# Period of the propagator of quantum harmonic oscillators

I found something that I'm confused with when calculating the propagator of harmonic oscillator. Using the energy representation, the propagator of a quantum harmonic oscillator can be expressed as : $$K(x_f,t_f;x_i,t_i)=\sum_ne^{-i\omega T(n+\frac{1}{2})}\psi_n(x_f)\psi_n^*(x_i)\tag{1}$$ where the $$\psi_n$$s are the wave functions of energy eigenstates, and $$T=t_f-t_i$$. Let $$T^\prime=T+\frac{2\pi}{\omega}$$, then $$K(T^\prime)=-K$$.
After some calculation, the final result of this propagator yields: \begin{aligned} &K\left(x_f, t_f ; x_i, t_i\right) \\ &=\left[\frac{m \omega}{2 \pi \hbar i \sin \omega T}\right]^{1 / 2} e^{\frac{i}{\hbar} \frac{m \omega}{2 \sin \omega T}\left[\left(x_f^2+x_i^2\right) \cos \omega T-2 x_f x_i\right]} \end{aligned}\tag{2} while $$K(T^\prime)=K$$ this time, rather than $$K(T^\prime)=-K$$. How did that happen?

Good observation. OP's eq. (2) should be amended with a metaplectic correction/Maslov index: There is a caustic at every half-period, which leads to a phase factor $$\exp\left(-\frac{i\pi}{2}\right)$$ jump, cf. e.g. my Phys.SE answer here.