Why does the charge operator in QED have an infinite vacuum term?

[Malamala]

Hello! When calculating the Hamiltonian operator in QED, in terms of creation and annihilation operators, we obtain an infinite term, which is considered irrelevant for most part (beside some renormalization problems) because we usually care about the difference in energies, so that would be just a shift. However, when computing the charge operator, coming from the Noether current, you still get an infinite term. What is the meaning of that? One can't really assume that the vacuum has an infinite charge. Thank you!

 

[DarMM]

Hello! When calculating the Hamiltonian operator in QED, in terms of creation and annihilation operators, we obtain an infinite term, which is considered irrelevant for most part (beside some renormalization problems) because we usually care about the difference in energies, so that would be just a shift. However, when computing the charge operator, coming from the Noether current, you still get an infinite term. What is the meaning of that? One can't really assume that the vacuum has an infinite charge. Thank you!

It just means the naive way of constructing the such operators is ill-defined. You can't multiple quantum fields like classical fields. All products need to be Wick Ordered.