It is well-known that if $S \gg \hbar$, then the classical path dominates the Feynman path integral. But is there some to show that if $S\gg\hbar$, then the particle's trajectory will approach the classical path?
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3$\begingroup$ the argument $ exp(iS/ \hbar ) $ oscillates heavily whenever $ \hbar \sim 0 $ in this case only the trajectory which satisfy $ \delta S =0 $ contributes to the path integral.. the other trajectories 'interfere' each other $\endgroup$– Jose Javier GarciaCommented Jul 15, 2012 at 21:58
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4$\begingroup$ This is only completely rigorously true if you add a small imaginary part to the time, so that high action paths are suppressed a little bit exponentially. This gives a cutoff which makes the Feynman integrals sensible mathematically, and it is always implicitly or explicitly used when you are doing a path integral. $\endgroup$– Ron MaimonCommented Jul 16, 2012 at 6:22
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4$\begingroup$ The particle path will not approach the classical path exactly, rather the contributions of the important trajectories in the sum will be no different than just looking at the classical trajectory. This is subtly different, because you can still describe a probability distribution over phase space in the classical limit of quantum mechanics, there is still some left-over residual superposition principle, but it's without interference. $\endgroup$– Ron MaimonCommented Jul 16, 2012 at 9:10
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$\begingroup$ The question is not clear albeit the next related questions:19417,32112,32237, suggest what has been demonstrated in the article. $\endgroup$– user11781Commented Dec 8, 2012 at 4:41
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$\begingroup$ More answers and references here: physics.stackexchange.com/q/32112/226902 physics.stackexchange.com/q/80825/226902 physics.stackexchange.com/q/384199/226902 $\endgroup$– QuilloCommented Dec 19, 2023 at 22:39
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Actually there is a very interesting approach followed by E.Gozzi and his students to express the transition probability in Classical Mechanics in terms of a path integral. He is currently studying the relation between MQ and CM using this approach.
You can take a look at his site : http://www-dft.ts.infn.it/~gozzi/, there you will find some interesting references.
answered Oct 1, 2012 at 15:26