Friday, June 24, 2016

Noncommutative Geometry

Before jumping into the topic for today, let me say a few words about Brexit: UK put a show of monstrous selfishness and hypocrisy: after colonizing half the globe, now they complain about immigrants?

Back to physics (and math). Last time I stated that geometry requires a generalization, so what does this all mean? There are many ways one can approach this, but let's do it in historical fashion and start with the duality:

Geometry - Algebra

It is informative to remember how ancient Greeks did geometry. For them everything (including the proofs) were a geometric construction with straightedge and compass and they had no concept of coordinates.

It was not until 1637 that geometry and algebra were married by Descartes with what we now call a Cartesian coordinate system. Subsequently mathematicians started realizing that geometry and algebra are nothing but distinct languages describing the very same thing. The first geometry theorem unknown to ancient Greeks was discovered in 1899 and the proof was done by purely algebraic arguments.

The power of algebra is higher than that of geometry because it is easier to formalize abstractions in algebra. In algebra one easily encounters noncommutativity and one example is operator non-commutativity in quantum mechanics. But if algebra is dual to geometry, what kind of geometric spaces would correspond to a non-commutative algebra? What does it mean that "the algebra of coordinates is non-commutative"?

The simplest example is that of a torus. Recall the old fashion arcade games where your character exits through the right side of the screen and re-enters though the left side? Similarly if you move past the top edge you re-emerge at the bottom. Topologically this is a torus. Now suppose you move in straight line in such a way that the ratio of your horizontal and vertical speeds is an irrational number. Slice the torus with trajectory lines respecting this ratio. What you get is a pathological foliation because all "measurable functions" are almost everywhere constant and there are no non-constant continuous functions. 

Other pathological examples are: the space of Penrose tilings, deformations of Poisson manifolds, quantum groups, moduli spaces, etc.

From the mathematical side, one can explain away all those pathological cases one by one, but this is missing the forest because of the trees. The duality from above now becomes:

Quotient spaces - Non Commutative Algebra

where we replace the commutative algebra of constant functions along the classes of an equivalence relation by the noncommutative convolution algebra of the equivalence relation.

Basically it all boils down to a generalization of measure theory. It is well known that the proper way to generalize measure theory is by von-Neumann algebras, and this is how quantum mechanics enters the picture (although historically non-commutative geometry arose from quantum mechanics and the work to classify von-Neumann algebras).

Next time we are going to dive deeper into non-commutative geometry and we will encounter the Dirac operator.


  1. First of all, Britons are generally welcoming towards the exotic migrants to the extent that is considered pathological e.g. in my country. To try to present them as some racist is unfair.

    Second of all, colonization differs from the current migration wave because it's in general the subordination by a more advanced civilization over the less advanced one while the opposite thing is taking place now. The logic of these two processes is completely different and opposite to each other so saying that they must have the same outcome or that it's hypocrisy to see the difference is just plain silly. Moreover, much of the migration-related dissatisfaction is vis-a-vis Eastern Europeans like you, not the Muslims, and Britain has never colonized Eastern Europe.

    You didn't really say much about the noncommutative geometry aside from a few purely verbal cliches, did you?

    1. -Farage was asked point blank about living next to a Romanian immigrant, and yes, he is a (very polished) scumbag fear-mongering racist (
      -we are having this discussion in english, not in romanian, or czech, or german, russian, chinese, languages. Modern immigration is part of globalization.
      -The first wave of immigration comes with a crime wave because criminals are very mobile and innovative, and the first to take advantage of open borders in hitting new targets. But this is a police matter. Blame the incompetent and unprepared police, not immigration.
      -the real fear is on opening the labor market because eastern Europeans are competitive in UK.

    2. Lubos,

      Did I give you enough material to write a pro brexit pro Farage post? Consider this. I don't know the details very well because I don't follow this area, but one day I heard on NPR about a sexual harassment scandal at some university on the west coast where a well known exoplanet researcher was I think dismissed from his position due to multiple allegations of inappropriate sexual behavior. Then the next day I read your post in defense of the guy because you claimed his intrinsic value was worth more than the grad student/junior researcher and if I recall correctly your point was that in capitalism money talks.

      Now on Brexit the markets has spoken. The fear of an imaginary invasion of low payed qualified plumbers from Romania wrecked the markets. So which one is right? Money talks or "do the right thing" on the basis of nationalism for UK (or sexual harassment in the astronomer case)? Seems to me that whichever position you pick you will contradict yourself.

    3. Concerning the language. I would very much prefer to communicate in Czech. But can you speak and write Czech? But I have been doing, am still doing things for which Czechia is too small a pond so I am forced to use the contemporary lingua franca which is English.

      The usage of English doesn't mean that I am obliged or we are obliged to accept anything that you call "globalization". Immigration and globalization are completely independent things and both of them may and their aspects may be rejected by sovereign nations and sometimes individuals. 98% of Czechs find any mass immigration into our country unacceptable and we have the unquestionable right and ability to prevent it, and so do other nations. Indeed, the criminal activity is a major reason for this attitude of ours.

      Eastern Europeans may be competitive at the British job market which may be good for the companies and their competitiveness - but it's not so good for the native British workers. That's indeed one of the reasons why picked Brexit. But they have the unquestionable right to make this decision - and even the right to have this reason, whether you like the reason or not. It's about the basic sovereignty of nations, the right of the citizens to dictate the rules on their territory.

  2. Your complaints are self-contradictory. You claim that the Britons are hypocritical when they dislike immigrants and then you give me a Guardian video where all the real-world people say that they're OK with a Romanian guy.

    Be sure that some people are OK with it, some people are not, like everywhere. Moreover, it's obviously not "racism" because you belong to the same race.

    I've written dozens of pro-Brexit and pro-Farage posts. I don't think that the material you are giving me is of a sufficient quality to be used anywhere on my blog or in my thinking about the world.

    What has spoken weren't markets but brainwashed people who had some pounds or stocks and their speaking is gonna be compensated by common sense in the near future.

    1. Not racism? Good one. See here:

    2. The fact that the Guardian also calls the opposition to Polish workers "racism" doesn't make it a correct term.

  3. I want to comment about quotient spaces and noncommutative geometry. You are sort of fiddling my tune here.

    The emotional aspect of the Brexit vote is reverberating around the world. What is fueling this is pretty standard stuff. You can even see it in Exodus of the Bible, where Pharoah Rameses, who knew not Joseph, is warned against the Israelites who are "swarming." The ancient Romans were all caught in kerfuffles over non-Romans, and the Bacon Rebellion in colonial America was fostered by hysterical concerns over freed slaves, native Americans and unbonded English servants living in emergent towns taking over from the "proper English." This is really standard stuff, it is not unique to our age, and it is usually people who have parochial or puerile concerns who are most often in fits over this stuff.

    People in southern England and around London voted more "Bremain," while those in the north, Liverpool, Manchester, York, Leeds etc voted Brexit. England is almost a sort of city state, with the state being England and the city London. The irony of Brexit is the London as a major financial capital can press on in the world without the EU. Northern England has been languishing for decades, and without the EU they are not likely to do well. By contranst outside the EU London can be a sort of island version of Switzerland. Scotland is already reconsidering their vote to remain UK, and I an certain that northern Ireland issues are going to heat up. As a result UK leaving EU might mean UK = Untied Kingdom being England holding onto a grumpy Wales, and northern England becoming more like Appalachia in the United States.

    The expression "hoisted by one's own petard" seems apt here.

  4. 48% of Brits voted to remain, 52% to exit. Please remember this.

    The problem is lack of border control, where someone like Merkel can allow into Germany 1 million Black people from outside of Europe to become German citizens, then moving into the UK and changing the ethnic identity there. Replacing White role models for White children with Black role models for Black children.

    The final straw for me was Whites becoming a minority in London, and a Black Pakistani-British Muslim major being voted in while Islamabad retains a Black, Muslim, Pakistani major with 0% White-British immigration. Not exactly in the spirit of equality of globalization and diversity, is it?

    1. The only quarter I will give along these lines is with Islam, which I distinguish from Muslims. After 9/11 I read the Koran, or a translation thereof, and I will have to confess that I found it to be a rather grim reading. In fact I would assess Islam as a highly ideological religion that has brutal and horrible instructions for believers to mete out against unbelievers or "Kafirs." The rise of Islam in the world is to me of concern, and it does not admittedly give me comfort to see it infiltrate into the west. Allah in the Koran does instruct Muslims to ultimately build a one world Islamic caliphate. I don't know if they can really do that, but trouble could result.

      This differs from Muslims who are for cultural and historical reasons believers of Islam, and how in the majority of cases are somewhat nominal believers. Muslims are human beings, and I am horrified by the likes of Donald Trump who wants extra-judicial actions against Muslims. This does not mean everyone who wants to become a citizen of the US, and I would presume by similar thinking the EU, has an automatic right to such. I would also say that just as Russian emigres were heavily vetted before granted entry during the cold war, that in some ways maybe the same should apply to Mulsims. However, extra-judicial actions against any group is inherently wrong.

      As for other races coming into a western country, I would say that in some ways this is our own fault. Back in the 80-90s there was a term, DINK (double income no kids) and with the rise of conservatism has been the rise of selfishness. In fact largely that is what conservativism has all been about; greed and the obsessive need to acquire money and wealth. So with this has been a big drop in birthrates by those of European ancestry. Anyone who is over the age of 40 and has not raised a family with children has absolutely no room to complain about the decline of their ethnic group. Then as a country has economic demands to fill employment requirements extra-national workers are inevitably imported. As I said in my previous post: Hoisted by your own petard.

      In general I find that sort of racialized, or in some ways racist, language to be disturbing. There are of course reasons to be concerned about cultures being heaped on top of other cultures, but it is wrong to frame this in language that denigrates other people.

  5. Sorry for the delay, it is only now that I am reading the comments and I can reply.

  6. On non-commutative geometry more is to follow but I am taking one step at a time. The father of the domain is Alain Connes, and where other saw pathologies, he saw a rich mathematical domain. In this end this will lead to a compact formulation of the Standard Model together with non-quantized general relativity.

    1. Connes released papers at the end of 2014 "Geometry and the Quantum: Basics" . The interesting thing is this has an 8-fold periodicity, which is a cyclicity of interest to me. His work is a projective system on two Clifford algebras. I extend this to a Grassmannian, where the 2 to 1 projectivization is related to the duality between OP^1 and OP^2. The Grassmannian, the elementary one

      P_2(V^{4,4} = SU(2,2)/SU(2)xSU(2)

      is a Hermitian symmetric space. It is also related to Arkani-Ahmed's amplituhedron and BCFW.

      Connes work, where the gamma matrices are the basis for the projection, leads to spheres that are quotient spaces, such as S^n = SO(n+1)/SO(n). This has some implications for AdS spaces as well, which are quotient SO(n,2)/SO(n,1). This duality is a general form of twistor theory or SUSY twistor-string. I think Connes work captures one of the fibrations. Arkani-Ahmed captures the other. Witten, which insight from Penrose, with string/twistor is somewhere in between.

  7. On Brexit, personally I don't care how UK voted. I know that the leading force behind brexit were bogus immigration fears from Eastern Europe (in particular from Romania) and I have a problem with that and with the politicians who stoke those fears for their own personal power grab.

    Before the second world war Romania was richer than Switzerland, and ended the war richer than it started. It was Soviet Union communism which ruined the country, and this was allowed to happen by Churchill who swapped with Stalin Greece for Romania for the post war spheres of influences.

    1. Brexit is disturbing for it reflects what I see as the ascendancy of fascistic ideology. The candidacy of Donald Trump is the American version of this rubbish. The rise of undue nationalism, xenophobia and what I see at ethnic hatred is something that could eat us all up in a few decades.

      I thought the big ethnic concern was with Syrian and other Muslim immigrants. The last time I was in London I have to admit Trafalgar was filled with Arab young men, seemingly for the most part to be doing nothing in particular. This is being translated into ethnic hatred, which is horribly in appropriate.

      Romania was an ally of Nazi Germany. There was even a meeting around 1940 between the king of Romania and Adolph Hitler. Hitler wanted Romania as a staging area for his Russian campaign. The Romainian army if I remember was guarding the flanks of the Germans at Stalingrad, and they were wiped out when the Soviets encircled Stalingrad. It seems a bit odd that Romania would have been wealthier. The Soviets came through there in autumn 1944, which I bet chewed the country up some, and you can be sure the Nazis stole everything they could steal there before they left. Also the oil fields in Ploesti were bombed by the allies in 1942, which probably did not help the Romanian economy.

    2. A little history. I don't recall King Michael to have ever met Hitler, but he was only a very young guy anyway with little power. The real power rested with a military guy: Antonescu who instituted a military dictatorship. And Antonescu did met with Hitler.

      Germany required the oil and grains of Romania, and they payed fairly for everything, unlike the Russians who not only stole everything they got their hands on, but they imposed official war reparation payments and moreover they made every factory in Romania a joined Soviet-Romania company where all the profits went to Russia. This system lasted until 1964 and this resulted in 2 decades of lost development after the war.

      On Stalingrad, it is true Romanians got wiped out during the soviet counteroffensive. Still, by the end of WWII, Romania had the fourth largest army in Europe after Russia, US, and UK. On August 23 1944, King Michael did a coup: he arrested Antonescu, made peace with the allies, and turn the Romanian army around to fight Germany. The only real damage the retreating Germans did was to bomb Bucharest the very next day.

      Before the war, Romania lost 3 provinces: majority of Transylvania to Hungary, a small south part to Bulgaria, and part of Moldova to Soviet Union. This was the result of the Hitler-Stalin pact (the Ribbentrop-Molotov agreement) where Germany and Russia cut Europe in 2 spheres of influences. Romania was pro France and pro UK but France got defeated, and the logical alliance in 1942 was with Hitler to get back the territory lost to Russia. Antonescu was urged by the politicians to stop at the original Romanian border, but he pressed on all the way to Stalingrad and this was his undoing. After the war Romania lost again that part of Moldova (which became the Soviet Republic of Moldova) but got back Transylvania.