Dirac Delta Constraint Question

Question

Given an integral of the form \begin{equation}\int d\bar z\, dz\, \delta (\bar z \cdot A \cdot z-b)\,f(\bar z,z)\end{equation} Where $z$ is a complex $n$ dimensional vector, and $A$ is an arbitrary, possibly complex matrix, how would you sort out the delta function so that it provides a constraint over each variable $z_{i}$ and $\bar z_{i}$ rather than a general constraint on the inner product? I guess it amounts to using a property analogous to $\delta(g(x))=\sum_{i}\frac{\delta (x-x_{i})}{|g'(x_{i})|}$ but I'm still unsure how to formalize that. Any thoughts?

Answer 1

Perhaps it will be useful for your application to think of it like this.

Define the degree 2, homogeneous, complex polynomial in 2n variables $p(z_1,z_2,...,\bar{z_1},...,\bar{z_n})$

$$\bar{z}Az = \sum_{i=1}^{n}\sum_{j=1}^{n}A_{ij}\bar{z_i}z_j = p(z,\bar{z})$$

So the condition in the delta that $\bar{z}Az = b$ can be thought of as simply integrating over the right domain. Namely, your integral is equal to the following:

$$\int d\bar z\, dz\, \delta (\bar z \cdot A \cdot z-b)\,f(\bar z,z) = \int_Vf(\bar z, z)d\bar z\,dz$$

Where $V \subset \mathbb{C}^{2n}$ is the solution set (algebraic variety) of $p(z,\bar z) -b$.

Answer 2

Here we consider the $n=1$ case in detail and give an outline for the case of higher $n$.

Case I: $n=1$

We have $$d\bar z d z \to 2 i dx dy \to 2 i r dr d\phi.$$ Let $A_{11}=a$. Then $$\begin{eqnarray*} \int d\bar z\, dz\, \delta (\bar z \cdot A\cdot z-b)\,f(\bar z,z) &=& 2i\int r dr d\phi \, \delta(a r^2 - b)f(r e^{-i\phi},re^{i\phi}). \end{eqnarray*}$$ But $$\delta(a r^2 - b) = \frac{\delta(r-\sqrt{b/a})}{2\sqrt{a b}}$$ and so $$\begin{eqnarray*} \int d\bar z\, dz\, \delta (\bar z \cdot A\cdot z-b)\,f(\bar z,z) &=& \frac{i}{a}\int d\phi \, f\left(\textstyle\sqrt{\frac{b}{a}}e^{-i\phi},\sqrt{\frac{b}{a}}e^{i\phi}\right). \end{eqnarray*}$$

Case II: Higher $n$

We assume that $A$ is hermitian, and so is diagonalizable by unitary transformation. Let $z = U \xi$, where $\bar z \cdot A\cdot z = \xi^\dagger U^\dagger A U \xi = \xi^\dagger D \xi$ and $D = \mathrm{diag}(\lambda_1,\ldots,\lambda_n)$. Without loss of generality assume $\lambda_1 \ne 0$. Then $$\begin{eqnarray*} \int d\bar z\, dz\, \delta (\bar z \cdot A\cdot z-b)\,f(\bar z,z) &=& \int d\bar \xi\, d\xi\, \delta (\bar \xi \cdot D\cdot \xi-b)\,f(\overline{U \xi},U\xi) \\ &=& \int d\bar \xi_1\, d\xi_1\, \left(\prod_{k=2}^n d\bar \xi_k\, d\xi_k\right) \delta \left(\lambda_1|\xi_1|^2+\sum_{k=2}^n \lambda_k|\xi_k|^2-b\right)\, f(\overline{U \xi},U\xi). \end{eqnarray*}$$ The integral over $(\bar \xi_1,\xi_1)$ is of the form of the $n=1$ case above.