Let me first use the opportunity to thank in writing all the participants of the Paris conference. From my side these several days of science meant a lot and the great atmosphere of friendship moved me so deeply.
I did my best to make some comments after the talks. In general it is quite difficult to have some real interaction in such circumstances since one rarely keeps in touch with the speakers after the conference. But I believe one useful side of a blog, like this one, can be to give at least a place where such comments can be written and perhaps even discussed.
Here is one example. Thanks to the great work of S. Popa, one can now control in many interesting concrete cases the algebra H(N) of correspondences which are "finite" on both sides, for a II_1 factor N. This was illustrated quite succesfully by S. Vaes. One idea which emerges then is that, exactly as in the BC-system coming from the Hecke algebra associated to an almost normal subgroup of a discrete group, the ratio of the left and right dimensions of correspondences should define a natural time evolution on the algebra H(N). This algebra is in fact defined over the rational numbers and it is a natural problem, then, to classify the KMS states, and compute the range of the rational subalgebra under zero temperature states. From a more general perspective, on the one hand recent developments have shown that the type III theory provides a natural analogue of the Frobenius in characteristic zero with a sophisticated way to take the "points over the algebraic closure of F_1". On the other hand, the theory of subfactors of Vaughan Jones is a striking extension of Galois theory to non-cocommutative group-like structures (like quantum groups, planar algebras etc...) and fits perfectly with the theory of correspondences. Time seems ripe now to merge the two sides (type III and subfactors), and in particular to explore possible relations with the other analogue of the Frobenius in characteristic zero coming from quantum groups at roots of unity.
Another very striking recent development was described in the talk of U. Haagerup on his joint work (I think it is with Magdalena Musat but am not sure, the paper is not out yet) on the classification of factors modulo isomorphism of the associated operator spaces. He gave an amazing necessary and sufficient condition for the class of the hyperfinite III_1 factor: that the flow of weights admits an invariant probability measure. (One knows that this holds for the von-Neumann algebra of a foliation with non-zero Godbillon-Vey class). This special case suggests that the general necessary and sufficient condition should be the "commensurability" of the flow of weights, and the idea of Mackey of viewing an ergodic flow as a "virtual subgroup" of the additive group R should be essential in developing the appropriate notion of "commensurability" for ergodic flows.
I was off at the beginning of the week for a short sobering trip in Sweeden (Atiyah's "Witten" talk always has a sobering effect) and heard a really interesting talk by Nirenberg which suggests that the Holder exponent 1/3 which enters as the limit of regularity for the winding number formula of Kahane corresponds to the 3= 2 + 1 of the periodicity long exact sequence in cyclic cohomology.
There is yet another conference taking place the whole week in paris, organized by Vincent Rivasseau.
Wednesday, April 25, 2007
Thursday, April 19, 2007
What is a noncommutative space? II
This is a continuation of a previous post where, to start the discussions, I quoted a portion of a recent interview with Alain (incidentally the full interview is now available and I highly recommend it to the readers of this blog). After a couple of busy weeks now I have some time to follow up on this important question. My aim is to gradually explore some of the themes discussed in the introduction to Alain's 1985 paper: "Noncommutative Differential Geometry". Without any doubts this will take many posts.
When the question ``what is a noncommutative space?" is put to a mathematician you may hear one of the following answers, among others:
1. A noncommutative space is just an associative algebra which may or may not be commutative,
2. A noncommutative space is just a C*-algebra which may or may not be commutative,
3. A noncommutative space is a von Neumann algebra which may or may not be commutative,
4. A noncommutative space is a `spectral triple',
5. A noncommutative space is an abelian category, possibly with some extra structure,
6. No one knows what a noncommutative space is.
I don't agree with item 6 and I would like to add that items 1-5 all have some elements of truth in them. The problem is that of course the very classic question " what is a space ?" is not so well defined and its answer depends on the context. So the best answer would be something like this: a space is a set endowed with some extra structure like a topology, a measure, a smooth structure, a metric, a sheaf, etc. etc...
A similar working definition can be adopted to noncommutative geometry and in fact this turns out to be essential for the whole subject. To start let us agree to call a not necessarily commutative associative algebra a noncommutative algebra. Then we can adopt the thesis:
``A noncommutative space is a noncommutative algebra possibly endowed with some extra structure"
This is a very powerful idea and can be fully justified as we shall see a bit later. For the moment
we should note that examples 1-4 in the above list all fall within the scope of this definition. For example a C* algebra is an algebra over the field of complex numbers equipped with an involution and a norm satisfying some conditions.
Proposal 5 however looks very different from the rest. This becomes relevant when we need to discuss morphisms between noncommutative spaces. Definitions 1-4 suggest that a morphism should be something like an algebra homomorphism. This is however too rigid and there may not be enough morphisms between noncommutative spaces in this rigid sense. This is in fact a purely noncommutative phenomenon to which we shall return later. For the moment it suffices to say that given an algebra A we can pass to the abelian category A-mod of representations of A. There is less information in A-mod than in A and in fact A-mod characterizes A only up to Morita equivalence but the topological information in A is not lost in this process. In many cases, like Hochschild homology, cyclic homology, and K-theory, it can be fully defined in terms of A-mod. More on this later.
Talking about a good definition of a noncommutative space inevitably brings up the issue of the role of definitions in the development of mathematics. The Orthodox view of mathematics as an exact science (generally held by non-mathematicians!) assumes that first come precise definitions of objects of study and then their main properties are explored by proving theorems
about them. In reality, however, it often happens that the main results of a subject, up to a certain point, are proved before a general agreement is reached on foundational issues. One can give so many examples of this but the development of differential geometry along with the notion of a manifold from Riemann to Weyl and Whitney is a case in point. Up to the early 1960's one of the main textbooks on differential geometry was Élie Cartan's classic, Geometry of Riemannian Spaces. Even in its second enlarged 1946 French edition Cartan says
'' The general notion of manifold is quite difficult to define with precision"
(see also this) and simply refrains from giving a general definition. He had much more important and interesting things to say!
Back to the main topic of this post, what we should do next is to justify our notion of a noncommutative space: why this is a reasonable idea and in what sense it extends the classical idea of a geometric space. This I will do in the next post.
When the question ``what is a noncommutative space?" is put to a mathematician you may hear one of the following answers, among others:
1. A noncommutative space is just an associative algebra which may or may not be commutative,
2. A noncommutative space is just a C*-algebra which may or may not be commutative,
3. A noncommutative space is a von Neumann algebra which may or may not be commutative,
4. A noncommutative space is a `spectral triple',
5. A noncommutative space is an abelian category, possibly with some extra structure,
6. No one knows what a noncommutative space is.
I don't agree with item 6 and I would like to add that items 1-5 all have some elements of truth in them. The problem is that of course the very classic question " what is a space ?" is not so well defined and its answer depends on the context. So the best answer would be something like this: a space is a set endowed with some extra structure like a topology, a measure, a smooth structure, a metric, a sheaf, etc. etc...
A similar working definition can be adopted to noncommutative geometry and in fact this turns out to be essential for the whole subject. To start let us agree to call a not necessarily commutative associative algebra a noncommutative algebra. Then we can adopt the thesis:
``A noncommutative space is a noncommutative algebra possibly endowed with some extra structure"
This is a very powerful idea and can be fully justified as we shall see a bit later. For the moment
we should note that examples 1-4 in the above list all fall within the scope of this definition. For example a C* algebra is an algebra over the field of complex numbers equipped with an involution and a norm satisfying some conditions.
Proposal 5 however looks very different from the rest. This becomes relevant when we need to discuss morphisms between noncommutative spaces. Definitions 1-4 suggest that a morphism should be something like an algebra homomorphism. This is however too rigid and there may not be enough morphisms between noncommutative spaces in this rigid sense. This is in fact a purely noncommutative phenomenon to which we shall return later. For the moment it suffices to say that given an algebra A we can pass to the abelian category A-mod of representations of A. There is less information in A-mod than in A and in fact A-mod characterizes A only up to Morita equivalence but the topological information in A is not lost in this process. In many cases, like Hochschild homology, cyclic homology, and K-theory, it can be fully defined in terms of A-mod. More on this later.
Talking about a good definition of a noncommutative space inevitably brings up the issue of the role of definitions in the development of mathematics. The Orthodox view of mathematics as an exact science (generally held by non-mathematicians!) assumes that first come precise definitions of objects of study and then their main properties are explored by proving theorems
about them. In reality, however, it often happens that the main results of a subject, up to a certain point, are proved before a general agreement is reached on foundational issues. One can give so many examples of this but the development of differential geometry along with the notion of a manifold from Riemann to Weyl and Whitney is a case in point. Up to the early 1960's one of the main textbooks on differential geometry was Élie Cartan's classic, Geometry of Riemannian Spaces. Even in its second enlarged 1946 French edition Cartan says
'' The general notion of manifold is quite difficult to define with precision"
(see also this) and simply refrains from giving a general definition. He had much more important and interesting things to say!
Back to the main topic of this post, what we should do next is to justify our notion of a noncommutative space: why this is a reasonable idea and in what sense it extends the classical idea of a geometric space. This I will do in the next post.
Friday, April 13, 2007
A new book on noncommutative geometry
It is almost ready! AN INVITATION TO NONCOMMUTATIVE GEOMETRY will be published this summer by World Scientific. The book contains expanded version of some of the lectures delivered during the International Workshop on Noncommutative Geometry at the Institute for Advanced Studies in Physics and Mathematics IPM, Tehran. This was a very successful workshop that attracted more than one hundred enthusiastic mathematics and physics grad students and faculty. Here is the program. There are a lot of pictures of this conference posted on the IPM website.
In a previous post we introduced two other new books on NCG that just appeared on the market this year.
In a previous post we introduced two other new books on NCG that just appeared on the market this year.
Friday, April 6, 2007
Birthday conference, the second week
The talks for the whole conference have been videotaped and together with the transparencies will be available soon. After a weekend break, the conference moved to the Institut Henri Poincare (IHP) at the heart of Paris on Monday. The talks on Monday were given by Dirk Kreimer ( Diffeomorphism invariance, locality and the residue: a physicist's harvest of a friend's work) ; Ali Chamseddine ( The little key to uncover the hidden noncommutative structure of space-time); Gianni Landi (Quantum Groups and Quantum Spaces are Noncommutative Geometries); Michel Dubois Violette (Moduli spaces for regular algebras) and myself (Hopf cyclic cohomology and noncommutative geometry: some new thoughts and a tribute to Alain).
On Tuesday the theme of the conference moved to von Neumann algebras and was a tribute to Alain Connes' immense legacy in the subject. The speakers were Anthony Wassermann (Non-commutative geometry and conformal field theory); Vaughan Jones (Operations on planar algebras and subfactors) ; Dietmar Bisch (Free product of planar algebras and inclusions of subfactors); Sorin Popa (Rigidity phenomena in von Neumann algebras of group actions); Stefan Vaes (Explicit computations of all bifinite Connes' correspondences for certain II_1 factors); and Dimitri Shlyakhtenko (Free entropy dimension, L^2 derivations and stochastic calculus).
Wednesday's talks were focused on the Baum-Connes conjecture, topology and C*-algebras. They were given by Vincent Lafforgue (Strengthening property (T)); Gennadi Kasparov (A K-theoretic index formula for transversally elliptic operators), Nigel Higson (The Baum-Connes conjecture and the Mackey analogy), Marc Rieffel (A new look at 'Matrix algebras converge to the sphere'); Guoliang Yu (Higher index theory of elliptic operators and noncommutative geometry); and Paul Baum (Noncommutative algebraic geometry and the representation theory of p-adic groups).
The talks on Thursday were given by Joachim Cuntz (C*-algebras associated with the ax+b-semigroup over N); Uffe Haggerup (Connes' classification of injective factors seen from a new perspective); Dan Voiculescu (Free analysis: relativistic quantum opportunities); Alain Connes (Thermodynamics of endomotives and the zeros of zeta); and Henri Moscovici (Spectral geometry of noncommutative spaces).
The last lecture on Thursday was delivered by Henri Moscovici , a long time collaborator and friend of Alain. Henri gave a nice survey of some of the developments in Connes' work in noncommutative geometry directly inspired by index theory, and the transverse geometry of foliations. As we know one of the offsprings has been the discovery of cyclic cohomology in 1981 and Hopf cyclic cohomology in 1998.
The conference on Thursday ended with a reception at Jussieu and the dedication of a birthday gift to Alain, a telescope to celebrate his far sighted visionary work!
There were four talks in the last day of the conference on Friday, all by former students of Alain.
Alain Valette talked about ``Proper isometric actions on Hilbert and Banach spaces" a piece of geometric group theory which is inspired by the Baum-Connes conjecture. Then George Skandalis talked about ``Holonomy groupoid and C*-algebra of a foliation" and showed that many aspects of the interaction between noncommutative geometry and foliation theory generalize to the set up of singular foliations.
At the end of the talk of Georges the whole audience gave him and the other organizers a big ovation for the splendid organizing job.
The talk of Marc Rosso was on ``quantum groups and algebraic combinatorics". The last talk of the conference was given by Jean-Benoit Bost who talked on "Diophantine approximation and noncommutative geometry". He sketched his conceptual geometric approach to Diophantine approximation as a theory of characteristic numbers in the context of Arakelov geometry and some intriguing analogies with noncommutative geometry and elliptic theory on noncommutative spaces.
The talks were a perfect testimony of the health of the subject of noncommutative geometry with so many interactions with other fields including operator algebras, physics, analysis, topology and number theory which is the most recent open frontier. The conference was a marvelous tribute to the breath and depth of Alain Connes' contributions to mathematics.
On Tuesday the theme of the conference moved to von Neumann algebras and was a tribute to Alain Connes' immense legacy in the subject. The speakers were Anthony Wassermann (Non-commutative geometry and conformal field theory); Vaughan Jones (Operations on planar algebras and subfactors) ; Dietmar Bisch (Free product of planar algebras and inclusions of subfactors); Sorin Popa (Rigidity phenomena in von Neumann algebras of group actions); Stefan Vaes (Explicit computations of all bifinite Connes' correspondences for certain II_1 factors); and Dimitri Shlyakhtenko (Free entropy dimension, L^2 derivations and stochastic calculus).
Wednesday's talks were focused on the Baum-Connes conjecture, topology and C*-algebras. They were given by Vincent Lafforgue (Strengthening property (T)); Gennadi Kasparov (A K-theoretic index formula for transversally elliptic operators), Nigel Higson (The Baum-Connes conjecture and the Mackey analogy), Marc Rieffel (A new look at 'Matrix algebras converge to the sphere'); Guoliang Yu (Higher index theory of elliptic operators and noncommutative geometry); and Paul Baum (Noncommutative algebraic geometry and the representation theory of p-adic groups).
The talks on Thursday were given by Joachim Cuntz (C*-algebras associated with the ax+b-semigroup over N); Uffe Haggerup (Connes' classification of injective factors seen from a new perspective); Dan Voiculescu (Free analysis: relativistic quantum opportunities); Alain Connes (Thermodynamics of endomotives and the zeros of zeta); and Henri Moscovici (Spectral geometry of noncommutative spaces).
The last lecture on Thursday was delivered by Henri Moscovici , a long time collaborator and friend of Alain. Henri gave a nice survey of some of the developments in Connes' work in noncommutative geometry directly inspired by index theory, and the transverse geometry of foliations. As we know one of the offsprings has been the discovery of cyclic cohomology in 1981 and Hopf cyclic cohomology in 1998.
The conference on Thursday ended with a reception at Jussieu and the dedication of a birthday gift to Alain, a telescope to celebrate his far sighted visionary work!
There were four talks in the last day of the conference on Friday, all by former students of Alain.
Alain Valette talked about ``Proper isometric actions on Hilbert and Banach spaces" a piece of geometric group theory which is inspired by the Baum-Connes conjecture. Then George Skandalis talked about ``Holonomy groupoid and C*-algebra of a foliation" and showed that many aspects of the interaction between noncommutative geometry and foliation theory generalize to the set up of singular foliations.
At the end of the talk of Georges the whole audience gave him and the other organizers a big ovation for the splendid organizing job.
The talk of Marc Rosso was on ``quantum groups and algebraic combinatorics". The last talk of the conference was given by Jean-Benoit Bost who talked on "Diophantine approximation and noncommutative geometry". He sketched his conceptual geometric approach to Diophantine approximation as a theory of characteristic numbers in the context of Arakelov geometry and some intriguing analogies with noncommutative geometry and elliptic theory on noncommutative spaces.
The talks were a perfect testimony of the health of the subject of noncommutative geometry with so many interactions with other fields including operator algebras, physics, analysis, topology and number theory which is the most recent open frontier. The conference was a marvelous tribute to the breath and depth of Alain Connes' contributions to mathematics.
Monday, April 2, 2007
Happy Birthday Alain!
I am in Paris since last Wednesday attending a conference in honor of Alain Connes. The conference started last Thursday at IHES by a talk by Michael Atiyah on "radical new thoughts on the foundations of physics". Atiyah has been proposing for a couple of years that perhaps one should abandon the idea of predicting the future using the formalism of differential equations and initial value problems. Somehow our short term memory of the past should play a role in predictions. This means that instead of a usual differential equation or PDE one should use a ``delayed differential equation". Other talks in the first day were by Manin (``cohomomorphisms and operads"); Katia Consani (``vanishing cycles: an adelic analogue"), Matilde Marcolli (``how noncommutative geometry looks at number theory").
The first talk on Friday was by Erling Stormer who gave a survey of entropy for operator algebras which was introduced in the seventies by Connes and Stormer in order to classify shifts of the II_1 hyperfinite factor. The second talk was by Alain who explained his approach to the standard model. He showed in particular how the work on renormalization and motivic Galois theory fits with the understanding of the extremely complex Lagrangian of gravity coupled with matter as unveiling the fine texture of space-time using the spectral action principle. He described the physics part of his forthcoming book (634 pages, joint with Matilde Marcolli) which is divided between number theory and physics half and half. At the end of the talk he explained the link between the two parts of the book based on the analogy between the electroweak phase transition in the standard model and the phase transitions which play a crucial role in the quantum statistical mechanics models involved in the approach to RH. In particular he proposed to extend the symmetry breaking to the full gravitational sector so that geometry appears only at low temperature as an emerging phenomenon.
The last talk on Friday was by Don Zagier who talked on `quantum modular forms'.
Friday ended with a wonderful piano concert (Chopin) and poetry session, with pianist Lydie Solomon at the piano and the poetess Nicole Barriere who read 20th century poetry and one of her poems.
I shall report soon on the lectures of this week.
The first talk on Friday was by Erling Stormer who gave a survey of entropy for operator algebras which was introduced in the seventies by Connes and Stormer in order to classify shifts of the II_1 hyperfinite factor. The second talk was by Alain who explained his approach to the standard model. He showed in particular how the work on renormalization and motivic Galois theory fits with the understanding of the extremely complex Lagrangian of gravity coupled with matter as unveiling the fine texture of space-time using the spectral action principle. He described the physics part of his forthcoming book (634 pages, joint with Matilde Marcolli) which is divided between number theory and physics half and half. At the end of the talk he explained the link between the two parts of the book based on the analogy between the electroweak phase transition in the standard model and the phase transitions which play a crucial role in the quantum statistical mechanics models involved in the approach to RH. In particular he proposed to extend the symmetry breaking to the full gravitational sector so that geometry appears only at low temperature as an emerging phenomenon.
The last talk on Friday was by Don Zagier who talked on `quantum modular forms'.
Friday ended with a wonderful piano concert (Chopin) and poetry session, with pianist Lydie Solomon at the piano and the poetess Nicole Barriere who read 20th century poetry and one of her poems.
I shall report soon on the lectures of this week.
Tuesday, March 20, 2007
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.
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.
Friday, March 16, 2007
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.
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.
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