Proof of cluster decomposition theorem I have looked through a few lits but can't find a proof for cluster decomposition theorem, which states that asymptotically separated apart operators cannot influence each other. Is there a formal proof?
 A: As suggested by Kyle, here is detail of the proof. This proof belongs to Streater and Wightman (Theorem 3-4 in their famous textbook). They prove such a form of the Clustering Decomposition Theorem:
If a is a space-like vector, then ($\mathscr W$ is the Wightman function)
$\mathscr{W}(x_1,\cdots,x_j,x_{j+1}+\lambda a,\cdots,x_{n}+\lambda a)\to\mathscr{W}(x_1,\cdots,x_j)\mathscr{W}(x_{j+1},\cdots,x_{n})$
as $\lambda\to\infty$, in the sense of convergence in $\mathscr S^\prime$ (tempered distribution).
In their proof, they assume that the theory has a non-zero mass gap $M$, which is not necessary. For convenience, they choose $a^2=-1$. Consider the Fourier transformation of 
$F_1=\mathscr{W}(x_1,\cdots,x_j,x_{j+1}+\lambda a,\cdots,x_{n}+\lambda a)-\mathscr{W}(x_1,\cdots,x_j)\mathscr{W}(x_{j+1},\cdots,x_{n})$,
$F_2=\mathscr{W}(x_{j+1}+\lambda a,\cdots,x_{n}+\lambda a,x_1,\cdots,x_j)-\mathscr{W}(x_1,\cdots,x_j)\mathscr{W}(x_{j+1},\cdots,x_{n})$.
They are zero unless $P=p_1+\cdots+p_j$ lies in forward and backward cones, respectively. It is a conclusion of the spectral conditions. The non-zero contribution of the first term at $P^\mu=0$ is cencelled by $-\mathscr{W}(x_1,\cdots,x_j)\mathscr{W}(x_{j+1},\cdots,x_{n})$. So $\tilde F_1\pm\tilde F_2=0$ unless $P^2\geqslant M^2$. From the locality condition, we know that $F_1$ and $F_2$ coincide up to sign for large enough $\lambda$ when $x_1,\cdots,x_n$ are fixed (when $x_i-x_k-\lambda a$ is space-like for any $i=1,\cdots,j$ and $k=j+1,\cdots,n$). The distribution theory tells us that $F_1\pm F_2$ is a finite sum of the derivatives of a polynomially bounded continuous function, $G$ say. Thus, for any test function $h\in\mathscr S$ ($\mathscr S$ is the space of the rapidly decreasing functions. See https://en.wikipedia.org/wiki/Schwartz_space for example. ),
$\int (F_1\pm F_2)(x_1,\cdots,x_n)h(x_1,\cdots,x_n)dx_1\cdots dx_n=\int D^mG(\lambda,x_1,\cdots,x_n)h(x_1,\cdots,x_n)dx_1\cdots dx_n$.
Because $G$ is polynomially bounded in all variables, if we denote the Euclidean norm (but not Minkowski) in ${\bf{R}}^{4n}$ (remember that $x_j$ is a point in spacetime, so it is 4 numbers in fact) as $R^2=\sum_j[(x_j^0)^2+({\bf{x}}_j)^2]$, we have 
$|G|\leqslant G_0\lambda^NR^Q$  for sufficiently large $\lambda$ and $R$. Here, both $N$ and $Q$ are positive integer by definition. Using local commutativity we can and will choose the $\pm$ sign so that $D^mG=0$ for $R<R_0$, where $R_0$ is a positive multiple of $\lambda$. This is because for each $(x_1,\cdots,x_n)$ inside the $4n$-dimension ball $R\leqslant R_0$, we can always find a $\lambda$ which vanishes $F_1\pm F_2$. But the ball is a compact set so there must be a maximum $\lambda_0$. So we know that  when $R<R_0$, $\lambda>\lambda_0$, the integrated function in the left-hand side is zero. Since $h$ could be any test function, it is obviously that $D^mG=0$ when  $R<R_0$, $\lambda>\lambda_0$. Now we find the $R_0$ for $\lambda_0$. To determine $R_0$ note that
$(x_i-x_k-\lambda a)^2=(x_i^0-x_k^0)^2-({\bf x}_i-{\bf x}_k)^2-\lambda^2-2\lambda(a^0(x_i^0-x_k^0)-{\bf a}\cdot({\bf x}_i-{\bf x}_k))$
So it is less than
$|x_i^0|^2+|x_k^0|^2+2|x_i^0||x_k^0|-\lambda^2+2\lambda(|a^0|(|x_i^0|+|x_k^0|)+|{\bf a}|(|{\bf x}_i|+|{\bf x}_k|))$
So for the points in the ball $R<R_0$, we have 
$(x_i-x_k-\lambda a)^2<4R_0^2-\lambda^2+4\lambda R_0(|a^0|+|{\bf a}|)$.
It is easy to see that for any fixed $\lambda$, if we choose the radiu $R_0$ of the ball to be $\frac{1}{8}[|a^0|+|{\bf a}|]^{-1}\lambda$, the right-hand side is $<0$. Thus, for any $\lambda$, the integral inside the ball $R<\frac{1}{8}[|a^0|+|{\bf a}|]^{-1}\lambda$ contributes 0 to the result. We need only check the contribution outside this ball, which is 
$|\int D^mG(\lambda,x_1,\cdots,x_n)h(x_1,\cdots,x_n)dx_1\cdots dx_n|=|\int_{R>R_0-\varepsilon} D^mG(\lambda,x_1,\cdots,x_n)h(x_1,\cdots,x_n)dx_1\cdots dx_n|$,
which is 
$|\int_{R>R_0-\varepsilon} G(\lambda,x_1,\cdots,x_n)D^mh(x_1,\cdots,x_n)dx_1\cdots dx_n|\leqslant\int_{R>R_0-\varepsilon}|G||D^mh|R^{4n-1}dR~d\Omega_{4n}$.
Because $h$ is a test function, for any $q$, $|D^mh|<cR^{-q}$ holds for some $c$ and all sufficiently large $R$. Then the right-hand side of the preceding inequality is bounded by 
$\int _{R>R_0}G_0c\lambda^NR^{Q-q+4n-1}dR~d\Omega_{4n}$
for all sufficiently large $\lambda$. When $q>Q+4n$, the integral gives 
$\displaystyle{\frac{4n\pi^{2n}}{\Gamma(1+2n)}\frac{G_0c\lambda^N}{(q-(Q+4n))R_0^{q-(Q+4n)}}=\frac{4n\pi^{2n}}{\Gamma(1+2n)}\frac{G_0c8^{q-(Q+4n)}(|a^0|+|{\bf a}|)^{q-(Q+4n)}}{(q-(Q+4n))\lambda^{q-(N+Q+4n)}}=\frac{c_1}{\lambda^p}}$.
So we have
$|\int (D^mG)h|<\displaystyle{\frac{c_1}{\lambda^p}}$.
Because $q$ could be any real number, for sufficient large $q$ ($q>N+Q+4n$), we see that the integral decreases faster than any inverse power of $\lambda$ when $\lambda$ goes to infinity. That means that the integral is a test function whose variable is $\lambda$. This is what is meant by 
$\lambda^p(F_1\pm F_2)(x_1,\cdots,x_j,x_{j+1}+\lambda a,\cdots,x_n+\lambda a)\to 0$   as   $\lambda\to \infty$
in the sence of convergence in $\mathscr S^\prime$.
The final step of the proof is to show that this equation holds for $F_1$ and $F_2$ separately, and here is where the spectral condition is used. Replace the $h$ of the preceding argument by a new $h_1$ which is defined by $\tilde h_1=\theta \tilde h$ ($\tilde{f}$ means the Fourier transformation of $f$) where $\theta$ is an infinitely differentiable function of the variable $P$, equal to 1 for $P^2\geqslant M^2$, $P^0>0$, and 0 for $P^0\leqslant0$. Clearly $h_1\in\mathscr S$ so the preceding argument works equally well for it. But $\int F_2h_1dx_1\cdots dx_n=0$ and $\int F_1h_1dx_1\cdots dx_n=\int F_1hdx_1\cdots dx_n$, so the Clustering Decomposition property is proved.               $\blacksquare$
Remarks: 
(1) The clustering decomposition property proved here shows that the correlation between large space-like distance separated subsystems decreases faster than any power of the inverse of the distance, when there is a non-zero mass gap in the theory. It can be show that it decreases exponentially and the damping factor depends on the mass gap $M$. If there are zero-mass particles, it will decrease as slowly as $1/\lambda^2$.
(2) The clustering decomposition property and the Reeh-Schlieder theorem (see https://en.wikipedia.org/wiki/Reeh–Schlieder_theorem for example) are obverse and reverse of a coin. The clustering decomposition theorem allows us to isolate a subsystem from the others which are large space-like distance separated. The Reeh-Schlieder theorem tells us that the correlation is universal. 
A: If you want a formal proof of the Clustering Decomposition Theorem, I think you may find it in 
"PCT, Spin and Statistic, and All That" by Streater and Wightman. You may also find it in 
"Introduction to Axiomatic Quantum Field Theory" by Bogolubov et al, 
"Mathematical Theory of Quantum Fields" by H. Araki, or 
"Methods of Modern Mathematical Physics, Vol III: Scattering Theory" by Reed and Simon. 
You may also read the original literatures by D. Ruelle and other researchers in 1960s. I hope this information helpful.
