Fubini-Study metric
John Baez
Todd Desiato <Todd...@san.rr.com> wrote:
>Does anyone feel like explaining what the Fubini-Study metric is?
>What is significant about it?
Do you know what complex projective space is - better known as
CP^n? If you do, then you'll be pleased to know that the
Fubini-Study metric is the metric on complex projective space
that has the maximal amount of symmetry. If you don't, you'd
better learn about complex projective space before messing with
the Fubini-Study metric.
Points in CP^n are, by definition, 1-dimensional subspaces of
C^{n+1}. If you use the usual way to measure distances in
C^{n+1}, you can cook an obvious way to measure distances in
CP^n. This is the Fubini-Study metric.
If the "obvious way" doesn't seem so obvious to you, maybe you
should warm up with real projective space - better known as RP^n.
Points in RP^n are, by definition, 1-dimensional subspaces of
R^{n+1}. These are easy to visualize: they are just lines
through the origin in Euclidean space. So to put a metric on
RP^n - the real analog of the Fubini-Study metric - we just need
to figure out a good way to measure the distance between lines
through the origin in Euclidean space.
There's a nice way to do this starting from the usual way to
measure the distance between points in Euclidean space. If
you have the knack for math, you may be able to figure out
the trick for yourself. Then the generalization to complex
projective space is straightforward. The formulas may look
a little scary, but the idea is not.
By the way, CP^n comes up all over the place in physics, so
the Fubini-Study metric is important all over the place:
quantum mechanics (where the space of states is CP^n),
nonlinear sigma models (where you have fields taking values
in CP^n), Kaluza-Klein theories (where the curled-up
dimensions can look like CP^n or some submanifold of CP^n),
and so on.
James Gibbons
> By the way, CP^n comes up all over the place in physics, so
> the Fubini-Study metric is important all over the place:
> quantum mechanics (where the space of states is CP^n),
> nonlinear sigma models (where you have fields taking values
> in CP^n), Kaluza-Klein theories (where the curled-up
> dimensions can look like CP^n or some submanifold of CP^n),
> and so on.
This also opens the door to Algebriac Geometry, because we have
projective varieties (certain subsets of CP^n), and it's
generalizations to schemes (ringed spaces). By the way, it
seems that we have a nice little cohomology theory (higher Chow
groups) defined on the category of varieties, with Chern map
defined into it,(from a K-theory on the category of varieties).
This mimics the situations in topology, differential geometry,
and C*-Algebras. Hence I think the categories: Var=(varieties),
Schem=(Schemes), Top=(topological spaces), Diff=(differentiable
manifolds) and C*-Alg=(C*-Algebras) fit into some kind of pattern
because similar things can be defined on all of them.
Does having a Model structure have anything to do with it?
Jim Gibbons