Finite spin blocks, spherical tensors and Clebsch–Gordan phases

A spin label is $j\in \{0,\frac{1}{2},1,\frac{3}{2},\dots \}$. Its magnetic labels are $m=-j,-j+1,\dots ,j$. The spin-$j$ block is $V_j={\mathbb{C}}^{2j+1}$ with orthonormal basis $e_{j,m}$ and operators $J_z^{(j)}{e}_{j,m}=m{e}_{j,m}$ and $J_{\pm }^{(j)}{e}_{j,m}=\sqrt{(j\mp m)(j\pm m+1)}{e}_{j,m\pm 1}$, with the out-of-range endpoint vector interpreted as zero. Put $J_x^{(j)}=(J_{+}^{(j)}+J_{-}^{(j)})\mathrm{/}2$, $J_y^{(j)}=(J_{+}^{(j)}-J_{-}^{(j)})\mathrm{/}(2i)$ and $J^{(j)2}=(J_x^{(j)}{)}^2+(J_y^{(j)}{)}^2+(J_z^{(j)}{)}^2$.

For an ordered coupling of spin labels $s,t$, use the ordered tensor product $V_s\otimes V_t$, its orthonormal product basis $e_{s,m}\otimes e_{t,n}$, and total operators $J_a^{\mathrm{t}\mathrm{o}\mathrm{t}}=J_a^{(s)}\otimes I+I\otimes J_a^{(t)}$ for $a=x,y,z$, with $J_{\pm }^{\mathrm{t}\mathrm{o}\mathrm{t}}=J_x^{\mathrm{t}\mathrm{o}\mathrm{t}}\pm i{J}_y^{\mathrm{t}\mathrm{o}\mathrm{t}}$. For each $j=\mathrm{\mid }s-t\mathrm{\mid },\mathrm{\mid }s-t\mathrm{\mid }+1,\dots ,s+t$, choose and record a unit highest-weight vector $u_j$ in the spin-$j$ summand: it has $J_z^{\mathrm{t}\mathrm{o}\mathrm{t}}{u}_j=j{u}_j$ and $J_{+}^{\mathrm{t}\mathrm{o}\mathrm{t}}{u}_j=0$. Set $v_{j,M}=(r!(2j)(2j-1)\cdots (2j-r+1){)}^{-1\mathrm{/}2}(J_{-}^{\mathrm{t}\mathrm{o}\mathrm{t}}{)}^r{u}_j$ for $M=j-r$ and $r=0,1,\dots ,2j$, using the empty product $1$ for $r=0$.

The Clebsch–Gordan coefficient $C_{s,m;t,n}^{j,M}$ is the coefficient of $e_{s,m}\otimes e_{t,n}$ in $v_{j,M}$, so $v_{j,M}={\sum }_{m,n}{C}_{s,m;t,n}^{j,M}{e}_{s,m}\otimes e_{t,n}$. Equivalently, with the supplier's inner product linear in its first argument, $C_{s,m;t,n}^{j,M}=⟨v_{j,M},e_{s,m}\otimes e_{t,n}⟩$. The coupling order and the choice of each $u_j$ are part of this convention. Replacing $u_j$ by $e^{i{\theta }_j}{u}_j$ multiplies every coefficient for that $j$ by $e^{i{\theta }_j}$; comparison with a table therefore requires matching its order and phases.

Let $H={⨁}_{\alpha \in A}{V}_{j_{\alpha }}$ and $K={⨁}_{\beta \in B}{V}_{{\mathrm{\ell }}_{\beta }}$ be finite direct sums of specified spin blocks over $\mathbb{C}$, with finite copy-index sets $A,B$; repeated spin labels designate distinct copies. The vector operations are coordinatewise, and the inner products are the sums of the block inner products. For $a=x,y,z$, define $J_a^H(h_{\alpha }{)}_{\alpha \in A}=(J_a^{(j_{\alpha })}{h}_{\alpha }{)}_{\alpha \in A}$ and $J_a^K(g_{\beta }{)}_{\beta \in B}=(J_a^{({\mathrm{\ell }}_{\beta })}{g}_{\beta }{)}_{\beta \in B}$. Set $J_{\pm }^H=J_x^H\pm i{J}_y^H$ and $J_{\pm }^K=J_x^K\pm i{J}_y^K$. For an integer $k\ge 0$, a rank-$k$ spherical tensor family from $H$ to $K$ is a family of linear maps $T_q:H\to K$ indexed by $q=-k,-k+1,\dots ,k$ satisfying $J_z^K{T}_q-T_q{J}_z^H=q{T}_q$ and $J_{\pm }^K{T}_q-T_q{J}_{\pm }^H=\sqrt{(k\mp q)(k\pm q+1)}{T}_{q\pm 1}$. Here $T_{k+1}=T_{-k-1}=0$. When $H=K$, these differences are ordinary commutators. No nonzero reduced matrix element or physical interpretation is included in this definition.

The finite-spin supplier constructs the stated matrices on each block and proves their ladder relations and $J^{(j)2}=j(j+1)I$. The ordered-register supplier supplies the product inner product and basis. The tensor-spin composition supplier gives one invariant spin-$j$ summand for each listed $j$ and a normalized ladder basis from a chosen unit highest-weight vector. Thus the finite choices and displayed positive normalization factors define every $v_{j,M}$ and its unique product-basis coefficients. Multiplying the chosen highest-weight vector by a unit complex scalar gives the stated phase change by linearity. The direct-sum supplier gives coordinatewise vector operations by taking $R=\mathbb{C}$ and the indicated finite families, while the linear-map supplier defines the maps $T_q$. The blockwise operators preserve each coordinate space and are linear because the spin-block operators are linear; therefore $J_a^K{T}_q$ and $T_q{J}_a^H$ are linear maps from $H$ to $K$, so each tensor-family equation is well typed. The sum of the block inner products is sesquilinear and positive definite because it is a finite sum of such pairings and is positive on every nonzero tuple. The zero family satisfies the equations, including at the endpoints. Finite spin matrices Tensor spin composition Ordered tensor registers, their inner product, and wire permutations The direct sum of an indexed family of modules Linear map between vector spaces over the same field