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?
[Physics] Imaginary time is to inverse temperature what imaginary entropy is to …
complex numbersentropytemperaturetimewick-rotation
Related Solutions
Lots of different ways to answer, but none of them can be too intuitive since imaginary time is, well, imaginary. But here is one attempt to make the result more or less self-evident.
The basic object to calculate in quantum statistical mechanics (in thermal equilibrium, in the canonical ensemble) is the partition function (with potential insertions if you want to calculate correlation functions):
$$Z= \operatorname{Tr}(e^{-\beta H})= \sum_\psi \langle \psi(0)|e^{-\beta H}|\psi(0) \rangle$$
where $H$ is the Hamiltonian and we have a sum over any complete set of states $\psi$, written in the Schrödinger picture at some fixed time which we take to be $t=0$. In that picture the time evolution of a state is
$$|\psi(t)\rangle = e^{-i t H}|\psi(0)\rangle$$
The basic observation now is that the Boltzmann factor $e^{-\beta H}$ can be regarded as an evolution of the state $\psi$ over imaginary time period $-i \beta$. Therefore we can write:
$$Z= \sum_\psi \langle \psi(0)|\psi(-i\beta) \rangle$$
This is now the vacuum amplitude (with possible insertions) which is the sum over all states $\psi$ in some arbitrary complete basis. Except that you propagate any final states with time $ i \beta$ with respect to the initial state. In other words however you choose to calculate your vacuum amplitude (or correlation function) — a popular method is a path integral — you have to impose the condition that the initial and final states are the same up to that imaginary time shift. This is the origin of the imaginary time periodicity.
Your equation is correct only if:$$\mathrm dQ = mc\,\mathrm dT$$ which is not generally true, indeed, common sense tells you that a change in temperature leads to conclusion that an object being heated up. But we do not encounter gases much in our life, which could be regarded as a general case. In reality your assumption is generally false, a good example would be a general gas process: $$\mathrm dQ =\mathrm dW +\mathrm dU$$ Which would take this form for an ideal gas: $$P\mathrm dV+\frac{3}{2}NR\,\mathrm dT = P\,\mathrm dV \,\,\,\text{ if } \,\,\,\mathrm dT=0$$ Now you can see that although temperature does not change, heat supply is still possible, at least mathematically. This process is called Isothermal heating, as might have guessed already. This could serve you as an example of non increasing temperature heating.
After that, you should see that $dQ = mc\mathrm dT$ is usually not the case. Because temperature is not the only extensive parameter for a gas, i.e. is not the only thing which determines the energy supply, i.e. is not the only thing which rises when heats comes in. Real equation for entropy of an ideal gas then would look like: $$\mathrm dS = \frac{1}{T}(P\,\mathrm dV + (3/2)NR\,\mathrm dT) = NR\frac{\mathrm dV}{V} + \frac{3}{2} NR \frac{\mathrm dT}{T}$$
Then $S$ would look like,
$$S = NR \ln{ \left( \left(\frac{V}{V_0} \right) \left( \frac{T}{T_0} \right)^{3/2} \right)}$$
for $N = \textrm{constant}$, namely your number of particles does not change, and $V_0$ and $T_0$ play the same role as $C$ in your answer. Bear in mind that this result is not to be used in your studies, since some modifications and generalizations are to be made first, but it is good enough for your understanding.
Best 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.