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I am looking to understand better what problem might come from the claimed non-invariance of the Bohr-Sommerfeld quantization, which Einstein criticizes in his article On the Quantum Theorem of Sommerfeld and Epstein (English translation). Specifically, Einstein writes

$$\int pdq = \int p\frac{dq}{dt}dt = nh.$$ The integral is to be extended here over one full period of movement; $q$ denotes the coordinate, $p$ the associated coordinate momentum of the system. Sommerfeld's work on the theory of spectra proves with certainty that in systems with several degrees of freedom several quantum conditions have to take the place of this single quantum condition; in general as many $l$ as the system has degrees of freedom. These $l$ conditions are, according to Sommerfeld, $$\int p_idq_i = n_ih.$$ As this formulation is not independent of the choice of coordinates, it can only be correct for a distinct choice of coordinates.

What it not clear to me is that what, specifically, might fail in Sommerfeld's formulation? Furthermore, when many sources talk about invariance in this context, do we mean invariance w.r.t. diffeomorphic coordinate transformations? So when someone might say that "See, the Sommerfeld's quantization scheme is not invariant" they mean to say that "See, the Sommerfeld's quantization scheme is not diffeomorphically invariant"?

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For the simplest example, consider 1D harmonic oscillator. The standard Hamiltonian is

$$ H = \frac{p^2}{2m} + \frac{1}{2}kq^2, $$ We define action

$$ I = \oint p dq\tag{*} $$ and express it using integral over time period: $$ = \int_0^T p \frac{\partial H}{\partial p}dt = \int_0^T \frac{~p^2}{m}dt. $$

Period $T$ is fixed by physical parameters of the system, but $p$ is not; it need not be equal to $m\dot{q}$, it can have any value, depending on the choice of canonical variables and Hamiltonian from an infinite set of possibilities which all produce the same equations of motion.

For example, we can describe the same system using different variables $$ p'=Gp $$ $$ q'=q $$ where $G$ is some big number, and with the Hamiltonian $$ H' = \frac{p'^2}{2Gm} + \frac{1}{2}G k q'^2. $$ It is easy to check that this Hamiltonian leads to the same equation of motion for $q$.

The definition of action (*) gives us now $$ I' = \oint p'dq' = \oint Gpdq = G I. $$

Thus the action definition in (*) is ambiguous, because action value does not depend only on physical variables of the system in the given inertial frame where the motion is periodic, but also on the choice of canonical variables used to describe this system, which is somewhat arbitrary.

The definition (*) becomes unambiguous only if we assume it applies only to a certain choice of canonical variables that is special, e.g. the Cartesian variables $x,p$ where $p = m\dot{x}$.

This reminds us a known fact from quantum theory, that quantization, in general, defines preferred set of canonical variables. A quantization procedure with one set is not equivalent to the same quantization with another set. For example, one cannot naively quantize momentum and position in spherical coordinates, one has to do so in Cartesian coordinates and only then one can analyze what that quantization means in the spherical coordinates.

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