Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Real valued path integral

  1. Apr 29, 2012 #1
    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?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Real valued path integral
  1. Path integrals (Replies: 5)

  2. The Path Integral (Replies: 0)

  3. Path integral (Replies: 2)

  4. The Path Integral (Replies: 15)

Loading...