Undergrad Is the Kroneker Delta Identity Used Correctly in this Paper?

[Safinaz]

TL;DR

A question about a Kroneker Delta identity

Is there a Kroneker Delta identity:

$$ \delta^3 (k) \delta^3 (k) = \frac{2\pi^2}{k^3} ~ \delta (k_1- k_2) $$?

Where k is a wave number. In this Paper: Equations 26 and 27, I think this identity is used to make $k_1=k_2$ and there is an extra negative sign.

 

[anuttarasammyak]

Could you tell us what is the relation among k, k_1 and k_2 ?
Assuming they have same dimension k, dimension of LHS is k^-6 that of RHS is k^-4. They do not coincide.

 

[vanhees71]

First of all you must clearly distinguish between a Kronecker (sic!) $\delta$ and Dirac $\delta$ distributions, which refers to discrete variables, i.e.,
$$\delta_{k_1 k_2}=\begin{cases} 1 &\text{for} \quad k_1=k_2, \\
0 &\text{for} k_1 \neq k_2. \end{cases}
$$
Then you have $\delta_{k_1k_2}^2=\delta_{k_1 k_2}$.

For Dirac-$\delta$ distributions the square doesn't make any sense. Whenever it occurs somewhere, the authors make a mistake. In the paper you linked, nowhere is such a non-sensical formula though!