Exploring Noncommutative Space in LQG: Gauge Networks and Spin Foams

[marcus]

This is an exciting development in LQG. They have a proposal for how to generalize the ideas of spin network and spin foam so that the network vertices are made of chunks of noncommutative space instead of ordinary space.

I'd be glad if anybody who's looked at the paper and wants to volunteer to explain any bits and pieces, or ask questions, would do so.

Basically it's just a matter of DIFFERENT LABELING of the vertices and edges of the network. A chunk of spectral (i.e. Alain Connes style) geometry is given by a rudimentary spectral triple which can be denoted by a pair (A,H) of an star-algebra A represented on a hilbertspace H. They can be finite dimensional and the fancier aspects of a spectral triple are assumed to vanish--so there is just this rudimentary label (A, H). That Alain Connes pair (A,H) is what labels a vertex in a Marcoli van Suijlekom "gauge network".

In usual LQG you have a network that is labeled by other stuff. There is an interpretation which Eugenio Bianchi (among others) has worked out where the vertex labels can be thought of as describing QUANTUM (i.e. fuzzy) POLYHEDRA. These polyhedra can't decide how their actual faces are shaped so they are blurry chunks of ordinary space.

So the difference now with the Marcolli-van Suijlekom version is the vertices of the network are labeled with blurry chunks of Alain Connes-type space. But very rudimentary because within each chunk the "Dirac operator" which serves as a substitute metric in spectral geometry is taken to be trivial.

=====semantic note======
Don't be put off by the mathematically correct term for "network" that they use. They call the network a QUIVER. Among mathematicians one often distinguishes between a directed graph (at most one edge between any pair of vertices) and a directed multigraph which can have several "arrows" or directed edges going between any pair of vertices. And some mathematicians call that a quiver.

But the LQG people were already using quivers as the basis for their spin networks---they just called the quivers by a different name. The LQG people have always been using LABELED QUIVERS to define the quantum states of geometry and to form an orthonormal basis for their Hilbert space of quantum states.

Personally I find the word "quiver" distasteful and I wish that the responsible mathematical authorities would provide a different name for directed graphs which can have multiple edges. I'm inclined to think we ought to be able to simply call them GRAPHS, as long as no confusion can arise. But I see the point---if you define a graph restrictively it will correspond to a matrix of zeros and ones---or if directed, to a matrix where the entries are -1, 0, or +1. And matrixes are the apple pie and motherhood of mathematics, so the restrictive definition of graph is forced by a mathematical sense of righteousness.

The less restrictive idea of a graph, or network, or "quiver" is two sets E and V with two maps called source and target, namely s:E→V and t:E→V

The basic message here is don't be put off by the fact that these authors, in the matter of a few terminologies, do not sound like ordinary physics folks. What they are talking about is real physics---it's just a few words like "quiver" and "functor" that sound a bit on the fancymath side.
==end of semantic note==

For me, square one of the paper comes near the top of page 10. The second paragraph there is where they define X the space of representations of a directed graph Γ in a label category C.
This label category is all the possible rudimentary chunks of noncommutative space. Crazy Alain Connes polyhedra. A "representation" is in effect a labeling. And there is a group G defined there on page 10 too, in the second paragraph. I think of this group as a kind of gauge equivalence group that is going to be factored out.

Now jump to the bottom of page 11 where they begin section 2.3 "Gauge Networks" with the words "The starting point for constructing a quantum theory is to construct a Hilbert space inspired by [a paper by Baez about spin networks]..." You can see them going for the L² space of square integrable functions that EVERYBODY uses except that it is the L² defined on this excellent space X and on X/G. This is cool and it was what was destined to happen

I have some other things to do but will try to get back to this later today. If you look at the Marcolli van Suijlekom paper (which I think is very important) please comment. I think there is a typo on page 28, in the conclusions section---will indicate later.
The link is January 3480------that is, http://arxiv.org/abs/1301.3480 .

 

[marcus]

van Suijlekom will be speaking at SISSA in April.
I expect the talk will be about this "spectralized" version of LQG. It's very interesting because it suggests a way to transplant Standard Model into the Loop picture.

But the SISSA workshop program has not been posted, so I don't know the title of his talk. I don't know how to pronounce his name either. He's Dutch. I imagine it is something like
"swoy le kum"

http://arxiv.org/abs/1301.3480
Gauge networks in noncommutative geometry
Matilde Marcolli, Walter D. van Suijlekom
(Submitted on 15 Jan 2013)
We introduce gauge networks as generalizations of spin networks and lattice gauge fields to almost-commutative manifolds. The configuration space of quiver representations (modulo equivalence) in the category of finite spectral triples is studied; gauge networks appear as an orthonormal basis in a corresponding Hilbert space. We give many examples of gauge networks, also beyond the well-known spin network examples. We find a Hamiltonian operator on this Hilbert space, inducing a time evolution on the C*-algebra of gauge network correspondences.
Given a representation in the category of spectral triples of a quiver embedded in a spin manifold, we define a discretized Dirac operator on the quiver. We compute the spectral action of this Dirac operator on a four-dimensional lattice, and find that it reduces to the Wilson action for lattice gauge theories and a Higgs field lattice system. As such, in the continuum limit it reduces to the Yang-Mills-Higgs system. For the three-dimensional case, we relate the spectral action functional to the Kogut-Susskind Hamiltonian.
30 pages

As a reminder: a "quiver" is simply the kind of directed graph that, in LQG, has always been used to define spin networks on, and has usually been called a directed graph. So what they are talking about here, although it sounds rather technical, is simply spin networks with a new type of label, a spectral or "Noncommutative Geometry" type of labeling on the nodes and links.

The convenient name they have chosen for this new type of spin network is "gauge network".
Here is Walter van Suijlekom's homepage:
http://www.math.ru.nl/~waltervs/

He is at Nijmegen. The Dutch have an interesting QG section of the physics department there. They have Renate Loll (Triangulations QG) and this guy (Spectral Geometry QG) and they recently gave a tenure-track appointment to Frank Saueressig (Reuter AS QG). A LQC guy named William Nelson is also there--formerly at Penn State and co-author with Ashtekar. Francesca Vidotto, also LQG, is there or at Utrecht, not sure which. They seem to be building up QG at Nijmegen.

http://www.math.sissa.it/workshop/quantum-geometry-and-matter
April 8 thru 12. SISSA is the Italian version of the Institute for Advanced Studies.
The workshop title is "Quantum Geometry and Matter".