Why is a general Green's function invariant under translations? I am struggling to understand Green's functions, as used in Quantum Field Theory. One of my main problems is that the source I have been reading has a definition which is certainly correct, but involves enough integrals to obscure any intuition I initially had. Could someone explain what a Green function $G^{(n)}(x_1,x_2,\dots,x_n)$ intuitively represents (or provide a reference to a readable explanation)? In particular, I am being told that the Green function enjoys the invariance $$G^{(n)}(x_1+y,x_2+y,\dots,x_n+y) = G^{(n)}(x_1,x_2,\dots,x_n)$$
Supposedly, this is very simple and intuitive, but I can't see it (other than going a few levels down in the definitions, which in not very enlightning). Could someone please explain this property?
(my usual disclaimer: I am not a physicist, but a mathematician who occasionally comes into contact with physics. Please forgive my lack of physical insight.)
 A: The intuition comes from Poincare-invariant scalar field theories, where a (Minkowski-signature) Green's function is the vacuum expectation value of a product of field operators:
$G(x_1,...,x_n) = \langle vac | \phi(x_1)...\phi(x_n)| vac\rangle.$
Here, I'm writing $\phi(x_i)$ for the observable which measures the value of a scalar field $\phi$ at a point $x_i \in \mathbb{R}^d$.  (This is a mild abuse of notation.  If we were being careful I would have to explain that fields are operator-valued distributions, and that the Green's function is a Schwarz kernel for the distribution in $n$ variables given by $f_1,...f_n\mapsto \langle vac | \phi(f_1)...\phi(f_n)|vac\rangle$.)
In such theories, the group of translations of $\mathbb{R}^d$ has a unitary representation $U$ on the Hilbert space where $|vac\rangle$ lives, and it acts by conjugation on operators on this Hilbert space.  In particular, translation by $y \in \mathbb{R}^d$ sends $\phi(x_i)$ to $U^*(y)\phi(x_i)U(y) = \phi(x_i + y)$.  These translation operators also leave the vacuum vector invariant. This is where the translation formula you asked about comes from.
$\langle vac | \phi(x_1)...\phi(x_n)| vac\rangle 
\\= \langle vac | U^*(y)\phi(x_1)...\phi(x_n) U(y)| vac\rangle \\= \langle vac | U^*(y)\phi(x_1)U(y)U^*(y)...U(y)U^*(y)\phi(x_n)U(y)| vac\rangle \\= \langle vac | \phi(x_1+y)...\phi(x_n+y)| vac\rangle$.
In the first equality, I used translation invariance of the vacuum: $U(y)|vac\rangle = |vac\rangle$.  In the second, I inserted $1 = U(y)U^*(y)$ between every pair of adjacent field operators.
Your source may be talking about Euclidean signature Green's functions.  These are obtained by analytically continuing in the variables $x_i$.
Generally, Streater & Wightman's book is a nice source for this stuff.  Also, David Kazhdan's contribution to the IAS QFT & Strings for Mathematicians book.
A: Roughly speaking, a Green's function measures the correlation between fields at different points of spacetime. The reason we care about them in particle physics is that they are related to scattering amplitudes by the LSZ formula. A scattering amplitude is just the transition amplitude from an $n$-particle initial state to an $m$-particle final state. 
To answer your specific question, a Green's function should respect the symmetries of the underlying theory. The easiest way to see this is by writing 
\begin{align*}
\langle F(\phi)\rangle=\int d\phi F(\phi)\exp(iS[\phi]),
\end{align*}
where $S$ is the action. Then if it's possible to define a quantum mechanical measure that respects a symmetry of the action, then the Green's functions should be invariant under that symmetry. For example, if the action is translation-invariant, then the Green's functions are also.
One should be careful about the measure, since the classical symmetry might be anomalous, meaning that it can't be carried over to the quantum theory. This is impossible for translations, since it's easy to come up with regulators that preserve translation invariance (dim reg, Pauli Villars, zeta function etc. etc.).
A: Well the green’s function is independent of a constant shift added to all of its arguments if and only if the corresponding operator is shift invariant. The simplest version of this situation is differential operator for one variable in that case the green function is function of difference of its two arguments.
