10
$\begingroup$

The Wick-Rotation rotates imaginary time into inverse temperature (as can be seen from its "rotating" the Schrödinger equation into the heat equation). Now since entropy is temperature's conjugate, I was wondering what its Wick-Rotation relates to?

$\endgroup$

1 Answer 1

7
$\begingroup$

This isn't a complete answer, but John Baez gave a pretty good treatment of this in a series of blog posts (part 1, part 2, part 3, part 4; arXiv paper with some more stuff).

Basically, he defines what he calls the "quantropy", which is just the classical entropy formula with $\beta$ replaced by $-i/\hbar$ and the energy replaced with the action. (Note that this is not at all the same as the von Neumann entropy.) The quantropy is essentially the Wick rotation of the entropy.

He then shows that finding a stationary point of the quantropy gives you the Schrödinger equation (and various other aspects of quantum mechanics), in much the same way that maximising the entropy gives you the Boltzmann distribution and much of the rest of classical statistical mechanics. It's quite interesting.

$\endgroup$
4
  • $\begingroup$ That's some very interesting reading you got there (I should have known to check John Baez on something like this...), though as you mention he relates $\beta$ with $-i/\hbar$ and not with $it$ as the Wick-Rotation does, or did I mix something up there? I guess I need to read that paper more thoroughly, haven't done any QFT for years :/ $\endgroup$ Oct 26, 2014 at 19:18
  • 1
    $\begingroup$ He replaces $\beta$ with $-i/\hbar$, but he also replaces the energy with the action, which at least has units of time times energy, so the $t$ is in there somehow. I have to admit I don't know much about Wick rotations myself, so I'm waving my arms a bit and hoping they'll turn out to be equivalent :) $\endgroup$
    – N. Virgo
    Oct 27, 2014 at 0:31
  • 2
    $\begingroup$ Thank you for the links. I came here from John Baez: Getting to the Bottom of Noether’s Theorem, so maybe that’s worth adding to the “Quantropy” reading list. $\endgroup$ Jul 1, 2020 at 22:58
  • $\begingroup$ @AdamChalcraft Nice read, thanks! $\endgroup$ Jul 2, 2020 at 9:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.