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I realise that we can't meaningfully answer the quesition "what is Planck's constant were zero" because it would break out physical models beyond recognition.

However, if we let Planck's constant approach zero asymptotically, is physics well-behaved?

For example, if you look at $E=hf$, clearly photons would carry asymptotically less and less energy, but qualitatively speaking this equation is stable. If the equation was instead something like $E=(h-\epsilon)f$ then we'd have a problem before $h$ reached zero.

So let $h$ become arbitrarily small but still greater than zero. What, if anything, breaks?

EDIT: to clarify, I am not asking what happens in the limit, I am asking if anything qualitatively changes if we take a finite step in the direction of the limit.

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    $\begingroup$ Heuristically this is the classical limit of quantum theory. Quantum interference disappears and the physics is governed by least action principle. In this world there are no photons because this is purely quantum concept, you have just classical electromagnetic field. $\endgroup$ – Blazej Jan 13 '17 at 8:52
  • $\begingroup$ I am not asking about what happens in the limit, but whether anything significant happens before the limit. In a sense -- are our physical models continuously consistent for arbitrarily-small-but-bigger-than-infinitesimal values of $h$? $\endgroup$ – spraff Jan 13 '17 at 10:52
  • $\begingroup$ Both observables and states act somewhat continuosly under a deformation of $\hbar $. $\endgroup$ – yuggib Jan 13 '17 at 12:31
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You ask what happens if $\hbar$ is small, but $\hbar$ is a quantity with a unit (that of angular momentum). The only meaningful question would be "what if $\hbar$ is small compared to [some other quantity with the same units].

However, there are no other fundamental quantities with the unit of angular momentum. $\hbar$ can not be "small" because $\hbar$ defines the scale with which we could determine "smallness" in the first place.

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  • $\begingroup$ Excellent point! The size of atoms scales with the Bohr radius, and so our size, the meter, and the pyramids are, ultimately, all sized by the Bohr radius and so h-bar. We ride together with it: we are it. The question might as well be: what if the universe were much larger? $\endgroup$ – Cosmas Zachos Jan 13 '17 at 16:00
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If h bar went to 0 then quantum mechanics wouldn't exist and all of the laws of nature would be classical. This would be devastating in a lot of ways. For example black bodies would exhibit ultra violet divergence in power emitted. Atoms would no longer be stable because electrons would spiral into the nucleus. A universe where h bar is 0 would be a very bleak one.

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