What is the meaning of the double complex integral notation used in physics?

Question

In Altland and Simons' condensed matter book, complex Gaussian integrals are introduced. Defining $z = x + i y$ and $\bar{z} = x - i y$, the complex integral over $z$ is
$$\int d(\bar{z}, z) = \int_{-\infty}^\infty dx \, dy.$$ In this way, any integral over $z$ can be done by just breaking into real and imaginary parts.

I'm confused about how one would actually use the notation on the left, as it is. It seems it must have some meaning besides just $dx \, dy$, or there should be no point in introducing it.

It is possible to break the double integral $\int d(\bar{z}, z)$ into two single complex integrals and do them individually? For example, if we were to write $$\int d(\bar{z}, z) = \int d \bar{z} \int dz$$ then what would the bounds of integration be? For the inner integral, isn't the value of $z$ determined by the value of $\bar{z}$? Alternatively, if we regard $z$ and $\bar{z}$ as independent, then where does the constraint $\bar{(z)} = \bar{z}$ come in? Should each of these integrals be regarded as regular integrals or contour integrals? If we don't break the integral into two, is $d(\bar{z}, z)$ some kind of area element? In that case, how do you do a complex surface integral?

Overall I don't understand what object $d(\bar{z}, z)$ is. What is it, and how do we integrate over it?

Answer 1

The complex notations$^1$ $$\int_{\mathbb{C}}\!\mathrm{d}z^{\ast}~\mathrm{d}z, \tag{1}$$ $$\int_{\mathbb{C}}\!\mathrm{d}^2z, \tag{2}$$ and similar notations, mean a real double integration $$N\iint_{\mathbb{R^2}} \!\mathrm{d}x~\mathrm{d}y\tag{3}$$ in the complex plane $\mathbb{C}\cong \mathbb{R}^2$ with coordinates $z=x+iy$, where $N$ is a conventional normalization factor that depends on the author.

$N=1$ convention:

• A. Altland & B. Simons, Condensed matter field theory, 2nd ed., 2010. See e.g. sentence above eq. (3.11).

$N=2$ convention:

• J. Polchinski, String Theory, Vol. 1, 1998. See e.g. eq. (2.1.7).

• R. Blumenhagen, D. Lust & S. Theisen, Basic Concepts of String Theory, 2012. See e.g. footnote on p. 85.

• D. Tong, Lectures on String Theory. See e.g. subsection 4.0.1.

$N=2i$ convention:

• J.H. Negele & H. Orland, Quantum Many-Particle Systems, 1998. See e.g. eq. (1.124).

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$^1$ Note that $z^{\ast}=x-iy$ denotes the complex conjugate variable. It is not an independent complex variable. In particular, the integration (1) is over $\mathbb{C}$. It is not over $\mathbb{C}^2$. See also e.g. this Math.SE post, this Phys.SE post and links therein.

Answer 2

This issue often arises in conformal field theory, when we may be interested in Euclidean field theory but analytically continue to $\mathbb C^2$. Suppose we have real, Euclidean coordinates $(x,y)$ and form the complex coordinates,

$$z = x+iy, \quad \bar z = x-iy.$$

It is easy to show that the metric $dx^2 + dy^2 = dz d \bar z$, that is, $g_{zz} = g_{\bar z \bar z} = 0$ and $g_{z\bar z} = g_{\bar z z} = \frac12$. From this we can deduce that the measure for integration is,

$$dz d \bar z = 2 dx dy$$

and thus there is a factor of two difference between $\int d^2 z$ and $\int d^2 x$. We can treat $z$ and $\bar z$ as totally independent which then extends us to $\mathbb C^2$. To return to $\mathbb R^2 \subset \mathbb C^2$, we must make the identification that $\bar z = z^\star$, that is, they are related by conjugation and are not independent.

Answer 3

I do not like this notation either, because it implies meaning that isn't there.

The authors define $$\int d(\bar{z}, z) \equiv \int_{-\infty}^\infty dx \, dy.$$ Note the $\equiv$ instead of an $=$ sign.

$d(z,\bar z)$ is literally the area element of the euclidean plane. The example at hand from the book is $$\int d(\bar{z}, z) e^{-\bar z w z} \equiv \int_{-\infty}^\infty dx \, dy e^{-x^2 w - y^2 w} =\sqrt{\frac{\pi}{w}}^2 $$

There are not two independent complex variables in play here. One is integrating functions from $\mathbb{C}\rightarrow\mathbb{C}$ seen as functions from $\mathbb{R}^2\rightarrow\mathbb{C}$.

If I had to guess I'd say they chose the notation because it might feel a little odd to write e.g.

$$\int dxdy\, e^{-\bar z w z}.$$

Maybe a less confusing choice of notation would be $d(z,\bar z) \equiv d\Re(z)d\Im(z) $ indicating integration over real and imaginary part separately.