Semantic distinction between "Partition Function" and "Generating Functional" in QFT?

Question

I am just now learning about these, and I have seen them defined as follows: The generating functional for a set of fields $\phi_i$ is defined by:

$$Z[J_i]=\int\mathcal{D}\phi_i e^{i(S[\phi_i]+\int J_i\phi_i)}$$

and the partition function is $Z[0]$, generating the vacuum bubbles. However, as I read more and more about this, I find the semantic distinction between "partition function" and "generating functional" upheld less and less. Many people, such as Wikipedia, equate the two and just call everything "the partition function".

Does anybody have any particular knowledge about how important the semantic distinction between the two is at higher levels (i.e. further down the line in QFT, or in certain lines of research, or...)?

Answer 1

As I bet you already know, these two names have perfectly good reasons.

• The quantity $Z[J]$ is called the generating functional because functionally differentiating it yields correlation functions. This is just like how the partition function $Z[\beta]$ in thermodynamics is a generating function for the cumulants of energy.
• The quantity $Z[0]$ is called the partition function because upon Wick rotation to imaginary time it becomes equal to the integral of $e^{- S_E/\hbar}$, so it can be interpreted as the partition function of a system with energy $S_E$.

Similarly, you can think of $Z[J]$ as a partition function for a thermodynamic system in a fixed external field, so $Z[J]$ can be called both names.

Your last question is primarily opinion-based, but my experience has been that in hep-th people generally get less consistent with terminology as they progress from undergrad to grad to postdoc to professor. (I'm sure it would be different in, e.g. mathematical physics.) All the intuition from all the different words gets melted into this big soup that they dip into freely. Which word they use just depends on how they're thinking about $Z[J]$ in the moment.