Undergrad Partition function of quantum mechanics

[PGaccount]

In quantum mechanics, we have the partition function Z[j] = e^-W[j] = ∫ e^{iS+ j_iOi}. The propagator between two points 1 and 2 can be calculated as

$\frac{\delta}{\delta j_1}\frac{\delta}{\delta j_2} Z = \langle O_1 O_2 \rangle$

The S in the path integral has been replaced by S → S + j_iOⁱ. Similarly we have what is also called the partition function in thermodynamics, Z[β] = e^-F = tr e^-βH, where F = E - TS is the free energy. The average energy can be calculated as

$\langle E \rangle = \frac{\partial Z}{\partial \beta}$

When you add heat to a gas at constant pressure, the change in the enthalpy H = U + PV is equal to the heat added. The gas has to expand to keep the pressure constant. The work PdV done by this expansion is automatically accounted for by the definition of H, which is analogous to the above formula for the transformation of S. What is the connection between these two types of partition functions?

 

[Demystifier]

The connection is mathematical, but I don't think that there is a deep physical connection. The mathematical connection is sometimes expressed concisely by saying that inverse temperature is an imaginary time.

 

[vanhees71]

To the contrary this similarity is crucial for many-body quantum field theory in thermal equilibrium. It's used in the socalled imaginary-time or Matsubara formulation of thermal field theory. The only difference to the vacuum case is that time becomes purely imaginary, $t=-\mathrm{i} \tau$ with $\tau \in (0,\beta)$ with $\beta=1/(k_{\text{B}} t)$ and the fields are subject to periodic (bosons) or antiperiodic (fermions) boundary conditions. You get the same Feynman rules for perturbation theory with some changes compared to the vacuum case: Instead of energy integrals you have sums over the Matsubara frequencies $\omega_k=2 \pi k_{\text{B}} T k$ (bosons) or $\omega_k = \pi(2 k+1) k_{\text{B}} T$ with $k \in \mathbb{Z}$ (I've chosen natural units with $\hbar=c=1$).

For details (relativistic thermal QFT), see

https://itp.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf