About integration measure in the completeness formula of coherent state

Question

In Negele's book "quantum many-particle systems", the completeness formula for coherent state is $$\int {\frac{{d{\phi ^*}d\phi }}{{2\pi i}}{e^{ - {\phi ^*}\phi }}} \left| \phi \right\rangle \left\langle \phi \right| = I\tag{1}$$ and it says the integration measure is $$\frac{{d{\phi ^*}d\phi }}{{2\pi i}} = \frac{{d\left( {{\mathop{\rm Re}\nolimits} \phi } \right)d\left( {{\mathop{\rm Im}\nolimits} \phi } \right)}}{\pi }.\tag{2}$$ How to understand this integration measure equality?

As I understand, If I view $\phi=x+iy$, then $${d\left( {{\mathop{\rm Re}\nolimits} \phi } \right)d\left( {{\mathop{\rm Im}\nolimits} \phi } \right)}=dxdy.\tag{3}$$ But

$$d{\phi ^*}d\phi = \left( {dx - idy} \right)\left( {dx + idy} \right) = d{x^2} + d{y^2}\tag{4}$$

I am pretty sure this understanding is wrong. What is the right way?

Answer 1

Concerning OP's last formula (4): Recall that there is an implicitly written antisymmetric wedge $\wedge$ in the integral measure. Therefore OP's last equality sign should read $$( \mathrm{d}x - i\mathrm{d}y )\wedge ( \mathrm{d}x + i\mathrm{d}y ) ~=~ 2i\mathrm{d}x\wedge\mathrm{d}y. \tag{4'}$$ See also e.g. this Phys.SE post.

Answer 2

This alternative derivation (without Grassman algebra) uses the transformation rule for integrals. We have an integral over $\mathbb {R}^2$ with the measure $dx\ dy $ (I dont care about $\pi$). We consider a change of variables $(x,y)\mapsto (z, z*):=(x+iy, x-iy) $ This is a diffeomprphism onto its image. (A subset of $\mathbb {C}^2$ seen as a 4d real vectorspace) . The Jacobian is $$ J=\begin{pmatrix} 1 & i \\ 1 & -i \end {pmatrix}$$. The "absolute value" (for real and complex part individually) of the determinant is $2i$, the factor we need.

Answer 3

I'm here to explain where that $\pi$ in eq(2) comes from:$$\begin{aligned}\int\left(\phi^{*}\right)^{n}\phi^{m}e^{-|\phi|^{2}}d\text{Re}\phi d\text{Im}\phi & =\int_{0}^{\infty}|\phi|^{n+m+1}e^{-|\phi|^{2}}d|\phi|\int_{0}^{2\pi}e^{i(m-n)\theta}d\theta\\ & =\pi n!\delta_{nm} \end{aligned} $$ $$\frac{1}{\pi}\int|\phi\rangle\langle\phi|d\text{Re}\phi d\text{Im}\phi=\sum_{n}|n\rangle\langle n|=1 $$ So actually the coherent state is over-complete and $\pi$ here is helping coherent state to disguise as a complete one. For more information one may refer to Sec2.4 of Quantum optics by Scully and Zubairy.
By the way as illustrated by Adomas Baliuka and Qmechanic♦, the relation of the coefficients in eq(1)&(2) is clear. They two arrive at the same end by different means because the appearance of the Jacobian is a natural property of the wedge product.