Let $\phi$ be the neutral, massive and free scalar field in $\mathbb{R}^4$. That is, $\phi$ is a tempered distribution whose values are unbounded operators on the Bosonic Fock space.

Note that the operator values of $\phi$ all have a common dense domain, which is sometimes called the *algebraic* Fock space

For any two state vectors $\Omega_1$ and $\Omega_2$ in the algebraic Fock space, let us consider the following tempered distribution on $\mathbb{R}^4 \times \mathbb{R}^4$: \begin{equation} F(x,y):=\bigl \langle \Omega_1, \phi(x)\phi(y) \Omega_2 \bigr \rangle + i \delta^4(x-y) \end{equation} where I have used the notation of ordinary functions. Here, $\langle, \rangle$ is the inner product of the Fock space.

Now, there are the notions of wave front sets and normal bundle, used by Lars Hörmander in his book "Analysis of Partial Differential Operators I" to show when it is possible to restrict a distribution to submanifolds, cf. Corollary 8.2.7.

For the above tempered distribution, the submanifold is the diagonal $\Delta$: \begin{equation} \Delta:= \bigl\{ (x,x) \mid x \in \mathbb{R}^4 \bigr \} \subset \mathbb{R}^4 \times \mathbb{R}^4 \end{equation} whose normal bundle $N(\Delta)$ is known to be \begin{equation} N(\Delta) = \bigl \{ (x,x,y,-y) \mid x,y \in \mathbb{R}^4 \bigr \} \end{equation}

Now, my questions are as follows:

Is the wave front set of $F(x,y)$ disjoint from $N(\Delta)$, so that $F \bigl \lvert_{\Delta}$ is well-defined in the sense of Corollary 8.2.7 of Lars Hörmander as mentioned above?

If so, does $F \bigl \lvert_{\Delta}$ coincide with the expectation value of the Wick product $\bigl \langle \Omega_1, : \phi^2 (x) : \Omega_2 \bigr \rangle$?

I think this is correct, but cannot find any direct reference for this issue. Could anyone please clarify for me?

Edit) Well, in the original definition of the Wick product, we should replace $i \delta(x-y)$ by $[\phi^{(+)}(x), \phi^{(-)}(y)]$. They have the same UV behavior, but it seems that the wave front set requires more than just the UV asymptotics.. So, let me revise my question.

Let $G(x,y):=\bigl \langle \Omega_1, \phi(x)\phi(y) \Omega_2 \bigr \rangle +\bigl \langle \Omega_1, [\phi^{(+)}(x), \phi^{(-)}(y)] \Omega_2 \bigr \rangle$ where $\phi^{(+)}$ is the creation part and $\phi^{(-)}$ is the annihilation part of $\phi$. Then,

I heard from one of the authors in the quoted paper below that this $G(x,y)$ is actually a "smooth function", so that its wave front set is empty. Is this true? How can one verify this?

Does $G \bigl \lvert_{\Delta}$ coincide with the expectation value of the Wick product given as $\bigl \langle \Omega_1, : \phi^2 (x) : \Omega_2 \bigr \rangle$? Therefore, do we conclude that $\bigl \langle \Omega_1, : \phi^2 (x) : \Omega_2 \bigr \rangle$ itself is a smooth function as well?

scalartwo-point distribution $\langle\Omega_1,\phi(x)\phi(y)\Omega_2\rangle$ with anoperator-valueddistribution $[\phi^{(+)}(x),\phi^{(-)}(y)]$ and equating the "result" to another,scalartwo-point distribution $G(x,y)$. $\endgroup$nullconormal directions to $\Delta_2$ are singular for $\omega_2$(x,y)$. That is precisely the point of 1.-2. in my answer and precisely the kind of information captured by the wave front set. $\endgroup$avoidmaking such identifications, otherwise one risks falling into mistakes such as the missing $\langle\Omega_1,\Omega_2\rangle$ factor I've pointed to in my previous comment. $\endgroup$4more comments