# Imaginary time is to inverse temperature what imaginary entropy is to …?

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?

## 1 Answer

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.

• 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 :/ – Tobias Kienzler Oct 26 '14 at 19:18
• 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 :) – Nathaniel Oct 27 '14 at 0:31