Dipole angular selection rules

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Use the notation of Harmonic polynomial dimension: $S^2=\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }=1\}$ with polar measure $\sigma$ (Sphere integrals in angle coordinates, Complex Lp classes and Euclidean test-function conventions), the Hilbert space $L^2(S^2,\sigma )$ with inner product linear in the first argument, the complete orthonormal family $(Y_{lm})$ (Associated Legendre function normalization, Sphere ONB), the blocks $V_l=\mathrm{span}\{Y_{l,-l},\dots ,Y_{ll}\}$ and the finite span $H$, the spaces $H_l^{\mathrm{p}\mathrm{o}\mathrm{l}}$ of complex harmonic homogeneous polynomials of degree $l$, and the polynomial representatives $h_{lm}\in H_l^{\mathrm{p}\mathrm{o}\mathrm{l}}$ of the $Y_{lm}$, characterized by ${\mathrm{r}\mathrm{e}\mathrm{s}}_l(h_{lm})=Y_{lm}$ where ${\mathrm{r}\mathrm{e}\mathrm{s}}_l(p)$ is the class of $p{\mathrm{\mid }}_{S^2}$ (homogeneous polynomial and homogeneous ideal, Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots ,x_n]$, The Laplacian of a $C^2$ function and of a $C^2$ vector field). Put $H_{-1}^{\mathrm{p}\mathrm{o}\mathrm{l}}:=\{0\}$, fix $l\ge 0$ and $m\in \{-l,\dots ,l\}$, write $\mathcal{H}:=h_{lm}$, and define the three polynomial multipliers $Q_0:=x_3,Q_{\pm }:=\frac{x_1\pm i{x}_2}{\sqrt{2}},$ so that $Q_q={\sum }_{i=1}^3{c}_{q,i}{x}_i$ with $c_{0,i}={\delta }_{i3}$ and $c_{\pm ,i}=({\delta }_{i1}\pm i{\delta }_{i2})\mathrm{/}\sqrt{2}$. Let $D_i$ and $L_z,L_3$ be as in Orbital self-adjoint generators, where $L_3=L_z$ and $L_z{Y}_{lm}=\mathrm{\hslash }m{Y}_{lm}$ (Sphere Laplacian diagonal domain). Then, for each $q\in \{-1,0,1\}$:

(a) (harmonic decomposition) the polynomial $G_q:={\displaystyle \frac{\sum _i{c}_{q,i}{\mathrm{\partial }}_i\mathcal{H}}{2l+1}}$ lies in $H_{l-1}^{\mathrm{p}\mathrm{o}\mathrm{l}}$ and $Q_q\mathcal{H}=R_q+r^2{G}_q,R_q:=Q_q\mathcal{H}-r^2{G}_q\in H_{l+1}^{\mathrm{p}\mathrm{o}\mathrm{l}},$ where $r^2(x)=x_1^2+x_2^2+x_3^2$; for $l=0$ one has $G_q=0$ and $R_q=Q_q\mathcal{H}$;

(b) (degree selection) the function $Q_q{\mathrm{\mid }}_{S^2}\cdot Y_{lm}$ lies in $V_{l+1}\oplus V_{l-1}$ (with $V_{-1}:=\{0\}$); consequently, for all $l^{\mathrm{\prime }}\ge 0$ and $m^{\mathrm{\prime }}\in \{-l^{\mathrm{\prime }},\dots ,l^{\mathrm{\prime }}\}$, $⟨Q_q{\mathrm{\mid }}_{S^2}{Y}_{lm},Y_{l^{\mathrm{\prime }}{m}^{\mathrm{\prime }}}⟩=0\text{whenever }{l}^{\mathrm{\prime }}\mathrm{\notin }\{l-1,l+1\};$

(c) (weight selection) as elements of $H\subset L^2(S^2,\sigma )$ one has $L_z\left(Q_q{\mathrm{\mid }}_{S^2}{Y}_{lm}\right)=\mathrm{\hslash }(m+q)\left(Q_q{\mathrm{\mid }}_{S^2}{Y}_{lm}\right),$ because ${\mathrm{\partial }}_{\phi }{Q}_q=iq{Q}_q$ on $S^2$ (Sine and cosine defined by their real power series, The complex exponential by its power series); hence in addition $⟨Q_q{\mathrm{\mid }}_{S^2}{Y}_{lm},Y_{l^{\mathrm{\prime }}{m}^{\mathrm{\prime }}}⟩=0\text{whenever }{m}^{\mathrm{\prime }}\mathrm{\ne }m+q;$

(d) (selection rules) combining (b) and (c): the angular matrix element $⟨Y_{l^{\mathrm{\prime }}{m}^{\mathrm{\prime }}},Q_q{\mathrm{\mid }}_{S^2}{Y}_{lm}⟩$ can be nonzero only if $l^{\mathrm{\prime }}=l+1\text{ and }{m}^{\mathrm{\prime }}=m+q,\text{or}l\ge 1,l^{\mathrm{\prime }}=l-1\text{ and }{m}^{\mathrm{\prime }}=m+q;$ in particular only $\mathrm{\Delta }l=\pm 1$ and $\mathrm{\Delta }m\in \{0,\pm 1\}$ occur with the component-dependent shift $\mathrm{\Delta }m=q$ for the multiplier $Q_q$. These are necessary conditions on the angular integrals only: no assertion is made that an allowed pair has a nonzero integral, and no absorption, emission or transition probability is asserted or computed.