In various papers that I have been reading about applying the Wightman axioms to conformal field theory, the authors write things like the following about the stress-energy tensor:
$$\int \mathrm{d}x^1[\Theta_{0\mu}(x^0,x^1),\phi(y)]=[P_\mu,\phi(y)]=-i\partial_\mu\phi(y).$$
Now, as $\Theta_{\mu\nu}$ is an operator-valued distribution, I don't see how we can integrate it over $1\notin \mathcal{S}(\mathbb{R}^2).$ In that case, how am I supposed to interpret this equation in a distributional sense?
Another question I have pertains to the stress energy tensor. Schottenloher's book mentions that it is contains a holomorphic part, i.e. some combination of its entries (which we'll call $T$) satisfies $\partial_zT=0$ distributionally. After it asserts this (in theorem 9.6), it states that we can write $$T(z)=\sum_{n\in\mathbb{Z}}L_nz^{-n-2}$$ with $$L_n = \frac{1}{2\pi i}\oint_{\vert \zeta\vert = 1} \frac{T(\zeta)}{\zeta^{n+1}}\mathrm{d}\zeta$$ generating a representation of the Virasoro algebra. Now, I don't really understand what it means to take a contour integral of a distribution, but it seems what I could do would be to use a sort of analog to Weyl's Lemma for Laplace's equation to conclude $T$ is actually a holomorphic function, and then just use standard complex analysis. The question then becomes: why is $T$ not holomorphic at the origin (which it obviously isn't as it has negative powers in its Laurent expansion)? Theorem 9.6 mentions nothing about the domain of holomorphicity. Is that way of thinking correct, or am I going wrong earlier?
This ties into a question I have about the answer to this question. They show that $Y(a,z)\vert 0\rangle$ is analytic for $\vert z\vert<1,$ but then they decompose it into $Y(a,z)=\sum_{n\in\mathbb{Z}} a^{(n)} z^{-n-1}.$ Assuming the $a^{(n)}\vert 0\rangle=0$ for $n\geq0,$ we get holomorphicity of $Y(a,z)\vert 0\rangle,$ but where do the negative powers in the expansion come from here?
