- #1

Hurkyl

Staff Emeritus

Science Advisor

Gold Member

- 14,950

- 19

Suppose we're work on some differentiable manifold M.

The basic object of study is a vector bundle on M with an algebraic structure. We would like to think of this as an algebra of operators, so we must find something upon which they can operate.

So, the simplest sort of thing that could serve as a basis state is a scalar field on M. (i.e. a complex valued function) (details of cosntructing Hilbert space not supplied -- I think that's something I can ponder independently)

So, it seems the natural sorts of things one would do to a scalar field to produce a scalar field would be:

(1) Add your favorite scalar field

(2) Multiply by your favorite scalar field

(3) Differentiate with respect to your favorite tangent vector field

So I can lift these operations to operators on the Hilbert space, and form some sort of algebra of operators.

If I'm not mistaken, (2) and (3) would give rise to operators corresponding to position and momentum according to some coordinate chart, so this would be sufficient for QM.

But we could operate on more interesting things. For instance, I could have a SU(1) valued function on M. With the appropriate connection, I can then differentiate these to get a su(1) valued field on M, but su(1) is just R, making it a scalar field. I guess something along these lines is how you're supposed to do electromagnetism?

Or, I could consider vector fields in my favorite vector bundle on M to be basis states. We produce scalar fields by applying a section of the dual vector bundle, but we might, first, want to do all sorts of fun vector operations like:

(1) Do some sort of linear transformation. (Apply a 1,1 tensor, if we're dealing with the tangent bundle!)

(2) Take the covariant derivative with respect to our favorite vector field.

(3) Some more that I don't know!

Or, I can use more exotic Lie Groups, and differentiate to get more exotic Lie Algebras, to which I can apply dual elements to get numbers. Is that what it would mean that color is SU(3)?