Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Sunday, August 12, 2007

Quote of the Fortnight

From D. R. Lunsford on Not Even Wrong:
Anthropocentrism is easily understood as the natural expression of the narcissistic era we inhabit.
It's good to see Lunsford back in the blogosphere after a long absence. Here is his 2003 paper on a (3,3) space (often cited by visitors to this blog, a number of whom are of course banned by the arxiv preprint server). The abstract states that the cosmological constant must vanish. Very sensible. When I told D. Wiltshire about this possibility back in 2002, in terms of varying $c$ and $\hbar$, he immediately scoffed and said that concordance nucleosynthesis ruled out such possibilities. Now he constantly receives the same reaction from many colleagues as he tries to convince them that Dark Energy is a false road (but of course I'm still crazy). As Louise Riofrio often tells us: concordes are prone to crash.


Blogger Matti Pitkanen said...

I agree in my own weird way that dark energy is a false road: actually only the more than obvious trick at the left side of G=T to bury the real problem under the rug. Quintessence is the same trick performed at the right hand side of same equation. Difficult crossword puzzles contain often some amazingly easy looking tips. You wonder for some time whether this is a trap but go on and type the obvious answer which is of course wrong.

Theoreticians believing in linear/exponential progress in science seem to forget that tough problems in physics are much nearer to very-high-dimensional crossword puzzles than proving a theorem from given premises or performing a long sequence of symbol manipulations.

August 12, 2007 5:05 PM  
Blogger Matti Pitkanen said...

String and Chern-Simons Lie 3-algebras was the title of a posting that I saw in n-Category Cafe. I didn't have patience to try to abstract the definition of this algebra from the arrow jungle and translate it to the simple arrowless language that I am used to speak.

Chern-Simons term defines in 3-D case (and only in this case) torsion part of connection as an antisymmetric 3-tensor. Its contraction with two vector fields defines a vector field giving a torsion term to the commutator of two vector fields using covariant derivative containing the torsion term. The torsion term is proportional to 3-D cross product of vectors locally and can therefore be regarded as SO(3) commutator with purely formal "Planck constant" proportional to Chern-Simons term.

I wondered whether the vector fields with this covariant derivative could define some kind of generalization of Lie algebra.
Now I am wondering whether the Chern-Simons-3-Lie algebra might have something to do with this.

Any comments?

I encountered some time ago the question whether

August 12, 2007 5:24 PM  
Anonymous Anonymous said...

Matti said: "... the torsion term is proportional to 3-D cross product of vectors locally ...
I [Mattti] wondered whether the vector fields with ... covariant derivative containing the torsion term ... could define some kind of generalization of Lie algebra ...".

Maybe that would only happen in certain special dimensions,
the only dimensions in which you have a non-trivial continuous vector cross-product of 2 vectors in R^n are n = 3 and n = 7,
the only dimension in which you have a non-trivial continuous vector cross-product of 2 vectors in C^n , with respect to a positive-definite Hermitian inner product in C^n , is n = 3.

See for example the chapter by Beno Eckmannn in the book Battelle Rencontres 1967 Lectures (Benjamin 19687).

Tony Smith

August 13, 2007 12:06 AM  
Blogger Matti Pitkanen said...

Yes Tony, this occurs only in D=3 and D=7 and could be used as argument in favor of dimension 3.

The commutator of vector fields modified by the addition of torsion term might indeed relate to 2-parallel transport, that is parallel transport of parallel transport along curve.

Consider diffeomorphisms induced by a flows defined by vector field X and parameterized by t. These flows take curves to curves and ordinary parallel transports to ordinary parallel transports.

Suppose that the flow induced diffeomorphism becomes identity for t= 1 and defines thus closed curves in the space of curves. Also parallel transport is mapped to itself in diffeomorphism since it is determined by the curve and metric connection alone. One can say that the connection defined by purely metric transfort is trivial in the space of curves.

The parallel transport by X of a parallel transport by vector field Y is defined by the map

Y-->ext(t[X,..]))Y= Y+[X,Y]+ [X,[X,Y]]/2!+.

In the presence of torsion [X,..] would be replaced by commutator containing the torsion term.

The flow which for t=1 takes curves back into themselves would no more yield the original parallel transport. 2-parallel transport in the space of curves would be non-trivial and C-S torsion would define a non-trivial 2-connection in the space of curves (strings).

This suggests that n-1-torsion corresponds to n-curvature quite generally. I think the category theorists might be saying something like this with their arrows.

August 13, 2007 1:30 AM  
Blogger Matti Pitkanen said...

Oops! In the formula for commutator exponential I forgot the powers of t.

August 13, 2007 1:32 AM  
Blogger L. Riofrio said...

Lunsford's paper is a gem, neat how it corresponds to the Maxwell equations. Great that he can prove a CC must vanish. Lunsford uses a refreshingly brief set of references: Weyl, Weinberg.

Your blog is still more entertaining, but Lunsford's comments in Not Even Wrong are hilarious!

August 13, 2007 6:17 AM  
Blogger nige said...

Louise, see

If civilization ever gets back on the path to progress and Lunsford gets a place on a Moon mission, maybe he'll be using the spacesuit that you're developing?

August 14, 2007 2:25 AM  
Anonymous a quantum diaries survivor said...

Quote of the fortnight ? You lazy bum! :)) Trying to work half as hard as the rest of us ? ... BTW you remind me I'm lagging behind by about six or seven weeks with my "say of the week" series...


August 14, 2007 9:53 AM  

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