What is the physical meaning of $W[J]=\frac{\hbar}{i}\ln Z[J]$?

Question

The quantity $Z[J]$ (which is the generating functional for all Green functions) physically represents the probability amplitude for a system to remain in the vacuum state. Can we find a similar physical meaning of the quantity $$W[J]=\frac{\hbar}{i}\ln Z[J]$$ (which is the generating functional for all connected Green functions)?

Answer 1

Recall that, by definition, $$ Z[J]=e^{-iW[J]}=\langle\Omega|e^{-iHT}|\Omega\rangle $$ where $H$ is the Hamiltonian of your system, and $\Omega$ is the vacuum state. Therefore, $W[J]$ can be interpreted as the energy of the vacuum $E_\Omega$ in the presence of a source $J$, where the origin is chosen at $E=0$ for $J=0$. In other words, $W[J]$ is how much more energy does the vacuum have when we turn on an external source $J$.

More precisely, and as per $Z[J]=e^{-iW[J]}$, you can think of $W$ as the Helmholtz free energy of the system.

For more details, see e.g. ref.1., §11.3, or ref.2., §6.1.2 (and Appendix 18). See also this PSE post.

References.

1. Peskin & Schroesder - An introduction To Quantum Field Theory.

2. Zinn-Justin - Quantum Field theory and Critical Phenomena.