# What does it mean for the Hamiltonian to not be bounded from below?

According to David Tong. In quantum field theory if you quantize the Dirac field using commutation relations instead of anti-commutation relations you end up an unbounded Hamiltonian from below, page 109.

What does the word unbounded Hamiltonian mean physically? And what does it mean mathematically if the Hamiltonian is not bounded from below? And why is a catastrophe for our field theory to have an unbounded Hamiltonian?

There are different ways to look at this problem. Let's start classically, for a free theory. Here, it is not really a problem if you have a Hamiltonian that is unbounded from below. Things change if we consider an interacting theory. Suppose you have a system $$S1$$ with Hamiltonian unbounded from below and a system $$S2$$ with Hamiltonian bounded from below, then an interaction between these two systems will excite $$S1$$ to a higher-energy state, and $$S2$$ to a lower-energy state, without violating conservation of energy. But this can happen indefinitely, leading to an instability.