View Single Post
Mar29-05, 10:15 AM
Sci Advisor
PF Gold
marcus's Avatar
P: 23,232
It is suggestive that you mention Ashtekar variables, and also mention the variables of BF theory (which Freidel tries to reform us so that we write EF thinking that it makes better sense than BF). Let me tell you what my sense of direction tells me. I listened to a (January, Toronto?) recorded talk by Vafa and I heard something ring in his voice when he said "form theories of gravity"----and I went back and looked at the current paper Dijkgraaf, Gukov, Neitzke, Vafa just to make sure. there was a sense of relief. it represents a hopeful general idea for him.

from my perspective, Ashtekar variables and BF are foremostly examples of "form theories" and there could be modifications and other "form theories" we dont know about yet. there is a mental compass needle pointing in this general direction.

it we want to play the game of making verbal (non-math) definitions for an intelligent reader, then it is important the ORDER we define the concepts and also the GOAL or where we are going. I think the direction is that we want to get to where we can say what a "form defined on a manifold" is, or to be more official we should always say "differential form" defined on a "differentiable manifold". So we need to say what the "tangent vectors" are at a point in a manifold.

the obstacle here is that these concepts are unmotivated, have too many syllables if you try to speak correctly, and seem kind of arbitrary and technical.

So I am thinking like this. the thing about a tangent vectors and forms is that they are BACKGROUND INDEPENDENT. All that means, basically, is that you dont have to have a metric. A background independent approach to any kind of physics simply means in practice that you start with a manifold as usual (a "continuum" you say Einstein liked to say) and you refrain from giving yourself a metric.

Well, how can you do physics on a manifold that (at least for now at the beginning) has no metric? What kind of useful objects can you define without a metric? Well, you do have infinitesimal directions because you have coordinates and you can take the derivative at any point, so at a microscopic level you do have a vectorspace of directions-----call them TANGENTS. and on any vectorspace one can readily define the dual space of linear functionals of the vectors-----things that eat the vectors up and give a number. The dual space of the tangents is called the FORMS.

and also the forms dont have to be number-valued, they can be "matrix" valued, one form can eat a tangent vector and produce therefrom not simply one number but 3 numbers or 4 numbers, or a matrix of numbers, but that is not quite right let's say it eats the tangent vector and produces not a number but an element of some Lie algebra. then it is a ALGEBRA-VALUED form.

now this already seems disgustingly complicated so let's see why it might appeal to Cumrun Vafa arguably the world's top string theorist still functioning as such.
I think it appeals to Cumrun Vafa because it is a background independent way to do physics. that is essentially what "form theory of gravity" means.

And string theorists have been held up for two decades by not having a background independent approach. And it JUST HAPPENS that the Ashtekar variables are forms, and the B and F of BF theory are forms, and (no matter what detractors say) Loop has been making a lot of progress lately, and Vafa says "hey, this might be the way to get background independence" and he creates a new fashion called "topological Mtheory" which is a way of focussing on forms and linking up with "form theories of gravity".

So maybe the point is not that this or that particular approach is good or not, but simply that one should work with a manifold sans metric, and do physics with the restricted set of tools that can be defined without a metric. And that means that, painfully abstract as it sounds, nightcleaner has to understand 3 things:

1. the tangent space at a point of a manifold is a vectorspace
2. any vectorspace has a dual space (the things that eat the vectors) and that dual space IS ITSELF a vectorspace.
3. the dual of the tangentspace is the forms and you can do stuff with forms.

Like, you can multiply two forms together (the cute "wedge" symbol), and you can construct more complicate forms that eat two vectors at once or that produce something more jazzy, in place of a number.

The hardest thing in the world to accept is that this is not merely something that mathematicians have invented to do for fun, a genteel and slightly exasperating amusement. The hardest thing to accept is that nature wants us to consider these things because it is practically the only thing you can do with a manifold that doesnt require a metric!

So instead of talking about BF theory or Ashtekar variables in particular, my compass is telling me to wait for a while and see if anyone is interested in "forms on a manifold" that is to say in the clunky polysyllabic language "differential forms defined on the tangent space of a differentiable manifold" UGH.

Also, selfAdjoint, you mentioned the word "bundle". Bundles may be going too far but they are in this general area of discussion, and there is also "connection"
A "connection" is a type of form. So if you understand "form" then you can maybe understand connection.

there is also this extremely disastrous thing that "form" is a misleading term. In real English it means "shape" but a differential form is not a shape at all. Richard being a serious fan of words will insist that it means shape. But no. Some frenchman happened accidentally to call a machine that eats tangent vectors and spits out numbers by the name "form" and so that is what it is called, even tho it is in nowise a shape. It is more like an incometax form, than it is a shape-form. And it is not like an incometax form either.

And as a final ace in the hole we can always say that Gen Rel is an example of a physical theory defined on a manifold without a metric. The metric is a variable that you eventually solve the equation to get. you start without a metric and you do physics and you eventually get a metric.
If there is any useful sense to Kuhntalk then this is a "paradigm". and when Vafa has a good word to say about "form theories of gravity" then this might be the kind of softening that accompanies a shift in perspective.