I consider particle in external magnetic field, ${\bf A}=(-yB,0,0)$ and find wave functions (may be up to normalization factors), $$\psi(x,y,z)=\sum_n\sum_s\int\frac{dp_xdp_z}{(2\pi)^2}f_s\left(eBy+p_x\right)e^{ip_xx+ip_zz},$$ where $f(eBy+p_x,n)$ is Hermite polynomial (up to numerical factors) and $n$ labels Landau's level and it is two-component spinor with eigenvalues $s=\pm 1$ ($\sigma_zf_s=sf_s$) . Then I consider the following quantitiy, $$G({\bf r}_1,t_1;{\bf r}_2,t_2)=\Psi^{\dagger}(x_2,y_,z_2,t_2)\Psi(x_1,y_1,z_1,t_1),\qquad \Psi=\psi e^{iE_nt},$$ where $E_n$ is an energy of particle. I would like to understand how should I define Wigner-Weyl transformation for this problem. Naively, I introduce "centered" coordinates ${\bf r}=({\bf r}_1+{\bf r}_2)/2$ and $\delta{\bf r}={\bf r}_1-{\bf r}_2$ and similarly for time variables. Then, I suppose that my function becomes $$G({\bf r},t;{\bf q},\omega_0)=\int\frac{d^3\delta{\bf r}}{(2\pi)^3}\int\frac{d\delta t}{(2\pi)}e^{i\delta{\bf r}\cdot{\bf q}}e^{-i\delta t\omega_0}G({\bf r}_1,t_1;{\bf r}_2,t_2).$$ However, I feel that it is not correct. Can someone provide references for this topic? I try to google it, but it was unsuccesful.
Question: how one should define Wigner-Weyl transformation for electron in external magnetic field?
