# Average of a vector over all directions

References: Anthony Zee, Einstein Gravity in a Nutshell, (Princeton University Press, 2013) – Chapter I.3, Problem 5.

In this problem, Zee gives us a 3-d vector ${\vec{p}}$ and asks us to find the quantity ${p^{i}p^{j}}$ averaged over the direction of ${\vec{p}}$. I wasn’t entirely clear what this question was asking, since once you’ve defined ${\vec{p}}$, surely its direction is fixed so how can you average over a fixed direction? I think what he means is: suppose you rotate ${\vec{p}}$ through all possible angles in 3-d while keeping its magnitude ${\left|\vec{p}\right|}$ fixed. What then is the average of ${p^{i}p^{j}}$ over all these rotations?

There is also a mistake in the formula he gives for the average. If ${\theta}$ and ${\phi}$ are the usual spherical angles, then the average of ${p^{i}p^{j}}$ is given by

$\displaystyle \left\langle p^{i}p^{j}\right\rangle =\frac{1}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}d\theta d\phi\sin\theta p^{i}p^{j} \ \ \ \ \ (1)$

(not ${\cos\theta}$ in the integral).

We can work out the integral by writing ${p^{i}}$ in rectangular coordinates in the usual way:

 $\displaystyle p^{x}$ $\displaystyle =$ $\displaystyle p\sin\theta\cos\phi\ \ \ \ \ (2)$ $\displaystyle p^{y}$ $\displaystyle =$ $\displaystyle p\sin\theta\sin\phi\ \ \ \ \ (3)$ $\displaystyle p^{z}$ $\displaystyle =$ $\displaystyle p\cos\theta \ \ \ \ \ (4)$

From symmetry, ${\left\langle p^{i}p^{j}\right\rangle =0}$ if ${i\ne j}$. If this isn’t obvious, suppose we’re looking at ${\left\langle p^{x}p^{y}\right\rangle }$, and we pick one direction specified by angles ${\theta_{1}}$ and ${\phi_{1}}$. Then the initial coordinates are

 $\displaystyle p_{1}^{x}$ $\displaystyle =$ $\displaystyle p\sin\theta_{1}\cos\phi_{1}\ \ \ \ \ (5)$ $\displaystyle p_{1}^{y}$ $\displaystyle =$ $\displaystyle p\sin\theta_{1}\sin\phi_{1} \ \ \ \ \ (6)$

If we then rotate ${\vec{p}}$ by keeping ${\theta_{1}}$ constant and rotating ${\phi}$ by ${2\left(\frac{\pi}{2}-\phi_{1}\right)}$ so that ${\phi_{2}=\phi_{1}+2\left(\frac{\pi}{2}-\phi_{1}\right)=\pi-\phi_{1}}$ then

 $\displaystyle p_{2}^{x}$ $\displaystyle =$ $\displaystyle p\sin\theta_{1}\cos\left(\pi-\phi_{1}\right)=-p\sin\theta_{1}\cos\phi_{1}=-p_{1}^{x}\ \ \ \ \ (7)$ $\displaystyle p_{2}^{y}$ $\displaystyle =$ $\displaystyle p\sin\theta_{1}\sin\left(\pi-\phi_{1}\right)=p\sin\theta_{1}\sin\phi_{1}=p_{1}^{y} \ \ \ \ \ (8)$

Thus this position for ${\vec{p}}$ cancels the original position in the average, so the overall average ${\left\langle p^{x}p^{y}\right\rangle =0}$. A similar argument applies to ${\left\langle p^{x}p^{z}\right\rangle }$ and ${\left\langle p^{y}p^{z}\right\rangle }$.

We could also work out these averages by direct integration. For example

 $\displaystyle \left\langle p^{x}p^{y}\right\rangle$ $\displaystyle =$ $\displaystyle \frac{1}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}d\theta d\phi\sin\theta\left(p\sin\theta\cos\phi\right)\left(p\sin\theta\sin\phi\right)\ \ \ \ \ (9)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{p^{2}}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}d\theta d\phi\sin^{3}\theta\sin\phi\cos\phi\ \ \ \ \ (10)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 0 \ \ \ \ \ (11)$

For the other terms, we have

 $\displaystyle \left\langle p^{x}p^{x}\right\rangle$ $\displaystyle =$ $\displaystyle \frac{1}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}d\theta d\phi\sin\theta\left(p\sin\theta\cos\phi\right)^{2}\ \ \ \ \ (12)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{p^{2}}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}d\theta d\phi\sin^{3}\theta\cos^{2}\phi\ \ \ \ \ (13)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{p^{2}}{3}\ \ \ \ \ (14)$ $\displaystyle \left\langle p^{y}p^{y}\right\rangle$ $\displaystyle =$ $\displaystyle \frac{1}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}d\theta d\phi\sin\theta\left(p\sin\theta\sin\phi\right)^{2}\ \ \ \ \ (15)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{p^{2}}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}d\theta d\phi\sin^{3}\theta\sin^{2}\phi\ \ \ \ \ (16)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{p^{2}}{3}\ \ \ \ \ (17)$ $\displaystyle \left\langle p^{z}p^{z}\right\rangle$ $\displaystyle =$ $\displaystyle \frac{1}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}d\theta d\phi\sin\theta\left(p\cos\theta\right)^{2}\ \ \ \ \ (18)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{p^{2}}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}d\theta d\phi\sin\theta\cos^{2}\theta\ \ \ \ \ (19)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{p^{2}}{3} \ \ \ \ \ (20)$

In summary

$\displaystyle \left\langle p^{i}p^{j}\right\rangle =\frac{1}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}d\theta d\phi\sin\theta p^{i}p^{j}=\frac{p^{2}}{3}\delta^{ij} \ \ \ \ \ (21)$