# Real valued path integral

1. Apr 29, 2012

### unchained1978

I've been studying the path integral approach to QM on my own, and trying to draw some analogies between the partition function of QM $$Z_{QM}=\int D\varphi e^{\frac{i}{\hbar}S[\phi]}$$ and that of statistical mechanics $$Z_{SM}=\displaystyle\sum\limits_{i=0}^N g_{i}e^{-\beta E_{i}}$$. The thing is I don't understand why there is an $i$ in $Z_{QM}$. I've gone through a derivation and it comes from the Unitary operator $\hat{U}=e^{-i\hat{H}t}$, but I don't see why this is necessary. On wikipedia, the explanation is that the $i$ comes from the jacobian of the complex projective space or something like that. I'm not quite satisfied with that definition. The reason I'm investigating this is because in one of Hawking's papers he calculates the entropy of various spacetimes, and one thing I noticed is that the entropy $S=k_{B}lnZ+\beta<E>$ is only defined when $lnZ$ is real, which requires that $iS[g]$ is also real, and therefore the action must be complex. But I don't quite understand this argument from an intuitive point. Could anyone give me a good description (or link to one) of why this $i$ appears at all?