I am a mathematician studying the dynamics of N-Body density matrix $\rho_{N}(x;y)$ for $n$ particle is defined by;

$$\rho_{N, t}^{(n)} (x_1,..,n_n; y_1,...,y_n) = \begin{cases} \int \rho_{N,t}(x_1,...,x_n,x_{n+1},...,x_N; y_1,...,y_n,x_{n+1},...,x_N) dx_{n+1}...d_{x_N},\,\,\,\,\, 1 \leq n \le N\\ 0,\,\,\,\,\, o.w. \end{cases}$$

and wigner transform for one particle is:

$$w_N(x;v):= \frac{1}{(3 \pi)^{3 N}} \int e^{- i v. y} \rho_N \left( x + \frac{y}{2}, x- \frac{y}{2}\right) dy.$$

I can not understand the obtain that

$$\int w_N(x,v) dx dv =1.$$

I am a mathematician studying the dynamics of the $N$-Body density matrix $\rho_{N}(x;y)$ for $n$ particles, defined by

$$\rho_{N, t}^{(n)} (x_1,..,n_n; y_1,...,y_n) = \begin{cases} \int \rho_{N,t}(x_1,...,x_n,x_{n+1},...,x_N; y_1,...,y_n,x_{n+1},...,x_N) dx_{n+1}...d_{x_N},\,\,\,\,\, 1 \leq n \le N\\ 0,\,\,\,\,\, o.w. \end{cases}$$

and the Wigner transform for one particle is:

$$w_N(x;v):= \frac{1}{(3 \pi)^{3 N}} \int e^{- i v. y} \rho_N \left( x + \frac{y}{2}, x- \frac{y}{2}\right) dy.$$

I cannot understand how to obtain that

$$\int w_N(x,v) dx dv =1.$$