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So: Is the close similarity of small waves crossing water of varying depths ("depth potentials") an example of an approximate gauge invariance?

If so, do other "only the surface dynamics matter" visualizations exist for more complicated gauge theories, ones that might help students understand them better?

Finally and more speculatively, would the depth-dependent forms of large waves (think surfing) have any useful tutorial parallels to broken symmetries?

(This question is mostly a sanity check. I like good tutorial visualizations, but only if they remain true to the original physics.)

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I'm not sure that I'm understanding the question completely, but: in a gauge transformation $\Psi\rightarrow e^{i\phi}\Psi$, isn't the phase supposed to be unobservable? The phase of a water wave is directly observable. – Ben Crowell May 31 '13 at 23:01
Yes, more detail of what exactly you are proposing would help. I'm not really sure what you're getting at here. I can think of plenty of examples of things which (approximately) don't depend on other things, but would never call gauge invariance, even approximately. – Michael Brown Jun 1 '13 at 5:19
Sorry, I think I confused the issue by mentioning water waves. Think of swimming in the ocean. Can you observe the depth of the ocean based on what you see at the surface? Generally, no; depth is not an unobservable variable. Yet if a new surface 1 meter lower suddenly appeared near where you are swimming, the potential difference between them would create a very real (and scary) flow. Similarly, someone enclosed within a highly charged metal sphere would see no change in local physics, yet would see a very real effect if that sphere were linked to another at lower electric potential. – Terry Bollinger Jun 2 '13 at 3:02
Hi @Terry Bollinger: Do you have a concrete (mathematically formulated) fluid-dynamical model in mind? – Qmechanic Jun 2 '13 at 16:43
I unfortunately mixed two ideas: Water surfaces as modeling the idea of a "locally flat" region of potential, and water waves for "breaking" symmetries. The first idea I'm pretty sure of as an accurate analogy (gravity potentials are real!) for explaining why interesting physics emerges from the interactions of varying potentials. The second idea "feels" like it could have merit for e.g. modeling how the very neat mass symmetry of u and d breaks down horribly in the next two generations, where some "depth limit" is reached and an "approximate" (er, is that even allowed?) gauge symmetry fails. – Terry Bollinger Jun 2 '13 at 17:05

There are no observable examples of gauge invariance. That's what gauge invariance means. It is a symmetry of the unobservable variables we use to simplify the expression of observable variables. A gauge transformation can change these unobservable variables, but not their observable combinations.

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user1504, please consider this: When you model the dynamics of small waves crossing the surface of a calm lake, do you include the gravitational energy potential of water on the lake surface as a variable? Of course not, because that particular variable does not produce an observable effect on your waves, at least not of a magnitude that is easily detectable at typical human scales of size and measurement. However, if you add a pipe that links two points on the surfaces of two lakes, the potential difference becomes very real and dramatically alters how the water surfaces behave near the pipe. – Terry Bollinger Jun 9 '13 at 20:20

No, surface waves are an example of depth-independence only when the depth is large compared to the wave length. So it is an example of asymptotic of a solution, not the true invariance.

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Thanks, and for both answers (@user1504 also) I of course realize that the water depth idea is at the very best just an everyday way of visualizing the concept of a gauge-invariant potential. I hope I am not confused on this point, but please whack me hard if I am: Gravity potential as represented in various heights of water really is gauge invariant, is it not? Also, I do like the "flatness" idea that a water surface conveys, which represents a locally shared value of that potential. – Terry Bollinger Jun 9 '13 at 2:02
@TerryBollinger: Dear Terry, the potential is not gauge invariant; it is the potential difference who is. A physical example of some sort of invariance is the relative velocity in Classical mechanics: $\mathbf{v}_1-\mathbf{v}_2$ is invariant with respect to Galilean transformations of velocity $\mathbf{v}_i\to\mathbf{v}_i+\mathbf{V}$, and the acceleration is invariant too. – Vladimir Kalitvianski Jun 9 '13 at 5:52
Vladimir, thanks. Well, I can honestly say you completely lost me on that one! Isn't the "gauge invariant variable" the one to which you can add any constant -- a symmetry transformation -- and still keep the same physics? Gravity potential in real bodies is of course not gauge invariant... but at human scales and geometries it feels invariant. So, if you do surface ripple experiments on Lake Titicaca in Peru (3,810 m above sea level) or on a fresh water enclosure in the Dead Sea (408 m below), the physics of surface ripples would seem very similar. But link the surfaces with a pipe and... – Terry Bollinger Jun 9 '13 at 19:51
@TerryBollinger: The force is height-invariant ($mg$) due to large Earth radius $R$. The potential is not. That is why you can fall and hurt yourself. – Vladimir Kalitvianski Jun 9 '13 at 22:13

Potential is not gauge invariant. The examples you point to - dropping the height of the water level, or linking a charged sphere to a sphere with lower potential - are all examples of you measuring a potential difference; those being the difference in gravitational potential you get when you drop one meter, the difference in electrostatic potential moving from one sphere to the other and so on. Potential differences are gauge invariant. You can measure phase differences but you can't measure phases.

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