Most correct intuition for vacuum energies in interacting and non-interacting field theories

Question

It seems like a lot of pop science gets thrown around when explaining vacuum energy.

For instance, our QFT lecturer descried the non-interacting field theory vacuum energy as 'purely Heisenberg Uncertainty Principle energy for particle-antiparticle pairs popping in and out of existence' while the interacting field theory vacuum as 'this combined with them being able to exchange virtual particles'.

How correct is this? Is there a more precise explanation of the differences between the non-interacting and interacting field theory vacuum states, and why, for instance, we can drop the vacuum energy contribution in non-interacting field theories but not in interacting field theories?

Is it true that the non-interacting vacuum is kind of irrelevant since it cannot affect anything?

Answer 1

The term “vacuum energies” is sometimes used to refer to the first order quantum corrections to a classical field theory, i.e. the order $\hbar$ corrections. Technically you need all orders to get the full correction, but often it is safe to assume that the first order corrections make the largest contribution. In e.g. QED you have to truncate the series expansion anyway, because it’s asymptotic.

In the perturbative scheme the first order correction corresponds more or less to all the Feynman diagrams containing a single internal loop. When referring to ‘being able to exchange virtual particles’ your lecturer is presumably referring to the internal lines of the relevant Feynman diagrams in the perturbative expansion.

For the latter question: you can get rid of the vacuum energy in the non-interacting case via normal ordering/Wick’s theorem - it’s just an infinite constant. In the interacting case this is more difficult because strictly speaking the interacting and non-interacting theories live in different Fock spaces (this is a bit hand wavey because it’s not so clear what the Fock space for an interacting theory looks like - at least not to me).