Why do we require quantum fields to vanish at infinity? 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.
 A: Quantum fields are tempered (operator-valued) distributions on space-time, because the machinery of QFT requires distributions that possess a Fourier transform - else you get no creators and annihilators. 
Their test functions are the Schwartz functions, which are the functions rapidly vanishing at infinity.
Since integration by parts for distributions is defined through the integration by parts when applied to any test function, and the boundary terms vanish there since the Schwartz functions are rapidly decreasing, all boundary terms for these distributions vanish.
Sometimes, you will encounter a quantum field configuration that seems to not vanish at infinity. This is usually due to a "classical background", i.e. the fields is $\phi = \phi_\text{cl} + \phi_\text{q}$, and only $\phi_\text{q}$ is the quantum field being (path)-integrated over, while $\phi_\text{cl}$ is a "vacuum" (not the true vacuum) around which the theory is considered.

One can also try to get the integrations by parts to vanish without requiring that fields vanish at infinity: Just take the theory inside a finite volume of spacetime and impose periodic boundary condition. For such periodic situations, integration by parts has no boundary terms. Take the limit of infinite extent of the volume to recover the usual theory.

A more physical, handwavy reasoning goes like this: We do not expect field configurations infinitely far away to influence what happens at finite spacetime coordinates, so we might as well say the field is zero there. This doesn't really work because functions don't have a "value at infinity", but rather a limiting behaviour, and one needs nice statements like the rapid decrease of Schwartz functions to really conclude that the boundary terms vanish.
