Wigner function as average value of parity

[naima]

I found two definitions of wigner function on space time. the first uses a Fourier transform of $\rho (q+ y/2,q-y/2)$
the second uses the Weyl transformation and parity operator $exp (i \pi \theta N)$
where N is the occupation number operator.
Could you give me a link which shows the equivalence of the definition?
thanks

 

[naima]

the first definition is
$w_\rho (q, p) = (2π\hbar)^{−1} \int <q − y/2|ρ|q + y/2> e^ {i \pi y/\hbar} dy$ (read Ballentine)
the other is
$w_\rho (q+i p) = 2 Tr(\rho D(p+iq) e^{i \pi a^\dagger a} D(-q -ip)$
(read Man'ko page 8)
I cannot prove that it is the same thing.

 

[naima]

I found the answer in this paper.

The proof needs the Campbell identity where the commutatot = constant.
$$Exp( i \frac{p_0 \hat {x}}{ \hbar}) Exp(i \frac{-x_0 \hat {p}}{ \hbar}) = Exp (i \frac{p_0 \hat x -x_0 \hat p + x_0 p_0/2 }{ \hbar} )$$
So
$$Exp( i \frac{p_0 \hat {x} - x_0 p_0/2}{ \hbar}) Exp(i \frac{-x_0 \hat {p}}{ \hbar}) = Exp (i \frac{p_0 \hat x -x_0 \hat p }{ \hbar} ) = D(\alpha]$$

$$D(\alpha) = Exp( i \frac{p_0 \hat {x} - x_0 p_0/2}{ \hbar}) Exp( x_0 \partial_x)$$
with
$$<x|D(\alpha)|\Psi> = Exp( i \frac{p_0 x - x_0 p_0/2}{ \hbar}) \Psi(x+x_0)$$

We have to compute
$$2 Tr(\rho D(x+ip) e^{i \pi a^\dagger a} D(-x -ip)$$
$$= 2 <\Psi|D(\alpha) e^{i \pi a^\dagger a} D^\dagger(\alpha)|\Psi>$$
It may be seen as a double sum of
$$= 2 <\Psi|D(\alpha)|y><y| e^{i \pi a^\dagger a}|x><x| D^\dagger(\alpha)|\Psi>$$
$$\int \int Exp(- i \frac{p_0 y - x_0 p_0/2}{ \hbar}) \Psi^*(y+x_0)<y| e^{i \pi a^\dagger a}|x> Exp( i \frac{p_0 x - x_0 p_0/2}{ \hbar}) \Psi(x+x_0)$$
It can be shown that $<y| e^{i \pi a^\dagger a}|x> = \delta(y+x)$ so after one integration we get (up to a normalizing constant) the formula of the first definition.