(4) A two-dimensional Gaussian integral in complex coordinates: $$ \begin{aligned} &\frac{1}{\pi} \int \exp \left(-\mu \alpha^{2}-\nu \alpha^{* 2}-z^{*} \alpha+z \alpha^{*}-|\alpha|^{2}\right) \mathrm{d}^{2} \alpha \\ &\quad=\frac{1}{\tau} \exp \left[-\left(\mu z^{2}+\nu z^{* 2}+z z^{*}\right) / \tau^{2}\right], \quad \tau=\sqrt{1-4 \mu \nu}, \quad \mu+\nu<1 . \end{aligned} $$

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Can anyone suggest some text reference where I can get to learn about the result and about Two-dimensional Gaussian integral in complex coordinates in general? Any Mathematical Physics text reference is preferred. I am having problems dealing with them in my study of Quantum Optics.

  • $\begingroup$ Welcome to math.SE: Please write equations in latex formatting instead of attaching images, this will help your question get better answers and avoid it being closed. $\endgroup$ Nov 22, 2021 at 11:37
  • $\begingroup$ Difficult to understand the connection with Fourier Transform. Is there a connection with quaternions like here arxiv.org/pdf/1406.1014.pdf ? $\endgroup$
    – Jean Marie
    Nov 22, 2021 at 12:06
  • $\begingroup$ I am not aware of the connection with quaternions, so I cannot comment on that. The kernel of the transformation here is $${e^{z\alpha^{*}-z^{*}\alpha}}$$ which transforms a function $$Q({\alpha},\alpha^{*})$$ to, lets say, $$P(z,z^{*})$$ which is the expression on the right side of the equation. $\endgroup$ Nov 22, 2021 at 12:27
  • $\begingroup$ I understand. I have never seen this. As it deals with a function of 2 complex variables, the usual notation for the differential element should have been $dz d\overline{z}$. It is a detail but it is interesting to know it if you are looking into maths books about "functions of several complex variables" like this one $\endgroup$
    – Jean Marie
    Nov 23, 2021 at 0:22
  • $\begingroup$ I too haven't seen this before. Can you give the source for this? $\endgroup$ Nov 23, 2021 at 8:46

1 Answer 1

  1. It follows from my Phys.SE answer here that OP's Gaussian integral becomes $$ \begin{align} I~:=~&\int_{\mathbb{R}^2} \! \frac{\mathrm{d}{\rm Re}\alpha \wedge \mathrm{d}{\rm Im}\alpha}{\pi}~ \exp\left\{-\mu \alpha^{2}-\nu \alpha^{* 2}-z^{*} \alpha+z \alpha^{*}-|\alpha|^{2}\right\}\cr ~=~&\int_{\mathbb{C}} \! \frac{\mathrm{d}\alpha^{\ast} \wedge \mathrm{d}\alpha}{2\pi i}~ \exp\left\{ -\frac{1}{2}A^TSA +Z^TA\right\}\cr ~=~&\sqrt{\frac{-1}{\det(S)}}\exp\left\{\frac{1}{2}Z^TS^{-1}Z \right\}\cr ~=~&\frac{1}{\tau} \exp \left\{-\frac{\mu z^2+\nu z^{* 2}+z z^{*}}{\tau^2}\right\}. \end{align}$$

  2. Here we have defined $$ \begin{align} A~:=~&\begin{pmatrix} \alpha \cr \alpha^{\ast} \end{pmatrix}, \cr Z~:=~&\begin{pmatrix} -z^{\ast} \cr z \end{pmatrix}, \cr J~:=~&\begin{pmatrix} 1 & i \cr 1 & -i \end{pmatrix}, \cr S~:=~&\begin{pmatrix} 2\mu & 1 \cr 1 & 2\nu \end{pmatrix}, \cr \tau^2~:=~&-\det(S), \cr S^{-1}~=~&\frac{1}{\tau^2}\begin{pmatrix} -2\nu & 1 \cr 1 & -2\mu \end{pmatrix}, \end{align}$$

  3. Moreover, the Gaussian integral $I$ is convergent if ${\rm Re}(J^TSJ)>0$ is positive definite.


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