It is well known that the propagator (kernel) of a simple harmonic oscillator is given by
$$
U\left(x_{b},T;x_{a},0\right)=\sqrt{\frac{m\omega}{2\pi i\hbar\sin\omega T}}\exp\left\{ \frac{im\omega}{2\hbar\sin\omega T}\left[\left(x_{a}^{2}+x_{b}^{2}\right)\cos\omega T-2x_{a}x_{b}\right]\right\}. \tag{1}
$$
I want to show explicitly that at times that are integer multiples of the period (i.e. $T=(2\pi/\omega) n$) the propagator becomes $\delta (x_b – x_a)$ while for odd multiples of the period (i.e. $T=(\pi/\omega)(2n+1)$) it's equal to $\delta(x_b + x_a)$.
Proving the first case seems straightforward. By defining
$$
\epsilon=\sqrt{-\frac{\hbar\sin \omega T}{im\omega}}
$$
we can rewrite the propagator as
$$
U\left(x_{b},T;x_{a},0\right)=\frac{1}{\epsilon\sqrt{2\pi}}\exp\left\{ -\frac{1}{2\epsilon^{2}}\left[\left(x_{a}^{2}+x_{b}^{2}\right)\cos\omega T-2x_{a}x_{b}\right]\right\}.
$$
Now, for $T\to2\pi n/\omega$ where $n$ is an integer, the propagator coincides with the definition of Dirac delta as a limit of a Gaussian:
$$
U\left(x_{b},T;x_{a},0\right)=\lim_{\epsilon\to0}\frac{1}{\epsilon\sqrt{2\pi}}\exp\left[-\frac{1}{2\epsilon^{2}}\left(x_{a}-x_{b}\right)^{2}\right]=\delta\left(x_{a}-x_{b}\right).
$$
So far, so good. However, when $T\to\frac{\pi}{\omega}\left(2n+1\right)$, we still have $\epsilon \to 0$ except now $\cos \omega T \to -1$ and thus
$$
U\left(x_{b},T;x_{a},0\right)=\lim_{\epsilon\to0}\frac{1}{\epsilon\sqrt{2\pi}}\exp\left\{ \frac{1}{2\epsilon^{2}}\left(x_{a}+x_{b}\right)^{2}\right\}.
$$
But now the exponent doesn't coincide with $\delta(x_a + x_b)$. Redefining $\epsilon$ such that it would behave appropriately inside the exponent leads to an overall imaginary phase, which isn't good either. What am I missing?
Best Answer
OP's troubles are (partly?) caused by the fact that OP's eq. (1) only holds for $0<T<\frac{\pi}{\omega}$. OP's eq. (1) lacks the caustics/metaplectic correction/Maslov index. The corrected formula $$\begin{align} K(x_b,T;x_a,0)~=~&\exp\left[-i\left(\frac{\pi}{4}+\frac{\pi}{2}\left[\frac{\omega T}{\pi}\right]\right)\right]\cr & \sqrt{\frac{m\omega}{2\pi \hbar|\sin\omega T|}} \exp\left\{\frac{i}{\hbar}S_{\rm cl}\right\}\end{align} $$ is known as Feynman-Souriau formula.
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