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This integral comes from equation (3.15) in an older paper I've been reading: $$ \int \mathrm{d} \Omega_k \, \delta\left(|\vec{k}|^2 - |\vec{k}+\vec{q}_1|^2\right) \delta\left(|\vec{k}|^2 - |\vec{k}-\vec{q}_2|^2\right) . $$ The integration over $\Omega_k$ refers to the angular coordinates of the vector $\vec{k}$. I'm most interested in the case of three spatial dimensions, for which $$ \int \mathrm{d} \Omega_k \to \int_0^{2\pi} d\theta \int_0^\pi d\varphi \sin \varphi, $$ where $\theta$ and $\varphi$ are the azimuthal and polar angles associated with $\vec{k}$. The result should depend on the lengths $|\vec{k}|$, $|\vec{q}_1|$, and $|\vec{q}_2|$, as well as the dot product $\vec{q}_1\cdot\vec{q}_2$.

The authors write that the integral is straightforward and don't give an explicit result, so I suspect that I'm missing something simple. I've tried a few strategies, such as using integral representations for the delta functions, but I haven't yet found an elegant approach.

wcw
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  • Using both $k$ and $\vec{k}$ makes the notation a little unclear. What is the $k$? Also what do you mean by $(\vec{k}+\vec{q}_1)^2$? It is equal to $|\vec{k}+\vec{q}_1|^2$? – mucciolo Jan 29 '18 at 22:01
  • Being used to physicists notation, I assume that $u^2 = |\vec{u}|^2$ and – md2perpe Jan 29 '18 at 22:03
  • I made some tweaks to the notation for clarity. Thanks for the comment. – wcw Jan 29 '18 at 22:10
  • Does $\text d\Omega_k$ indicate integration over some sphere ${ |k| = c }$? – Calvin Khor Jan 29 '18 at 22:14
  • @CalvinKhor: yes, that's right. – wcw Jan 29 '18 at 22:21
  • I don't have time to make a proper attempt, but I feel like the difficulty should be in finding some change of variables to allow one of the diracs to depend only of one of the angular coordinates, so that it can be treated via the usual formulas eg. https://math.stackexchange.com/questions/276583/dirac-delta-function-of-a-function . Has this been your experience? – Calvin Khor Jan 29 '18 at 22:43
  • The best choice of coordinate system I've come up with so far involves aligning one of the q-vectors with the z-axis and the other in the x-z plane. This does "allow one of the diracs to depend only of one of the angular coordinates". I can post my progress in this direction a bit later. – wcw Jan 29 '18 at 22:56
  • I am unable to see how it is possible that in the Lebesgue sense this integral is not equal to $0$ or $\infty$. Am I missing something? – mucciolo Jan 29 '18 at 23:23
  • @mucciolo I don't know what this integral is but I don't think it's infinite. It's a more complicated version of $\iint_{\mathbb R^2} \delta(x)\delta(y) dxdy$, right? – Calvin Khor Jan 30 '18 at 09:48
  • @CalvinKhor Not really, since a sphere is a compact set. This integral will evaluate to $\infty$ if $q_1=0$ or $q_2=0$. Furthermore $\iint_{\mathbb R^2} \delta(x)\delta(y) dxdy = 0$ in the Lebesgue sense because it differs of the zero function only in a null set. However I am aware that in some way it is possible to attribute the value of $1$ to it. Which makes me wonder what kind of integral is that. – mucciolo Jan 30 '18 at 11:09
  • @mucciolo the integral $\int_U \delta(x) dx$ is more commonly written in modern mathematics(i.e. not physics) as $$\int_U d\delta = \delta(U) = \begin{cases} 1 & 0 \in U \ 0 & \text{otherwise} \end{cases}$$. The measure whose null sets are relevant is not the Lebesgue measure. – Calvin Khor Jan 30 '18 at 11:11
  • @CalvinKhor Oh! This really clarifies the things! Many thanks! Could you suggest a reference with a discussion on that? – mucciolo Jan 30 '18 at 11:43
  • @mucciolo not really, I cobbled what I know from a variety of resources. Knowing measure theory is a good start, you may also want to look for books on distribution theory. If I knew a good reference for working with integrals written in the physics style, I'd have written an answer :) – Calvin Khor Jan 30 '18 at 11:46

1 Answers1

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This isn't a full solution (yet), but it's the closest I've come so far. I still think there is likely a more elegant approach. I will use the convention that $k = |\vec{k}|$.

Coordinate system (three dimensions)

Choose a spherical coordinate system for $\vec{k}$ such that $\vec{q}_1$ coincides with the positive $z$-axis, and let $\theta$ and $\varphi$ represent the azimuthal and polar angles associated with $\vec{k}$. Then, $$ \vec{k} \cdot \vec{q}_1 = k \, q_1 \cos \varphi . $$

Rotate the coordinate system about the z-axis until $\vec{q}_2$ lies on the $x>0$ portion of the $x$-$z$ plane. Let $\alpha$ represent the angle between $\vec{q}_2$ and the $z$-axis. Then, using the formula for a dot product in spherical coordinates,

$$ \vec{k} \cdot \vec{q}_2 = k \, q_2 \left[ \sin \varphi \sin \alpha \cos \theta + \cos \varphi \cos \alpha \right]. $$

The original integral may now be written $$ \int_0^{2\pi} d\theta \int_0^\pi d\varphi \, \sin \varphi \,\, \delta\left(f(\varphi)\right) \delta\left(g(\theta,\varphi)\right) $$

with

$$ f(\varphi) = 2k \, q_1\cos\varphi + q_1^2 $$

and

$$ g(\theta,\varphi) = 2kq_2 \left[ \sin \varphi \sin \alpha \cos \theta + \cos \varphi \cos \alpha \right] -q_2^2. $$

Evaluating the integral over $\varphi$

Using the formula suggested by Calvin Khor, we can say that

$$ \delta\left(f(\varphi)\right) \to \frac{\delta(\varphi-\varphi_0)}{2k q_1 |\sin \varphi_0|} $$

with $\varphi_0 = \arccos \left(-q_1 / 2k\right)$, which is valid for $q_1 \le 2k$. We can now evaluate the integral over $\varphi$, obtaining $$ \frac{1}{2kq_1} \int_0^{2\pi} d\theta \, g(\theta,\varphi_0) $$

If $q_1 > 2k$, the integral evaluates to $0$.

Evaluating the integral over $\theta$

Work in progress...

wcw
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