I've searched a little bit on this site to find:
- (a) What does it mean for the Hamiltonian to not be bounded from below?
- (b) Why should energy have a lower bound but not an upper bound?
- (c) Importance of Hamiltonian being bounded from below
- (d) Why do we always require the potentials to be bounded from below?
- (e) Is it possible to tell whether a potential is unbounded using only perturbation theory?
- (f) Boundedness of a Hamiltonian and when does a Hamiltonian have a spectrum?
- (g) Lagrangian with a negative kinetic term
which are more or less related but not specific to my interest.
The common argument against "no ground state" is instability that system would be in an infinite decay process. But intuitively I could imagine there are false vacua separated so far away that each tunneling event would take a very long time to be expected to happen. Say, the expected time is longer than age of our universe. Then it seems ok to imagine an "ultimate Hamiltonian" without a true vacuum but only an infinite series of false vacua.
If such an unbounded Hamiltonian can be made compatible with cosmology data or other experiment, then I imagine the usual bounded assumption is just a convention to make something more convenient? Can we tell if there is a true vacuum, at least in ideal experiment?
More specifically, I wonder if (and how) this relate to
Non-perturbative effects, such as instanton. As in perturbative view the answer seems to be clear that we can think our local minimum as the only minimum in many cases.
Mathematical definition of a QFT. Does boundedness play an important role?
