Katia’s last post ended with a provocative question motivated by Grothendieck’s description in Récoltes et Semailles of the “heart of the heart” of arithmetic geometry, namely the theory of motives. Her question was formulated like this:

——–What is the “heart of the heart” of noncommutative geometry?——-

I’ll try to explain here that there is a definite “supplément d’âme” obtained in the transition from classical (commutative) spaces to the noncommutative ones. The main new feature is that “noncommutative spaces generate their own time” and moreover can undergo thermodynamical operations such as cooling, distillation etc…

This opens up completely new ways of handling geometric spaces and our work with Matilde Marcolli and Katia Consani is just one example of potential applications to number theory. It is closely related to the Riemann zeta function and is very close in spirit to Grothendieck’s ideas on motives so that it is not out of place in the present discussion of Katia’s question.

The story starts by a qualitative distinction between spaces which comes from the classification (by von Neumann) of noncommutative algebras in types I, II and III. The commutative spaces are all of type I. When encoding a space X by an algebra A of (complex valued) functions on X one uses some structure on X to restrict the class of functions (e.g. to smooth functions on a smooth space) and the above distinction between types uses the coarsest possible structure which is the measure theory. The corresponding algebras (called von Neumann algebras) are quite simple to characterize abstractly: they are commutants in Hilbert space of some unitary representation.

Since one can take the direct sum of algebras A and B, one can mix algebras of different types. More precisely any von Neumann algebra decomposes uniquely as an integral of algebras which cannot be decomposed further and are called factors. A factor is a von Neumann algebra whose center is as small as it can be, namely is reduced to the complex numbers. The factors of type I are Morita equivalent to the complex numbers, and thus a type I factor really corresponds to the classical notion of “point” in a space X.

To understand geometrically what factors of type II and III look like, it is useful to describe the (von Neumann) algebra A associated to the leaf space of a foliated manifold: (V,F). An element T of A assigns to each leaf an operator in the Hilbert space of square integrable functions on the leaf, and it makes sense to say that T is bounded, measurable, or zero almost everywhere. The algebraic operations are done leaf per leaf, and the algebra of bounded measurable elements modulo the negligible ones is a von Neumann algebra. The simplest example corresponds to the foliation whose leaf space is the noncommutative torus. It is the foliation of the two torus by the equation “dy= a dx” in flat coordinates. The corresponding von Neumann algebra is a factor when “a” is irrational and this factor is not of type I but of type II. To obtain type III examples one can take any codimension one foliation whose Godbillon-Vey invariant does not vanish. The integrable subbundle F defining a codimension one foliation is the orthogonal of a one form v and integrability gives dv as the wedge product of v by a one form w. The Godbillon-Vey invariant is the integral over V of the wedge product of w by dw when V is compact oriented of dimension three. In essence the form w is the logarithmic derivative of a transverse volume element and the GV invariant is an obstruction to finding a holonomy invariant tranverse volume element ie one which does not change when one moves along a leaf keeping track of the way the nearby leaves are developing.

More generally the factors of type II are those which possess a trace and those of type III are those which are neither of type I nor of type II. In the foliation context, a holonomy invariant tranverse volume element allows one to integrate the ordinary trace of operators and this yields a trace on the von Neumann algebra of the foliation.

Until the work of the Japanese mathematician Minoru Tomita, very few positive results existed on type III factors. The key result of Tomita is that a cyclic and separating vector v for a factor A in a Hilbert space H generates a one parameter group of automorphisms of A by the following recipee: one considers the modulus square S*S of the closable operator S which sends xv to S(xv)=x*v for any x in A, and then raises it to the purely imaginary power “it”. Tomita showed that the resulting unitary operator normalizes A and hence defines an automorphism of A. One obtains in this way a one parameter group of automorphisms of A associated to the choice of a cyclic and separating vector v. He also showed that the phase J of the above closable operator S yields an antiisomorphism of A with its commutant A’ which coincides with JAJ. In his account of Tomita’s work, Takesaki characterized the relation between the state defined by the cyclic and separating vector v and the one parameter group of automorphisms of Tomita as the Kubo-Martin-Schwinger (KMS) condition, which had been formulated in C*-algebraic terms by the physicists Haag, Hugenholtz and Winnink.

——–What is the “heart of the heart” of noncommutative geometry?——-

I’ll try to explain here that there is a definite “supplément d’âme” obtained in the transition from classical (commutative) spaces to the noncommutative ones. The main new feature is that “noncommutative spaces generate their own time” and moreover can undergo thermodynamical operations such as cooling, distillation etc…

This opens up completely new ways of handling geometric spaces and our work with Matilde Marcolli and Katia Consani is just one example of potential applications to number theory. It is closely related to the Riemann zeta function and is very close in spirit to Grothendieck’s ideas on motives so that it is not out of place in the present discussion of Katia’s question.

The story starts by a qualitative distinction between spaces which comes from the classification (by von Neumann) of noncommutative algebras in types I, II and III. The commutative spaces are all of type I. When encoding a space X by an algebra A of (complex valued) functions on X one uses some structure on X to restrict the class of functions (e.g. to smooth functions on a smooth space) and the above distinction between types uses the coarsest possible structure which is the measure theory. The corresponding algebras (called von Neumann algebras) are quite simple to characterize abstractly: they are commutants in Hilbert space of some unitary representation.

Since one can take the direct sum of algebras A and B, one can mix algebras of different types. More precisely any von Neumann algebra decomposes uniquely as an integral of algebras which cannot be decomposed further and are called factors. A factor is a von Neumann algebra whose center is as small as it can be, namely is reduced to the complex numbers. The factors of type I are Morita equivalent to the complex numbers, and thus a type I factor really corresponds to the classical notion of “point” in a space X.

To understand geometrically what factors of type II and III look like, it is useful to describe the (von Neumann) algebra A associated to the leaf space of a foliated manifold: (V,F). An element T of A assigns to each leaf an operator in the Hilbert space of square integrable functions on the leaf, and it makes sense to say that T is bounded, measurable, or zero almost everywhere. The algebraic operations are done leaf per leaf, and the algebra of bounded measurable elements modulo the negligible ones is a von Neumann algebra. The simplest example corresponds to the foliation whose leaf space is the noncommutative torus. It is the foliation of the two torus by the equation “dy= a dx” in flat coordinates. The corresponding von Neumann algebra is a factor when “a” is irrational and this factor is not of type I but of type II. To obtain type III examples one can take any codimension one foliation whose Godbillon-Vey invariant does not vanish. The integrable subbundle F defining a codimension one foliation is the orthogonal of a one form v and integrability gives dv as the wedge product of v by a one form w. The Godbillon-Vey invariant is the integral over V of the wedge product of w by dw when V is compact oriented of dimension three. In essence the form w is the logarithmic derivative of a transverse volume element and the GV invariant is an obstruction to finding a holonomy invariant tranverse volume element ie one which does not change when one moves along a leaf keeping track of the way the nearby leaves are developing.

More generally the factors of type II are those which possess a trace and those of type III are those which are neither of type I nor of type II. In the foliation context, a holonomy invariant tranverse volume element allows one to integrate the ordinary trace of operators and this yields a trace on the von Neumann algebra of the foliation.

Until the work of the Japanese mathematician Minoru Tomita, very few positive results existed on type III factors. The key result of Tomita is that a cyclic and separating vector v for a factor A in a Hilbert space H generates a one parameter group of automorphisms of A by the following recipee: one considers the modulus square S*S of the closable operator S which sends xv to S(xv)=x*v for any x in A, and then raises it to the purely imaginary power “it”. Tomita showed that the resulting unitary operator normalizes A and hence defines an automorphism of A. One obtains in this way a one parameter group of automorphisms of A associated to the choice of a cyclic and separating vector v. He also showed that the phase J of the above closable operator S yields an antiisomorphism of A with its commutant A’ which coincides with JAJ. In his account of Tomita’s work, Takesaki characterized the relation between the state defined by the cyclic and separating vector v and the one parameter group of automorphisms of Tomita as the Kubo-Martin-Schwinger (KMS) condition, which had been formulated in C*-algebraic terms by the physicists Haag, Hugenholtz and Winnink.

The key result of my thesis (in 1972) is that the class modulo inner automorphisms of the Tomita automorphism group is in fact

**independent**of the choice of the (faithful normal) state that is used in its construction. Needless to say it is this uniqueness that allows to define**invariants**of factors. The simplest is the subgroup T(A) of R which is formed of the**periods,**namely the set of times t for which the corresponding automorphism is inner. This, together with the spectral invariant S(A), led me to the classification of type III factors into subtypes III_s for s in [0,1] and the reduction from type III to type II and automorphisms done in my thesis except for the case III_1 which was later completed by Takesaki. All of this goes back to the beginning of the seventies and will suffice for this first heart beat. It is only the beginning of a long saga which is far from over hopefully, and whose main theme is this mysterious generation of an intrinsic “time” that emerges from the noncommutativity of a von Neumann algebra. Exactly as manifolds come with a natural “smooth” measure class, a noncommutative space X generally gives rise to a von Neumann algebra A which encodes the natural measure class on X. It is thus a totally new feature of the noncommutative world that the corresponding time evolution is well defined and gives a canonical homomorphism:where the second line gives the definition of the group of outer automorphisms Out(A) of A as the quotient of the group Aut(A) of automorphisms by the normal subgroup Int(A) of inner automorphisms (which are obtained by conjugating by a unitary element of the algebra A).

When I last tried to understand the statement that type III factors come equipped with a canonical time evolution, I tried to see if there are any known physical examples where one sees this canonical time evolution realized as the actual time evolution.

My understanding was that the time evolution induced by a KMS state, which is a “thermal” state v, is that given by the Hamiltonian evolution with respect to which it is thermal.

But that crucially involves the inner part of the evolution automorphism. Right? I mean, for the true, physical time evolution, it does seem to matter precisely what the inner part of the evolution automorphism is.

After all, in the standard quantum mechanical context of finitely many degrees of freedom, time evolution is

entirelyinner.So, while I do find it very interesting that there is a crucial outer part for evolution in type III factors, and while I appreciate that this is an invariant of that type III factor, I am a little unsure about to which extent I can fairly think of that outer part alone as the time evolution.

It does not seem true that with only this outer part known the physically relevant time evolution, in the ordinary sense, is actually already specified.

Rather — it seems to me but please correct me — this outer part is a certain interesting twist on top of the ordinary time evolution “by conjugation with the exponentiated Hamiltonian” which occurs for infinite-dimensional systems. Something like a quantum anomaly in the time evolution, or the like.

Please correct me, I am not an expert on this.

Dear Urs

What you need to understand is that all the interesting stuff here occurs when the number of degrees of freedom involved is infinite. A typical example is quantum statistical mechanics (such as a spin system on a lattice). Systems occuring in quantum field theory, in the examples related to prime numbers are all involiving infinitely many degrees of freedom and are most often of type III. Very simple quantum mechanical systems are of type I, of course and the deeper structure does not appear there. It has nothing to do with anomalies.

Alain Connes wrote:

“What you need to understand is that all the interesting stuff here occurs when the number of degrees of freedom involved is infinite.”

Thanks for your reply.

Yes, I understand that. I mentioned the finite DOF case just to highlight that inner automorphisms matter for physical time evolution.

I understand that for infinitely many DOFs there may be an outer part of the time evolution and that this is canonical for type III factors.

But I am asking: in which sense does this allow us to speak of canonical time evolution, given that the — physically relevant — inner part is not canonically determined?

Dear Urs

The time evolution is as “canonical” as it can be since any noncommutative algebra has inner automorphisms. Moreover one can show that the time evolution belongs to the center of Out(A)!

If you take very simple examples as the lattice case you will find that an inner automorphism essentially ony affects what happens on finitely many lattice sites. In a simple translation invariant product situation, the hamiltonian (which generates the time evolution we are talking about) is an infinite sum of contributions of lattice sites and its essence is unaltered by a perturbation coming from finitely many terms in the sum.

It is the fact that the sum is infinite and does not belong to the algebra of observables that creates the type III behavior.

You can slightly perturb this time evolution by an inner automorphism but its overall global action on the algebra of observables will remain essentially unaltered, since it will only be changed on finitely many of the degrees of freedom. Put in other words this “time evolution” of the algebra is taking place overall, on all degrees of freedom, whereas inner automorphisms only control a total of finitely many such degrees of freedom!

You need to carefully study various examples, including foliations, the set of primes, or the case of QFT to appreciate what is going on… (and I need to get some sleep at this point)…

These are likely very dumb questions?

How are degrees of freedom [DOF] related to dimensions [D]?

Are there constraints?

For example, is a DOF the trajectory from source to target [optimal or not], constrained in and by 3D Euclidean space and time-D, from start to finish?

The trajectory from target to source would appear to be noncommutative over the same constraints?

Alain Connes explained:

“inner automorphisms only control a total of finitely many such degrees of freedom!”

Thanks, that’s very helpful to know.

Where can I find this spelled out in detail for some simple examples? Is this in your recent book?

Dear Urs

It is not really nicely spelled out anywhere, so the best is to understand the basic idea in an example without entering in the technicalities. Consider the spin system on an infinite lattice. The algebra of observables is the inductive limit of the finite dimensional algebras that come from tensor products of matrix algebras over finite subsets of the lattice. By construction these only involve finitely many lattice sites at a time. Thus an inner automorphism–since it is implemented by a unitary element of the algebra– really only “sees” finitely many degrees of freedom.

First thank you for your post,very exciting and so I would like to ask you some qualitative questions.

First you said that commutative space are encoded by a Von Neumann algebra of type I, and so I would like to know if in your noncommutative standard model you use Von Neumann algebra of type II or III. Or Von Neumann algebra must appear only at the quantization level of the noncommutative standard model and then would emerge quantum space time. “Then” your one parameter map canonicaly associated to a Von Neumann algebra would be time. But already time is relative at the classical level and “then” in my intuitive deduction it seems that it is absolute,well, but in what reference frame. In fact I have some difficulty to swallow the name time to your delta map associated to a Nc Von Neumann agebra.

(Sorry if my questions are quite fuzzy and reckless, I hope that is not to remote to the spirit of this blog)

Dear Anonymous

The space-time which allows to recover the Sandard Model coupled to gravity is of type I, since it is the product of a manifold M by a finite space F i. e. a space whose algebra of coordinates is finite dimensional. It is not at this level that we expect to get “emergent time” but rather at the level of the algebra of observables in QG. The origin of this idea comes from Carlo Rovelli who –completely independently from the KMS story– had found by reflecting about basic philosophical issues in QG that the “time we feel” (as opposed to a time coordinate in space-time) should be of thermodynamical nature and should be tied up to a thermal state: the heat bath of the relic photon radiation which breaks naturally Lorentz invariance. The real thing now is to put one’s hands on a good model for an algebra of spectral observables in QG. Some ingredients towards this are explained at the end of our forthcoming book with Matilde Marcolli. But I’d rather tell the story in one of the coming “heart beats” rather than explain it in a comment…

Hello, Alain Connes. (I think we once had dinner together along with some other people at the Legal Sea Foods restaurant at MIT after you gave a talk there.)

In this paper, the Russian mathematicians Vershik and Yakubovich give a representation of a continuous (hyperfinite) partition lattice in both the lattice of projectors of the hyperfinite factor type II_1 and in the (profinite) continuous geometry of von Neumann:

http://citeseer.ist.psu.edu/271103.html

AC, since I am not a physicist and since you are a genius with von Neumann algebras I would like to ask you if you see any potential role within mathematical physics for the hyperfinite factor type II_1 (or a role for any other hyperfinite or profinite math such as the above) ? Thanks.

Right now, the only things that come to my mind are a question from the string theorist Raphael Bousso and a conjecture from the string theorist Edward Witten. R. Bousso asked if string theory could be discretized in the following sense:

Is there a sequence of theories with finite dimensional Hilbert spaces such that string theory emerges in the infinite dimensional limit?

Also, see E. Witten’s conjecture about U(infinity)-valued Chan Paton factors in this paper:

http://arxiv.org/abs/hep-th/0007175

(By the way and for what it’s worth, both von Neumann algebras and continuous lattices can be described by category theory, and I suspect it is possible to also describe von Neumann’s continuous geometry via category theory.)

Dear Professor Connes,

Do you take requests for expository pieces with respect to older work?

I would love to have a post or series of posts explaining renormalization from the basics, and what is special about your approach. I am deeply confused about what wild fundamental groups are and why Riemann-Hilbert problems should play a role in renormalization.

This question may be naive, so I apologize in advance for that.

It is mostly based on John Baez’s week 175 where he said:

“… it’s easy to construct a type II1 factor. Start with the algebra of 1 x 1 matrices, and stuff it into the algebra of 2 x 2 matrices … This doubles the trace, so define a new trace on the algebra of 2 x 2 matrices which is half the usual one. Now keep doing this, doubling the dimension each time, using the above formula to define a map from the 2^n x 2^n matrices into the 2^(n+1) x 2^(n+1) matrices, and normalizing the trace on each of these matrix algebras so that all the maps are trace-preserving. Then take the union of all these algebras… and finally, with a little work, complete this and get a von Neumann algebra! …

the hyperfinite II1 factor is a kind of infinite-dimensional Clifford algebra.

But the Clifford algebra of 2^n x 2^n matrices is secretly just another name for the algebra generated by creation and annihilation operators on the fermionic Fock space over C^(2n) …”.

On the possibility that an underlying reason that the union/completion of 2^n x 2^n complex matrices worked nicely enough to produce the II1 factor

is because complex Clifford algebras have periodicity 2,

my question is

Would real Clifford algebra periodicity 8 allow you to construct a generalized II1 factor from unions and completions of tensor products of the real Clifford algebra Cl(8) of 16×16 real matrices ?

Tony Smith

This might appear quite unrelated to the heart bit, but we usually start from the set of rationals as a dense subset of completion of its algebraic closure, i.e. arriving finally at the complex numbers. It might be very interesting to start from a dense subset of transcendental numbers; not just rational multiples of a given transcendental number, t, but choosing from the uncountably many t’s, each t_i of which is transcendental over the extension field Q(t_1, …, t_(i-1), t_(i+1), …). Then from this dense totally transcendental subset we must recover the rationals which might actually give us yet another intuition for the existence of p-adic numbers and Galois theory on them. It dawned upon me when trying to study Littlewood’s conjecture explained by Akshay Venkatesh in the recent ams bulletin article of jan 2008. Thus, if so, is there such a natural transcendental dense subset including e and pi or is it purely a probablistic matter to choose the t’s?