The Lie Derivative and Mathematical Physics

[ajb]

John Stewart in his book "Advanced General Relativity" on page 23 states;

 

"The Lie derivative is the most important and least understood concepts in mathematical physics."

 

I have no idea what he meant by that as the Lie derivative of any tensor density is well know. In fact all tensor densities on a manifold N can be understood not just as twisted sections of various vector bundles but as scalar densities on a larger manifold M (in fact a supermanifold). Thus all we need is the Lie derivative of a scalar density.

 

 

By definition this is simply

 

[math]L_{X}f = \lim_{\epsilon \rightarrow 0} \frac{\phi^{*}f-f}{\epsilon}[/math]

 

where [math]X \in \mathfrak{X}(M)[/math] and [math]\phi : M \rightarrow M[/math] such that in local coordinates [math]x^{i} \mapsto \overline{x}^{i}= x^{i} + \epsilon X^{i}[/math]

 

The pullback of any scalar density of weight [math]w [/math] is defined as

 

[math]\phi^{*}f (x) = f(\overline{x}(x))J^{-w}[/math]

 

where [math]J[/math] is the Jacobian of the diffeomorphism.

 

The definition I gave above is in fact very workable once you know [math]M[/math]. For example differential forms on [math]N[/math] are functions on the supermanifold [math]M = \Pi TN[/math] which has local coordinates [math]\{x,dx \}[/math] where [math]dx[/math] are anticommuting. You can quickly recover the homotopy formula for the Lie derivative of differential forms.

 

I invite comments on Stewart 's quote.

[fredrik]

I don't know what he means in the context of GR. But to give another coment I find alot of the geometrical thinking to be unsatisfactory, when you try to recast it into a information view, and question each construct for it's epistemological status relative a real inside observer.

 

In classical deterministic and objective reality thinking there are no problems, but the question is, if we consider two - bounded memory - inside observers, how do they inform themselves about this manifold and any other things much of the classical formalism depends on? Is it obvious that they come to the same conclusions?

 

This probably isn't what he meant though, but it's a general issue I see. I would like to reformulate differential geometry itself, in terms of information. I think that's what we need.

 

There are many things, classically one imagines that the observer can cover the manifold and do measurements with rods and clocks, but he may not have the memory to store all this information - that's only one problem.

 

I'm not aware of any matematical formalisms that realises this desire? Are you? I think this is part of what we might need to understand QG, at least that's how I see it.

 

/Fredrik

 

Somehow, the way I picture it, the observers information "forms" something that might be a manifold, but a highly dynamical one, and that is subject to change in response to interactions with the environment. Where the dynamics on this manifold, represents the observers subjective expectations of his environment.

 

But in that picture the manifold is subjective, not objective. And then one can imagine that a particular observer A can behave differently than what B would suspect by projecting A onto his expectations. And this is because A:s internal degrees of freedom is unknown. This unpredictability probably would not be favourable for anyone, so one might expect interactions to exert a selective pressure for finding a state of maximum agreement.

 

This might explain why most of the time, today, evolution has come so far that these strange things are neglectable.

 

If we take Einsteins idea to be that the laws of physics must be found in a form that is the same regardless of the observer. But Einstein considered chosing and observer, as choosing a reference frame. But what about the observers microstructure, and other constraining qualities? ie. maybe the diffeomorphisms doesn't generate alla observers.

 

This seems to me like a very natural direction of synthesis of GR and QM. But I think we need new math, or at least a new applied formalism, that unifies information theory (both information storate/compression AND communication theory) with differental geometry.

 

Maybe that's something for you to fix ajb, since I understnad you are studying in this direction? )

 

/Fredrik

 

This may sound similar to string theory in the sense that one considers a simplest possible case of a p-manifold beeing an "inside observer" living in say an 11-manifold.

 

But the question I see as physically relevant is how that 11-manifold "looks like", from the point of view of a 1-manifold, and to what extent the 11-manifold is "speakable" from the POV of an inside observer?

 

This also also connects to the holographic ideas that information contained in a "volume" of space can be represented by information living on the boundary.

 

I associate a generalisation of this boundary to simlpy be the "communication channel" between an observer and it's environment. Clearly anything speakable is constrained but the "capacity" of the communication channel, and the conceptual step from "surface area" and "channel capacity" is clearly not large.

 

Even thouhg I think they are sniffing alot of interesting sutff, the line of reasoning in string theory (judged by my incomplete understanding), never appealed to me and it doesn't seem sound, unless reinterpreted and reformulated to the extent that it almost gets sometihng else and isn't string theory anymore.

 

/Fredrik