The Wikipedia page on the Spin-Statistics theorem states that
In relativity, there are no local fields that are pure creation operators or annihilation operators.
In this answer at SE Phys, in a different context, it is stated
[...] express a given field operator as a sum of positive- and negative-frequency terms, called creation and annihilation operators. (These operators are necessarily non-local in space.)
Why are creation and annihilation operators necessarily non local in space in QFT?
In condensed matter we routinely use operators like $c^\dagger(x)$ which create particles at a point in space. Why does this break down when we move to a relativistic theory?
