Fourier transform with complex variables

[naima]

I found this formula in a paper:
$$\int exp( \frac{x1 + i x2}{ \sqrt 2} \eta^* - \frac{x1 - i x2}{ \sqrt 2}<br /> \eta) D(\eta)/ \pi d^2 \eta$$
the author calls it the Fourier transform of D.
It is the first time thar i see this formula.
How common is this notation? Can we use it without problem?

 

[Dr. Courtney]

I've not seen this notation before.

I tend to stick with the standard notations.

 

[jasonRF]

I haven't seen this either - could you give us a reference?

 

[Mark44]

This part is especially confusing to me: $D(\eta)/ \pi d^2 \eta$. I have no idea what it means.

 

[naima]

the link is here (eq 40):
http://arxiv.org/abs/quant-ph/0112110

I can follow a part of the calculus.with $\eta = (q + ip)/ \sqrt 2$
we get $\int \int exp(i (x_2 q - x_1 p)) D(q,p) dq dp$
So it would be twice a Fourier transform?

 

[blue_leaf77]

When you define
$$
Z = \frac{x_1 + i x_2}{ \sqrt 2} \eta^*
$$
you can write
$$
\exp( \frac{x_1 + i x_2}{ \sqrt 2} \eta^* - \frac{x_1 - i x_2}{ \sqrt 2}\eta) = \exp( Z-Z^*) = \exp( 2i\Im[Z]) = \exp( 2i(-x_1\xi_2+x_2\xi_1)/\sqrt{2})
$$
justifying the calling of Fourier transform. In other words, it's just mathematical manipulation of the exponent

So it would be twice a Fourier transform?

It's a 2D Fourier transform.
.

 

[naima]

I see here:
http://arxiv.org/abs/1510.02746
that the wigner function of a density matrix (with a Fourier transfom) can be written
$W_\rho(\alpha) = Tr [ \rho U(\alpha)]$
We will retrieve the density matrix with an inverse Fourier transform.
But i do not see why we have
$\rho = \int W(\alpha) U(\alpha) d \alpha$ up to a $2 \pi$ coefficient
could you help me?
the second formula is in
http://arxiv.org/abs/quant-ph/0112110