# An approachable example of a field with a "mass gap"

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Preamble: I have come to believe that alot of difficulties in explaining physics to people of all levels comes from the relatively mundane idea of a wave equation with a mass gap $$\left(-\partial^2_t +\nabla^2 -m^2\right)\phi = 0$$ or more generally a field that does not have propagating modes in some frequency band. Being able to demonstrate this behavior would be useful in explaining the differences between conductors and insulators, the difference between the Higgs field and the Higgs boson and other things - it would even be useful explaining gapless fields like EM to be able to refer to a gapped field.

However despite being an extremely common and tame phenomenon I can't think of any approachable examples of gapped wave equations. I can't think of anything where I could say, even to a junior undergraduate "its a gapped field, just like X". Nor can I think of a system I could show a video of, or a demonstration that would give intuitive understanding of gapped fields.

So my question is: does anyone know of a "gapped system" that would be useful for pedagogical purposes? That is, a system that has a continuum of propogating frequencies and a continuum of non-propogating frequencies.

The best I can think of is the following experiment: by scattering debris on the bottom of a pan filled with shallow water, you can localize the surface waves, but I assume the high frequency waves should still propagate. I don't like this for a variety of reasons, least of which is you are using Anderson localization to explain a mass gap.

To reiterate I am looking for something which can give understanding - so something intuitive or something that can be played with until it becomes intuitive. I know there are lots of common things that are gapped (metals and EM radiation for example), but I can't think of anything that is pedagogically useful.

This post imported from StackExchange Physics at 2014-06-21 08:57 (UCT), posted by SE-user BebopButUnsteady
retagged Jun 21, 2014
I didn't quite understand in what sense the simple massive Klein-Gordon equation you started with is "harder" than your pan-surface-water-waves story – I didn't actually understand the latter story and why it's claimed to be important for the mass gap pedagogy. ;-)

This post imported from StackExchange Physics at 2014-06-21 08:57 (UCT), posted by SE-user Luboš Motl
@LubosMotl: I said it was the best I could come up with, not that it was any good :). The supposed advantage over the KGE is not that it is simpler, but that it is less abstract. Wouldn't it be nice if you could explain the properties of the KGE by reference to something you could actually touch? BTW, it actually seems that you can make a solid-state-like dispersion in water through the much more obvious idea of putting a periodic structure on the bottom of a shallow pan. This would give you bands by necessity.

This post imported from StackExchange Physics at 2014-06-21 08:57 (UCT), posted by SE-user BebopButUnsteady
Hi, it must be better for someone to think about an example one can touch. I feel much more intimately familiar with a simple dispersion relation for a massive particle even though I arguably "can't touch it". It is actually not clear to me what you mean by "touching" - it seems you mean it very literally. We can't touch a Higgs boson with our hands, it burns etc., but a Higgs boson is an observed massive scalar particle, too. Quite generally, I feel that the concept of a "mass gap" is so theoretically abstract that it makes no sense to compare it to some childishly tangible examples.

This post imported from StackExchange Physics at 2014-06-21 08:57 (UCT), posted by SE-user Luboš Motl

Wave spectrum in a cavity or a wave guide starts form a non zero frequency, which is somewhat analogous to the mass gap.

The term "mass gap" looks as redundancy. Or "an energy gap", either "a finite rest mass" are better, no?

All stable and even meta-stable massive particles/atoms/molecules are good as examples.

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The trivial example: consider a line of masses connected to each other with springs. This is an ungapped system, translation invariance means that you have arbitrarily low-frequency waves that move the springs back and forth. To get gapped behavior, you now have a rigid bar parallel, and attach each mass by a spring to the nearest point on the bar. That's good enough to answer the question.

But this is not the best pedagogical explanation for what "gapped" and "ungapped" mean, it just gives you basic intuition. The best pedagogical method is to look at the behavior of the correlation functions in a gapped and ungapped system. The gapped system has correlation functions that fall off exponentially in imaginary time, the ungapped system has correlation functions that fall off as a power. So in the Ising model, the gapped behavior is everywhere except at the phase transition point, and the phase transition point is ungapped.

The question is probably motivated by the Clay problem, and this is more intricate. The issue here is that gauge fields are naively massless, because there can be no "spring" restoring you to zero gauge field, because the concept of "zero gauge field" is not gauge invariant. This means that the particles are massless in the Lagrangian. But these theories are confining, and so end up being gapped anyway, because the low-energy limit is completely different from the high-energy Lagrangian. To understand this qualitatively, you need to understand the flow of coupling to infinity in the infrared, and that this means that the lattice version of the theory is flowing to a Lagrangian which is constant. A constant Lagrangian means that all the links are completely statistically independent of the other links, and this means that the correlation function is localized at one point. That's what happens when correlation functions fall off exponentially, in the infrared, the Lagragian flows to something ultra-local.

answered Jun 22, 2014 by (7,535 points)
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An interesting example I can think of would be the theory of superconductivity: the energy spectrum of cooper pairs contains a mass gap. In an effective scalar field theory description, the mass gap is related to the mass of a Higgs-like particle. I suppose that this is an example people can relate to, since superconductivity is, albeit theoretically not that easy to grasp, something that people have definitely heard of.

This post imported from StackExchange Physics at 2014-06-21 08:57 (UCT), posted by SE-user Frederic Brünner
answered Jun 20, 2014 by (1,120 points)

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