I have heard that in QFT, the vacuum has zero four-momentum:
$$P^\mu |\Omega \rangle = 0.$$
However, I also know that the vacuum has vacuum energy, i.e.
$$ \langle \Omega | H | \Omega \rangle = E_0 \neq 0.$$
In textbooks, we patch up the latter equation by subtracting an infinite constant, and are told not to worry about it. In any case, the vacuum _has_ to have zero four-momentum, as a fundamental axiom. 

I am confused since [a lot of answers](https://physics.stackexchange.com/questions/22468/what-are-the-calculations-for-vacuum-energy) on this site talk about vacuum energy as a real thing that has a definite value. If it's real, then why can we subtract it off to get $P^\mu |\Omega \rangle = 0$? If it's not real, why do we talk about it as if it is?