Time

I will try to describe in loose terms the steps that lead to the emergence of time from noncommutativity in operator algebras. This hopefully will answer the questions of Paul and Sirix (at least in parts) and of Urs.
First I'll explain the basic formula due to Tomita that associates to a state L a one parameter group of automorphisms. The basic fact is that one can make sense of the map x --> s(x)= L x L^{-1} as an (unbounded) map from the algebra to itself and then take its complex powers s^{it}.
To define this map one just compares the two bilinear forms on the algebra given by L(xy) and L(yx) . Under suitable non-degeneracy conditions on L both give an isomorphism of the algebra with its dual linear space and thus one can find a linear map s from the algebra to itself such that

L(yx)=L(xs(y)) for all x and y.

One can check at this very formal level that s fulfills s(ab)=s(a)s(b) :

L(abx)=L(bxs(a))=L(xs(a)s(b))

Thus still at this very formal level s is an automorphism of the algebra, and the best way to think about it is as x --> L xL^{-1} where one respects the cyclic ordering of terms in writing Lyx=LyL^{-1}Lx=LxLyL^{-1}. Now all this is formal and to make it "real" one only needs the most basic structure of a noncommutative space, namely the measure theory. This means that the algebra one is dealing with is a von-Neumann algebra, and that one needs very little structure to proceed since the von-Neumann algebra of an NC-space only embodies its measure theory, which is very little structure. Thus the main result of Tomita (which was first met with lots of skepticism by the specialists of the subject, was then succesfully expounded by Takesaki in his lecture notes and is known as the Tomita-Takesaki theory) is that when L is a faithful normal state on a von-Neumann algebra M, the complex powers of the associated map s(x)= L x L^{-1} make sense and define a one parameter group of automorphism s_L of M.

There are many faithful normal states on a von-Neumann algebra and thus many corresponding one parameter groups of automorphism s_L . It is here that the two by two matrix trick (Groupe modulaire d’une algèbre de von Neumann, C. R. Acad. Sci. Paris, Sér. A-B, 274, 1972) enters the scene and shows that in fact the groups of automorphism s_L are all the same modulo inner automorphisms!

Thus if one lets Out(M) be the quotient of the group of automorphisms of M by the normal subgroup of inner automorphisms one gets a completely canonical group homomorphism from the additive group R of real numbers

\delta: R--> Out(M)

and it is this group that I always viewed as a tantalizing candidate for "emerging time" in physics. Of course it immediately gives invariants of von-Neumann algebras such as the group T(M) of "periods" of M which is the kernel of the above group morphism. It is at the basis of the classification of factors and reduction from type III to type II + automorphisms which I did in June 1972 and published in my thesis (with the missing III _1 case later completed by Takesaki).

This "emerging time" is non-trivial when the noncommutative space is far enough from "classical" spaces. This is the case for instance for the leaf space of foliations such as the Anosov foliations for Riemann surfaces and also for the space of Q-lattices modulo scaling in our joint work with Matilde Marcolli.

The real issue then is to make the connection with time in quantum physics. By the computation of Bisognano-Wichmann one knows that the s_L for the restriction of the vacuum state to the local algebra in free quantum field theory associated to a Rindler wedge region (defined by x_1 > + - x_0) is in fact the evolution of that algebra according to the "proper time" of the region. This relates to the thermodynamics of black holes and to the Unruh temperature. There is a whole literature on what happens for conformal field theory in dimension two. I'll discuss the above real issue of the connection with time in quantum physics in another post.

What is a noncommutative space?

When we started this blog we promised to gradually build a dictionary (see here and here ) of concepts in use in NCG (= noncommutative geometry). We started with the following list

Commutative .................................................Noncommutative
functions f: X \to C .................operators on Hilbert space; elements of an algebra
pointwise multiplication fg.....................................ab (composition)
range of a function................................spectrum of an operator
Complex variable................Operator on Hilbert space
Real variable..........................Self-adjoint operator

The very first entry of this dictionary however should be about the idea of a noncommutative space. So what is a `noncommutative space', really? Let me quote here an excerpt from Alain's interview with George Skandalis to appear soon in
the EMS (European math society) journal:

"Question: What is noncommutative geometry? In your opinion, is "noncommutative geometry" simply a better name for operator algebras or is it a close but distinct field?

answer: Yes, it’s important to be more precise. First, noncommutative geometry for me is this duality between geometry and algebra, with a striking coincidence between the algebraic rules and the linguistic ones. Ordinary language never uses parentheses inside the words. This means that associativity is taken into account, but not commutativity, which would permit permuting the letters freely. With the commutative rules my name appears 4 times in the cryptic message a friend sent me recently: « Je suis alenconnais, et non alsacien. Si t’as besoin d’un conseil nana, je t’attends au coin annales. Qui suis-je ? »

Somehow commutativity blurs things. In the noncommutative world, which shows up in physics at the level of microscopic systems, the simplifications coming from commutativity are no longer allowed. This is the difference between noncommutative geometry and ordinary geometry, in which coordinates commute. There is something intriguing in the fact that the rules for writing words coincide with the natural rules of algebraic manipulation, namely associativity but not commutativity. Secondly, for me, the passage to noncommutative is exactly the passage from a completely static space in which points do not talk to each other, to a noncommutative space, in which points start being related to each other, as isomorphic objects of a category. When some points are related to each other, they will be represented by matrices on the algebraic side, exactly in the same way as Heisenberg discovered the matrix mechanics of microscopic systems. One does not go very far if one remains at this strictly algebraiclevel, with letter manipulations... and the real point of departure of noncommutative geometry is von Neumann algebras. What really convinced me that operator algebras is a very fertile field is when I realized –because of the 2 by 2 matrix trick – that a noncommutative operator algebra evolves with time! It admits a canonical flow of outer automorphisms and in particular it has “periods”! Once you understand this, you realize that the noncommutative world instead of being only a pale reflection, a meaningless generalization of the commutative case, admits totally new and unexpected features, such as this generation of the flow of time from noncommutativity. However, I don’t identify noncommutative geometry with operator algebras; this field has a life of its own. New phenomena are discovered and it is very important to study operator algebras per se -I have spent a large part of my life doing that. But on the other hand, operator algebras only capture certain aspects of a noncommutative space, and the “only” commutative von Neumann algebra is L∞[0; 1]! To be more specific, von Neumann algebras only capture the measure theory, and Gelfand’s C*-algebras the topology. And there are many more aspects in a geometric space: the differential structure and crucially the metric. Noncommutative geometry can be organized according to what qualitative feature you look at when you analyze a space. But, of course, as a living body you cannot isolate any of these aspects from the others without destroying its integrity. One aspect on which I worked with greatest intensity in recent times is a shift of paradigm which is almost forced on you by noncommutativity: it bears on the metric aspect, the measurement of distances. This is where the Dirac operator plays a key role. Instead of measuring distances effectively by taking the shortest path from one point to another, you are led to a dual point of view, forced upon you when you are doing non-commutative geometry: the only way of measuring distances in the noncommutative world is spectral. It simply consists of sending a wave from a point a to a point b and then measuring the phase shift of the wave. Amusingly this shift of paradigm already took place in the metric system, when in the sixties the definition of the unit of length, which used to be a concrete metal bar, was replaced by the wavelength of an atomic spectral line. So the shift which is forced upon you by noncommutative geometry already happened in physics. This is a typical example where the noncommutative generalization corresponds to an abrupt change even in the commutative case."

I was planing to continue with a detailed analysis of the question, but I think it is important to stop right now and answer some questions. I would particularly encourage students and others to come online and pose their questions, comments and remarks about issues discussed so far.

Le rêve mathématique

I guess one possible use of a blog, like this one, is as a space of freedom where one can tell things that would be out of place in a "serious" math paper. The finished technical stuff finds its place in these papers and it is a good thing that mathematicians maintain a high standard in the writing style since otherwise one would quickly loose control of what is proved and what is just wishful thinking. But somehow it leaves no room for the more profound source, of poetical nature, that sets things into motion at an early stage of the mental process leading to the discovery of new "hard" facts. Grothendieck expressed this in a vivid manner in Récoltes et semailles :

"L'interdit qui frappe le rêve mathématique, et à travers lui, tout ce qui ne se présente pas sous les aspects habituels du produit fini, prêt à la consommation. Le peu que j'ai appris sur les autres sciences naturelles suffit à me faire mesurer qu'un interdit d'une semblable rigueur les aurait condamnées à la stérilité, ou à une progression de tortue, un peu comme au Moyen Age où il n'était pas question d'écornifler la lettre des Saintes Ecritures. Mais je sais bien aussi que la source profonde de la découverte, tout comme la démarche de la découverte dans tous ses aspects essentiels, est la même en mathématique qu'en tout autre région ou chose de l'Univers que notre corps et notre esprit peuvent connaitre. Bannir le rêve, c'est bannir la source - la condamner à une existence occulte"

I shall try to involve on the post of Masoud about tilings and give a heuristic description of a basic qualitative feature of noncommutative spaces which is perfectly illustrated by the space T of Penrose tilings of the plane. Given the two basic tiles : the Penrose kites and darts (or those shown in the pictures), one can tile the plane with these two tiles (with a matching condition on the colors of the vertices) but no such tiling is periodic. Two tilings are the same if they are carried into each other by an isometry of the plane. There are plenty of examples of tilings which are not the same. The set T of all tilings of the plane by the above two tiles is a very strange set because of the following:
"Every finite pattern of tiles in a tiling by kites and darts does occur, and infinitely many times, in any other tiling by the same tiles''.
This means that it is impossible to decide locally with which tiling one is dealing. Any pair of tilings can be matched on arbitrarily large patches and there is no way to tell them apart by looking only at finite portions of each of them. This is in sharp contrast with real numbers for instance since if two real numbers are distinct their decimal expansions will certainly be different far enough. I remember attending quite long ago a talk by Roger Penrose in which he superposed two transparencies with a tiling on each and showed the strange visual impression one gets by matching large patches of one of them with the other... he expressed the intuitive feeling one gets from the richness of these "variations on the same point" as being similar to "quantum fluctuations". A space like the space T of Penrose tilings is indeed a prototype example of a noncommutative space. Since its points cannot be distinguished from each other locally one finds that there are no interesting real (or complex) valued functions on such a space which stands apart from a set like the real line R and cannot be analyzed by means of ordinary real valued functions. But if one uses the dictionary one finds out that the space T is perfectly encoded by a (non-commutative) algebra of q-numbers which accounts for its "quantum" aspect. See this book for more details.
In a comment to the post of Masoud on tilings the question was formulated of a relation between aperiodic tilings and primes. A geometric notion, analogous to that of aperiodic tiling, that indeed corresponds to prime numbers is that of a Q-lattice. This notion was introduced in our joint work with Matilde Marcolli and is simply given by a pair of a lattice L in R together with an additive map from Q/Z to QL/L. Two Q-lattices are commensurable when the lattices are commensurable (which means that their sum is still a lattice) and the maps agree (modulo the sum). The space X of Q-lattices up to commensurability comes naturally with a scaling action (which rescales the lattice and the map) and an action of the group of automorphisms of Q/Z by composition. Again, as in the case of tilings the space X is a typical noncommutative space with no interesting functions. It is however perfectly encoded by a noncommutative algebra and the natural cohomology (cyclic cohomology) of this algebra can be computed in terms of a suitable space of distributions on X, as shown in our joint work with Consani and Marcolli.

There are two main points then, the first is that the zeros of the Riemann zeta function appear as an absorption spectrum (ie as a cokernel) from the representation of the scaling group in the above cohomology, in the sector where the group of automorphisms of Q/Z is acting trivially (the other sectors are labeled by characters of this group and give the zeros the corresponding L-functions).
The second is that if one applies the Lefschetz formula as formulated in the distribution theoretic sense by Guillemin and Sternberg (after Atiyah and Bott) one obtains the Riemann-Weil explicit formulas of number theory that relate the distribution of prime numbers with the zeros of zeta.
A first striking feature is that one does not even need to define the zeta function (or L-functions), let alone its analytic continuation, before getting at the zeros which appear as a spectrum. The second is that the Riemann-Weil explicit formulas involve rather delicate principal values of divergent integrals whose formulation uses a combination of the Euler constant and the logarithm of 2 pi, and that exactly this combination appears naturally when one computes the operator theoretic trace, thus the equality of the trace with the explicit formula can hardly be an accident.
After the initial paper an important advance was done by Ralf Meyer who showed how to prove the explicit formulas using the above functional analytic framework (instead of the Cauchy integral).
This hopefully will shed some light on the comment of Masoud which hinged on the tricky topic of the use of noncommutative geometry in an approach to RH. It is a delicate topic because as soon as one begins to discuss anything related to RH it generates some irrational attitudes. For instance I was for some time blinded by the possibility to restrict to the critical zeros, by using a suitable function space, instead of trying to follow the successful track of André Weil and develop noncommutative geometry to the point where his argument for the case of positive characteristic could be successfully transplanted. We have now started walking on this track in our joint paper with Consani and Marcolli, and while the hope of reaching the goal is still quite far distant, it is a great incentive to develop the missing noncommutative geometric tools. As a first goal, one should aim at translating Weil's proof in the function field case in terms of the noncommutative geometric framework. In that respect both the paper of Benoit Jacob and the paper of Consani and Marcolli that David Goss mentionned in his recent post open the way.

I'll end up with a joke inspired by the European myth of Faust, about a mathematician trying to bargain with the devil for a proof of the Riemann hypothesis. This joke was told to me some time ago by Ilan Vardi and I happily use it in some talks, here I'll tell it in French which is a bit easier from this side of the atlantic, but it is easy to translate....

La petite histoire veut qu'un mathématicien ayant passé sa vie à essayer de résoudre ce problème se décide à vendre son âme au diable pour enfin connaître la réponse. Lors d'une première rencontre avec le diable, et après avoir signé les papiers de la vente, il pose la question "L'hypothèse de Riemann est-elle vraie ?" Ce à quoi le diable répond "Je ne sais pas ce qu'est l'hypothèse de Riemann" et après les explications prodiguées par le mathématicien "hmm, il me faudra du temps pour trouver la réponse, rendez vous ici à minuit, dans un mois". Un mois plus tard le mathématicien (qui a vendu son âme) attend à minuit au même endroit... minuit, minuit et demi... pas de diable... puis vers deux heures du matin alors que le mathématicien s'apprête à quitter les lieux, le diable apparaît, trempé de sueur, échevelé et dit "Désolé, je n'ai pas la réponse, mais j'ai réussi à trouver une formulation équivalente qui sera peut-être plus accessible!"

the second issue of Journal of Noncommutative Geometry

The second issue of JNCG is out. Take a look.....

in medieval architecture, signs of noncommutative geometry?

MOSAIC SOPHISTICATION A quasi-crystalline Penrose pattern at the Darb-i Imam shrine in Isfahan, Iran

A few days ago I noticed this article in NYT science section that reports on a recent paper by Lu and Steinhardt in Science (see here and here for the full article; thanks to `thomas1111'). Their abstract says: ``The conventional view holds that girih (geometric star-and-polygon, or strapwork) patterns in medieval Islamic architecture were conceived by their designers as a network of zigzagging lines, where the lines were drafted directly with a straightedge and a compass. We show that by 1200 C.E. a conceptual breakthrough occurred in which girih patterns were reconceived as tessellations of a special set of equilateral polygons ("girih tiles") decorated with lines. These tiles enabled the creation of increasingly complex periodic girih patterns, and by the 15th century, the tessellation approach was combined with self-similar transformations to construct nearly perfect quasi-crystalline Penrose patterns, five centuries before their discovery in the West''.

Interestingly enough the occurrence of quasi periodic tilings in old Persian art was also extensively commented on, last year, in Alain and Matilde's article ``A walk in the noncommutative garden" (see Section 9 on tilings). The first four pics are from their article. (see also lieven le bruyn’s weblog where the NYT article is commented at). We look forward to comments by people in NCG, operator algebras, and those working on quasi periodic crystals.