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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:?

  1. The set of all field configurations makes a (very complicated) manifold.
  2. We do perturbation theory about a background field configuration corresponding to a point on this manifold.
  3. 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?

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At the non-rigorous/intuitive level, OP's observations are spot on. To facilitate such thinking, physicists often use DeWitt's condensed notation, where a field $\phi^{\alpha}(x)$ is written as $\phi^{i}$, while pretending that $i=(\alpha,x)$ is an index of a local coordinate $\phi^{i}$ for some differential manifold.

The problem is that the space of all field configurations is typically an infinite-dimensional space, while ordinary differential geometry is usually only discussed in the context of finite-dimensional manifolds.

Thus strictly speaking, one would have to master/study/develop an infinite-dimensional mathematical version of differential geometry to make OP's picture precise/rigorous.

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