The orbital Casimir identity: the restrictions of the orbital generators to the degree-l blocks are spin-l models

Given: The Axiom of Countable Choice; the sphere $S^2$ with polar measure $\sigma$; the family $(Y_{lm})$, the blocks $V_l$, the finite span $H$; the operators $L_1,L_2,L_3$; the diagonal operator $A$; a degree $l\ge 0$; indices $m,i$; a vector $F\in H$.

Orbital generators. Each $L_i$ is self-adjoint on $L^2(S^2,\sigma )$; for $F\in V_l$ with polynomial representative $p_F$ one has $L_{i}F={\mathrm{r}\mathrm{e}\mathrm{s}}_l(-i\mathrm{\hslash }{D}_i{p}_F)$ with $D_{i}p=(x\times \mathrm{\nabla }p{)}_i$, so $L_i{V}_l\subseteq V_l$ and $L_{i}H\subseteq H$; on $H$ one has $L_3=L_z$ with $L_3{Y}_{lm}=\mathrm{\hslash }m{Y}_{lm}$, and $L_i{L}_j-L_j{L}_i=i\mathrm{\hslash }{\sum }_k{\epsilon }_{ijk}{L}_k$ for all $i,j$ (Orbital self-adjoint generators clauses (a),(b),(c)).

Sphere Laplacian and the basis of a block. $A$ is self-adjoint with $A{Y}_{lm}=l(l+1)Y_{lm}$, the finite span $H$ is a core of $A$, and the $Y_{l,-l},\dots ,Y_{ll}$ are orthonormal in $L^2(S^2,\sigma )$ (Sphere Laplacian diagonal domain clauses (a),(b), Sphere ONB); hence $\dim V_l=2l+1$ (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Orthonormal families, complete orthonormal systems and Hilbert bases).

Finite restrictions are self-adjoint. If a self-adjoint operator $T$ on a Hilbert space leaves a finite-dimensional subspace $W$ invariant, then $T{\mathrm{\mid }}_W$ is a self-adjoint endomorphism of $W$: for $x,y\in W$ one has $⟨T{\mathrm{\mid }}_{W}x,y⟩=⟨Tx,y⟩=⟨x,Ty⟩=⟨x,T{\mathrm{\mid }}_{W}y⟩$ because $Ty\in W$ and $T$ is symmetric on its domain (Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space, Symmetric, self-adjoint and essentially self-adjoint operators, Linear subspace of a vector space).

Block decomposition of a finite angular momentum triple. If a nonzero finite-dimensional complex inner product space $V$ carries self-adjoint endomorphisms $J_1,J_2,J_3$ with $[J_1,J_2]=i{J}_3$ and its cyclic companions, then there are $r\ge 1$ and $j_1,\dots ,j_r\in \frac{1}{2}\mathbb{N}$ with ${\sum }_{\nu }(2j_{\nu }+1)=\dim V$ and an orthonormal basis $(v_{\nu ,k})$ of $V$ such that, with $J_{\pm }:=J_1\pm i{J}_2$ and $J^2:=J_1^2+J_2^2+J_3^2$, $J_3{v}_{\nu ,k}=(j_{\nu }-k)v_{\nu ,k},J_{+}{v}_{\nu ,k}=\sqrt{k(2j_{\nu }-k+1)}{v}_{\nu ,k-1},J_{-}{v}_{\nu ,k}=\sqrt{(k+1)(2j_{\nu }-k)}{v}_{\nu ,k+1},J^2{v}_{\nu ,k}=j_{\nu }(j_{\nu }+1)v_{\nu ,k};$ moreover $\dim \ker (J_3-MI)$ equals the number of blocks with $j_{\nu }\in M+\mathbb{Z}$ and $j_{\nu }\ge M$, and the number of blocks with $j_{\nu }=M$ equals $\dim \ker (J_3-MI)-\dim \ker (J_3-(M+1)I)$, for $M\in \frac{1}{2}\mathbb{N}$; on each block the map $v_{\nu ,k}\mapsto e_{j_{\nu }-k}$ is a unitary equivalence of $(J_1,J_2,J_3)$ with the spin-$j_{\nu }$ matrices, and $J^2$ is self-adjoint and commutes with $J_1,J_2,J_3$ (Finite spin-block decomposition clauses (a),(b),(d), Finite spin matrices clause (b)).

Countable sums of weights. The spectrum of $L_3$ on the finite space $V_l$ consists of the numbers $\mathrm{\hslash }m$, $m=-l,\dots ,l$, each with multiplicity one, because $(Y_{lm}{)}_m$ is an orthonormal basis of $V_l$ consisting of eigenvectors by [F1], [F2] (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, The induced length is a norm).