Does the free electron theory have electron-positron pair creation?

In QED, photon number is not conserved and photons can produce electron-positron pairs in the vacuum.

But if we take away electromagnetism and have a pure electron theory. Does this still produce electron-positron pairs in the vacuum?

i.e. starting with a single electron, in the free theory is there zero chance of detecting two electrons and a positron at a later time? There is no Feynman diagram that can do this....

On the other hand I hear a lot about electron-poistron pairs being created out of the vacuum. I can understand this would happen near a black hole if a high energy graviton decayed into an electron-posistron pair.

In the free theory even if electron-posistron pairs were created and destroyed would they have any impact if they can't interact with ordinary electrons via electromagnetism?

Any ideas?

• In free theories there is no creation nor annihilation of anything, precisely because they are free. I don't understand the question. In pure electron theories, the leading interactions are given by non-renormalisable terms a la Fermi. Is this what you had in mind? – AccidentalFourierTransform Aug 22 '17 at 16:48
• Why did you not put this as an answer if this is the answer? – zooby Aug 22 '17 at 16:54
• I did not put that as an answer because it is not an answer: I'm asking for clarification, mostly because I don't really understand the question. – AccidentalFourierTransform Aug 22 '17 at 16:55
• If I understood what I was saying I wouldn't need an answer.... – zooby Aug 22 '17 at 17:17

If you forget about the $U(1)$ gauge group of QED you can certainly take a theory of fermions (e.g. electrons) and add all sorts of coupling terms (so long as these are sensible terms respecting Poincaré and global $U(1)$) that will give you vacuum graphs and pair creation-annihilation. For example, you could have a quartic coupling of the form $(\bar{\psi}\psi)^2$ in your theory of electrons and positrons. In a free theory, there would be no interactions (by definition), and no pair-creation or annihilation.