Importance of Hamiltonian being bounded from below

Question

In perturbation theory one doesn't really explore much of the global properties of Hamiltonian. So I think PT is not affected by whether a Hamiltonian is bounded below. Yet in the linear potential or the phi cubed theory various sources complain about the fact that the spectrum is not bounded below. Why does this matter?

Answer 1

There are various reasons, not limited to

1. Defining the free theory about which perturbation theory will be carried out. What if the states of the interacting theory look nothing like those of the free theory? This could happen if the interacting theory is well defined but the free theory is not (or the other way round) due to run away solutions / foundational instability caused by states of infinitely negative energy.

   1b. Runaway solutions caused when the perturbation is turned on will generally lead to perturbative series that don't make sense (cubic deformation of harmonic oscillator, for example), if the new ground state will have infinitely negative energy then how could any finite order in perturbation theory take you there?

   2. The Gell-Mann / Low theorem relates the ground state of the interacting theory to that of the free theory and assumes that the spectrum is bounded from below.

On the other hand, it is indeed the case that most perturbative calculations will essentially insensitive to these details and it becomes more of an academic or philosophical point!