Deriving Bundle Map for Tangent Space on Sphere | QM and Phase Space Topology

[deferro]

Homework Statement

Trying to derive a bundle map for a tangent space on a sphere. This is in line with some online courses in QM and phase space topology. I'm not doing an assignment as such (I'm a postgrad though).
This is also in line with keeping up with tensor calculus (and the symmetry of space+time), since QM is way easier in that case.

Homework Equations

a is a 2 deg vertex $$a(\phi_i \bar \phi_j)$$. i,j are state indices; i indexes a null-vertex in the graph G(V,E), the vertex is aligned as a line (a geodesic) on the sphere, w/ edges at +/- infinity.
$$\phi_i$$ recovers $$\bar \phi_j$$ at a, the crossing in G.
a = (p + ic) where c is the color at the crossing indexed by j.

This is the trivial fiber over the base space S' and G: G -> B needs a continuous sectional function which is smooth over E the inner measure, M the manifold (sphere), F the fiber space and B the base space or map.

The Attempt at a Solution

Write a bivector form for the vertices/crossings and derive the section and projective functions that way?

ed: s'ok, I think I just realized it's a matter of a swapping for vertex-edge pairs from one to the other space, and so on...(doh!)

 

[mighty2000]

Hello,

I'm not sure I fully understand the question, but it seems like you are trying to derive a bundle map for a tangent space on a sphere in the context of quantum mechanics and phase space topology. It's great that you are keeping up with tensor calculus and exploring the symmetry of space and time in relation to QM.

To address your attempt at a solution, it may be helpful to first define some terms and clarify the problem a bit more. What do you mean by a "bivector form" for the vertices/crossings? Are you looking for a specific mathematical expression or equation? Also, what do you mean by "section and projective functions"? How do they relate to the bundle map you are trying to derive?

In general, a bundle map is a continuous function that maps points in one bundle to points in another bundle, preserving the structure of the bundles. In this case, it seems like you are trying to map points on the sphere (the base space) to points in the tangent space bundle. It may be helpful to first define the tangent space bundle and its structure before attempting to derive a bundle map.

I hope this helps and if you can provide some more information or clarification, I would be happy to try and assist further. Good luck with your work!