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 \begin{equation}Z_{QM}=\int D\varphi e^{\frac{i}{\hbar}S[\phi]}\end{equation} and that of statistical mechanics \begin{equation}Z_{SM}=\displaystyle\sum\limits_{i=0}^N g_{i}e^{-\beta E_{i}}\end{equation}. The thing is I don't understand why there is an [itex]i[/itex] in [itex]Z_{QM}[/itex]. I've gone through a derivation and it comes from the Unitary operator [itex]\hat{U}=e^{-i\hat{H}t}[/itex], but I don't see why this is necessary. On wikipedia, the explanation is that the [itex]i[/itex] 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 [itex]S=k_{B}lnZ+\beta<E>[/itex] is only defined when [itex]lnZ[/itex] is real, which requires that [itex]iS[g][/itex] 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 [itex]i[/itex] appears at all?(adsbygoogle = window.adsbygoogle || []).push({});

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Real valued path integral

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - Real valued path | Date |
---|---|

I Are the energy fluctuations in space real or virtual? | Jan 18, 2018 |

I Virtual and Real particles | Jan 4, 2018 |

Expectation value of a real scalar field in p state | Apr 28, 2015 |

What makes expectation values real? | Feb 7, 2015 |

Mega quick question: must the Lagrangian (density) be real valued? | May 23, 2014 |

**Physics Forums - The Fusion of Science and Community**