the geometry with the modular theory for noncommutative tori.
2. It was used in great advances on the Baum-Connes conjecture, on coarse geometry, and on higher index theory.
3. It was used in the geometrization of the pseudo-differential calculi using smooth groupoids.
4. It was used in the development of Hopf cyclic cohomology.
5. It was used in the increasing the role of topological cyclic homology in number theory, and of the lambda operations in archimedean cohomology.
6. It was used in the understanding of the renormalization group as a motivic Galois group.
7. It was used in the development of quantum field theory on noncommutative spaces.
8. It was used in the discovery of a simple equation whose irreducible representations correspond to 4-dimensional spin geometries with quantized volume and give an explanation of the Lagrangian of the standard model coupled to gravity.
9. It was used in the discovery that very natural toposes such as the scaling site provide the missing algebra geometric structure on the noncommutative adele class space underlying the spectral realization of zeros of L-functions.
In additional of other new usages such as:
- It was used in the study of strings and membranes.
- It was used in the review of the gravitational field to noncommutative models of space-time.
- It used special emphasis is placed on the case which could be considered as the noncommutative analog of a parallelizable space-time.
- It studies a rigid relation between the noncommutative structure of the space-time on the one hand and the nature of the gravitational field which remains as a ‘shadow’ in the commutative limit on the other.
https://www.alainconnes.org/docs/book94bigpdf.pdf
http://www.its.caltech.edu/~matilde/coll-55.pdf
https://arxiv.org/pdf/1910.10407.pdf
https://cds.cern.ch/record/845575/files/corfu016.pdf
Regards
Dear Dr. Naceur Selmane,
Your question is very broad, however, there are some important aspects to your question. So, the description of the theory of called non-commutative geometry rests on two essential points:
1. The existence of many natural spaces for which the classical set-theoretic tools of analysis, such as measure theory, topology, calculus, and metric ideas lose their pertinence, but which correspond very naturally to a non-commutative algebra.
2. The extension of the classical tools, such as to measure theory, topology, differential calculus, and Riemannian geometry, to the noncommutative situation. This extension involves, of course, an algebraic reformulation of the above tools,
but passing from the commutative to the non-commutative case is never straightforward. On the one hand, completely new phenomena arise in the non-commutative case, such as the existence of canonical time evolution for a non-commutative measure space.
As for its importance, it is used in many aspects, including the following:
1. It was used in the interplay of the geometry with the modular theory for noncommutative tori.
2. It was used in great advances on the Baum-Connes conjecture, on coarse geometry, and on higher index theory.
3. It was used in the geometrization of the pseudo-differential calculi using smooth groupoids.
4. It was used in the development of Hopf cyclic cohomology.
5. It was used in the increasing the role of topological cyclic homology in number theory, and of the lambda operations in archimedean cohomology.
6. It was used in the understanding of the renormalization group as a motivic Galois group.
7. It was used in the development of quantum field theory on noncommutative spaces.
8. It was used in the discovery of a simple equation whose irreducible representations correspond to 4-dimensional spin geometries with quantized volume and give an explanation of the Lagrangian of the standard model coupled to gravity.
9. It was used in the discovery that very natural toposes such as the scaling site provide the missing algebra geometric structure on the noncommutative adele class space underlying the spectral realization of zeros of L-functions.
In additional of other new usages such as:
- It was used in the study of strings and membranes.
- It was used in the review of the gravitational field to noncommutative models of space-time.
- It used special emphasis is placed on the case which could be considered as the noncommutative analog of a parallelizable space-time.
- It studies a rigid relation between the noncommutative structure of the space-time on the one hand and the nature of the gravitational field which remains as a ‘shadow’ in the commutative limit on the other.
https://www.alainconnes.org/docs/book94bigpdf.pdf
http://www.its.caltech.edu/~matilde/coll-55.pdf
https://arxiv.org/pdf/1910.10407.pdf
https://cds.cern.ch/record/845575/files/corfu016.pdf
Regards
Electrons have a permanent magnetic dipole moment. A simple magnetic dipole interaction works very well for interactions down to nuclear distances. Magnetic dipole binding is a good approximation to the more complete multipole models. And that, in turn, is a version or substitute for the full nonlinear Schrodinger solutions at very close distances. And that is just the nonlinear wave equation in 3D. And that is just an approximation to the quark gluon soup dynamics.
My point is that the magnetic dipole interaction, as simple as it is, is vector, time dependent interaction. It is "non-commutative", but when working with real geometries and interactions, simple models often fail.
Try solving the interaction of these magnetic dipole pairs, with only their magnetic dipole terms, at "nuclear" distances. You have to add rotation to get most of the bound states.
electron-electron, electron-positron, proton-antiproton, proton-proton, electron-proton, proton-neutron, neutron-neutron, electron-neutron (difficult), positron-proton, antiproton-positron. Most any particle with a "permanent" dipole moment has bound solutions.
The particle-antiparticle pairs are very interesting as they have no charge, no magnetic dipole moment. They are electromagnetically "invisible", meaning not easily detectable by electromagnetic sensors. But they have mass. I thought they would be good candidates for dark matter, and a more satisfying solution to matter-antimatter asymmetry.
I am busy tracking the global status of Covid-19, and just took a short break to read some relaxing ideas and papers. I am concerned that it has just entered the developing countries, after swiftly reaching the countries with faster global, regional and local transportation. One thing I am trying to trace, is if the US lock down, which triggered massive use of online delivery services might have been a fast mixing process that put the virus in to most counties.
There are slums, refugee camps, prisons, shanty towns and other high density, at-risk groups. New York had/has a horrible time, and they are in an "advanced" region. Think of the problems where there are methods for finding clusters, than "detection by death". So every city in the developing or poorer countries has to face the infection now. They are "lucky" because they have younger populations. But often the health care systems, while sometimes excellent, can be quickly overwhelmed.
Cell phones and the Internet might be a help. Since there are apps for reporting new cases and outbreaks. But that is also problematic because many cannot afford them, the apps are not in their language, and Elon Musk global access in not available, and likely not affordable to the most at risk.
You can sit here and talk about asymmetric interactions, and the "terribly" complex problem of keeping track of the order and consequences of two symbols placed on a line of paper mathematics where AB is not the same as BA. BFD!
Take a look. The expected death rate in any country will depend on its age and sex specific population structure, and can be estimated by their experience with similar viral and bacterial respiratory. But at a personal, social and economic level, it is shutting down the global economy, whole countries could go bankrupt, and you still have time to count angels.
The death rate for some infected and not treated is close to one percent. If the rate is smaller than that, then there are many more infected to produce the streams of cases and deaths we are observing.
Now my efforts at tracking and intervening are feeble, as I have become. I am not even going to bother trying to convince the owners and operators of this commercial site to care and do anything to organize the considerable skills, intelligence, influence and connections of the people who chat and gather here. I am just a fairly private person. I would much rather work on gravitational detectors, or simple nuclear reaction notations and methods for future engineers. I enjoy a good simulation, and dearly love noise statistics in most any form.
In December 1985, Bill Trayfors, deputy at USAID Africa Bureau, came to me when I was finishing up the the Economic and Social Database. He said, "We lost somewhere between one and two million people to famine in Africa." And, paraphrasing, "we now want you to help set up a monitoring system" so it does not happen again. So I spent two and a half years setting up the Famine Early Warning System (FEWS.net) which is still operating, and I presume working, since I have not heard of millions dying in Africa lately, except for AIDS and many other things that are apparently "too complex" for the human species to solve.
I am a fool to even try. I even heard "Just let them all die, good riddance".
So, if you guys can break yourselves away from this world-shaking conversation. Can you get up off your asses and help?
About 7000 people a day are dying. That is a stable equilibrium that the developed countries and countries with better health systems and communications has reached. There is no easy detection method, though NYC did a survey to estimate they have 2.5 million who were exposed. In the world, it is something like 15 million who are currently infectious. Most people under 40 have minor to no symptoms, but still carry the virus to others. It is separating people from their families.
Take a look. The videos I made are boring and technical. I wander all over trying to think of everything that is likely to happen. For the Internet Foundation I focus on data flows and inadequate sharing of models and data in the world, and particularly in the research communities.But I cannot even persuade web site owners to make their websites physically and business process accessible. Let alone actually content accessible.
The world has almost infinite capabilities - if people can work in groups of millions or hundreds of million -- using lossless, accountable, reliable, stable, effective and responsive Internet methods. I call it "not hard, just tedious, requiring meticulous care and effort".
World Covid Status Report 23 Apr 2020 with Pie Chart Showing Infected Not Treated
https://youtu.be/jHjGcGsuSsg
That magnetic binding approximation works. If people would quit going "hoo-ha, oh oh oh!" about "nuclear" energy. Or quit beating the non commutative dead horse, the world could have the nuclear rockets and power modules that Elon Musk needs for solar system exploration. The human species can do anything it puts is mind to, but its best and brightest let millions die. I have to count myself in that group (not the best and brightest, just the not trying to do something part).
I have almost no time left. Think about PDF format. It is suppose to be a way for people to share knowledge but it lock the contents so that only humans can read it by eyeball. The equations are not computer enabled equations, the tables not computer enabled data tools, the graphs are not computer enabled visualizations. All the "information" on the Internet is not immediately usable. We don't have interchangeable models that can be quickly and reliably connected into larger and large constrained optimization models that can solve a dinky little virus problem.
This is NOT a hard scientific problem, it is a" human scientists not sharing and putting their models in a form where they can be used in a crisis or for the human species" problems.
If anyone knows Elon Musk ask him to let whole countries use his satellites for free. "for the duration". If anyone knows who is actually making decisions at Google, ask them to index and professionally curate ALL pages found that mention "covid" "coronavirus" "corona virus" and any connected memes. Since that means the whole internet, put out 'meme profiles" for everything. And quit using Wikipedia without giving links and profiles to the original author. Wikipedia/Wikimedia get your A in gear and get all those "ink on paper" equations, models, diagrams, charts, graphs, datasets and datastreams that people are only chatting about and make them useful and complete tools for solving real problems and for working out new industries and solutions.
I could go on for hours. You can watch my tired videos. You can laugh at my stupid mistakes and feeble efforts. But please do "something".
Richard Collins, Director, The Internet Foundation
Noncommutative operators arise in Yang-Mills fields. Such fields are in the Quantum Cromodynamics (quarks, glue particles) and in the electroweak theory (mesons). They have a noncommutative Lie algebra. You can interpret it in a geometric way (there is a formal similarity between connections (like Levi-Civita commection in General Relativity and quantum gauge fields) and talk of noncommutative geometry.
First of all, it is about noncommutative operators, not noncommutative geometry.
It means dynamical variable (represented by operators) not simultneously measurable. Ie. [x, p(x)] =i hbar is the best known case.
In the classical arena, the rotation of a cube by 90 degrees about x and y axis, is
noncommutive, and you get noncommutative Poisson brackets, the anologue of
commutators. [L(x), L(y)]= i hbar L(z) gives noncommutative relations for angular momentum, associated with the rotations.
[A,B] = AB-BA , A and B operators, is the commutator.
This is the more physics, QM answer.
Non commutivity is associated with uncertainty. It is hard to get common sense.
What would you do noncommutative arithmetic if 2 times 6 were not 6 time 2?
Non commutivity is dangerous, unless in a very specific framework, QM.
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