Building space with quantum chunks (Bianchi Doná Speziale)

[marcus]

The nodes that have been used in LQG to build spatial geometries (almost from the start) turn out to be quantum polyhedra. MTd2 spotted this paper yesterday:

==quote==
http://arxiv.org/abs/1009.3402
Polyhedra in loop quantum gravity
Eugenio Bianchi, Pietro Doná, Simone Speziale
32 pages, many figures
(Submitted on 17 Sep 2010)
"Interwiners are the building blocks of spin-network states. The space of intertwiners is the quantization of a classical symplectic manifold introduced by Kapovich and Millson. Here we show that a theorem by Minkowski allows us to interpret generic configurations in this space as bounded convex polyhedra in Euclidean space: a polyhedron is uniquely described by the areas and normals to its faces. We provide a reconstruction of the geometry of the polyhedron: we give formulas for the edge lengths, the volume and the adjacency of its faces. At the quantum level, this correspondence allows us to identify an intertwiner with the state of a quantum polyhedron, thus generalizing the notion of quantum tetrahedron familiar in the loop quantum gravity literature. Moreover, coherent intertwiners result to be peaked on the classical geometry of a polyhedron. We discuss the relevance of this result for loop quantum gravity. In particular, coherent spin-network states with nodes of arbitrary valence represent a collection of semiclassical polyhedra. Furthermore, we introduce an operator that measures the volume of a quantum polyhedron and examine its relation with the standard volume operator of loop quantum gravity. We also comment on the semiclassical limit of spinfoams with non-simplicial graphs."
==endquote==

To put it differently, we have already seen the CDT program get interesting results just by gluing tetrahedra together in different ways (and doing the 4D analog of that, as well.)
You get curvature, using tet building blocks, the same way you get curvature in a 2D surface made of identical equilateral triangles---where you can put more or less than 6 around a given point and putting exactly 6 makes it flat there. CDT introduces quantum rules for gluing the tets, and so it gets an uncertain quantum geometry--one which has turned out to be very interesting.

So by analogy we can ask what about LQG? Does it have something like building blocks? If so, how should we imagine them?

 

[marcus]

One of the main differences between CDT and LQG is that the latter uses directed GRAPHS as a method of truncation. The graph gives a recipe for how chunks are joined to each other, and provides a concrete measure of complexity. You formulate the theory for each given graph, then pass to the limit with larger and more complicated graphs.

For example here is a graph consisting of 2 nodes joined by 4 links: ([])
To complete the description we give each link a direction, north or south in this picture.

You can interpret that graph topologically as a recipe for making the hypersphere S³. Take two tetrahedra, say a red and a blue, each having 4 triangular faces. Pair up the faces--each red Δ is paired with a blue Δ--and identify them ("glue" them together.)

In our familiar 3D world you couldn't actually glue them unless they were made of stretchy material. But thinking topologically, you can identify a red triangle with a blue. So if a small explorer in the red tet comes to a boundary triangle he will exit and pass into the blue at the corresponding point. Like PacMan who goes out at the right top and comes in at the left bottom of the screen.

The way LQG is constructed ( http://arxiv.org/abs/1004.1780 ) you build a Hilbert space H_Γ for each directed graph Γ. Then let the complexity of the graphs to to infinity.
Limiting to one finite graph and studying the quantum geometry on that graph (with all its subgraphs) makes the theory tractable---so you can handle it.

In a way, this is analogous to using a CUTOFF with a perturbation series in ordinary field theory---except the cutoff parameter is not a number or an energy, but a graph.
==================

The primitive quantum states that LQG builds on graphs are called "spin networks" and consist of "nodes" and "links". The nodes basically represent quantities of VOLUME and the links represent quantities of AREA where neighbor chunks touch. It's not that simple but that's the idea.

The links are labeled with SU(2) reps, called "spins", and the nodes are labeled with "intertwiner" mappings which connect the incoming reps with outgoing reps. This is a bit of technical math. You have to learn something about group representations, and in particular about the representations of the group SU(2).

====================

The intertwiners at a given node do themselves form a separate small Hilbert space. It is a modest finite-dimensional thing, simpler than the Hilbert for the whole graph. What it seems to me to amount to is the quantum states associated with that one node.

What Bianchi Doná Speziale seem to have done is this. They defined the idea of a quantum polyhedron with F faces. F here is just some number. Like if F = 4 we are talking about a tetrahedron because a tet has 4 faces.
They constructed a Hilbert space of quantum states of a random geometrical object.

Then they found that this Hilbert of quantum polyhedra (this uncertain indeterminant fuzzy convex F-faced object) was the same as the relevant Hilbert space of intertwiners, involving the appropriate number of in-and-out links.

Personally, just to have a name, I call this graph (()) the "pumpkin". And often LQG people call this graph (|) the "theta graph" because if you rotate it 90 degrees it resembles the Greek letter Theta Θ.
Graphs for more complicated and detailed geometries would be difficult to show typographically, but we can get some use out of these very simple ones.

ΓΔΘ

 

[marcus]

Reference to another paper by Doná and Speziale came up in other thread, here:
https://www.physicsforums.com/showthread.php?p=2891547#post2891547

They wrote a fairly good Introduction to (spin-network type of) LQG. It even mentions this polyhedra paper! At the time this paper was not yet finished, so they cited it as work "in preparation."

BTW in this polyhedra paper they use an algorithm developed by Jean Lasserre. Lasserre is a Harley-Davidson biker whose specialty is optimization/control and who is a research director at one of France's National Centers for Scientific Research (CNRS-Toulouse). I guess his general field is applied mathematics and information technology.
He developed a computer algorithm which will completely describe a polyhedron if you merely specify the unit normals of the faces and their heights from a central point.
http://homepages.laas.fr/lasserre/
This website shows several rather beautiful motorcycles, and lists Lasserre's scientific publications.

One of the things that Bianchi Doná Speziale (BDS) accomplish in this article is to show how to go from a listing of the face areas and unit normals to a complete description of the possible polyhedra.

One of their results is an equivalence mapping between the space of all polyhedra with F faces a corresponding Hilbert space of intertwiners with valence F. This means that the intertwiners which label the nodes of any LQG spin network can be be identified as polyhedra.

That equivalence they sketch already on page 2 or the paper, and then they take some of the following pages to prove it. So one can think of the node of a spin network (at least symbolically) as a polyhedron.
Another result they get is a way to calculate, from a given intertwiner, what is the volume of its associated polyhedon.