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The exponential factor is a phase factor, where a change in the exponent represents a rotation in the complex plane. Because $\xi^2$ multiplies a very large number (both $\epsilon$ and $\hbar$ being very small), when $\xi$ is not small any $d\xi$ makes the rotation "fast" compared to the corresponding change in the other factor (the phase is proportional to ...


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I) Apart from a full proof of the Gutzwiller's formula in the context of the Feynman path integral (FPI), then OP is essentially asking if the FPI knows about the metaplectic correction/Maslov index and caustics? The physics lore is that when the FPI is set up and interpreted properly, it does contain these semiclassical phase factors. II) In practice, let ...



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