**Context:** I am reading a physics paper [Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime](https://arxiv.org/abs/gr-qc/0103074) which applies the notion of the wave front set to operator-valued distributions in a globally hyperbolic spacetime. In particular, the paper considers vector-valued distributions $W\_n$, whose wave front set is (eq. 9)


\begin{equation}
\text{WF}(W\_n)=\left\{ (x\_1,k\_1,\dots,x\_n,k\_n)\in(T^\*M)^n \backslash \left\{0\right\}|k\_i\in V^-\_{x\_i},i=1,\dots,n \right\},
\end{equation}  

where $V^\pm\_{x\_i}$ denotes the future/past lightcone in the tangent space of $x\_i\in M$. It is then claimed that we will get well-defined products between these $W\_n$ distributions and distributions whose wave front sets are subsets of 


\begin{equation}
G\_n(M,g)\equiv (T^\* M)^n \backslash \left(\bigcup\_{x\in M} (V\_x^+)^n\cup\bigcup\_{x\in M} (V\_x^-)^n \right),
\end{equation}
where $g$ denotes the spacetime metric.


It is then asserted that the wave front set of the distribution 
\begin{equation}
t(x\_1,\dots,x\_n)\equiv f(x\_1)\delta (x\_1,\dots,x\_n) \tag{eq. 16} \label{t}
\end{equation}
is a subset of $G\_n(M,g)$ because its wave front set is 
\begin{equation}
\text{WF}(t)=\left\{(x,k\_1,\dots,x,k\_k)\in (T^\*M)^k \backslash \left\{0\right\}|\sum\_i k\_i =0 \right\}. \tag{\*} \label{WFt}
\end{equation}


**Questions:**


1) It is not clear to me why the wave front set of \eqref{t} is given by \eqref{WFt}. More descriptively, taking the Fourier transform of, e.g., $\delta(x\_1,x\_2)=\delta (x\_1)\otimes \delta (x\_2)$, I get 
\begin{equation}
\delta(e^{ix\_1\cdot k\_1},e^{ix\_2\cdot k\_2})=(e^{i0\cdot k\_1},e^{i0\cdot k\_2})=(1,1),
\end{equation}
which suggests to me that the wave front set of \eqref{WFt} is instead $(0,k\_1,\dots,0,k\_n)$ for $k\_i\in \mathbb{R}^n$


2) Even if I were accept that the wave front set of \eqref{t} is given by \eqref{WFt}, it is not clear to me how the condition $\sum\_i k\_i=0$ implies that this wave front set is a subset of $G\_n(M,g)$. In particular, suppose $i=2$ and thus $k\_1+k\_2=0 \rightarrow k\_1=-k\_2$. Now, if $k\_1\in{V^+\_x}$, then $k\_2\in{V^-\_x}$ which would seem to imply that this wave front set is not a subset of $G\_n(M,g)$.



Wave front set of vector-valued Dirac delta distribution
(1) A careful reading of the paper will reveal that the notation $\delta(x\_1,\ldots,x\_n)$ is precisely the $\delta$-distribution supported on the diagonal of the $n$-fold Cartesian product $M \times \cdots \times M$. If we look at (a possibly small portion of) the diagonal as covered by the coordinates $(x\_1,\ldots,x\_n)$, then $\delta(x\_1,\ldots,x\_n) = \delta(x\_2-x\_1) \cdots \delta(x\_n-x\_1)$, or any other equivalent form, rather than $\delta(x\_1-0)\cdots \delta(x\_n-0)$. It is not hard to see that, on $(\mathbb{R}^{\dim M})^n$, the Fourier transform of the $\delta$-distribution supported on the diagonal is $\delta(\sum\_{i=1}^n k\_i)$. From this you can read off that for a general manifold $M$ has the form (\*),
$$
 \mathrm{WF}(\delta(x\_1,\ldots,x\_n)) = \{ (x,k\_1,\ldots, x,k\_n) \in (T^\*M)^n \setminus \{0\} \mid x\in M, \sum\_{i=1}^n k\_i = 0 \} .
$$


(2) Over the diagonal of $M^n$, a non-zero element of $(T^\*M)^n \setminus G\_n(M,g)$ is of the form $(x,k\_1,\ldots, x,k\_n)$ with either all $k\_i$ in $V\_x^+$ or all $k\_i$ in $V\_x^-$. In particular, by the convexity of the cones, either $\sum\_{i=1}^n k\_i$ in $V\_x^+$ or in $V\_x^-$, meaning that this sum could not be zero. In other words, the condition $\sum\_{i=1}^n k\_i = 0$ forces $\mathrm{WF}(\delta(x\_1,\cdots,x\_n))$ to be a subset of $G\_n(M,g)$, as claimed in the paper.


