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Fourier transform with complex variables

  1. Jan 25, 2016 #1

    naima

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    I found this formula in a paper:
    [tex]\int exp( \frac{x1 + i x2}{ \sqrt 2} \eta^* - \frac{x1 - i x2}{ \sqrt 2}
    \eta) D(\eta)/ \pi d^2 \eta[/tex]
    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?
     
  2. jcsd
  3. Jan 25, 2016 #2
    I've not seen this notation before.

    I tend to stick with the standard notations.
     
  4. Jan 25, 2016 #3

    jasonRF

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    I haven't seen this either - could you give us a reference?
     
  5. Jan 25, 2016 #4

    Mark44

    Staff: Mentor

    This part is especially confusing to me: ##D(\eta)/ \pi d^2 \eta##. I have no idea what it means.
     
  6. Jan 26, 2016 #5

    naima

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    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?
     
  7. Jan 26, 2016 #6

    blue_leaf77

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    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
    It's a 2D Fourier transform.
    .
     
    Last edited: Jan 26, 2016
  8. Jan 30, 2016 #7

    naima

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    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
     
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