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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?

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    $\begingroup$ 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. $\endgroup$
    – d_b
    Sep 21 at 19:02
  • $\begingroup$ @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... $\endgroup$ Sep 22 at 17:53

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