Classical fields, like the electrical field must vanish at infinity, because otherwise their energy would be infinite. This can be used in computations to exclude certain solutions.
In quantum mechanics, the wave function must be normalized, because of the probalistic interpretation. (The probability for finding the particle, can't be bigger than $100$%). A wave function spreading out all over space can't be normalized.
In quantum field theory, it's a commenly used trick to integrate by parts and neglect the boundary term "because fields vanish at infinity"? At first sight, this sounds reasonable and logical, but I was never able to nail it really down. Whats the exact reason, we can or must assume that quantum fields vanish at infinity?
Take for example the electron field, which is responsible for the creation and destruction of electrons. Why shouldn't there be electrons everywhere? I know that we have a probabilistic interpretation in qft, too. Nevertheless, I can't put it together why this means here that our quantum field must vanish.