What is a symmetry of the generating functional, and what is the significance? I cannot find a definition for a symmetry of the generating functional in Quantum Field Theory:
$$ Z[J] = \int \mathrm d \mu \, \exp\left\lbrace i S[J] \right\rbrace \, .$$
I know it's a simple question, but I cannot find a definition. What is the definition of a `symmetry' of this object? And what is its significance?
The generating functional $Z[J]$ is a function of $J$ that spits out a complex number. The only technically sound definition of a symmetry would be a map $f : J \mapsto J'$ which leaves $Z[J]$ invariant, i.e. $Z[J] = Z[f(J)]$. But that doesn't seem to have anything to do with the usual meaning of symmetry in this context.
 A: Dual transformations : Let $\phi \to g\cdot\phi$ be a (non anomalous) symmetry of the theory in the usual way (with $g$ belonging to some group $G$):
$$S[\phi] = S[g\cdot\phi] \quad \text{and}\quad \mathcal D\phi = \mathcal D(g\cdot \phi)$$
Then, we can find a transformation $J\to g\cdot J$ such that $\int J\cdot \phi = \int  (g\cdot J) \cdot (g\cdot \phi)$. Performing the change of variable in the generating functional, we get :
\begin{align}
Z[J] &= \int\mathcal D\phi e^{iS[\phi]  - i\int J\cdot \phi} \\
&= \int\mathcal D\phi e^{iS[g\cdot \phi] - i\int (g\cdot J) \cdot (g\cdot \phi)} \\
&= \int\mathcal D[g^{-1}\cdot \phi] e^{iS[\phi] - i\int(g\cdot J) \cdot\phi} \\
&= \int\mathcal D[\phi] e^{iS[\phi] - i\int(g\cdot J) \cdot\phi}\\
&=Z[g\cdot J]
\end{align}
A few examples

*

*for translation symmetry $\phi(x) \to \phi(x-a)$ the dual transformation is $J(x) \to J(x-a)$

*for a $U(1)$ symmetry, $\phi(x) \to e^{i\alpha} \phi(x)$, the dual transformation is $J(x) \to e^{-i\alpha}J(x)$

*for a $U(n)$ symmetry $\phi^a(x) \to {U^a}_b \phi^b(x)$, the dual transformation is $J_a(x) \to J_b(x) {(U^{-1})^b}_a$
A: *

*Yes, OP is right. Symmetries in terms of the underlying fields $\phi^k$ is in the generating functional/partition function/path integral
$$Z[J]~=~\int \!{\cal D}\phi \exp\left(\frac{i}{\hbar}\left( S[\phi]+J_k\phi^k\right)\right)  $$ transcripted into symmetries of the sources $J_k$.


*More generally, the same thing happens for correlation functions
$$\langle F \rangle_J ~:=~ \frac{1}{Z[J]}\int \! {\cal D}\phi~\exp\left(\frac{i}{\hbar}\left(S[\phi] +J_k\phi^k \right)\right)F[\phi] .$$


*Note that since the fields $\phi^k$ might not transform as tensorial objects, the $\phi^k$-symmetries may get encoded in non-trivial manners in the $J$-picture. For this reason, one seldomly tries to classify symmetries of a theory starting from the $J$-picture alone, which seems to be at the core of OP's question. Symmetries are typically identified in the $\phi$-picture, and then transcribed to the $J$-picture. Also keep in mind that sources $J_k$ are usually put to zero at the end of the calculations.


*Similarly, the $J_k$-symmetries get in turn transcripted into symmetries of the classical fields $\phi^k_{\rm cl}$ in the 1-PI effective action $\Gamma[\phi_{\rm cl}]$.


*For an example see e.g. my Phys.SE answer here.


*There might also be symmetries involving external parameters.
A: The generating function $Z[J]$ produces all the correlation functions of theory. The correlation functions contain all the symmetries of theory. However, these symmetries may be hidden in the generating function.
Since the correlation functions are produced by the number of functional derivatives, we can represent the generating function function crudely as
$$ Z[J] = Z[0]+J\langle \phi \rangle +\tfrac{1}{2}J^2\langle \phi^2 \rangle 
+\tfrac{1}{3!}J^3\langle \phi^3 \rangle + ... , $$
with some integral contractions between the $J$'s and correlation functions that are not shown and also normalizations not shown. The symmetries are inside the correlation functions $\langle \phi^n \rangle$. The $J$'s are just tags with which we can extract the appropriate correlation function. Their transformation would therefore not have any role to play. Apart from the formal role of $J$'s to serve as a device to extract the appropriate correlation function, they have no physical significance and therefore do not take part in any symmetry transformations.
