Dynamics in non-commutative geometry models

[Physics Monkey]

I was reminded of this point by a recent discussion in the "Classification of manifolds ..." thread.

The question is this:

As I understand it, the NCG framework requires a Riemannian manifold. Given this, at best one could hope to obtain a Euclidean theory, right? So is it fair to say that the NCG framework does not include real dynamics?

I note that even in field theory there are many non-trivial issues about going from imaginary to real time (so that it is, as a practical matter, not possible to do so). In quantum gravity the situation is, I think, much worse.

Other related questions:

In the Riemannian framework, can the NCG program describe actions which are not real (so that the "Boltzmann weight" is not positive)?

Are there "non-relativistic" NCG models, where time is conventional but the spatial geometry is treated in an interesting way. I think I know of models which are roughly like this, e.g. quantum Hall physics, but perhaps its well studied in some other context?

 

[martinbn]

In NCG the Riemannian manifold is a figure of speech. What is meant is a C* algebra, a representation on a Hilbert space and so on, but the algebra is not some algebra of function on a manifold. As far as I know there isn't (yet) a notion of a geometric object to play the role of the manifold.

 

[arivero]

In NCG the Riemannian manifold is a figure of speech. What is meant is a C* algebra, a representation on a Hilbert space and so on, but the algebra is not some algebra of function on a manifold. As far as I know there isn't (yet) a notion of a geometric object to play the role of the manifold.

It is a bit more advanced, martin. The trick is that the conmutator [D,f] gives you the derivative of the function, and bounding it to be <1, you can use this condition to define a distance, Lipschitz distance. So the C* plus a operator D plus some axioms, give you a metric manifold. If you want just a topological manifold, you ask ||D||=1, ie all the eigenvalues of modulus one.

When the C* algebra is boolean, the algebra IS an algebra of function on some traditional manifold.

 

[marcus]

I don't recall ever seeing that the definition of NCG needs a differential manifold. Please let me know if I missed something.

Of course in special cases, for particular applications, you can build the C* algebra using a piece which consists of functions defined on a manifold. That is used in one type of NCG construction, called "almost commutative". You see that in the current approach to realizing the standard particle model in NCG.

 

[marcus]

As I understand it, the NCG framework requires a Riemannian manifold...

This is what I don't get. As I recall from a seminar with some Marc Rieffel students a few years back the axiomatic development of NCG does not involve any manifold at all. If you take the functions defined on a manifold you get a commutative C* algebra. And it certainly is an option to incorporate such a commutative piece in some particular NCG setup. But you don't have to, its not part of the basic rules.

What I recall is the general framework was defined at an abstract *algebra level, sans manifold. So what you say here, and what Torsten said in other thread, sounds weird to me.
Hope someone can explain.

 

[Eredir]

This is what I don't get. As I recall from a seminar with some Marc Rieffel students a few years back the axiomatic development of NCG does not involve any manifold at all. If you take the functions defined on a manifold you get a commutative C* algebra. And it certainly is an option to incorporate such a commutative piece in some particular NCG setup. But you don't have to, its not part of the basic rules.

What I recall is the general framework was defined at an abstract *algebra level, sans manifold. So what you say here, and what Torsten said in other thread, sounds weird to me.
Hope someone can explain.

A commutative C* algebra is isomorphic to the algebra of continuous functions on some space (here I'm skipping the technical details), so it provides only topological information.

One of the ideas of Connes' approach to non-commutative geometry is that, by adding an operator D satisfying some properties, you can also recover the metric aspect of a Riemannian manifold from a spectral point of view.

Now the question is: what kind of Riemannian manifolds can you reconstruct when you have a commutative algebra and these extra ingredients? The answer is that you get a Riemannian spin manifold and that the operator D is the Dirac operator, which can be defined for a spin manifold.

Of course what one gets depends on the conditions one requires, and indeed they can be weakened to obtained more general manifolds at the cost of introducing some extra ingredients.

 

[marcus]

A commutative C* algebra is isomorphic to the algebra of continuous functions on some space ...

Now the question is: what kind of Riemannian manifolds can you reconstruct when you have a commutative algebra ...

Hi Eredir (I like the name reversed Italian :-)),

this confirms what I remember namely there is no manifold involved in the general Connes NCG setup. And the axioms on the Hilbertspace H and the C* algebra of operators A and the operator D are chosen so that if the algebra A happens to be commutative then what you get is what would be expected from functions defined on a manifold. But that simply shows that the NCG setup is a valid generalization.

It does not require a manifold. But the geometry of a manifold is captured by the COMMUTATIVE algebra of functions defined on it. In fact the "points" of the manifold are realized abstractly as the maximal ideals of the algebra. The set of algebra elements which vanish at that "point". So, as you say, if you find in a Connes setup that the algebra is commutative (which it will not be in general) then you can expect to be able to work backwards and construct a manifold which the algebraic setup MIGHT have come from.

But that is just a side issue---confirming that the generalized differential geometry includes the more limited/primitive case of manifold geometry which it is generalizing from. The NCG setup, in general, is manifold-free.

BTW did you happen to look at the paper of Suijlekom and Marcolli? The title was something about "Gauge Networks". They could have said "spectral networks", I think, because they were talking about the analogous thing to spin networks with spectral-geometry labels instead of spin labels.

 

[Physics Monkey]

I know that in general there is no actual manifold, but in the other thread I mentioned, it was asserted that for some class of models the choice of an actual riemannian space is still relevant. Furthermore, in Connes' book, I do also recall that he made some distinction about the importance of Minkowski-like vs Euclidean-like.

But apart from this, the general question remains. To what extent is the NCG program producing a dynamical theory or some kind of Euclidean theory? I would think this question definitely makes sense without gravity and is probably also meaningful with gravity.

 

[marcus]

Hi Brian! It's exciting to find that you are interested in spectral geometry (as well as the other stuff you do, your papers discussed in Atyy's thread etc.). Have you glanced at the Suijlekom Marcolli paper on what they call "Gauge Networks"?

Here is a sample from the introduction, in case anyone might be intrigued.
==quote from Marcolli Suijlekom==
1. Introduction
We develop a formalism of gauge networks that bridges between three apparently different notions: the theory of spin networks in quantum gravity, lattice gauge theory, and the almost-commutative geometries used in the construction of particle physics models via noncommutative geometry.
The main idea behind the spin networks approach to quantum gravity is that a space continuum is replaced by quanta of space carried by the vertices of a graph and quanta of areas, representing the boundary surface between two adjacent quanta of volume, carried by the graph edges. The metric data are encoded by holonomies described by SU(2) representations associated to the edges with intertwiners at the vertices, [1], [2].
On the other hand, in the noncommutative geometry approach to models of matter coupled to gravity, one considers a non-commutative geometry that is locally a product of an ordinary 4-dimensional spacetime manifold and a finite spectral triple. A spectral triple, in general, is a noncommutative generalization of a compact spin manifold, defined by the data (A, H, D) of an involutive algebra A with a representation as bounded operators on a Hilbert space H, and a Dirac operator, which is a densely defined self-adjoint operator with compact resolvent, ...
==endquote==

 

[arivero]

http://arxiv.org/abs/math/0610418 is the relevant work for Riemannian Manifolds in NCG. There is a prequel, http://arxiv.org/abs/math-ph/9903021, and a follow up, http://arxiv.org/abs/1109.2196 The development runs along the thesis work of S Lord, who has withdrawn his earlier results, so I guess 1109.2196 is the state of the art.

Still, one must remember that Diff or Riemannian is a class more restricted that just Topological Manifolds, and so they need a lot more of axiomatic. For Topological, see the Red Book, free in Connes website.

 

[marcus]

Some of that stuff was quite interesting, thanks for the links! I learned that some mathematicians have begun calling a certain type of algebraic structures "noncommutative manifold" when it is NOT a manifold in the normal sense. Not a topological space with overlapping coordinate patches and a differential structure.

An algebraic structure can be dubbed (by these specialists) a "noncommutative Riemannian manifold" if it has what they think are the right algebraic properties.

 

[marcus]

In NCG the Riemannian manifold is a figure of speech. What is meant is a C* algebra, a representation on a Hilbert space and so on, but the algebra is not some algebra of function on a manifold. As far as I know there isn't (yet) a notion of a geometric object to play the role of the manifold.

Yes! In the papers that Arivero linked they have this purely *-algebraic concept of a "non-commutative Riemannian manifold", which is NOT constructed as functions on a genuine manifold or anything like that. So calling it "manifold" really is, as you said, a figure of speech!

The closest thing to the "geometric object to play the role of" that you were talking about is the GRAPH ON WHICH FINITE SPECTRAL TRIPLES LIVE in the Marcolli-Suijlekom paper (on the first quarter MIP poll). Strictly speaking it is a graph where there can be several directed links between any pair of vertices. Some people call this a "quiver" to distinguish from where there can only be one or zero links between any two vertices.
I do not make this distinction---it is just a directed graph (possibly with multiple links).

LQG teaches us that such a graph (labeled with whatever known geometric info) can take the place of a manifold---is a kind of "truncation" of a manifold, to some finite d.o.f.

Now in NCG it is the finite spectral triple add-on that (in the current version) realizes the standard model! Currently, in Connes standard model the algebra is a cross of a commutative (i.e. manifold based) piece with a finite dimensional non-manifold piece.

The authors M&S had what I think is a beautiful new idea: replace the manifold-based piece by a GRAPH. And label the graph with finite non-manifold spectral triples.

Let the vertices be labeled with triples (A, H, λ) and let the edges be labeled by MORPHISMS. That is by spectral triple mappings. I guess you could call them "triplo-morphisms". So what you have is a representation of the graph in the category Finite Triples, or something vaguely like that.

Here is how M&S introduce these ideas. It might be a good non-manifold way to do geometry, and a home for particle theory at the same time:

====quote Marcolli Suijlekom January 2013 paper====

A spectral triple, in general, is a noncommutative generalization of a compact spin manifold, defined by the data (A, H, D) of an involutive algebra A with a representation as bounded operators on a Hilbert space H, and a Dirac operator, which is a densely defined self-adjoint operator with compact resolvent, satisfying the compatibility condition that commutators with elements in the algebra are bounded. In the finite case, both A and H are finite dimensional: such a space corresponds to a metrically zero dimensional noncommutative space. A product space of a finite spectral triple and an ordinary manifold (also seen as a spectral triple) is known as an almost-commutative geometry. There is a natural action functional, the spectral action, on such spaces, whose asymptotic expansion recovers the classical action for gravity coupled to matter, where the matter sector Lagrangian is determined by the choice of the finite noncommutative space, [5], [6], [7], [8].

Just as the notion of a spin network encodes the idea of a discretization of a 3-manifold, one can consider a similar approach in the case of the almost-commutative geometries and “discretize” the manifold part of the geometry, transforming it into the data of a graph, with finite spectral triples attached to the vertices and morphisms attached to the edges. This is the basis for our definition of gauge networks, which can be thought of as quanta of noncommutative space. While we mostly restrict our attention to the gauge case, where the Dirac operators in the finite spectral triples are trivial, the same construction works more generally. We show that the manifold Dirac operator of the almost-commutative geometry can be replaced by a discretized version defined in terms of the graph and of holonomies along the edges.

==endquote==
http://arxiv.org/pdf/1301.3480v1.pdf

 

[marcus]

I see that on page 9 at beginning of section 2.2 a manifold is employed in some minor role.

 

[marcus]

Torsten and I were talking about NCG models in another thread. I would like to reply and continue HERE because it is more on-topic. In fact what we were talking about is precisely dynamics in non-commutative geometry models.

I'm very excited by the Suijlekom-Marcolli paper which introduces the concept of a "gauge network", have you looked at the paper?

They are aiming at the Connes standard model without any commutative part of the algebra! All the spectral triples are purely non-manifold and also they are finite dimensional i.e. in A,H,D the algebra A is finite dimensional and so is the hilbertspace.

I remembered to read the paper in January but now I read it again. In the paper there was a clear motivation. Everything looks very interesting. But I'm not surprised that the authors obtained the Higgs sector. A spin network with a NCG at every vertex...
One thing troubles me: the spin network was usually constructed to be related to the triangulation of the 3-manifold. I know of only one approach to simplify a triangulatio of a 3-manifold: use a 2-complex (a so-called spine of a 3-manifold). It is known that every 3-manifold can be constructed by using spines but nothing less (including no 1-complex like a spin network).
But I have to think more carefully about it.

The two-complex generalization of a graph( i.e. a one-complex) is called a foam.
The DYNAMICS of changing states of geometry (described e.g. by labeled graphs called spin networks) is described by labeled two-complexes dubbed spin FOAMS.
So in the new Marcolli Suijlekom picture, where we have gauge networks, the dynamics might be described by what they call gauge foams. They propose a definition for gauge foams.

See page 28 for the definition of gauge foam.

A gauge foam (as you could expect) has its vertices labeled by spectral triples (A, H, λ)

 

[marcus]

Basically what Marcolli&Suijlekom are doing is to take the LQG spin-network setup and substitute NCG labels for the spin labels. Likewise with spin-foams. The motive is to handle geometry and MATTER.

Recall that in LQG, spin network nodes are quanta of volume, and the links are quanta of area where the adjacent volumes meet.
Spin network describes quantum state of 3d geometry.
Spin foam describes dynamics of changing geometry.

Now, M&S use the word "gauge" to designate the NCG labeling they are switching over to, that uses finite dimensional spectral triples The basic idea is:
Gauge network describes quantum state of 3d geometry and MATTER.
Gauge foam describes dynamics of changing geometry and MATTER.

==quote Marcolli&Suijlekom==
4.3. A proposal for gauge foams.
We propose a noncommutative generalization of spin foams as higher-dimensional analogues of spin networks. The construction of gauge foams is such that taking a “slice” of the spin foam at a given “time” will then produce a gauge network. With this in mind, it is intuitively clear how gauge foams encode the dynamics of quantum noncommutative spaces, while gauge networks give the kinematics.
A natural way to arrive at spin foams is by computing the partition function for lattice gauge fields [17], expressing probability amplitudes as sums over spin foams (on a fixed graph/lattice). Their computation is essentially based on the fact that the path integral depends only on the ‘plaquette product’ of group elements assigned to the four edges of a plaquette. We already encountered this before in Theorem 12.
For simplicity, we only propose a definition for closed gauge foams. The generalization to gauge foams between two gauge networks is straightforward, and can be done as in [10]. In the following, by a two-complex we mean a simplicial complex with two-dimensional faces, one-dimensional edges, and zero-dimensional vertices, endowed with the usual boundary operator ∂, which assigns to a face the formal sum of its boundary edges with positive or negative sign according to whether the induced orientation from the face agrees or not with the orientation of the corresponding edge.
==endquote==
The definition that follows involves several equations which, to spare me the trouble of copying in LaTex I hope you will read in the original paper:
http://arxiv.org/pdf/1301.3480v1.pdf

I think the Dutch name is pronounced "SWEE LE KUM", anybody know for sure?

 

[Physics Monkey]

Dear all,

Thanks for your replies so far. However, I think we've gotten a little sidetracked about the manifold/no manifold issue. The central question remains as to what kind of theory does the NCG framework produce, e.g. real time dynamical, a euclidean theory, or something else entirely.

Presumably all this talk about the non-commutative standard model implies some limit in which one must approximately recover qft + gravity, so we must be able to answer what sort of formulation of qft + gravity (e.g. real time or euclidean) is obtained in this context.

 

[marcus]

...what kind of theory does the NCG framework produce, e.g. real time dynamical, a euclidean theory, or something else entirely...[?]

==quote from abstract summary of Marcolli Suijlekom==
...We find a Hamiltonian operator on this Hilbert space, inducing a time evolution on the C*-algebra of gauge network correspondences.
==endquote==

In general I believe dynamics in NCG framework, because it has the so called "spectral action" boils down to a Lagrangian. IIRC in application to Standard Model the Lagrangian, at least formally, includes both gravity and matter. I will check that later, have to go out and run some errands now. Arivero doubtless knows with certainty and perhaps may answer before I get back.

 

[Physics Monkey]

==quote from abstract summary of Marcolli Suijlekom==
...We find a Hamiltonian operator on this Hilbert space, inducing a time evolution on the C*-algebra of gauge network correspondences.
==endquote==

In general I believe dynamics in NCG framework, because it has the so called "spectral action" boils down to a Lagrangian. IIRC in application to Standard Model the Lagrangian, at least formally, includes both gravity and matter. I will check that later, have to go out and run some errands now. Arivero doubtless knows with certainty and perhaps may answer before I get back.

OK, good, I didn't notice this.

They speak about both Hamiltonians and a euclidean-like actions. The euclidean statements I think I am comfortable with, but I don't understand the meaning of this hamiltonian acting on the algebra. Is this a conventional quantum theory with time evolution or something else, because the euclidean lattice theory is indeed not a quantum theory with conventional time evolution.

 

[marcus]

OK, good, I didn't notice this.

They speak about both Hamiltonians and a euclidean-like actions. The euclidean statements I think I am comfortable with, but I don't understand the meaning of this hamiltonian acting on the algebra. Is this a conventional quantum theory with time evolution or something else, because the euclidean lattice theory is indeed not a quantum theory with conventional time evolution.

I'd be interested in Arivero's response. MarcolliSuijlekom's language is suggestive (but this is only their first paper). They start by recalling some things about the "almost commutative" construction:

==quote==
A product space of a finite spectral triple and an ordinary manifold (also seen as a spectral triple) is known as an almost-commutative geometry. There is a natural action functional, the spectral action, on such spaces, whose asymptotic expansion recovers the classical action for gravity coupled to matter, where the matter sector Lagrangian is determined by the choice of the finite noncommutative space, [5], [6], [7], [8].

Just as the notion of a spin network encodes the idea of a discretization of a 3-manifold, one can consider a similar approach in the case of the almost-commutative geometries and “discretize” the manifold part of the geometry, transforming it into the data of a graph, with finite spectral triples attached to the vertices and morphisms attached to the edges.
...
...
We show that the manifold Dirac operator of the almost-commutative geometry can be replaced by a discretized version defined in terms of the graph and of holonomies along the edges.

In lattice gauge theory, the Wilson action defined in terms of holonomies recovers, in the continuum limit, the Yang–Mills action, [9]. We show that the spectral action of the [discretized] Dirac operator on a gauge network recovers the Wilson action with additional terms that give the correct action for a lattice gauge theory with a Higgs field in the adjoint representation, [11], [12].
==endquote==

spectral action → Wilson action → Y-M action

I assume that because this is the first paper of a research line there must be unfinished business, and the above is a kind of sketch of a program with probably some missing pieces to be filled in.
(Like the "euclidean" business you mentioned, that Arivero also talked about. The Riemannian instead of pseudo-Riemannian issue.) But FWIW here is what they seem to be saying. And I have a lot of respect for both authors so I tend to take it seriously.

The almost-commutative geometry (a conventional manifold piece product with noncommutative piece) spectral action "recovers the classical action".
They say they can "discretize" the manifold-part of the almost-commutative geometry, as a graph.
Then they say they can "replace the Dirac operator by a discretized version" defined on the graph.
And that the [discretized] Dirac operator "recovers the Wilson action".

http://arxiv.org/pdf/1301.3480v1.pdf
"GAUGE NETWORKS IN NONCOMMUTATIVE GEOMETRY"

 

[marcus]

One way to think of this paper is that the non-trivial mathematical core is the Artin–Wedderburn_theorem
http://en.wikipedia.org/wiki/Artin–Wedderburn_theorem.
Any finite dimensional C* algebra is a direct sum of MATRIX algebras over the complex numbers.

This acquires significance from the Connes idea that you can inject matter into geometry by crossing the geometry with a finite dimensional C* algebra. (exactly what Artin-Wedderburn classified.)

They also say they were inspired by a John Baez paper:
"Spin Network States in Gauge Theory" http://arxiv.org/abs/gr-qc/9411007
Another along the same lines which they do not cite is:
"Spin Networks in Nonperturbative Quantum Gravity"
http://arxiv.org/abs/gr-qc/9504036

So their basic idea is to choose a graph as a finite d.o.f. truncation of geometry, and put finite dimensional spectral triples (chunks of noncommutative geometry+matter) at every node of the graph. Beyond that you can more or less just follow your nose and define whatever is needed so that things make sense.

 

[Physics Monkey]

I would still like to understand better the physics here. In particular, in the paper http://arxiv.org/abs/hep-th/0610241 on page 3 below equation 1.4 the authors state that the full euclidean action is recovered in some asymptotic sense. So I continue to wonder to what extent the ncg-enriched standard model is a dynamical theory.