For a non-interacting vacuum of QFT we have $$\hat{P}^\mu|0\rangle=0$$ where $\hat{P}^\mu$ is the $4$- vector momentum operator having eigenvalue and eigenvector given by $$\hat{P}^\mu|p\rangle=p^\mu|p\rangle$$ Does this relation also hold for an interacting QFT vacuum $|\Omega\rangle$ i.e. $$\hat{P}^\mu|\Omega\rangle=0$$
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4See the answer here: https://physics.stackexchange.com/q/239640/ – Andrew Dec 15 '20 at 05:42
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6Does this answer your question? Does the vacuum in QFT have nonzero energy or not? – Krup'a Dec 15 '20 at 08:59
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1Short answer is yes, this is one of the Wightman axioms – Poincare-invariance of the vacuum. – Prof. Legolasov Dec 15 '20 at 21:37