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)

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