What Are the Definitions of Wilson Loops and Spin Networks in Physics?

[marlon]

Hi, cam anyone give me a clear definition of the following two things

1) Wilson loop
2) Spin network (invented by Penrose, right?)

What do these quantities fysically mean?

PS : can anyone explain me what the term LOOP means in Loop quantum gravity. To what does it refer ?

regards
marlon

 

[jtolliver]

Spin networks are graphs(In the sense of graph theory) whose edges are labeled by spins and whose vertices are labeled by intertwining operators(mappings between the representations). For instance a vetex can have 3 edges coming into it, 2 of them having spin 1/2, and 1 having spin 1. In that case the intertwining operator could be the mapping from the spin 1/2 representations to the spin 1 representation, given by $\psi_1^+ \beta \gamma^\mu \psi_2$.

 

[marcus]

Hi, cam anyone give me a clear definition of the following two things

1) Wilson loop
2) Spin network (invented by Penrose, right?)

What do these quantities fysically mean?

PS : can anyone explain me what the term LOOP means in Loop quantum gravity. To what does it refer ?

regards
marlon

http://www.wordiq.com/definition/Wilson_loop

marlon, let's think of this far-fetched analogy. suppose you have a particle that can be located in the interval 0 to 1. Call this the "configuration space".
then you make L²[0,1] the Hilbert space of sqr. integrable cx valued functions psi(x) defined on interval [0,1] and you call this the "state space"

here is the analogy: in quantum gravity the configuration space is all the possible geometries that the universe can have----it could be all the possible metrics on some manifold

but it could also be all the possible connections on the manifold.
Let's take the configuration space to be the set X of all the connections x.

Now we have to define FUNCTIONS ON X THE SPACE OF CONNECTIONS.
We have to think of a function psi(x) which, given a connection x, gives you back some complex number psi(x).

Mr. Wilson, in the 1970s, working on a different problem in field theory, invented a laughably simple way of defining a function on a space of connections. You just choose a LOOP and a character of the group G of the connection. And you just do the "holonomy" which is you run around the loop with that connection giving you group elements and you take the representation of the group element, or rather the character, and you get a complex number.

I'm being vague about whether you specify a group character or a group representation or a Lie Algebra representation even. You just specify a loop and whatever it takes to get a number when you travel around that loop.

So this is now a function psi(x) and it sort of feels out the geometry by feeling what the connection does as you run around loops. It is a really good start for a STATE SPACE defined on the configuration space of all possible geometries.

And if you take linear combinations of all the possible Wilson loops then that is a fine hilbert space, or at least you can get a hilbert space from it by a little work.

then the trouble is, the multiple combinations of loops are too many, there is a lot of redundancy and they are not linear independent. So how do you get a BASIS for the hilbert space.

that is where spin networks come in. Instead of loops you take graphs or networks and label the links by representations of the gauge group.
A basis can be constructed out of these things. It gets rid of some redundancy because there are a bunch of different combinations of Wilson loops that lead to the same spin network if you plaster them together.

Unfortunately I have to go! When I have more time I will try to get some links to introductory articles about LQG that do a better job than I have done here just off the top of my head. But I don't have time to get you authoritative links right now. So I just post this as a beginning of a response. Must go. (a Berthold Brecht play this afternoon!)

 

[marlon]

http://www.wordiq.com/definition/Wilson_loop

marlon, let's think of this far-fetched analogy. suppose you have a particle that can be located in the interval 0 to 1. Call this the "configuration space".
then you make L²[0,1] the Hilbert space of sqr. integrable cx valued functions psi(x) defined on interval [0,1] and you call this the "state space"

First of all, thanks for your reply marcus, i can see why you are the physics expert of 2003. But if have some difficulties understanding your analogy. I studied physics but i never took advanced classes in stuff like topology. Can you please elaborate (after your play ofcourse) on this state-space-thing. The way I see it , it is a bunch of functions psi(x) that are integrable right? What is this cx you are referring to ?

 

[marlon]

http://www.wordiq.com/definition/Wilson_loop

here is the analogy: in quantum gravity the configuration space is all the possible geometries that the universe can have----it could be all the possible metrics on some manifold

but it could also be all the possible connections on the manifold.
Let's take the configuration space to be the set X of all the connections x.

Now we have to define FUNCTIONS ON X THE SPACE OF CONNECTIONS.
We have to think of a function psi(x) which, given a connection x, gives you back some complex number psi(x).

Mr. Wilson, in the 1970s, working on a different problem in field theory, invented a laughably simple way of defining a function on a space of connections. You just choose a LOOP and a character of the group G of the connection. And you just do the "holonomy" which is you run around the loop with that connection giving you group elements and you take the representation of the group element, or rather the character, and you get a complex number.

Now i am lost. What are these connections ? Are they functions that connect between two points on a manifold ? Or are they just the arguments of such functions like x in y=f(x). Perhaps i am getting this completely wrong, if that is the case sorry, but i would really like to learn these things. I have looked on the encyclopedia on the net but they do not bring much clarity though...

What is a holonomy and what is a character of the group G of the connection. I really don't get anything of this, sorry. Would it be possible to explain these mystery things to me?


regards
marlon

 

[marcus]

I didnt realize that you would first want a definition of a connection.
sorry

If anyone else is reading this thread and wants to help out,
what would you think about recommending this online book
as a reference?

http://xxx.lanl.gov/abs/math-ph/9902027

Preparation for Gauge Theory
George Svetlichny 97 pages.

"Class lecture notes at a beginning graduate level on the mathematical background needed to understand classical gauge theory. Covers group actions, fiber bundles, principal bundles, connections, gauge transformations, parallel transport, curvature, covariant derivatives, pseudo-riemannian manifolds, lagrangians, clifford algebras, spin bundles, and the Dirac operator. Requires an elementary knowledge of groups, manifolds, lie groups and algebras, and mutlilinear algebra."

marlon, I am not now sure what reading is best to suggest
I would like to find on the web some reading that defines a connection for you and gives some intuition about it.

this book by Svetlichney is much more than you need. On the other hand I cannot now think of an online source that tells you only what you need.

The connection is a regular part of differential geometry
You can think of it as a mechanism for transporting tangent vectors along a path from one point of a manifold to another.
If you specify a path from point A to point B, the connection allows you to compute how a tangent vector will turn as you move from A to B.
The connection idea is not special to LQG but belongs to several fields of differential geometry and physics.

I will try to suggest something---we should check Eric Weisstein's mathworld and wikipedia to see what they have.

 

[marlon]

hi marcus,

i looked at your reference and i can say that i am used to most of it's content. I now "remember" the concept of these connections. This may sound strange but it is just that i am used to another name. I took most of my classes in Dutch at the university, so sometimes i will have some difficulties with the English terms. But thanks to your text i will cope.


I am still reading on LQG.

thanks a lot and until later

marlon

 

[marlon]

ok ,marcus (how was the Brecht play ? were you attending or performing) let me see if i am getting this straight.

Doing the holonomy means the following : In order to check out the properties of a manifold (let's say a sphere) you take a groupelement, eg a vektor.

Then we perform a sort of parallel transport along some cirkle on the sphere (that's the Wilsonloop right ?) and we look how the connection between the original vektor and the vektor we are transporting varies. By studying this variation we can deduce info of the manifold we are on, ofcourse without being able to see the entire manyfold at once. This is like saying how can i check whether the Earth is flat or round while my local space is flat, right?

Them vektors we are transporting are parallel-vectors to the original one. So is it possible to describe a manifold by only using the tangent-base of this manifold.

We can use fiberbundels in order to go from the manifold to the tangent space, so that all parallel vectors in the manifold will be projected on a different subspace of the tangent-space. So that different points of the tangent-space correspond to different parallel-classes of vektors of the original manifold. Is this vision somewhat correct ?

regards
marlon

 

[marcus]

ok ,marcus (how was the Brecht play ? were you attending or performing) let me see if i am getting this straight.

attending. The troup of idealistic young actors was inventive and skillful
so they made the play (Chalk Circle) a dramatically effective performance and my wife said she enjoyed it----but to be frank I was not thrilled.

Doing the holonomy means the following : In order to check out the properties of a manifold (let's say a sphere) you take a groupelement, eg a vektor.

Then we perform a sort of parallel transport along some cirkle on the sphere (that's the Wilsonloop right ?) and we look how the connection between the original vektor and the vektor we are transporting varies. By studying this variation we can deduce info of the manifold we are on, ofcourse without being able to see the entire manyfold at once. This is like saying how can i check whether the Earth is flat or round while my local space is flat, right?

Yes, I agree. To discover the curvature of the Earth one can parallel transport a tangent vector around a loop and when one gets back it is very apt to be pointing in a different direction.

Them vektors we are transporting are parallel-vectors to the original one. So is it possible to describe a manifold by only using the tangent-base of this manifold.

We can use fiberbundels in order to go from the manifold to the tangent space, so that all parallel vectors in the manifold will be projected on a different subspace of the tangent-space. So that different points of the tangent-space correspond to different parallel-classes of vektors of the original manifold. Is this vision somewhat correct ?

I think it is very much correct.

this might help too. Marcus Gaul in Physics at Uni Munich has done an introduction to LQG with Carlo Rovelli. This talks about connections
(the set of all possible connections being the LQG configuration space
and loop holonomies defining the states) but it only says what you need
to know. It may be easier to read:
http://arxiv.org/gr-qc/9910079
especially pages 9 and 10---"Basic Definitions" section.

here also is a quick one-page discussion of parallel transport
showing a picture of holonomy on a 2-sphere
http://www.maths.adelaide.edu.au/people/mmurray/line_bundles/node8.html
this would not be new to you, Marlon, but maybe some other reader
would like it. there are links to other basic definitions

Your questions motivate me to try to collect some links to online
basic definitions that could help read Smolin's "Invitation to LQG" article.

 

[marlon]

marcus, you the man...

this first link is a masterpiece. Wanna know why ?

"Giving a difficult explanation to something difficult is easy
Giving an easy explanation to something difficult is difficult."

Thanks a lot, you give me a lot to read that is worth it during the holidays


regards
marlon