Confusing concepts in proof of spherical addition theorem

In

section 4, pages 6..9 is a proof of the spherical harmonics addition theorem.

Page 8 has eq.(25), an application of Laplace series:

$$Y_{lm}^*(\theta,\phi) = \sum_{m'=-l}^{l} B_{mm'} Y_{lm'}(\gamma,\beta).$$

This is just very confusing. I sort of get the idea of what $Y_{lm}^*(\theta,\phi)$ is. It is simply the usual spherical harmonic but just relating it to the vector $\vec{r}$ i.e. if we play around with $\vec{r}$ and let it point all different ways, and create a function that outputs the usual spherical harmonic tortoise shell values as $\vec{r}$ moves around..

But what is $Y_{lm'}(\gamma,\beta)$ on the right hand side? When you offset $\vec{r}$ from $\vec{r}'$ which is how you get $\gamma$ ($\gamma$ being defined as the angle between $\vec{r}$ and $\vec{r}'$ on page 7) then I guess we can define a spherical harmonic type function over the coordinates $(\gamma,\beta)$ analogously, but $\beta$ is arbitrary! When we line up $\vec{r}'$ with the $+z$ axis (creating a new coordinate system) we pick some $x$ and $y$ axes to go with the new coordinate system, but that choice is arbitrary and determines $\beta$.

Can anyone explain more sensibly what's going on here?

• Yes $\beta$ is arbitrary'', but if you continue to follow his proof you see that he integrates over the solid angle, so $\beta$ integrates out in the end. – Bill N Sep 8 '15 at 17:58

What he's really trying to say in eq.(25),

$$Y_{lm}^*(\theta,\phi) = \sum_{m'=-l}^{l} B_{mm'} Y_{lm'}(\gamma,\beta)$$,

is "Given the complex conjugate of an e.g. $l=l_0=10,m=m_0=+7$ tortoise shell (spherical harmonic), there exists a set of 21 coefficients (the $B$s) that constitute a linear combination of the 21 tortoise shells of $l_0=10$ but TILTED, such that this linear combination of TILTED tortoise shells sums to the original 'right side up' spherical harmonic $Y_{lm}^*(\theta,\phi)$."

This is a result of Laplace series.

Fine, but then he says "Now set $\gamma \rightarrow 0$ and we can solve for $B_{m_0=+7,m'=0}$" (because all the $B$s with $m' \neq 0$ go away).

But by his definition (see page 7), to say $\gamma=0$ means the right hand side $Y$s are not tilted at all with respect to the left hand side $Y$!

So the statement

${Y^*}_{l_0=10}^{m_0=+7} = B_{m_0=+7,m'=0} \sqrt{\dfrac{2l+1}{4\pi}}$

(see his work after eq.(27) on page 8) does not make sense to me. How could a $m_0=+7$ spherical harmonic = a constant?