Diffeomorphism invariance

[Mike2]

Main Question or Discussion Point

I know what a diffeomorphism is. But what is diffeomorphism invariance? And why is it important in physics?
Thanks.

 

[Atheist]

Can only guess here: Since a coordinate transformation is a diffeomorphism I´d say diffeomorphism invariance is invariance under coordinate transformations.

Importance in physics thus would be obvious: Ideally, physical processes are invariant under coordinate transformations (would be a bit unpractical when physical processes depend on the coordiante system) so it´s a requirement for physical meaningfull equations.

 

[mathwonk]

A concept is a diffeomorphism invariant if it is unchanged under diffeomorphism. For a physicist that probably means essentially unchanged under coordinate transformation.

E.g. it has recently been proved in algebraic geometry that rationality of surfaces is a diffeomorphism invariant.

E.g. a complex algebraic surface that is diffeomorphic to a projective plane is also algebraically birationally equivalent to the projective plane. Thus "rationality" is a diffeomorphism invariant, for surfaces.

 

[marcus]

...

E.g. it has recently been proved in algebraic geometry that rationality of surfaces is a diffeomorphism invariant.
...

sounds intriguing! do you happen to know if the paper is online
or if some website has an informative discussion of it?

BTW wonk, General Relativity (you may know all this, I haven't been reading your posts so don't know your familiarity with physics) is
an important example of a diffeo-invariant physical theory.

Since you could already be well-informed on these matters I wont venture to be more explicit except to say that solutions to the Einstein equation are diffeo invariant.

 

[mathwonk]

My new friend, I am never offended to have something overexplained so please never hold back with me.

I apologize for not checking back here more recently but the result i mentioned I believe was due to my old friend Bob Friedman, and I will find a source for you.

 

[mathwonk]

here you go:

http://arxiv.org/abs/alg-geom/9404010

 

[Mike2]

Can only guess here: Since a coordinate transformation is a diffeomorphism I´d say diffeomorphism invariance is invariance under coordinate transformations.

Importance in physics thus would be obvious: Ideally, physical processes are invariant under coordinate transformations (would be a bit unpractical when physical processes depend on the coordiante system) so it´s a requirement for physical meaningfull equations.

Would this be the same as "covariant" expressions?

 

[mathwonk]

Just a guess here, but in math parlance, i.e. category theory, one asks how quantities transfer under any mapping at all, not just invertible mappings like diffeomeorphisms and coordinate transformations.

So the terms "covariant" and "contravariant" are used to distinguish quantities that transfer in the same direction as the map from quantities that transfer in the opposite direction. Unfortunately people in different areas of math disagree as to which etrm which emans which. I.e. differential geometers and hence physicists are all alone in using "covariant" for opposite direction transforming quantities, whereas everyone else uses "contravariant" for that.

Now if you are only interested in invertible transformations like coordinate changes it does not matter quite as much, since you can always use the inverse transformation to transfer your quantity, but it is till extremely confusing.