Exploring Lie Groups and Their Use in Physics

[marlon]

Here is a nice question

I know that exponentiating elements of a Lie-Algebra gives you back an element of the Lie-Group. These Lie-algebra-elements generate the Lie-Group transformations. Like the Galilei-group, these Lie-groups are used in theoretical fysics as the great START, I mean they contain the transformations under which all the fysical interactions (equations) have to be invariant. Like the Galilei-transformation for Newtonian Mechanics...

Now, my question concernes the background independence of formalisms like LQG, for example. On a certain manifold you can study it's structure by parallel transporting elements of the Lie Algebra, like vektors of a tangent vektor-space. The reason why we take just these vektors, is because the Lie-Algebra provides us with differentials and that's always nice in fysics.

Now we take a vektor and transport it around some loop on the manifold. Exponentiation the differential motion of this vektor when one step allong the loop is taken gives us back an element from the Lie-Group that represents the total movement the vektor made allong the transport. For example a rotation of 90 degrees when taken around the loop.

Is this vision correct, or is this science fiction. I think it must be ok, just wanting to check it.

AAAh, can someone give me a PRACTICAL example (i know the theoretical definition) of the use of a fiber. Can these trnaformations be used to go from a manifold to a tangent space ?

regards
marlon

 

[matt grime]

I think you're getting the right idea. What you're talking about is, I think, the holonomy group of the manifold.

A *practical* use of fibres? hmm, I'm not sure I can think of what practical might mean here. I know of several uses of them theoretically, ie in pure mathematics, and of results about them, if they'd be any help.

 

[matt grime]

http://math.ucr.edu/home/baez/week149.html

here's a starting link that mentions fibers, cohomology and might lead to a bit of physics, if that'll do.

 

[marlon]

Matt, thanks for the help.

I don't want to get to deep into the math but is it possible to show me an example of some element taken along a loop and study the way it changes on the way. So can you give me some kind of "exercise-example" if this holonomy-thing. I understand what this all means, yet i never saw a real example of the use of all this.

I am referring to something like the way affine connections are introduced in GTR...

regards
marlon

 

[matt grime]

I suppose the simplest example I can think of is the sphere S^2.

Imagine a vector (arrow) at the north pole and tangential pointing in the plane of 0 degrees longitude, transport it down that line of longitude to the equator, transport along the equator a quarter of the way round, now transport back up to the north pole and it's at 90 degrees to the angle it started out from. Since the tangent space at a point is a plane, we've defined a rotation, element of SO(2), doing this.

I don't really know much about holonomies; this is something I've picked up by osmosis.

 

[matt grime]

Incidentally have you seen this explanation of what a lie algebra (of vector fields) means:

imagine that the vectors of the vector field are drawn on the manifold's surface (assume it's locally 2d), so there's a "mesh of" field lines. the bracket xy-yx measures how much these lines "fail to commute", xy-yx means, sort of go in the x direction, then the Y, then back along the x, then back along the y, so you're doing a little traversal around a "square" in the mesh. if you end up where you started from then xy-yx=0, otherwise you're a little skewed by the curvature of the surface, loosely.

 

[marlon]

I suppose the simplest example I can think of is the sphere S^2.

Imagine a vector (arrow) at the north pole and tangential pointing in the plane of 0 degrees longitude, transport it down that line of longitude to the equator, transport along the equator a quarter of the way round, now transport back up to the north pole and it's at 90 degrees to the angle it started out from. Since the tangent space at a point is a plane, we've defined a rotation, element of SO(2), doing this.

I don't really know much about holonomies; this is something I've picked up by osmosis.

Hmmm, thanks but I knew this already. What I meant was to ask for an mathematical example. I mean i want to see a real parallell transport executed on some manifold in terms of differential geometry.
Just like in GTR and the introduction of affine connections.

regards
marlon

ps don't worry, i found a text that solves my problem

 

[marlon]

Incidentally have you seen this explanation of what a lie algebra (of vector fields) means:

imagine that the vectors of the vector field are drawn on the manifold's surface (assume it's locally 2d), so there's a "mesh of" field lines. the bracket xy-yx measures how much these lines "fail to commute", xy-yx means, sort of go in the x direction, then the Y, then back along the x, then back along the y, so you're doing a little traversal around a "square" in the mesh. if you end up where you started from then xy-yx=0, otherwise you're a little skewed by the curvature of the surface, loosely.

Can you give me a reference to this text, please

regards
marlon

 

[matt grime]

sorry, i can't, this appeared as a throw away comment in the first lecture i attended on lie algebras, and was followed by a comment along the lines of "we're algebraists, so now we're happy these objects have some interest to the more applied bits of maths, let's carry on and ignore that aspect of it" and proceeded to give the (better, for i am still an algebraist) algebraists definitioni of tangent space (dual numbers)

 

[humanino]

Hey Marlon ! I see I need not worry about your Lie algebras anymore, because matt certainly knows math better than I do. Anyway, I can give you a few references for physicists : you know I am not back home, I don't have access to my books, so maybe this is going to be inaccurate.

Frontiers In Physics edited a marvelous "Lie Algebras in particle physics" by Howard Georgi

Cambridge edited another excellent one, "Symmetries, Lie Algebras and Representations : A Graduate Course for Physicists" by Jürgen Fuchs, et al (Cambridge Monographs on Mathematical Physics)

These are two references I think are what you need. Hope you have access to university library.

 

[humanino]

I should have thought about another reference for your current concerns Marlon. I just recently bought Wald's book on general relativity. Of course, it is not directly linked to gauge theories, but there is an excellent and short introduction to Lie derivative, and how it is linked to killing vector fields in one of the appendices to the book. This should clarify to you the difference between connection and metric, especially the fact that there is a canonical connection associated to the metric, as you referred once to me. I can give the exact reference, because I do have the book right here. Damn my mind is slow...

Robert M. Wald, "General relativity" (1984), the University of Chicago Press
ISBN 0-226-87033-2 (pbk)

...paper back is always less expensive
So, check out the appendix C : Maps of manifolds, Lie derivative, and killing vector fields
(If you want Marlon I can email you a summary) The point is that, I advise you to read the two previous books, but that will require you maybe ten times more efforts than going through Wald's appendix.

 

[rick1138]

1. Exponentiating the members of the algebra only gives the LOCAL structure of the group. In most cases the global topology of the group manifold is lost, the exceptions being when said toplology is trivial.
2. Get a copy of the book "Dictionary on Lie Algebras and Superalgebras" by Frappat et al. It is beautiful.

I have noticed the algebraic prejudices of Lie theorists myself - I was thought of creating pictures of group manifolds. Many thought I was mad.

 

[mathwonk]

I guess everyone knows that the lie algebra is the tangent space to the lie group at the origin, and exponentiation is just a projection of that space down onto the group. Thus in case of a torus, i.e. n dimensional doughnut, the projection is surjective and seems to completely capture the topology of the group as a quotient space of the tangent space, which may be a (possibly the only) "non trivial" global example. Of course it depends on the definition of the word "trivial".

As I recall, Spivak explains the connection between curvature and translates of vectors.

 

[marlon]

yes, thanks for all the answers guys...

regards
marlon

 

[mathwonk]

Marlon,

as usual spivak volume 2 explains curvature as a commutator of differential operators, and i believe even draws the picture of parallel transport of vectors around a little parallelogram. this is standard explanation and may be in almost any elementary book on diffrential geometry. I'll try also to find another reference. ah yes: try Lecture Notes on Elementary Topology and Geometry, I. M. Singer, J. A. Thorpe