Changing dummy integration variables for Lorentz measures Let's say we have a double integral in spacetime,
\begin{equation}
\int d^4 x_1 d^4 x_2 f(x_1, x_2)= \int d^3 \vec{x}_1 d^3 \vec{x}_2 \int d x_1^0 d x_2^0\,\, f(x^0_1, x^0_2,\vec{x}_1, \vec{x}_2)
\end{equation}
Where $x_1$, $x_2$ are 4-vectors in Minkowski space. On the r.h.s. are we allowed to change the dummy variables $\vec{x}_1\leftrightharpoons \vec{x}_2$ without exchanging the time variables also, $x_1^0, x_2^0$?
 A: You can certainly relabel your integration variables however you'd like, but that doesn't get you anything all by itself.  Dummy variables in an integral are essentially placeholders, and the pen strokes you use to represent them are arbitrary.

How about if I have something like this in the integrand $e^{i(p_1x_1+p_2x_2)}+e^{i(p_1x_2+p_2x_1)}$, where $p_1, p_2$ are 4-momentum and $p_1x_1....$ etc are the inner products between energy-momentum 4-vector and space-time 4-vector in Minkowski space?

You can perform a trivial relabeling of your variables if you'd like, but all that's going to do is muddle the interpretation of the various quantities.  For example, looking at the first term, you would have
$$\exp[i(p_1^0 x_1^0 + \vec p_1 \cdot \vec x_2 + p_2^0 x_2^0 + \vec p_2 \cdot \vec x_1)]\equiv  \exp[i(p_1 \tilde x + p_2 \tilde y)]$$
where $\tilde x\equiv(x_1^0,\vec x_2)$ and $\tilde y\equiv (x_2^0,\vec x_1)$. In other words, if you relabel $\vec x_1\leftrightarrow \vec x_2$, you're simply choosing to make $\vec x_2$ the position of particle $1$ and $\vec x_1$ the position of particle 2.  This is confusing and probably pointless, but not illegal.
A: Yes, you are allowed. The product $d^3 \vec{x}_1 d^3 \vec{x}_2$ is just the dot product of two position vectors. This is commutative. So  $d^3 \vec{x}_1 d^3 \vec{x}_2=d^3 \vec{x}_2 d^3 \vec{x}_1$. Likewise, the time differentials can be interchanged. These are just scalars.
