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In Shankar's QM book pg. 224, it was said that the contributions $Z=e^{iS[x(t)]/\hbar}$ add constructively near the classical path $x_{cl}(t)$ since the action $S[x_{}(t)]$ is stationary here.

As we move away from the classical path $x_{cl}(t)$, destructive interference sets in and the contributions $Z=e^{iS[x(t)]/\hbar}$ cancel each other out.

It was also stated that one can crudely say coherence is lost once the phase differs from the stationary value $S[x(t)]/\hbar \equiv S_{cl}/\hbar$ by about $\pi$.

How does this specific value of $\pi$ come about? Why can we say that if the phase difference is larger than $\pi$, destructive intereference occurs?

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TaeNyFan
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    There is nothing special about the phase angle $\pi$. It's just that when the classical action deviates from stationary action by few multiples of $\hbar$, the destructive interference sets in due to vigorous oscillation of the phase factor. Since $\hbar$ is such a small quantity, a slight deviation from stationary action will be a large multiple of $\hbar$ – KP99 Jun 03 '22 at 06:55

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Pay attention to the word crudely. The value $\pi$ is not a precise estimate, just a typical value for phases out of phase. Shankar is implying that when the path integral sum up a neighborhood of paths $\gamma$ whose phases $S[\gamma]/\hbar$ differ by the order of $\pi$, then coherence is lost and destructive interference sets in.

See also e.g. this related Phys.SE post.

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