- #1
kakarotyjn said:Thank you quasar,but I'm sorry I still can't understand the connection between the way we construct E and the transition function[tex]g_{\alpha \beta}[/tex].In E, (x,y) is equivalent to (x,g_{\alpha \beta} by definition.But how does this equivalent relation induce that g_{\alpha \beta} is the transition function?Why there is an equivalent relation?
In order to prove g_{\alpha \beta} is the transition function,we need to find fiber-preserving homeomorphisms [tex]\psi_\alpha[/tex] and [tex]\psi_\beta[/tex] for [tex]g_{\alpha \beta}(x)=\psi_\alpha \psi_\beta^{-1}[/tex]
Thank you very much!
A fiber bundle is a mathematical construct used to describe the topological structure of a space. It consists of a base space, a total space, and a projection map that maps points in the total space to points in the base space.
Transition functions describe how the local coordinate systems of a fiber bundle are related to each other. They are used to understand how the bundle changes as it moves from one point to another in the base space.
The structure group of a fiber bundle is the group of transformations that preserve the bundle's local coordinates. The transition functions are elements of this group, and they determine the structure of the bundle.
One example of a fiber bundle is a Möbius strip. The base space is a circle, the total space is a cylinder, and the projection map maps each point on the cylinder to its corresponding point on the circle. The transition function for this bundle is a half-twist, as the local coordinate systems change when moving from one side of the strip to the other.
Fiber bundles are used in physics to describe physical fields and their interactions. For example, the electromagnetic field can be described as a fiber bundle with the base space being spacetime, the total space being the set of all possible electromagnetic fields, and the projection map mapping each point in spacetime to its corresponding electromagnetic field. This allows for a deeper understanding of the structure and behavior of physical systems.