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Today I was doing some QFT homework and in one of them they ask me to calculate the Harmonic Oscillator propagator, which, as you may know is:

[tex]W(q_2,t_2 ; q_1,t_1) = \sqrt{\frac{m\omega}{2\pi i \hbar \sin \omega (t_2-t_1)}} \times \exp \left(\frac{im\omega}{2\hbar \sin \omega (t_2-t_1)}\left[ (q_1^2+q_2^2)\cos \omega (t_2-t_1)-2q_2q_1\right]\right)[/tex]

So, the last question of the problem is explaining why do we have singularities at [itex]t_2-t_1=n\pi / \omega[/itex].

I've been searching for it on books and I've found something about the "caustics", which I've no idea what is it...

I understand that when [itex]t_2-t_1=0 [/itex] it has to diverge, because the particle hasn't moved from its place as the time hasn't passed, but why it happens every half period?

It seems to me that if the function diverges at those points then we are sure of which is the position of the particle at those times, and I thought that there had to be an uncertainty for [itex]t_2-t_1 \neq 0 [/itex] (I don't know if I'm explaining well enought what I think hahahhaa....)

Thanks in advance!

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# Singularities in the harmonic oscillator propagator

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