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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.

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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. ;-) –  Luboš Motl Jun 28 '13 at 7:10
    
@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. –  BebopButUnsteady Jun 28 '13 at 7:19
    
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. –  Luboš Motl Jun 28 '13 at 7:31
    
Please change $-\partial_t^2+\nabla^2$ to $\eta_{\mu\nu}\partial_{\mu}\partial_{\nu}$ or $\Box$. Nobody treats space and time separately in Relativity/QFT. –  Dimensio1n0 Jul 4 '13 at 4:23
    
@dimension10: I study condensed matter not relativistic QFT, and I am equally, if not particularly interested, in non relativistic dispersions. Although if you know of a relativistic dispersion at pedagogically useful scales I would love to hear it. –  BebopButUnsteady Jul 4 '13 at 5:11
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