Well the argument I presented shows that at least one state has a negative energy density at any given point (I suppressed the dependence on position, T = T (x)

Well the argument I presented shows that at least one state has a negative energy density at any given point (I suppressed the dependence on position, T = T (x) )— but the argument does not prove that this state has lower total energy than the vacuum because there could be lots of positive energy density somewhere else! In fact this has to be true, because as you correctly pointed out the hamiltonian has to be bounded below, and with some assumptions about the field theory you can show that the vacuum has 0 energy and every other state has higher total energy. You can even put constraints on how far away the positive energy density is from the negative energy density. These identities historically went by the strangely non-descriptive name of Quantum Inequalities, and there’s a nice modern proof of a similar result in conformal field theory due to Blanco and Casini [0]. Take the Casimir state as an example. In the usual setup there’s negative energy density between two perfectly conducting planes. Either the setup is unstable and the plates will attract, or else something is holding the plates apart which takes work (a source of positive energy density). It’s a bit unclear to me how to think about the former type of states, they’re unstable and so they can’t be energy eigenstates. Another instance of the Casimir effect is when you take a 2D CFT on a cylinder, in which case the normal vacuum state on flat space maps to a state of negative energy on the cylinder under the relevant conformal transformation. That’s a form of the Casimir effect, the energy has dropped to something negative whereas on flat space it was zero (from the algebraic perspective it arises from the conformal anomaly), but now that’s the new vacuum state and the new (negative) lower bound on the energy. [0] https://arxiv.org/abs/1309.1121 https://arxiv.org/abs/1309.1121