# Wigner-Weyl transformation for particle in magnetic field

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?

• Thank You, Prof. Zachos, I will check these references. To be specific, I am dealing with Wigner-Weyl transform for $G^{-+}$ Green function (if it makes sense) of electron in external magnetic field (3+1 space-time). – Artem Alexandrov Dec 5 '19 at 11:18