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I think your error is in assuming that $E_{n+1} - E_{n}$ is proportional to $n$. At least, I assume you assumed it; it's the only way I can see that you could go from the statement $$E_{n+1} - E_n \propto E_n^{-1/2}$$ to the statement $$E_n \propto n^{-2}.$$ Really, what the first proportionality above implies is that  \frac{dE}{dn} \propto ...

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It's tacky to answer my own question, but I think I've found a paper by Klauder and Alicki which explicitly discusses it. The crux of the paper is contained in the statement below: As a result we are able to show that the canonical coherent-state propagator based on a large class of generally non-Gaussian fiducial vectors admits a rigorous representation, ...

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This is an interesting question. I'm rather fond of coherent state phase-space path integrals, but their rigorous aspects are quite tricky (particularly issues of operator ordering). I'm not an expert on the proper measure theory, but it's interesting that the semi-classical analysis also has continuous-> discontinuous trajectory feature. Take as the ...

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One reason for the box is the Fourier expansion of field in stable macroscopic condition (thermal radiation, cavity oscillations) works well only for finite volume. For infinite volume, the Fourier integral of such stationary field is problematic, because the field function is not L2 integrable.

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Quantizing in a finite volume is not specific to the electromagnetic field, and it is not a necessity, neither for the electromagnetic field nor for any other. It is generally more well-behaved to quantize in a finite volume because no infrared-like divergences appear from allowing arbitrarily low momenta (since no arbitrarily long wavelengths fit into the ...

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I believe that you must study (not simply reading but trying to prove) that beautiful story of angular momentum quantization starting from the non-commutativity of the coordinate and momentum of a particle along any axis. The following is a starting point (found in any introductory book on Quantum Mechanics). In Classical Mechanics the angular momentum ...

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Comments to the question (v9): If we ignore the overall normalization, then OP correctly applies the Dirac-Bergmann$^1$ method, which leads to second-class constraints.$^2$ Normally the Majorana Lagrangian (1) is defined with a factor $\frac{1}{2}$ in front. Then there will be no factor $\frac{1}{2}$ in the anti-commutator relation (9), see e.g. Ref. 2. ...

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Comments to the question (v3): The $X^-$ coordinate has (a part from a zero mode) been integrated out in the light-cone (LC) formalism. The above mentioned LC Hamiltonian cannot fully address questions about the $X^-$ coordinate. To get the well-known expansion of $X^-$ as a sum of zero and oscillator modes including the sought-for $\alpha^-_0$ mode term, ...

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