First of all forgive me if this question is not well suited for this forum: it is motivated by physics however after all my concerns are mathematical so I hope it would be appropriate to post it here. I'm interested in understanding the proof of Wightman reconstruction theorem: the standard literature where this proof can be found is the book by Streater and Wightman ,,PCT, Spin and Statistics and all that''. However I found several problems in this proof and I would be grateful if someone can clarify those issues.
First problem which I have is the equality $(f,g)=\overline{(g,f)}$ (see page 119). Authors claim that this equality follows from hermicity condition namely that $W_n(f_1 \otimes f_2 \otimes ... \otimes f_n)=\overline{W_n(f_n \otimes f_{n-1} \otimes ... \otimes f_1)}$ for simple tensors. However in the definition of this scalar product we sum terms of the form $W_n(\overline{f_k} \otimes g_{n-k})$ and I don't see why are we allowed to swap the first $k$ ,,variables'' with the other $n-k$ (we are allowed only to reverse all variables). Should we assume something more about our Wightman distributions?
The second problem which I have is somewhat similar:
On page 120 it is stated that $(\varphi(\overline{h})f,g)=(f,\varphi(h)g)$ i.e. the formal adjoint of $\varphi(h)$ is $\varphi(\overline{h})$. However I would rather say that the formal adjoint multiplies by $\overline{h}$ from the right as opposed to $\varphi(h)$ which multiplies from the left by $h$. But after all-do we need to check this equality in order to construct quantum field theory? There is no requirment for the form of $\varphi(h)^*$ in the Wightman axioms (or am I wrong? Maybe it has something to do with the fact whether we want to construct neutral or charged field etc?)
And the last one
On page 125 it is claimed that the fact that the support of the (joint) spectral measure for $\{U(a,I)\}_{a \in \mathbb{R}^4}$ lies in/on the future light cone is an immediate consequence of Axiom b) (spectral conditions) on page 117. However it is not clear for me how one can reverse the argument given on page 109 where it is proven that once we start with the quantum field obeying Wightman axioms then this spectral condition is satisfied. Roughly speaking on page 109 they show that $4$-dimensional Fourier transform of $W$ is zero (in other words that Fourier transform applied to one variable is zero) which implies that the $4n$-dimensional Fourier transform is $0$-I don't see how to reverse this argument.
I will be grateful if someone could shed some light upon my issues.
