How to complete the evaluation of green's function for harmonic oscillator in second quantized notation?

1. For the given Hamiltonian, $H=\sum a^{\dagger}_{i}a_{i}$, how one could evaluate the green's function using path integral approach?

While evaluating in the coherent state path integral approach I end up with the following,

$$G(\phi_{f}^{*}, t_{f};\phi_{i},t_{i})=\int [\prod_{i=1}^{N-1}d\phi^{*}_{i}d\phi_{i}] \exp\Bigg[-\phi_{f}^{*}\phi_{f}+\dfrac i\hbar\Bigg(\int_{t_{i}}^{t_{f}}dt\Bigg\{i\hbar\phi^{*}\dfrac {d\phi}{dt}-\omega\phi^{*}\phi\Bigg\}\Bigg]$$ However, To evaluate this I have started to integrate this in the discretized notation $$\int \dfrac{d\phi_{1}^{*}d\phi_{1}}{\pi}\int\dfrac{d\phi_{2}^{*}d\phi_{2}}{\pi}\int\dfrac{d\phi_{3}^{*}d\phi_{3}}{\pi}......\int\dfrac{d\phi_{N-1}^{*}d\phi_{N-1}}{\pi}\exp\Bigg(-\phi_{f}^{*}\phi_{f}\Bigg)\exp\Bigg[\dfrac{i}{\hbar}\epsilon\sum_{i=1}^{N-1}\Bigg(\phi_{i}^{*}\dfrac{\phi_{i}-\phi_{i-1}}{\epsilon}-\omega\phi_{i}^{*}\phi_{i-1}\Bigg)\Bigg]$$. To get the green's function I have expanded the summation and collected the $$\phi_{1},\phi_{1}^{*}$$ terms to complete the integral.$$\int\dfrac{d\phi_{1}^{*}d\phi_{1}}{\pi}\exp\Bigg[\epsilon\dfrac{i}{\hbar}\Bigg\{\Bigg(\phi^{*}_{1}\dfrac{\phi_{1}-\phi_{0}}{\epsilon}-\omega\phi_{1}^{*}\phi_{0}\Bigg)+\Bigg(\phi^{*}_{2}\dfrac{\phi_{2}-\phi_{1}}{\epsilon}-\omega\phi_{2}^{*}\phi_{1}\Bigg)\Bigg\}\Bigg]$$ However, I do not know to complete the integral to get a pattern so that this whole integrals would be expressed only in terms of initial and final boundary conditions.I do not know how to proceed this further as I don' the limits of this integral. Can some one help me in completing this integral for Bosons.

• This is a gaussian integral, which can be performed exactly. Same thing for the whole path integral, in fact. – Adam May 17 '17 at 7:56
• Do I have to transform this variables to real polar variables $r$ and $\theta$ to evaluate. Can you please suggest some reference to complete this evaluation ? What may be the answer to this evaluation ? – user135580 May 17 '17 at 8:22
• Have a look here lptmc.jussieu.fr/files/chap_fi.pdf , Eq1.848 and following – Adam May 17 '17 at 8:25
• Thank you so much yet, the link does not open. Can you please send the link or file to the following email id vigneshwaran@imsc.res.in – user135580 May 17 '17 at 10:30