# Fourier transform with complex variables

Gold Member
I found this formula in a paper:
$$\int exp( \frac{x1 + i x2}{ \sqrt 2} \eta^* - \frac{x1 - i x2}{ \sqrt 2} \eta) D(\eta)/ \pi d^2 \eta$$
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
Gold Member
I've not seen this notation before.

I tend to stick with the standard notations.

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

Mark44
Mentor
This part is especially confusing to me: ##D(\eta)/ \pi d^2 \eta##. I have no idea what it means.

Gold Member
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
Homework Helper
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.
.

Last edited:
Gold Member
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