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Why is the partition function
$$Z[J]=\int\ \mathcal{D}\phi\ e^{iS[\phi]+i\int\ d^{4}x\ \phi(x)J(x)}$$
also called the generating function?
Is the partition function a q-number or a c-number?
Does it make sense to talk of a partition function in classical field theory, or can we define partition functions only in quantum field theories?
Is the source $J$ a q-number or a c-number?
It is called a generating function, because one can use it to generate $n$-point functions with the aid of functional derivatives with respect to the source $J$. For instance, one can compute the two point function as follows: $$ \langle\phi(x_1)\phi(x_2)\rangle = \frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} Z[J]_{J=0} . $$ The result would be a $c$-number. Hence, the generating function itself is also a $c$-number.
Formally one can treat classical field theories also with the aid of such generating functions, provided that, if you set $J=0$, you recover the original theory.
These generating functions are based on the path integral approach in which fields can be interpeted as $c$-numbers as apposed to the $q$-numbered operator-valued fields used in the second quantization approach. As a result, the source $J$ is also interpeted as a $c$-number.
