I am studying GR right now, and one interesting thing I learned about vectors is that they are defined to have the same properties as derivatives.
With this in mind, can I make a differential geometric interpretation of ordinary perturbation theory used in quantum field theory in the following way:?
- The set of all field configurations makes a (very complicated) manifold.
- We do perturbation theory about a background field configuration corresponding to a point on this manifold.
- The perturbations are like expanding functions out to first order -- can be viewed like tangent vectors
Am I on some track to enlightenment, or is this a dead-end train of thought?