Sunday, December 17, 2017

Physics and Geometry

Returning to gauge theory, physics is best understood in terms of geometry. The area is vast and I am considering how to best explain the key concepts in the most intuitive way. Today I want to start with the broad picture to present the relevant mathematical landscape. We need to consider two kinds of transformations:
  • transformations in space-time
  • gauge transformations of physical fields.
The mathematical machinery involved uses fiber bundles and Cartan's language of differential forms. There are two key differential forms: the connection 1-form and the curvature 2-form. Another essential ingredient is that of parallel transport. In terms of physics parallel transport corresponds to the transport of physical information. When we do parallel transport around a closed loop the final state is in general different than the initial state.

Here is some trivial example from ordinary high-school geometry. You are walking on Earth along the equator  and you carry with you an arrow which points North. You walk 1/4 circumference of the Earth when you decide to walk all the way to North Pole. At any point during your journey you keep the orientation of your arrow at time \(t\) parallel to the orientation of your arrow at time \(t+\Delta t\). Initially the arrow was perpendicular to the direction of travel, and when you started going North the arrow is parallel to the direction of travel. Once at North Pole you deice to take the shortest path to your starting point. What is the orientation of your arrow when you get back? Try this with a pencil and a ball.

The parallel transport on a closed loop can be used to define curvature. In terms of physics, gauge theory teaches us that:


When discussing gauge theory, the natural language is that of bundles. On bundles, one starts with product bundles but then one proceeds to general bundles obtained by gluing together product bundles. For Standard Model the product bundles are enough, but for gauge theory on curved space-time (string theory, quantum gravity) you need to use the most general bundles.

One problem arises then: how to relate what different observers see?

Changes in observers are described by cocycles. Cocycles depend on both the topology of the space-time manifold and the structure of the gauge group. The deviation of vector bundles from from product bundles is measured by the so-called characteristic classes (Chern, Euler, Pontryagin, Stiefel-Whitney, Thom classes).

To explain the math machinery of all this is a very ambitious project and I am not sure how far along I can carry the series but I will try. The geometry involved is very beautiful (at least to me). Please stay tuned.

No comments:

Post a Comment