# Relation between quantum and classical mass gaps

We say a QFT has a mass gap if the spectrum of the mass operator $$M:=\sqrt{P_\mu P^\mu}$$ is bounded below by some $$\Delta >0$$.

I will define a $$\textit{classical}$$ field theory to have a mass gap if the Fourier transformed field, call it $$\phi(p)$$, is constrained by the EOMs to have support only when $$p^2 > \Delta$$, for some $$\Delta > 0$$.

I am interested in how these notions relate to each other when we quantise a classical field.

One implication seems easy: if a classical field has mass gap $$\Delta$$, then the quantised theory also has mass gap $$\Delta$$. This is because (at least formally, before we regulate) the quantum field satisfies the same EOM as the classical field. So the states $$|p\rangle:=\phi(p)|0\rangle$$ are only nonzero when $$p^2>\Delta$$. I believe gapped-ness follows. Although I've not thought much about what happens when we regulate, I don't see it causing problems (correct me if this is wrong).

But if the quantised theory has a mass gap, must the classical theory also have had a mass gap? We can't reverse the argument above, since $$\phi(p)$$ having support on $$p^2<\Delta$$ is not enough to conclude that $$\phi$$ creates particles with $$p^2 < \Delta$$ - instead, we could have $$\phi(p)|0\rangle = 0$$ whenever $$p^2 < \Delta$$ (recall that $$\phi(p)|0\rangle$$ is either an eigenstate of $$P^\mu$$, $$\textit{or}$$ is simply the zero state). So it seems the quantum field could conspire to have a mass gap even if the classical theory had no mass gap.

Are there any examples of non-gapped classical fields which develop a mass gap when quantised? If so, is there a "deep reason" for why the quantum field conspires to be gapped?

• Indeed there is one quite famous example of a quantum theory with a mass gap not present in the classical theory, namely QCD (or Yang-Mills more generally). If you can figure out the "deep reason" why this is the case (and state it rigorously), there are a million dollars waiting for you.
– d_b
Sep 21 at 19:02
• @d_b Are there any simpler examples than Yang-Mills? For instance, I seem to recall hearing that weakly-coupled $\phi^4$ theory is gapped when quantised - but then again, perhaps the classical theory is also gapped... Sep 22 at 17:53