I would say they are not entirely the same, but it depends on the context. First the definitions:
the Wigner transform of an operator $\hat{A}$ is defined as $$\tilde{W}\left[\hat{A}\right]=\int dz\left[e^{\mathbf{i}pz/\hbar}\left\langle x-z/2\right|\hat{A}\left|x+z/2\right\rangle \right]$$ and this is a strange function. You see that on the left, the operator is projected onto a real-space representation, then Fourier transformed. You may find more details (especially the link with the Weyl transform) on the wonderful review by Hillery, M., O’Connel, R. F., Scully, M. O. & Wigner, E. P. Distribution functions in physics: Fundamentals, Phys. Rep. 106, 121–167 (1984) which is unfortunately beyond a paywall.
the Wigner transform of the density operator $\hat{\rho}=\left|\Psi\right\rangle \left\langle \Psi\right|$ is then naturally defined as the Wigner transform $$W\left(p,x\right)=\int dz\left[e^{\mathbf{i}pz/\hbar}\left\langle x-z/2\right|\hat{\rho}\left|x+z/2\right\rangle \right]$$ and it is coined Wigner function in that context.
the Green function is not an operator, it is a correlation function, defined as $G\left(x_{1},x_{2}\right)=\left\langle \hat{T}\left[\hat{a}\left(x_{1}\right)\hat{a}^{\dagger}\left(x_{2}\right)\right]\right\rangle $ where $\hat{T}$ is the time-ordering operator, $\hat{a}$ is the (fermionic or bosonic) destruction operator, and $\left\langle \cdots\right\rangle $ represents the averaging process: it could be $\left\langle \cdots\right\rangle =\left\langle N\right|\cdots\left|N\right\rangle $ if you're working with number states $\left|N\right\rangle$, or $\left\langle \cdots\right\rangle =\text{Tr}\left\{ e^{-\beta H}\cdots\right\} /\text{Tr}\left\{ e^{-\beta H}\right\} $ if you're working with thermal averaging ($\beta=\left(k_{B}T\right)^{-1}$ is an inverse temperature in that case), ... Note there are other conventions for the Green functions, but it does not matter here. The Fourier transform of the Green function reads $$G\left(p,x\right)=\int dz\left[e^{\mathbf{i}pz/\hbar}G\left(x-z/2,x+z/2\right)\right]$$ and it looks like a Wigner transform of the Green function, but it should be more appropriate to call it a Fourier transform of the Green function when you choose $x_{1,2}=x\mp z/2$ for the components. In condensed matter theory, $G\left(p,x\right)$ is often called a mixed-Fourier Green function (the full Fourier transform would have given $G\left(p_{1},p_{2}\right)$ instead) or a quasi-classical Green function for the reason to come.
In the limit $\hbar/\tilde{p}\tilde{x}\ll1$ (called quasi-classical limit), with $\tilde{p}\tilde{x}$ the phase-space exploration of the system, the equation of motion of the quasi-classical Green function is the Boltzmann's (transport) equation. The quasi-classical Green functions are not normalised, so they can not be interpreted (whatever it means) as quasi-probability distribution.
As far as I remember, the quasi-classical equation of motion for the Wigner function is not the Boltzmann's one, but the Liouville's one: the collision term is absent, since there is no self-energy method associated with the density matrix. One needs to work with the Lindblad equation for the density matrix, whereas the self-energy method is sufficient when you work with the quasi-classical Green function. Other method to deal with open systems when working with the density matrix is the so-called stochastic method, see e.g. Walls, D. F. & Milburn, G. J. Quantum optics (Springer-Verlag, 1994).
To conclude, note I've put the time under the carpet in the above explanation. That's for a good reason: time is always more complicated to deal with in the Wigner-Weyl transform, especially in the quasi-classical limit and with the Green functions method. The use of the Wigner function is not a big problem when time is taken into account. Of course dealing with the Lindblad equation is not a simple issue... but that's an other story :-)
This post imported from StackExchange Physics at 2014-08-22 05:05 (UCT), posted by SE-user FraSchelle
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