In a general QFT we say that the vacuum state $| \Omega \rangle$ is a state that is invariant under the poincare action, that is $U(\Lambda , a) | \Omega \rangle = |\Omega \rangle$ (wightmann axioms).
In particular consider just an infinitesimal spacetime translation (no boost) then $U( \mathbb{I} , a )| \Omega \rangle = \big(\mathbb{I} - ia.P + \mathcal{O}(a^2) \space\big)| \Omega \rangle = | \Omega \rangle $ $\Rightarrow$ $P^{\mu} |\Omega \rangle = 0$.
However, often the idea of a zero point energy is discussed, we say quantum fields fluctuate even in a vacuum because the heisenberg uncertainity principle says the 'fields can not stay still' and this yields a non-zero vacuum energy density.
Also by considering the zero point energy of our fields we predict the casimir effect which has been expeirmentally verified.
My question is, how do these two ideas relate? On the face of it, it seems contradictory. We must have the vacuum with zero energy else the theory would violate the principle of special relativity, yet it is often said an interacting QFT holds a zero point energy.