show that the product of two delta functions δ(x)δ(y) is invariant under rotation around the origin.

Question

Show that the product of two delta functions $\delta{(x)}$$\delta{(y)}$ is invariant under rotation around the origin. This is a problem from Zee's textbook on Gravity on page 51.

The book was speaking about Generators $J_x, J_y, J_z$ of a rotation group SO(3), and writing A as a linear combination of these Generators ($A = \theta_xJ_x + \theta_yJ_y+\theta_zJ_z$) and then making the claim that rotations about an angle $\theta$, $R(\theta)$ can be written as $e^A$. Probably this has something to do with $e^X = \sum_{n=0}^\infty \frac{X^n}{n!}$. I also only know that $$\delta(x) =\begin{cases} \infty & \text{ if } x=0\\ 0 & \text{ otherwise}\end{cases}$$ s.t. $\int_{-\infty}^{\infty}f(t)\delta(t)dt=f(0)$

But it also brought up $R(\theta)$ as the plain rotation matrix from elementary linear algebra.

Answer 1

Presumably you want to show it is invariant under 2D rotations (because it isn't rotationally invariant in 3D). So for any smooth compactly supported test function $\varphi$, we have $$ \int_{-\infty}^\infty \! \int_{-\infty}^\infty \varphi(x,y) \delta(x) \delta(y) \, dy \, dx = \varphi(0,0) .$$ Now suppose that $R = \begin{bmatrix} c & -s \\ s & c \end{bmatrix}$ is any 2D-rotation, with $c = \cos\theta$ and $s = \sin\theta$. You want to show that $$ \delta(cx - sy) \delta(sx + cy) = \delta(x)\delta(y) .$$ So consider $$ I = \int_{-\infty}^\infty \! \int_{-\infty}^\infty \varphi(x,y) \delta(cx - sy) \delta(sx + cy) \, dy \, dx .$$ Make the substitution $\xi = cx - sy$, $\eta = sx + cy$. Note that $x = c\xi+s\eta$, $y = -s\xi + c \eta$, and that $dx\,dy = d\xi \, d\eta$, the last equality coming from the fact that the determinant of $R$ is 1. Therefore $$ I = \int_{-\infty}^\infty \! \int_{-\infty}^\infty \varphi(c\xi+s\eta,-s\xi + c \eta) \delta(\xi) \delta(\eta) \, d\eta \, d\xi = \varphi(c0+s0,-s0 + c 0) = \varphi(0,0) .$$

Answer 2

Hint: One approach is to approximate $$ \delta(x)=\phi_n(x)=ne^{-\pi n^2x^2} $$ You will need to show that $\int_{\mathbb{R}}\phi_n(x)\,\mathrm{d}x=1$ and that for any $\varepsilon\gt0$, $\lim\limits_{n\to\infty}\int_{|x|\ge\varepsilon}\phi_n(x)\,\mathrm{d}x=0$.

Then show $$ \int_{\mathbb{R}^2}f(x,y)\,\phi_n(x)\,\phi_n(y)\,\mathrm{d}x\,\mathrm{d}y $$ is unchanged under the change of variables $x=u\cos(\theta)-v\sin(\theta)$ and $y=u\sin(\theta)+v\cos(\theta)$.