# 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?

• The connection lives on the bundle, if you want the one on spacetime which we typically refer to also as the connection but understand it as the field, this is actually the pullback of the connection by a section. Mar 13, 2021 at 19:07
• Since you mentioned "operator valued"... This question might be easier or harder to answer depending on which definition of QFT you want to use. A formulation directly in terms of operators on a Hilbert space might not be the easiest one. It might be easier using the formulation that I cited in another question, which is expressed in terms of partition/transition functions instead of operators. Does the question require using a formulation that directly uses operators, or are other formulations allowed? Mar 13, 2021 at 19:59
• It is not important how the QFT is formulated. So the formulation does not have to relay on operators. But it should be capable of describing a real physical theory like quantum electrodynamics. Mar 14, 2021 at 13:30