Global mathematical structure of QFT

"Classical" gauge theories (e.g. electrodynamics combined with quantum mechanics) have the following global description:

1. $A_{mu}$ is a connection in a principle bundle
2. The matter fields are sections in an associated bundle

If one considers now "quantum" gauge theories (e.g. quantum electrodynamics) what is the global structure of the theory?

The four potential $A$ and the matter fields are now operator valued. Is it still possible to describe them as section of some kind of bundle?