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Qm Math Orbital And Spin

1 · Prerequisites

2 · Summary

The orbital-and-spin layer of the non-relativistic quantum mechanics mathematical prerequisites, all of it mathematics: the finite-dimensional spin- $s$ matrices of $s u (2)$ , their symmetric-power and tensor representations and block decompositions, the Tsirelson bound for two binary sharp models with its singlet attainment, the Legendre and associated Legendre equations with the normalized spherical functions, their completeness in $L^{2} (S^{2}, σ)$ , the diagonal sphere Laplacian and weight operator with explicit domains, the harmonic-polynomial decomposition of polynomial spaces and of the isotropic oscillator levels, the strongly continuous unitary rotation action on $L^{2} (S^{2}, σ)$ , its self-adjoint orbital generators with the angular-momentum commutators and rotational domain identification, and the dipole angular selection rules. Every item carries domain: mathematics and depends only on mathematical suppliers, published supersets or items of qm-math-oscillator; nothing here asserts a physical interpretation of spin, measurement, transition rates or empirical evidence, and the companion page qm-math-orbital-and-spin-examples stays mathematics as well.

The spin layer begins with Finite spin matrices: for $s \in \frac{1}{2} N$ the standard matrices $J_{z} e_{m} = m e_{m}$ and $J_{\pm} e_{m} = \sqrt{(s \mp m) (s \pm m + 1)} e_{m \pm 1}$ on $C^{2 s + 1}$ satisfy $J_{+}^{*} = J_{-}$ ; the matrices $J_{z}$ , $J_{x} = (J_{+} + J_{-}) / 2$ and $J_{y} = (J_{+} - J_{-}) / (2 i)$ are self-adjoint, obey the $s u (2)$ commutators and have Casimir $J^{2} = s (s + 1) I$ . For $s = \frac{1}{2}$ , $J_{i} = \frac{1}{2} σ_{i}$ for $i = x, y, z$ , with the multiplication law $σ_{i} σ_{j} = δ_{i j} I + i \sum_{k} ε_{i j k} σ_{k}$ . Symmetric-power SU2 representation identifies the symmetric power of the spin- $\frac{1}{2}$ module with the spin- $\frac{n}{2}$ module inside the $n$ -fold tensor power; Finite spin-block decomposition splits any self-adjoint $s u (2)$ triple on a finite-dimensional space into spin blocks with the standard ladders and reads the multiplicities off $\dim \ker (J_{z} - M I)$ ; and Tensor spin composition composes two spins, $V_{s} \otimes V_{t} = ⨁_{j = ∣ s - t ∣}^{s + t} W_{j}$ with each $W_{j}$ unitarily equivalent to the spin- $j$ model and an explicit ladder construction. These finite-dimensional results feed Tsirelson bound: for bounded self-adjoint involutions the Bell operator $C = A_{0} \otimes (B_{0} + B_{1}) + A_{1} \otimes (B_{0} - B_{1})$ satisfies $C^{2} = 4 I - [A_{0}, A_{1}] \otimes [B_{0}, B_{1}]$ and $∣ ⟨ C ψ, ψ^{'} ⟩ ∣ \leq 2 \sqrt{2} ∥ ψ ∥ ∥ ψ^{'} ∥$ , the bound is attained by the singlet with the standard settings, and the explicitly specified common-outcome $\pm 1$ model obeys the CHSH bound $2$ instead.

The angular layer is analytic. Legendre Rodrigues equation develops the Rodrigues polynomials $P_{l}$ and the Legendre equation and orthogonality; Sphere integrals in angle coordinates identifies the polar surface measure with the chart measure $\sin θ d θ d φ$ and computes $σ (S^{2}) = 4 π$ ; Associated Legendre function normalization establishes the associated functions $w_{l m}$ , their equation and the normalized spherical functions $Y_{l m} = b_{l m} w_{l m} (\cos θ) e^{i m φ} / \sqrt{2 π}$ for $0 \leq m \leq l$ , with $Y_{l, - m} = (- 1)^{m} \overline{Y_{l m}}$ for $1 \leq m \leq l$ ; Weighted angular completeness proves completeness of $(w_{l m})_{l}$ in the weighted space; and Sphere ONB assembles the complete orthonormal family $(Y_{l m})$ of $L^{2} (S^{2}, σ)$ . Sphere Laplacian diagonal domain defines the self-adjoint diagonal operators $A Y_{l m} = l (l + 1) Y_{l m}$ and $L_{z} Y_{l m} = ℏ m Y_{l m}$ on explicit square-summability domains with the finite span as a common core, matching the sphere differential expressions away from the poles and reducing the blocks $V_{l}$ . Harmonic polynomial dimension connects the sphere to polynomials: $P_{l} = H_{l} \oplus r^{2} P_{l - 2}$ , $\dim H_{l} = 2 l + 1$ , the $Y_{l m}$ are restrictions of explicit harmonic homogeneous polynomials, and the isotropic oscillator level $M$ contains exactly the orbital degrees $M, M - 2, \dots \geq 0$ , each once. Strongly continuous rotation action constructs the unitary action $U (R) F = F \circ R^{- 1}$ with $U (R S) = U (R) U (S)$ , its preservation of the harmonic blocks with intertwining against the restriction isomorphisms, and the norm-continuity of every orbit map with the truncation estimate. Orbital self-adjoint generators extracts the self-adjoint generators $L_{i}$ of the coordinate-axis rotation groups, proves $L_{i} = - i ℏ (x \times \nabla)_{i}$ on the blocks, the commutators $[L_{i}, L_{j}] = i ℏ \sum_{k} ε_{i j k} L_{k}$ on the core, the identification $L_{3} = L_{z}$ , and the unitary identification of the three domains by explicit quarter turns. Finally Dipole angular selection rules expands $Q_{q} h_{l m}$ into harmonic homogeneous pieces of degrees $l + 1$ and $l - 1$ and proves that the angular matrix element $⟨ Y_{l^{'} m^{'}}, Q_{q} Y_{l m} ⟩$ can be nonzero only for $l^{'} = l \pm 1$ , $m^{'} = m + q$ : necessary conditions only, with no transition-probability claim. All items are draft mathematics under construction; the physical orbital and spin pages consume these results as stated mathematical premises and add their own postulates and interpretation.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

PropositionStatement: AI-adaptedProof: AI-adaptedOpen item page →

Finite spin matrices

Statement

Let $s \in \frac{1}{2} N = {0, \frac{1}{2}, 1, \frac{3}{2}, \dots}$ , put $N : = 2 s \in N$ and let $V_{s} : = C^{N + 1}$ carry the standard inner product $⟨ z, w ⟩ = \sum_{k = 0}^{N} z_{k} \overline{w_{k}}$ , linear in the first argument (Real and complex inner product spaces, with the inner product linear in the first argument). Write $(e_{m})_{- s \leq m \leq s}$ for the standard basis, indexed so that $e_{m}$ is the vector with entry $1$ in position $k = s - m$ and entry $0$ elsewhere; thus $⟨ e_{m}, e_{n} ⟩ = 1$ if $m = n$ and $⟨ e_{m}, e_{n} ⟩ = 0$ otherwise. Define $C$ -linear maps $J_{z}, J_{+}, J_{-} \in {End}_{C} (V_{s})$ by their values on this basis (Linear map between vector spaces over the same field):

$J_{z} e_{m} : = m e_{m}, J_{+} e_{m} : = \sqrt{(s - m) (s + m + 1)} e_{m + 1}, J_{-} e_{m} : = \sqrt{(s + m) (s - m + 1)} e_{m - 1},$

where $e_{s + 1} : = 0$ and $e_{- s - 1} : = 0$ , and put

$J_{x} : = \frac{1}{2} (J_{+} + J_{-}), J_{y} : = \frac{1}{2 i} (J_{+} - J_{-}), J^{2} : = J_{x}^{2} + J_{y}^{2} + J_{z}^{2} .$

Then:

(a) $J_{z}$ , $J_{x}$ and $J_{y}$ are self-adjoint endomorphisms of $V_{s}$ (Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space), and $J_{+}^{*} = J_{-}$ , $J_{-}^{*} = J_{+}$ (The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ );

(b) on $V_{s}$ the commutators are

$[J_{z}, J_{+}] = J_{+}, [J_{z}, J_{-}] = - J_{-}, [J_{+}, J_{-}] = 2 J_{z}, [J_{x}, J_{y}] = i J_{z}, [J_{y}, J_{z}] = i J_{x}, [J_{z}, J_{x}] = i J_{y};$

(c) $J^{2} = J_{-} J_{+} + J_{z}^{2} + J_{z} = J_{+} J_{-} + J_{z}^{2} - J_{z} = s (s + 1) I$ , and $[J^{2}, J_{z}] = [J^{2}, J_{\pm}] = 0$ ;

(d) the matrix elements of the raising operator are $∥ J_{+} e_{m} ∥^{2} = (s - m) (s + m + 1)$ and $∥ J_{-} e_{m} ∥^{2} = (s + m) (s - m + 1)$ , and $⟨ J_{+} e_{m}, e_{m + 1} ⟩ = \sqrt{(s - m) (s + m + 1)}$ , which is nonzero exactly when $m < s$ ;

(e) for $s = \frac{1}{2}$ , identify $V_{\frac{1}{2}} = C^{2}$ with the basis ordered as $(e_{\frac{1}{2}}, e_{- \frac{1}{2}})$ and let

$σ_{x} : = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}), σ_{y} : = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array}), σ_{z} : = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$

be the Pauli matrices. Then $J_{i} = \frac{1}{2} σ_{i}$ for $i = x, y, z$ , $σ_{x}^{2} = σ_{y}^{2} = σ_{z}^{2} = I$ , $σ_{x} σ_{y} = i σ_{z}$ , $σ_{y} σ_{z} = i σ_{x}$ , $σ_{z} σ_{x} = i σ_{y}$ , and consequently $σ_{i} σ_{j} + σ_{j} σ_{i} = 2 δ_{i j} I$ and $σ_{i} σ_{j} = δ_{i j} I + i \sum_{k} ε_{i j k} σ_{k}$ for all $i, j$ , where $δ_{i j} = 1$ if $i = j$ and $0$ otherwise, and $ε_{i j k} = 0$ when two of $i, j, k$ coincide, and otherwise $ε_{i j k}$ is the sign of the permutation $(x, y, z) \mapsto (i, j, k)$ ; the sum ranges over $k \in {x, y, z}$ . The same convention with indices $1, 2, 3$ uses the order-preserving relabelling $(1, 2, 3) = (x, y, z)$ .

All statements are finite-dimensional linear algebra: no physical interpretation of $J_{x}, J_{y}, J_{z}$ as angular momentum, and no measurement postulate, is asserted here.

Facts & Assumptions

Given: $s = \frac{N}{2}$ with $N \in N$ ; the space $V_{s} = C^{N + 1}$ with its standard inner product; the standard basis $(e_{m})_{- s \leq m \leq s}$ ; the maps $J_{z}, J_{\pm}, J_{x}, J_{y}, J^{2}$ displayed in the Statement.

[F1]

The standard basis is orthonormal: $⟨ e_{m}, e_{n} ⟩ = δ_{m n}$ in the sense of the Statement, and every $z \in V_{s}$ has the unique expansion $z = \sum_{m} ⟨ z, e_{m} ⟩ e_{m}$ with coefficients $⟨ z, e_{m} ⟩$ ; a linear map is determined by, and may be prescribed by, its values on a basis (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Linear combination of a finite list, and the span $span (S)$ as the smallest linear subspace containing $S$ , Linear map between vector spaces over the same field).

[F2]

For $T \in End (V_{s})$ the adjoint $T^{*}$ is the unique endomorphism with $⟨ T z, w ⟩ = ⟨ z, T^{*} w ⟩$ for all $z, w$ ; $T$ is self-adjoint when $T = T^{*}$ , equivalently when $⟨ T z, w ⟩ = ⟨ z, T w ⟩$ for all $z, w$ (The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ , Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space).

[F3]

Complex conjugation is additive, satisfies $\overline{z w} = \overline{z} \overline{w}$ , $\overline{\overline{z}} = z$ and $z \overline{z} = ∣ z ∣^{2} \geq 0$ , and $z \overline{z} = 0$ only for $z = 0$ ; also $i^{2} = - 1$ and $\overline{i} = - i$ (The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F4]

For real $t \geq 0$ the square root symbol denotes the unique nonnegative $r$ with $r^{2} = t$ ; the numbers $(s - m) (s + m + 1)$ and $(s + m) (s - m + 1)$ are nonnegative integers for every $m \in {- s, - s + 1, \dots, s}$ , and they vanish exactly at the endpoints $m = s$ , respectively $m = - s$ , where an endpoint vector is killed (The natural numbers $N$ (von Neumann)).

[F5]

Matrix multiplication of the displayed $2 \times 2$ matrices is entrywise-finite multiplication; the identities $σ_{x}^{2} = σ_{y}^{2} = σ_{z}^{2} = I$ , $σ_{x} σ_{y} = i σ_{z}$ , $σ_{y} σ_{z} = i σ_{x}$ , $σ_{z} σ_{x} = i σ_{y}$ are checked by direct multiplication using [F3].

Proof

1.1F1F2F3F4algebra

[Well-definedness; (a)] By [F4] the coefficients are nonnegative reals and the endpoint conventions are compatible, so prescribing the values $J_{z} e_{m}, J_{\pm} e_{m}$ on the basis defines unique linear maps $J_{z}, J_{+}, J_{-} \in End (V_{s})$ by [F1]; $J_{x}, J_{y}, J^{2}$ are then defined by the displayed formulas and are linear. For all basis vectors, $⟨ J_{z} e_{m}, e_{n} ⟩ = m δ_{m n} = ⟨ e_{m}, J_{z} e_{n} ⟩$ , and $⟨ J_{+} e_{m}, e_{n} ⟩ = \sqrt{(s - m) (s + m + 1)} δ_{n, m + 1}$ , while $⟨ e_{m}, J_{-} e_{n} ⟩ = \sqrt{(s + n) (s - n + 1)} δ_{m, n - 1}$ , which at $n = m + 1$ equals $\sqrt{(s + m + 1) (s - m)} = \sqrt{(s - m) (s + m + 1)}$ , and both sides vanish otherwise; hence $⟨ J_{+} e_{m}, e_{n} ⟩ = ⟨ e_{m}, J_{-} e_{n} ⟩$ and $⟨ J_{-} e_{m}, e_{n} ⟩ = ⟨ e_{m}, J_{+} e_{n} ⟩$ for all $m, n$ , so $J_{+}^{*} = J_{-}$ and $J_{-}^{*} = J_{+}$ by [F1], [F2]. Taking adjoints in the definitions of $J_{x}, J_{y}$ and using $(J_{+} + J_{-})^{*} = J_{-} + J_{+} = J_{+} + J_{-}$ and $(\frac{1}{2 i} (J_{+} - J_{-}))^{*} = \frac{1}{\overline{2 i}} (J_{+} - J_{-})^{*} = \frac{1}{- 2 i} (J_{-} - J_{+}) = \frac{1}{2 i} (J_{+} - J_{-})$ gives $J_{x}^{*} = J_{x}$ and $J_{y}^{*} = J_{y}$ . This is (a).

1.2F1F4algebra

[(b), ladder relations] For every basis vector, $J_{z} (J_{+} e_{m}) = J_{z} \sqrt{(s - m) (s + m + 1)} e_{m + 1} = (m + 1) J_{+} e_{m} = J_{+} (J_{z} e_{m}) + J_{+} e_{m}$ , so acting on the basis and using linearity, $[J_{z}, J_{+}] = J_{+}$ ; similarly $J_{z} (J_{-} e_{m}) = (m - 1) J_{-} e_{m} = J_{-} (J_{z} e_{m}) - J_{-} e_{m}$ gives $[J_{z}, J_{-}] = - J_{-}$ . Moreover, using the endpoint conventions, $J_{+} J_{-} e_{m} = (s + m) (s - m + 1) e_{m}$ and $J_{-} J_{+} e_{m} = (s - m) (s + m + 1) e_{m}$ , so $[J_{+}, J_{-}] e_{m} = [(s + m) (s - m + 1) - (s - m) (s + m + 1)] e_{m} = 2 m e_{m} = 2 J_{z} e_{m}$ , and by linearity $[J_{+}, J_{-}] = 2 J_{z}$ .

1.3F1F4algebra

[(d)] For $- s \leq m \leq s$ , orthonormality [F1] and the definitions give $∥ J_{+} e_{m} ∥^{2} = (s - m) (s + m + 1)$ and $∥ J_{-} e_{m} ∥^{2} = (s + m) (s - m + 1)$ , and $⟨ J_{+} e_{m}, e_{m + 1} ⟩ = \sqrt{(s - m) (s + m + 1)}$ ; the coefficient vanishes exactly when $m = s$ by [F4].

2.1step 1.2F3algebra

[(b), remaining commutators] From $J_{x} = \frac{1}{2} (J_{+} + J_{-})$ , $J_{y} = \frac{1}{2 i} (J_{+} - J_{-})$ and the relations $[J_{+}, J_{-}] = 2 J_{z}$ , $[J_{-}, J_{+}] = - 2 J_{z}$ of step 1.2, expand the products bilinearly: $[J_{x}, J_{y}] = \frac{1}{4 i} ([J_{+}, J_{+}] - [J_{+}, J_{-}] + [J_{-}, J_{+}] - [J_{-}, J_{-}]) = \frac{1}{4 i} (0 - 2 J_{z} - 2 J_{z} - 0) = \frac{- 4 J_{z}}{4 i} = i J_{z}$ , because $1 / i = - i$ by [F3]. Likewise $[J_{z}, J_{x}] = \frac{1}{2} ([J_{z}, J_{+}] + [J_{z}, J_{-}]) = \frac{1}{2} (J_{+} - J_{-}) = i J_{y}$ , and $[J_{z}, J_{y}] = \frac{1}{2 i} ([J_{z}, J_{+}] - [J_{z}, J_{-}]) = \frac{1}{2 i} (J_{+} + J_{-}) = - i J_{x}$ , which is $[J_{y}, J_{z}] = i J_{x}$ . This completes (b).

2.2step 1.2F1algebra

[(c)] The definitions give $J_{x}^{2} + J_{y}^{2} = \frac{1}{2} (J_{+} J_{-} + J_{-} J_{+})$ ; combining this with $[J_{+}, J_{-}] = 2 J_{z}$ from step 1.2 yields $J^{2} = J_{-} J_{+} + J_{z}^{2} + J_{z} = J_{+} J_{-} + J_{z}^{2} - J_{z}$ . Direct expansion on each basis vector gives $(J_{-} J_{+} + J_{z}^{2} + J_{z}) e_{m} = [(s - m) (s + m + 1) + m^{2} + m] e_{m} = s (s + 1) e_{m}$ and $(J_{+} J_{-} + J_{z}^{2} - J_{z}) e_{m} = [(s + m) (s - m + 1) + m^{2} - m] e_{m} = s (s + 1) e_{m}$ , using $(s \mp m) (s \pm m + 1) = s^{2} + s - m^{2} \mp m$ ; hence $J^{2} = s (s + 1) I$ by [F1]. Since $J^{2}$ is a scalar multiple of the identity, it commutes with $J_{x}, J_{y}, J_{z}$ and with $J_{\pm} = J_{x} \pm i J_{y}$ , giving (c).

3.1F3F5algebra∎

[(e), Pauli case] Let $s = \frac{1}{2}$ , so $N = 1$ and the basis is $(e_{\frac{1}{2}}, e_{- \frac{1}{2}})$ . From the definitions, $J_{z} e_{\pm \frac{1}{2}} = \pm \frac{1}{2} e_{\pm \frac{1}{2}}$ , $J_{+} e_{- \frac{1}{2}} = e_{\frac{1}{2}}$ , $J_{+} e_{\frac{1}{2}} = 0$ , $J_{-} e_{\frac{1}{2}} = e_{- \frac{1}{2}}$ , $J_{-} e_{- \frac{1}{2}} = 0$ ; reading these actions as matrices in the ordered basis $(e_{\frac{1}{2}}, e_{- \frac{1}{2}})$ gives $J_{z} = \frac{1}{2} σ_{z}$ , $J_{+} = \frac{1}{2} (σ_{x} + i σ_{y})$ and $J_{-} = \frac{1}{2} (σ_{x} - i σ_{y})$ , whence $J_{x} = \frac{1}{2} (J_{+} + J_{-}) = \frac{1}{2} σ_{x}$ and $J_{y} = \frac{1}{2 i} (J_{+} - J_{-}) = \frac{1}{2} σ_{y}$ . The displayed matrices satisfy $σ_{i}^{*} = σ_{i}$ by inspection. Direct multiplication using [F3] and [F5] gives $σ_{x}^{2} = σ_{y}^{2} = σ_{z}^{2} = I$ , $σ_{x} σ_{y} = i σ_{z}$ , $σ_{y} σ_{z} = i σ_{x}$ , $σ_{z} σ_{x} = i σ_{y}$ ; taking adjoints of the three off-diagonal products and using that the $σ_{i}$ are Hermitian with $\overline{i} = - i$ gives $σ_{j} σ_{i} = - σ_{i} σ_{j}$ for $i \neq j$ , so $σ_{i} σ_{j} + σ_{j} σ_{i} = 2 δ_{i j} I$ and $σ_{i} σ_{j} = δ_{i j} I + i \sum_{k} ε_{i j k} σ_{k}$ for all ordered pairs $i, j \in {x, y, z}$ . For $i = j$ , every $ε_{i i k}$ is zero, and the product formula reduces to $σ_{i}^{2} = I$ ; for $i \neq j$ , only the remaining coordinate $k$ has nonzero $ε_{i j k}$ , with sign positive in the three cyclic orders and negative in their reversals, exactly matching the products just computed. This is (e).

The displayed relations are those of the standard $(2 s + 1)$ -dimensional spin representation; the proposition alone asserts only the finite matrix identities written in its Statement.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Symmetric-power SU2 representation

Statement

Fix an integer $n \geq 0$ , let $V : = C^{2}$ carry the standard inner product linear in the first argument, with orthonormal basis $(e_{+}, e_{-})$ where $e_{+} : = (1, 0)$ and $e_{-} : = (0, 1)$ (Real and complex inner product spaces, with the inner product linear in the first argument), and let $V^{\otimes n}$ be the $n$ -fold algebraic tensor power with its finite string basis (The elementary tensors of two bases form the product basis of the tensor product) made orthonormal. Use $S_{n} = Sym ({0, \dots, n - 1})$ as in The finite symmetric group $S_{n}$ , one-line notation, and cycle notation. For $n \geq 1$ define the bijection $b_{n} : {0, \dots, n - 1} \to {1, \dots, n}$ by $b_{n} (a) = a + 1$ , and for $π \in S_{n}$ put $\hat{π} : = b_{n} \circ π \circ b_{n}^{- 1}$ . Define $U_{π}$ to be the place action of $\hat{π}$ in Commuting symmetric-group and linear actions on a tensor power: explicitly, for $x_{1}, \dots, x_{n} \in V$ , $U_{π} (x_{1} \otimes \dots \otimes x_{n}) = x_{π^{- 1} (0) + 1} \otimes \dots \otimes x_{π^{- 1} (n - 1) + 1}$ , extended linearly. Use the diagonal action $W^{\otimes n}$ of $W \in M_{2} (C)$ and the operators $Δ (T) : = \sum_{j = 1}^{n} 1^{\otimes (j - 1)} \otimes T \otimes 1^{\otimes (n - j)}$ . Put

$P_{+} : = \frac{1}{n!} \sum_{π \in S_{n}} U_{π}, S_{n} : = ran P_{+} \subseteq V^{\otimes n}, S U (2) : = {W \in M_{2} (C) : W^{*} W = I_{2}, \det W = 1},$

where $W^{*} : = {\overline{W}}^{T}$ is the conjugate transpose of $W$ and $I_{2}$ the identity matrix (Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix). For $W \in S U (2)$ define $U_{n} (W) : = W^{\otimes n} ∣_{S_{n}}$ . For $k \in {0, \dots, n}$ let $s_{k}$ be the sum of the elementary tensors $e_{ε_{1}} \otimes \dots \otimes e_{ε_{n}}$ over all $2^{n}$ strings $(ε_{1}, \dots, ε_{n}) \in {+, -}^{n}$ containing exactly $k$ entries $-$ , and put $v_{k} : = {(\binom{n}{k})}^{- 1 / 2} s_{k}$ .

For $n = 0$ use $V^{\otimes 0} = C$ , the empty string $s_{0} = v_{0} = 1$ , $0! = (\binom{0}{0}) = 1$ , the trivial group $S_{0}$ , $U_{π} = P_{+} = I_{C}$ , $W^{\otimes 0} = I_{C}$ for every $W$ , and $Δ (T) = 0$ . These conventions define $S_{0} = C$ and the trivial representation $U_{0}$ , with $J_{x} = J_{y} = J_{z} = 0$ .

Then:

(a) $P_{+}$ is the orthogonal projection onto the closed subspace $S_{n}$ , $U_{π}$ and $W^{\otimes n}$ commute for every $π \in S_{n}$ and every $W$ , and $W^{\otimes n} (S_{n}) \subseteq S_{n}$ ; the assignment $W \mapsto U_{n} (W)$ is well defined on all of $M_{2} (C)$ , and $U_{n} (W^{'}) U_{n} (W) = U_{n} (W^{'} W)$ holds for all $W, W^{'}$ ;

(b) $S U (2)$ is a subgroup of the multiplicative monoid of $M_{2} (C)$ with identity $I_{2}$ ; for $W \in S U (2)$ the operator $U_{n} (W)$ is unitary with inverse $U_{n} (W^{- 1}) = U_{n} (W)^{*}$ , so $U_{n} : S U (2) \to U (S_{n})$ is a group homomorphism (Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces);

(c) the vectors $v_{0}, \dots, v_{n}$ form an orthonormal basis of $S_{n}$ ; in particular $S_{n}$ is $(n + 1)$ -dimensional;

(d) the operators $J_{i} : = \frac{1}{2} Δ (σ_{i}) ∣_{S_{n}}$ , $i = x, y, z$ , where $σ_{x}, σ_{y}, σ_{z}$ are the Pauli matrices of Finite spin matrices, are self-adjoint and satisfy, with $J_{\pm} : = J_{x} \pm i J_{y}$ , $J^{2} : = J_{x}^{2} + J_{y}^{2} + J_{z}^{2}$ ,

$J_{z} v_{k} = (\frac{n}{2} - k) v_{k}, J_{+} v_{k} = \sqrt{k (n - k + 1)} v_{k - 1}, J_{-} v_{k} = \sqrt{(k + 1) (n - k)} v_{k + 1},$

with $v_{- 1} : = 0$ and $v_{n + 1} : = 0$ ; moreover $[J_{z}, J_{\pm}] = \pm J_{\pm}$ , $[J_{+}, J_{-}] = 2 J_{z}$ , $[J_{i}, J_{j}] = i \sum_{k} ε_{i j k} J_{k}$ and $J^{2} = \frac{n}{2} (\frac{n}{2} + 1) I_{S_{n}}$ ; the map $v_{k} \mapsto e_{n / 2 - k}$ is a unitary equivalence of $(J_{x}, J_{y}, J_{z})$ with the spin- $\frac{n}{2}$ matrices of Finite spin matrices;

(e) for each $i$ the curve $A_{i} (t) : = \cos (t / 2) I_{2} - i \sin (t / 2) σ_{i}$ ( $t \in R$ ) lies in $S U (2)$ , satisfies $A_{i} (s) A_{i} (t) = A_{i} (s + t)$ and $A_{i}^{'} (0) = - \frac{i}{2} σ_{i}$ , and $t \mapsto V_{i} (t) : = U_{n} (A_{i} (t))$ is a strongly continuous one-parameter unitary group on $S_{n}$ (Strongly continuous one-parameter unitary group) whose infinitesimal generator $G_{i}$ (Infinitesimal generator of a unitary group) is $G_{i} = - i J_{i}$ on all of $S_{n}$ ; equivalently $\frac{1}{t} (V_{i} (t) v - v) \to - i J_{i} v$ for every $v \in S_{n}$ ;

(f) $U_{n}$ is strongly continuous on $S U (2)$ : if $W_{m} \to W_{0}$ in $M_{2} (C)$ with all $W_{m} \in S U (2)$ and $W_{0} \in S U (2)$ , then $∥ U_{n} (W_{m}) - U_{n} (W_{0}) ∥ \to 0$ in the operator norm on $S_{n}$ .

This is finite-dimensional linear algebra: the theorem constructs a unitary representation of $S U (2)$ and its generators. It does not assert that $S U (2)$ or any spin observable describes a physical particle, and it makes no measurement or statistics claim.

Facts & Assumptions

Given: $n \geq 0$ ; the space $V = C^{2}$ with its orthonormal basis $(e_{+}, e_{-})$ ; the tensor power $V^{\otimes n}$ ; the place permutations $U_{π}$ and diagonal actions $W^{\otimes n}$ ; the projection $P_{+}$ ; the subspace $S_{n}$ ; the strings $s_{k}$ and vectors $v_{k}$ ; the group $S U (2)$ ; the Pauli matrices $σ_{x}, σ_{y}, σ_{z}$ ; the operators $Δ (T)$ and $J_{i}$ .

[F1]

For $n \geq 1$ , iterating The elementary tensors of two bases form the product basis of the tensor product gives the finite basis of all $2^{n}$ strings $e_{ε_{1}} \otimes \dots \otimes e_{ε_{n}}$ ; for $n = 0$ the basis is $1 \in C$ . Define the inner product by $⟨ \sum c_{ε} e_{ε}, \sum d_{ε} e_{ε} ⟩ = \sum c_{ε} \overline{d_{ε}}$ . Expansion on elementary tensors gives the product inner product. The finite tensor product is thus the standard finite-dimensional coordinate inner product space; no completion or countable choice is used. The universal multilinear property is Finite iterated tensor products represent multilinear maps independently of parenthesization.

[F2]

For $n \geq 1$ , conjugation by $b_{n}$ is a bijection from the zero-indexed $S_{n}$ of The finite symmetric group $S_{n}$ , one-line notation, and cycle notation to the group on tensor positions ${1, \dots, n}$ of Commuting symmetric-group and linear actions on a tensor power; it preserves composition because $\hat{π} \hat{σ} = b_{n} π σ b_{n}^{- 1} = \hat{π σ}$ , and ${\hat{π}}^{- 1} (j) = π^{- 1} (j - 1) + 1$ for $1 \leq j \leq n$ . Thus the stated $U_{π}$ are well-defined linear maps by [F1], permute the orthonormal string basis, are unitary, satisfy $U_{π} U_{σ} = U_{π σ}$ and have $U_{π}^{*} = U_{π^{- 1}}$ . There are $n!$ permutations: prescribing the images of $0, \dots, n - 1$ successively leaves $n, n - 1, \dots, 1$ possibilities. In the finite average $P_{+} = (1 / n!) \sum_{π} U_{π}$ , inversion and left or right multiplication by a permutation reindex the summands. Thus $P_{+}^{*} = P_{+}$ , $U_{π} P_{+} = P_{+} U_{π} = P_{+}$ and $P_{+}^{2} = P_{+}$ ; its range consists exactly of the invariant vectors, since every invariant vector $v$ satisfies $P_{+} v = v$ . For $n = 1$ , $\hat{π} (1) = 1$ and $U_{π} = I_{V}$ ; for $n = 0$ , all identities follow from the separate empty-tensor conventions.

[F3]

On the spanning string basis, expansion of matrix entries gives $(S \otimes T) (S^{'} \otimes T^{'}) = (S S^{'}) \otimes (T T^{'})$ , and evaluation of the inner products of [F1] gives $(S \otimes T)^{*} = S^{*} \otimes T^{*}$ . Thus a tensor product of unitary matrices is unitary. Applying the same $W$ to every factor commutes with any place permutation, since both rules send $v_{1} \otimes \dots \otimes v_{n}$ to the tensor of the same permuted list $W v_{j}$ ; this identity is also valid for $n = 0$ (Commuting symmetric-group and linear actions on a tensor power).

[F4]

For a bounded operator $L$ on a finite-dimensional Hilbert space, the operator norm is $∥ L ∥ = \sup_{∥ z ∥ \leq 1} ∥ L z ∥$ , and $∥ L ∥ \leq \sum_{j} ∥ L e_{j} ∥$ for every orthonormal basis $(e_{j})$ ; a linear map with finite-dimensional domain is bounded, and $∥ L z ∥ \leq ∥ L ∥ ∥ z ∥$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces). In orthonormal product coordinates, writing $z = \sum_{b} z_{b} \otimes e_{b}$ gives $∥ (S \otimes I) z ∥^{2} = \sum_{b} ∥ S z_{b} ∥^{2} \leq ∥ S ∥^{2} \sum_{b} ∥ z_{b} ∥^{2}$ ; similarly $∥ I \otimes T ∥ \leq ∥ T ∥$ . Since $S \otimes T = (S \otimes I) (I \otimes T)$ and $∥ L M z ∥ \leq ∥ L ∥ ∥ M ∥ ∥ z ∥$ , this proves $∥ S \otimes T ∥ \leq ∥ S ∥ ∥ T ∥$ , and iteration supplies the tensor estimates below.

[F5]

$W^{*} W = I_{2}$ means that $W$ is a unitary operator on $C^{2}$ ; in orthonormal bases the matrix of the adjoint is the conjugate transpose, and $\det (W^{*}) = \overline{\det W}$ , $\det (W^{'} W) = \det W^{'} \det W$ (In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix, Coordinate columns $[v]_{B}$ and matrices $[T]_{B}^{C}$ of linear maps relative to ordered bases, For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ , Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

[F6]

The Pauli matrices satisfy $σ_{i}^{*} = σ_{i}$ , $σ_{x}^{2} = σ_{y}^{2} = σ_{z}^{2} = I_{2}$ , $σ_{x} σ_{y} = i σ_{z}$ , $σ_{y} σ_{z} = i σ_{x}$ , $σ_{z} σ_{x} = i σ_{y}$ ; also $σ_{+} + σ_{-} = 2 σ_{x}$ and $σ_{+} - σ_{-} = 2 i σ_{y}$ for $σ_{+} : = σ_{x} + i σ_{y}$ , $σ_{-} : = σ_{x} - i σ_{y}$ , and $σ_{+} e_{-} = 2 e_{+}$ , $σ_{+} e_{+} = 0$ , $σ_{-} e_{+} = 2 e_{-}$ , $σ_{-} e_{-} = 0$ , $σ_{z} e_{\pm} = \pm e_{\pm}$ (Finite spin matrices).

[F7]

The sine and cosine addition identities hold (The addition formulas for sine and cosine). Sine and cosine are differentiable with $\sin^{'} = \cos$ , $\cos^{'} = - \sin$ , and $\sin^{2} + \cos^{2} = 1$ ; derivatives of sums, scalar multiples and compositions are computed by the algebra of derivatives and the chain rule (The derivatives of sine and cosine are cosine and minus sine, A function differentiable at $c$ is continuous at $c$ , Parity and the Pythagorean identity for sine and cosine, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set).

[F8]

For integers $0 \leq k \leq n$ , $(\binom{n}{k}) = n! / (k! (n - k)!) > 0$ and $(\binom{n}{k}) = (\binom{n}{n - k})$ and $(\binom{n}{k - 1}) / (\binom{n}{k}) = k / (n - k + 1)$ for $1 \leq k \leq n$ ; the ratio follows by cancelling the factorials using $k! = k (k - 1)!$ and $(n - k + 1)! = (n - k + 1) (n - k)!$ . Each string with $k$ minus signs corresponds bijectively to its set of minus positions, so there are $(\binom{n}{k})$ such strings and the string sums split accordingly ( $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ , The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ , The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ , A finite sum in a commutative monoid indexed by an arbitrary finite set).

[F9]

Complex conjugation is additive and multiplicative, $i^{2} = - 1$ , $\overline{i} = - i$ , and $\overline{z^{- 1}} = {\overline{z}}^{- 1}$ for $z \neq 0$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive, The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ ).

[F10]

For $s = \frac{n}{2}$ the operators $J_{z}, J_{\pm}, J_{x}, J_{y}, J^{2}$ of Finite spin matrices act on their orthonormal basis $e_{m}$ , $- s \leq m \leq s$ , by $J_{z} e_{m} = m e_{m}$ , $J_{+} e_{m} = \sqrt{(s - m) (s + m + 1)} e_{m + 1}$ , $J_{-} e_{m} = \sqrt{(s + m) (s - m + 1)} e_{m - 1}$ , with endpoint conventions $e_{s + 1} = e_{- s - 1} = 0$ , have self-adjoint $J_{x}, J_{y}, J_{z}$ and $J_{-} = J_{+}^{*}$ , satisfy $[J_{z}, J_{\pm}] = \pm J_{\pm}$ , $[J_{+}, J_{-}] = 2 J_{z}$ and $J^{2} = s (s + 1) I$ , and have $∥ J_{+} e_{m} ∥^{2} = (s - m) (s + m + 1)$ .

Proof

1.1givenF10algebra

[The zero tensor power] For $n = 0$ , the conventions give $P_{+} = I$ , $S_{0} = C v_{0}$ , $U_{0} (W) = I$ , and $J_{i} = 0$ . Clauses (a)-(c) follow immediately; all ladder and Casimir formulas in (d) vanish, and $v_{0} \mapsto e_{0}$ identifies the zero operators with the spin-zero model [F10]. Each $V_{i} (t) = I$ has derivative zero and is strongly continuous, and $U_{0}$ is constant, proving (e)-(f); the subgroup and $A_{i}$ assertions, independent of $n$ , are proved below. For the tensor and basis calculations in the remaining steps assume $n \geq 1$ .

1.2F2F3algebra

[The symmetric subspace and invariance] By [F2] $P_{+}$ is an orthogonal projection and $U_{π} P_{+} = P_{+}$ ; hence $P_{+} v = v$ exactly for $v \in S_{n}$ , and $S_{n} = \ker (I - P_{+})$ is closed because the finite matrix $I - P_{+}$ is continuous. For $W \in M_{2} (C)$ and $π \in S_{n}$ , [F3] gives $W^{\otimes n} P_{+} = \frac{1}{n!} \sum_{π} W^{\otimes n} U_{π} = \frac{1}{n!} \sum_{π} U_{π} W^{\otimes n} = P_{+} W^{\otimes n}$ , so $v \in S_{n}$ implies $P_{+} (W^{\otimes n} v) = W^{\otimes n} P_{+} v = W^{\otimes n} v$ , i.e. $W^{\otimes n} (S_{n}) \subseteq S_{n}$ ; therefore $U_{n} (W) : = W^{\otimes n} ∣_{S_{n}}$ is a well-defined linear map on $S_{n}$ for every $W$ .

1.3F6F7F9algebra

[The curves $A_{i}$ ] Using $I_{2}^{*} = I_{2}$ , $σ_{i}^{*} = σ_{i}$ , $σ_{i}^{2} = I_{2}$ , [F7] and [F9]: $A_{i} (t)^{*} A_{i} (t) = \cos (t / 2)^{2} I_{2} + \sin (t / 2)^{2} σ_{i}^{2} - i \cos (t / 2) \sin (t / 2) (σ_{i} - σ_{i}^{*}) = (\cos^{2} + \sin^{2}) (t / 2) I_{2} = I_{2}$ , and direct computation of the three $2 \times 2$ determinants from the displayed matrices gives $\det A_{i} (t) = \cos^{2} (t / 2) + \sin^{2} (t / 2) = 1$ ; so $A_{i} (t) \in S U (2)$ . The identities $\cos (s + t) = \cos s \cos t - \sin s \sin t$ and $\sin (s + t) = \sin s \cos t + \cos s \sin t$ , applied with $σ_{i}^{2} = I_{2}$ , give $A_{i} (s) A_{i} (t) = A_{i} (s + t)$ . Differentiating the displayed formula at $t = 0$ with [F7] and [F9] gives $A_{i}^{'} (0) = - \frac{i}{2} σ_{i}$ .

2.1step 1.2F3F5algebra

[Representation law and unitarity] For $W, W^{'} \in M_{2} (C)$ , [F3] gives $(W^{'})^{\otimes n} W^{\otimes n} = (W^{'} W)^{\otimes n}$ on $V^{\otimes n}$ , hence on $S_{n}$ by step 1.2 and restriction: $U_{n} (W^{'}) U_{n} (W) = U_{n} (W^{'} W)$ . If $W \in S U (2)$ , then $W$ is unitary by [F5], [F6]; by [F3] $W^{\otimes n}$ is unitary on $V^{\otimes n}$ , and for $v \in S_{n}$ , $∥ U_{n} (W) v ∥ = ∥ W^{\otimes n} v ∥ = ∥ v ∥$ with $W^{\otimes n}$ mapping $S_{n}$ bijectively onto $S_{n}$ (its inverse is $(W^{- 1})^{\otimes n}$ restricted, where $W^{- 1} = W^{*} \in S U (2)$ ); hence $U_{n} (W)$ is unitary with $(U_{n} (W))^{*} = U_{n} (W^{- 1})$ . This gives the operator statements of (a) and (b).

2.2step 1.2F1F2F8algebra

[The basis $v_{k}$ and the dimension] Fix $k$ and let $w = e_{ε_{1}} \otimes \dots \otimes e_{ε_{n}}$ have exactly $k$ minus entries. In $n! P_{+} w = \sum_{π} U_{π} w$ every string with exactly $k$ minus entries occurs once for each of the $k! (n - k)!$ permutations carrying the pattern of $w$ to it, so $n! P_{+} w = k! (n - k)! s_{k}$ and $P_{+} w = {(\binom{n}{k})}^{- 1} s_{k}$ by [F8]; since the $2^{n}$ elementary tensors span $V^{\otimes n}$ [F1], this shows $S_{n} = ran P_{+} \subseteq span {s_{0}, \dots, s_{n}}$ , while $s_{k} \in S_{n}$ because each $U_{π}$ permutes the strings with $k$ minus entries, so $P_{+} s_{k} = s_{k}$ . Distinct elementary tensors are orthonormal [F1], and two summands from $s_{k}$ and $s_{l}$ with $k \neq l$ are distinct elementary tensors, so $⟨ s_{k}, s_{l} ⟩ = δ_{k l} (\binom{n}{k})$ by [F8]; hence $v_{0}, \dots, v_{n}$ are orthonormal and span $S_{n}$ , an orthonormal basis of the $(n + 1)$ -dimensional space $S_{n}$ .

2.3step 1.2F2F3F6algebra

[The operators $Δ (σ_{i})$ and their commutation] For $T, T^{'} \in M_{2} (C)$ the place sums satisfy $[Δ (T), Δ (T^{'})] = Δ ([T, T^{'}])$ and $Δ (T)^{*} = Δ (T^{*})$ : expanding the definitions, the summands at different places commute and the summand at a common place contributes $[T, T^{'}]$ , while the adjoint of a place sum is the place sum of the adjoints [F3]. Consequently $[Δ (σ_{i}), Δ (σ_{j})] = Δ ([σ_{i}, σ_{j}])$ ; and $Δ (σ_{i})$ commutes with every $U_{π}$ because permuting places preserves the multiset of places, so by [F2] each $Δ (σ_{i})$ preserves $S_{n}$ , and its restriction $J_{i} = \frac{1}{2} Δ (σ_{i}) ∣_{S_{n}}$ is self-adjoint.

3.1step 2.1F5F9algebra

[ $S U (2)$ is a subgroup] If $W, W^{'} \in S U (2)$ , then $(W W^{'})^{*} (W W^{'}) = W^{' *} W^{*} W W^{'} = W^{' *} W^{'} = I_{2}$ and $\det (W W^{'}) = \det W \det W^{'} = 1$ , so $W W^{'} \in S U (2)$ ; $I_{2}^{*} I_{2} = I_{2}$ and $\det I_{2} = 1$ , so $I_{2} \in S U (2)$ ; and for $W \in S U (2)$ one has $W^{*} W = W W^{*} = I_{2}$ (multiplying $W^{*} W = I_{2}$ by $W^{- 1}$ on the right gives $W^{*} = W^{- 1}$ , whence $W W^{*} = I_{2}$ ) and $\det W^{*} = \overline{\det W} = 1$ , so $W^{- 1} = W^{*} \in S U (2)$ . Hence $S U (2)$ is a subgroup, and $U_{n}$ is a group homomorphism by step 2.1.

3.2step 2.1step 2.3step 1.3F1F3F4F6F7algebra

[Generator of $t \mapsto U_{n} (A_{i} (t))$ ] By step 2.1, $V_{i} (t) : = U_{n} (A_{i} (t))$ satisfies $V_{i} (s) V_{i} (t) = V_{i} (s + t)$ and $V_{i} (0) = I$ , and each $V_{i} (t)$ is unitary. The telescoping identity $W^{\otimes k} - W_{0}^{\otimes k} = \sum_{j = 1}^{k} (W^{\otimes (j - 1)} \otimes (W - W_{0}) \otimes W_{0}^{\otimes (k - j)})$ , valid by [F3] with $W^{\otimes 0} : = I$ , gives $∥ W^{\otimes k} - W_{0}^{\otimes k} ∥ \leq k \max (∥ W ∥, ∥ W_{0} ∥)^{k - 1} ∥ W - W_{0} ∥$ by [F4]; with $W = A_{i} (t)$ , $W_{0} = A_{i} (t_{0})$ and $∥ A_{i} (\cdot) ∥ = 1$ , the explicit formula for $A_{i}$ and [F7] show that $∥ V_{i} (t) v - V_{i} (t_{0}) v ∥$ is bounded by $n ∥ A_{i} (t) - A_{i} (t_{0}) ∥ ∥ v ∥$ for all $v \in S_{n}$ by the operator-norm bound; so $V_{i}$ is a strongly continuous one-parameter unitary group. For the derivative, write $A_{i} (t) = I_{2} + t A_{i}^{'} (0) + r (t)$ with $∥ r (t) ∥ \leq ∣ t ∣ η (t)$ and $η (t) \to 0$ by [F6], [F7] and step 1.3; the same telescoping expansion with $W = A_{i} (t)$ , $W_{0} = I_{2}$ , using $∥ A_{i} (t) ∥ \leq 1$ and $∥ r (t) ∥ \leq ∣ t ∣ η (t)$ , gives $A_{i} (t)^{\otimes n} - I_{2}^{\otimes n} = t \sum_{j = 1}^{n} 1^{\otimes (j - 1)} \otimes A_{i}^{'} (0) \otimes 1^{\otimes (n - j)} + R (t)$ with $∥ R (t) ∥ \leq n ∣ t ∣ η^{'} (t)$ and $η^{'} (t) \to 0$ as $t \to 0$ ; restricting to $S_{n}$ and using step 2.3 and $A_{i}^{'} (0) = - \frac{i}{2} σ_{i}$ of step 1.3 gives $\frac{1}{t} (V_{i} (t) v - v) \to - \frac{i}{2} Δ (σ_{i}) v = - i J_{i} v$ for every $v \in S_{n}$ . Hence the infinitesimal generator is $G_{i} = - i J_{i}$ with domain $D (G_{i}) = S_{n}$ .

3.3step 2.2step 2.3F1F6F8algebra

[Actions on the basis vectors] Fix $0 \leq k \leq n$ . On an elementary tensor with $k$ minus entries, $Δ (σ_{z})$ acts by $# + - # - = n - 2 k$ , and by linearity $Δ (σ_{z}) s_{k} = (n - 2 k) s_{k}$ , so $J_{z} v_{k} = (n / 2 - k) v_{k}$ . For $σ_{+} = σ_{x} + i σ_{y}$ one has $σ_{+} e_{+} = 0$ and $σ_{+} e_{-} = 2 e_{+}$ by [F6]. If $1 \leq k \leq n$ , each target string with $k - 1$ minus entries in $Δ (σ_{+}) s_{k}$ occurs with coefficient $2$ from each of its $n - k + 1$ plus positions, so $Δ (σ_{+}) s_{k} = 2 (n - k + 1) s_{k - 1}$ and $J_{+} v_{k} = (n - k + 1) \sqrt{(\binom{n}{k - 1}) / (\binom{n}{k})} v_{k - 1} = \sqrt{k (n - k + 1)} v_{k - 1}$ by [F8]. For $k = 0$ , every factor is $e_{+}$ , so $J_{+} v_{0} = 0$ , agreeing with the stated $v_{- 1} = 0$ convention. Similarly, if $0 \leq k < n$ , $σ_{-} e_{-} = 0$ and $σ_{-} e_{+} = 2 e_{-}$ give $Δ (σ_{-}) s_{k} = 2 (k + 1) s_{k + 1}$ , hence $J_{-} v_{k} = (k + 1) \sqrt{(\binom{n}{k + 1}) / (\binom{n}{k})} v_{k + 1} = \sqrt{(k + 1) (n - k)} v_{k + 1}$ by [F8]. For $k = n$ , every factor is $e_{-}$ , so $J_{-} v_{n} = 0$ , agreeing with $v_{n + 1} = 0$ .

4.1step 3.3F10algebra

[Commutators and identification with the spin- $n / 2$ model] By steps 2.3 and 3.3, $J_{z}, J_{\pm}$ satisfy $[J_{z}, J_{\pm}] = \pm J_{\pm}$ and $[J_{+}, J_{-}] = 2 J_{z}$ on the basis $(v_{k})$ ; substituting $J_{x} = \frac{1}{2} (J_{+} + J_{-})$ , $J_{y} = \frac{1}{2 i} (J_{+} - J_{-})$ and expanding gives $[J_{x}, J_{y}] = i J_{z}$ , $[J_{y}, J_{z}] = i J_{x}$ , $[J_{z}, J_{x}] = i J_{y}$ . Since $v_{k} \mapsto e_{n / 2 - k}$ maps the basis of $S_{n}$ onto the basis of the spin- $\frac{n}{2}$ space of Finite spin matrices, and by step 3.3 the three operators have there exactly the action prescribed in that proposition, the described map is a unitary equivalence intertwining $(J_{x}, J_{y}, J_{z})$ with the spin- $\frac{n}{2}$ matrices and carrying $J^{2}$ to $s (s + 1) I$ with $s = n / 2$ ; in particular $J^{2} = \frac{n}{2} (\frac{n}{2} + 1) I_{S_{n}}$ . This proves (a)-(d) except the strong-continuity clause.

5.1step 1.1step 1.2step 2.1step 3.1step 3.2F3F4algebra∎

[Strong continuity] Let $W_{m} \to W_{0}$ in $M_{2} (C)$ . By the telescoping identity of step 3.2 with [F3], $∥ W_{m}^{\otimes n} - W_{0}^{\otimes n} ∥ \leq n \max (∥ W_{m} ∥, ∥ W_{0} ∥)^{n - 1} ∥ W_{m} - W_{0} ∥$ , and $\max (∥ W_{m} ∥, ∥ W_{0} ∥)$ is bounded because $∥ W_{m} ∥ \to ∥ W_{0} ∥$ by [F4] and convergence of the entries; hence $∥ W_{m}^{\otimes n} - W_{0}^{\otimes n} ∥ \to 0$ . Restricting the operators to the invariant subspace $S_{n}$ can only decrease the operator norm, so $∥ U_{n} (W_{m}) - U_{n} (W_{0}) ∥ \to 0$ , proving (f); in particular $U_{n}$ is strongly continuous on $S U (2)$ , since norm continuity implies strong continuity by [F4]. This completes (a)-(f).

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Finite spin-block decomposition

Statement

Let $V$ be a nonzero finite-dimensional complex inner product space, with inner product linear in the first argument (Real and complex inner product spaces, with the inner product linear in the first argument, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis), and let $J_{x}, J_{y}, J_{z} \in End (V)$ be self-adjoint operators satisfying (Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space)

$[J_{x}, J_{y}] = i J_{z}, [J_{y}, J_{z}] = i J_{x}, [J_{z}, J_{x}] = i J_{y} .$

Put $J_{+} : = J_{x} + i J_{y}$ , $J_{-} : = J_{x} - i J_{y}$ and $J^{2} : = J_{x}^{2} + J_{y}^{2} + J_{z}^{2}$ . Then:

(a) $J_{+}^{*} = J_{-}$ , $J_{-}^{*} = J_{+}$ , $[J_{z}, J_{\pm}] = \pm J_{\pm}$ , $[J_{+}, J_{-}] = 2 J_{z}$ , $J^{2} = J_{-} J_{+} + J_{z}^{2} + J_{z} = J_{+} J_{-} + J_{z}^{2} - J_{z}$ , and $J^{2}$ is self-adjoint, commutes with $J_{x}, J_{y}, J_{z}, J_{\pm}$ , and satisfies $⟨ J^{2} v, v ⟩ = ∥ J_{x} v ∥^{2} + ∥ J_{y} v ∥^{2} + ∥ J_{z} v ∥^{2} \geq 0$ for every $v \in V$ ;

(b) there are an integer $r \geq 1$ , numbers $j_{1}, \dots, j_{r} \in \frac{1}{2} N = {0, \frac{1}{2}, 1, \dots}$ with $\sum_{ν = 1}^{r} (2 j_{ν} + 1) = \dim V$ , and an orthonormal basis $(v_{ν, k})$ , $ν \in {1, \dots, r}$ , $k \in {0, 1, \dots, 2 j_{ν}}$ , of $V$ such that

$J_{z} v_{ν, k} = (j_{ν} - k) v_{ν, k}, J_{+} v_{ν, k} = \sqrt{k (2 j_{ν} - k + 1)} v_{ν, k - 1}, J_{-} v_{ν, k} = \sqrt{(k + 1) (2 j_{ν} - k)} v_{ν, k + 1},$

$J^{2} v_{ν, k} = j_{ν} (j_{ν} + 1) v_{ν, k}, J_{x} v_{ν, k} = \frac{1}{2} (J_{+} + J_{-}) v_{ν, k}, J_{y} v_{ν, k} = \frac{1}{2 i} (J_{+} - J_{-}) v_{ν, k},$

with $v_{ν, - 1} : = 0$ and $v_{ν, 2 j_{ν} + 1} : = 0$ ; each summand $V_{ν} : = span {v_{ν, 0}, \dots, v_{ν, 2 j_{ν}}}$ is $J_{x}, J_{y}, J_{z}$ -invariant, the decomposition $V = ⨁_{ν = 1}^{r} V_{ν}$ is orthogonal, and the map $v_{ν, k} \mapsto e_{j_{ν} - k}$ is a unitary equivalence of $(J_{x}, J_{y}, J_{z})$ on $V_{ν}$ with the spin- $j_{ν}$ matrices of Finite spin matrices;

(c) let $λ$ be the largest eigenvalue of $J_{z}$ (it exists and is real), and let $u \neq 0$ satisfy $J_{z} u = λ u$ . Then $λ \geq 0$ , $J_{+} u = 0$ , and with $w_{k} : = J_{-}^{k} u$ one has $w_{k} \neq 0$ for $0 \leq k \leq 2 λ$ , $w_{k} = 0$ for $k = 2 λ + 1$ , $2 λ \in N$ , and $∥ w_{k} ∥^{2} = k! (2 λ) (2 λ - 1) \dots (2 λ - k + 1) ∥ u ∥^{2}$ for $0 \leq k \leq 2 λ + 1$ ; the vectors $w_{k} / ∥ w_{k} ∥$ , $0 \leq k \leq 2 λ$ , form the block of (b) with $j = λ$ ; consequently $λ \in \frac{1}{2} N$ and the largest eigenvalue of $J_{z}$ on each block is one of the numbers $j_{1}, \dots, j_{r}$ ;

(d) for every $M \in \frac{1}{2} N$ the number of blocks with $j_{ν} \in M + Z$ and $j_{ν} \geq M$ equals $\dim \ker (J_{z} - M I)$ , and the number of blocks with $j_{ν} = M$ equals $\dim \ker (J_{z} - M I) - \dim \ker (J_{z} - (M + 1) I)$ ; in particular the multiset ${j_{1}, \dots, j_{r}}$ is determined by the pair $(V, J_{z})$ together with the relations, and the spectrum of $J_{z}$ is the multiset union of the numbers $j_{ν} - k$ , $0 \leq k \leq 2 j_{ν}$ (a block with parameter $j$ contributes its weights only to the lattice coset $j + Z$ ).

No physical interpretation of the operators as angular momentum, spin or observables is asserted; the theorem is a classification of finite-dimensional triples satisfying the displayed commutation relations.

Facts & Assumptions

Given: a nonzero finite-dimensional complex inner product space $V$ ; self-adjoint $J_{x}, J_{y}, J_{z}$ satisfying the three commutation relations; the derived operators $J_{\pm}$ and $J^{2}$ .

[F1]

Adjoints are conjugate-linear, $(T^{*})^{*} = T$ , $(T + T^{'})^{*} = T^{*} + T^{'}$ , $(T T^{'})^{*} = T^{' *} T^{*}$ , and a restriction of a self-adjoint operator to a closed invariant subspace is self-adjoint; self-adjoint operators are normal (The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ , Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space).

[F2]

Every self-adjoint endomorphism of a nonzero finite-dimensional complex inner product space has a real eigenvalue and a unit eigenvector, so its set of eigenvalues is a nonempty finite subset of $R$ and has a largest element (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

[F3]

For a self-adjoint $T$ and eigenvectors $u, v$ with $T u = λ u$ , $T v = μ v$ and $λ \neq μ$ one has $⟨ u, v ⟩ = 0$ : $(λ - μ) ⟨ u, v ⟩ = ⟨ T u, v ⟩ - ⟨ u, T v ⟩ = 0$ ; the orthogonal complement of an invariant subspace is invariant for a self-adjoint operator; and $⟨ T^{2} v, v ⟩ = ∥ T v ∥^{2}$ for self-adjoint $T$ (Orthogonality and the orthogonal complement, Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space, Real and complex inner product spaces, with the inner product linear in the first argument).

[F4]

The norm satisfies $∥ v ∥^{2} = ⟨ v, v ⟩ \geq 0$ , $∥ v ∥ = 0$ iff $v = 0$ , and $⟨ T v, w ⟩ = ⟨ v, T^{*} w ⟩$ for all $v, w$ (The induced length is a norm, The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ ).

[F5]

An orthogonal decomposition $V = W \oplus W^{⊥}$ has $\dim V = \dim W + \dim W^{⊥}$ , and a finite orthogonal family of nonzero vectors is linearly independent (Orthogonality and the orthogonal complement, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis, Linear subspace of a vector space).

[F6]

The matrices of Finite spin matrices for $s = j$ act on an orthonormal basis $e_{m}$ , $- j \leq m \leq j$ , by the formulas displayed in the Statement with $s = j$ , are self-adjoint, satisfy the same commutation relations, and have $J^{2} = j (j + 1) I$ (Finite spin matrices).

[F7]

For integers $0 \leq k \leq m$ the products $k! (2 λ) (2 λ - 1) \dots (2 λ - k + 1)$ vanish if and only if $2 λ$ is a nonnegative integer strictly less than $k$ ; sums of dimensions over a finite orthogonal direct sum add (The natural numbers $N$ (von Neumann), A finite sum in a commutative monoid indexed by an arbitrary finite set, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis).

Proof

1.1F1F3F4algebra

[The derived algebra] Taking adjoints of $J_{\pm} = J_{x} \pm i J_{y}$ with $J_{x}^{*} = J_{x}$ , $J_{y}^{*} = J_{y}$ and the conjugation laws gives $J_{+}^{*} = J_{-}$ and $J_{-}^{*} = J_{+}$ . The commutation relations give $[J_{z}, J_{\pm}] = [J_{z}, J_{x}] \pm i [J_{z}, J_{y}] = i J_{y} \pm i (- i J_{x}) = \pm (J_{x} \pm i J_{y}) = \pm J_{\pm}$ and $[J_{+}, J_{-}] = [J_{x} + i J_{y}, J_{x} - i J_{y}] = - i [J_{x}, J_{y}] + i [J_{y}, J_{x}] = 2 J_{z}$ . Direct expansion gives $J_{-} J_{+} = J_{x}^{2} + J_{y}^{2} - J_{z} = J^{2} - J_{z}^{2} - J_{z}$ and $J_{+} J_{-} = J_{x}^{2} + J_{y}^{2} + J_{z} = J^{2} - J_{z}^{2} + J_{z}$ , hence $J^{2} = J_{-} J_{+} + J_{z}^{2} + J_{z} = J_{+} J_{-} + J_{z}^{2} - J_{z}$ . Since $J_{\pm}$ and $J^{2}$ are polynomials in the self-adjoint $J_{i}$ , [F1] gives $J^{2 *} = J^{2}$ . Finally $[J^{2}, J_{x}] = [J_{y}^{2}, J_{x}] + [J_{z}^{2}, J_{x}] = J_{y} [J_{y}, J_{x}] + [J_{y}, J_{x}] J_{y} + J_{z} [J_{z}, J_{x}] + [J_{z}, J_{x}] J_{z} = - i (J_{y} J_{z} + J_{z} J_{y}) + i (J_{z} J_{y} + J_{y} J_{z}) = 0$ , and cyclically $[J^{2}, J_{y}] = [J^{2}, J_{z}] = 0$ , so $J^{2}$ commutes with $J_{x}, J_{y}, J_{z}$ and hence with $J_{\pm}$ ; and $⟨ J^{2} v, v ⟩ = ⟨ J_{x}^{2} v, v ⟩ + ⟨ J_{y}^{2} v, v ⟩ + ⟨ J_{z}^{2} v, v ⟩ = ∥ J_{x} v ∥^{2} + ∥ J_{y} v ∥^{2} + ∥ J_{z} v ∥^{2} \geq 0$ by [F3], [F4]. This proves (a).

2.1step 1.1F2F3F4algebra

[A highest vector] By [F2] $J_{z}$ has a largest eigenvalue $λ \in R$ and a unit eigenvector $u$ with $J_{z} u = λ u$ , where $u \neq 0$ because $∥ u ∥ = 1$ . From $[J_{z}, J_{+}] = J_{+}$ of step 1.1, $J_{z} (J_{+} u) = J_{+} J_{z} u + J_{+} u = (λ + 1) J_{+} u$ , so $J_{+} u$ is either $0$ or an eigenvector of $J_{z}$ with eigenvalue $λ + 1$ ; maximality of $λ$ forces $J_{+} u = 0$ . Applying step 1.1 to $u$ gives $⟨ J^{2} u, u ⟩ = ⟨ (J_{+} J_{-} + J_{z}^{2} - J_{z}) u, u ⟩ = ⟨ J_{+} J_{-} u, u ⟩ + λ^{2} - λ$ , while $⟨ J^{2} u, u ⟩ \geq λ^{2}$ by (a); equivalently, since $J_{+} u = 0$ , $J_{+} J_{-} u = J_{+} J_{-} u - J_{-} J_{+} u = [J_{+}, J_{-}] u = 2 J_{z} u = 2 λ u$ and therefore $⟨ J^{2} u, u ⟩ = 2 λ + λ^{2} - λ = λ (λ + 1) \geq ∥ J_{z} u ∥^{2} = λ^{2}$ , so $λ \geq 0$ .

3.1step 1.1step 2.1F3F4F5algebra

[The string and its length] Define $w_{k} : = J_{-}^{k} u$ for $k \in N$ , so $w_{0} = u$ . Inductively, $J_{z} w_{k} = (J_{-} J_{z} - J_{-}) J_{-}^{k - 1} u = (λ - k) w_{k}$ by step 1.1, and using $J_{+} J_{-} = J^{2} - J_{z}^{2} + J_{z}$ , the fact that $J^{2}$ commutes with $J_{-}$ and $J^{2} u = λ (λ + 1) u$ from step 2.1, one computes $J_{+} w_{k} = J_{+} J_{-} w_{k - 1} = (J^{2} - J_{z}^{2} + J_{z}) w_{k - 1} = [λ (λ + 1) - (λ - k + 1)^{2} + (λ - k + 1)] w_{k - 1} = k (2 λ - k + 1) w_{k - 1}$ for $k \geq 1$ . Pairing with $w_{k - 1}$ and using [F4] gives the recursion $∥ w_{k} ∥^{2} = k (2 λ - k + 1) ∥ w_{k - 1} ∥^{2}$ , hence $∥ w_{k} ∥^{2} = k! (2 λ) (2 λ - 1) \dots (2 λ - k + 1) ∥ u ∥^{2}$ by induction, the case $k = 0$ being the empty product. If $w_{k} \neq 0$ for every $k$ , the vectors $w_{k}$ are nonzero eigenvectors of $J_{z}$ for the pairwise distinct eigenvalues $λ, λ - 1, λ - 2, \dots$ ; any $\dim V + 1$ of them form an orthogonal family of nonzero vectors, hence are linearly independent by [F3] and [F5], which is impossible in the finite-dimensional space $V$ . So there is a least $m \geq 1$ with $w_{m} = 0$ , and then $0 = ∥ w_{m} ∥^{2} = m (2 λ - m + 1) ∥ w_{m - 1} ∥^{2}$ forces $2 λ = m - 1 \in N$ .

4.1step 1.1step 2.1step 3.1F3F6algebra

[The normalized block] Put $2 j : = m - 1 = 2 λ$ with $j = λ \in \frac{1}{2} N$ , and for $0 \leq k \leq 2 j$ set $v_{k} : = w_{k} / ∥ w_{k} ∥$ , which is defined because $w_{k} \neq 0$ for $k < m = 2 j + 1$ and unit by construction. The eigenvectors $w_{k}$ have distinct eigenvalues $j, j - 1, \dots, - j$ of the self-adjoint $J_{z}$ , so they are pairwise orthogonal by [F3]; hence $v_{0}, \dots, v_{2 j}$ is an orthonormal family. The action formulas follow from steps 1.1, 2.1 and 3.1: $J_{z} v_{k} = (j - k) v_{k}$ ; $J_{-} v_{k} = ∥ w_{k + 1} ∥ / ∥ w_{k} ∥ v_{k + 1} = \sqrt{(k + 1) (2 j - k)} v_{k + 1}$ for $k < 2 j$ and $J_{-} v_{2 j} = 0$ ; $J_{+} v_{k} = k (2 j - k + 1) ∥ w_{k - 1} ∥ / ∥ w_{k} ∥ v_{k - 1} = \sqrt{k (2 j - k + 1)} v_{k - 1}$ for $k \geq 1$ and $J_{+} v_{0} = 0$ ; and $J^{2} v_{k} = λ (λ + 1) v_{k} = j (j + 1) v_{k}$ because $J^{2}$ commutes with $J_{-}$ . The span $V_{λ} : = span {v_{0}, \dots, v_{2 j}}$ is therefore invariant under $J_{\pm}$ and $J_{z}$ , hence under $J_{x}, J_{y}, J_{z}$ , and [F6] identifies the action with the spin- $j$ model, giving $J_{x} v_{k} = \frac{1}{2} (J_{+} + J_{-}) v_{k}$ and $J_{y} v_{k} = \frac{1}{2 i} (J_{+} - J_{-}) v_{k}$ .

5.1step 4.1F1F3F5algebra

[The invariant complement] Let $W : = V_{λ}$ . If $w \in W^{⊥}$ and $w^{'} \in W$ , then for each $i$ the invariance of $W$ and self-adjointness of $J_{i}$ give $⟨ J_{i} w, w^{'} ⟩ = ⟨ w, J_{i} w^{'} ⟩ = 0$ , so $J_{i} w \in W^{⊥}$ ; thus $W^{⊥}$ is invariant under $J_{x}, J_{y}, J_{z}$ . The restriction of each $J_{i}$ to $W^{⊥}$ is self-adjoint by [F1], the commutation relations hold there, and $V = W \oplus W^{⊥}$ is an orthogonal direct sum with $\dim V = \dim W + \dim W^{⊥}$ by [F5]; if $W^{⊥} \neq {0}$ , the restricted triple again satisfies the hypotheses of the Statement.

6.1step 2.1step 3.1step 4.1step 5.1F5F6F7algebra

[Exhaustion and the decomposition] Iterate step 5.1: as long as the orthogonal complement of the blocks already produced is nonzero, step 2.1 supplies a largest eigenvalue and step 4.1 a further nonzero invariant block of dimension $2 j_{ν} + 1 \geq 1$ , orthogonally split off, decreasing the dimension of the complement by $2 j_{ν} + 1$ . Since the dimension is a nonnegative integer, the process terminates after some finite number $r \geq 1$ of blocks with complement ${0}$ , yielding an orthogonal decomposition $V = ⨁_{ν = 1}^{r} V_{ν}$ and an orthonormal basis $(v_{ν, k})$ as in (b); the basis is orthonormal as a union of the block bases because the blocks are pairwise orthogonal, and $\sum_{ν} (2 j_{ν} + 1) = \dim V$ by additivity of dimensions in orthogonal direct sums [F5], [F7]. Each block action is the spin- $j_{ν}$ action by step 4.1 and [F6], which proves (b); step 3.1 applied to a block identifies the largest eigenvalue of $J_{z}$ on that block as $j_{ν}$ , completing (c) except for $λ \geq 0$ , proved in step 2.1.

7.1step 6.1algebra∎

[Multiplicity count] Let $M \in \frac{1}{2} N$ and consider the eigenvalue- $M$ eigenspace of $J_{z}$ . A block with parameter $j$ contributes the basis vector with $k = j - M$ exactly when $j \in M + Z$ and $0 \leq M \leq j$ , and contributes nothing in all other cases (its weights are $j, j - 1, \dots, - j$ , all in the coset $j + Z$ ); hence $\dim \ker (J_{z} - M I) = # {ν : j_{ν} \in M + Z, j_{ν} \geq M}$ by the orthonormal basis of step 6.1. Applying this at $M$ and at $M + 1$ , both in the same coset $M + Z$ , and subtracting gives $# {ν : j_{ν} = M} = \dim \ker (J_{z} - M I) - \dim \ker (J_{z} - (M + 1) I)$ . The multiset ${j_{1}, \dots, j_{r}}$ is therefore determined by $J_{z}$ (using the fixed relations to know that the decomposition exists), and the spectrum statement follows by collecting the eigenvalues $j_{ν} - k$ of the blocks; the coset qualifier in (d) excludes nothing, since blocks outside $M + Z$ contribute no vector to either eigenspace. This proves (d).

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Tensor spin composition

Statement

Let $s, t \in \frac{1}{2} N$ , write $L_{s} : = {- s, - s + 1, \dots, s}$ and $L_{t} : = {- t, - t + 1, \dots, t}$ , and let $(V_{s}, J_{x}^{(s)}, J_{y}^{(s)}, J_{z}^{(s)})$ and $(V_{t}, J_{x}^{(t)}, J_{y}^{(t)}, J_{z}^{(t)})$ be the spin models of Finite spin matrices with orthonormal bases $(e_{m}^{(s)})_{m \in L_{s}}$ and $(e_{n}^{(t)})_{n \in L_{t}}$ and the ladders $J_{\pm}^{(s)} : = J_{x}^{(s)} \pm i J_{y}^{(s)}$ , $J_{\pm}^{(t)} : = J_{x}^{(t)} \pm i J_{y}^{(t)}$ and Casimirs $J^{(s) 2}$ , $J^{(t) 2}$ (The elementary tensors of two bases form the product basis of the tensor product). On the tensor product $V_{s} \otimes V_{t}$ put

$J_{i}^{t o t} : = J_{i}^{(s)} \otimes I_{V_{t}} + I_{V_{s}} \otimes J_{i}^{(t)} (i = x, y, z), J_{\pm}^{t o t} : = J_{x}^{t o t} \pm i J_{y}^{t o t}, J^{t o t 2} : = (J_{x}^{t o t})^{2} + (J_{y}^{t o t})^{2} + (J_{z}^{t o t})^{2} .$

Then:

(a) each $J_{i}^{t o t}$ is self-adjoint, the triple $J_{x}^{t o t}, J_{y}^{t o t}, J_{z}^{t o t}$ satisfies the commutation relations of Finite spin-block decomposition, and $J^{t o t 2} = J^{(s) 2} \otimes I_{V_{t}} + I_{V_{s}} \otimes J^{(t) 2} + 2 \sum_{i} J_{i}^{(s)} \otimes J_{i}^{(t)}$ is self-adjoint and commutes with $J_{x}^{t o t}, J_{y}^{t o t}, J_{z}^{t o t}$ ;

(b) there is an orthogonal decomposition $V_{s} \otimes V_{t} = ⨁_{j = ∣ s - t ∣}^{s + t} W_{j}$ into $J^{t o t}$ -invariant subspaces, one for each $j \in {∣ s - t ∣, ∣ s - t ∣ + 1, \dots, s + t}$ , with $\dim W_{j} = 2 j + 1$ , on each of which $(J_{x}^{t o t}, J_{y}^{t o t}, J_{z}^{t o t})$ is unitarily equivalent to the spin- $j$ model of Finite spin matrices; consequently $\sum_{j = ∣ s - t ∣}^{s + t} (2 j + 1) = (2 s + 1) (2 t + 1)$ ;

(c) for every $M \in \frac{1}{2} Z$ the $M$ -eigenspace of $J_{z}^{t o t}$ has dimension $N (M) : = # {m \in L_{s} : M - m \in L_{t}} = {\begin{cases} \max (0, \min (s, M + t) - \max (- s, M - t) + 1), & M - (s + t) \in Z, \\ 0, & M - (s + t) \notin Z . \end{cases},$ and the identity $(e_{m}^{(s)} \otimes e_{n}^{(t)})_{m, n}$ is an orthonormal basis of $V_{s} \otimes V_{t}$ consisting of eigenvectors of $J_{z}^{t o t}$ with eigenvalues $m + n$ ;

(d) the number of summands with parameter equal to $M$ in the decomposition of (b) is $N (M) - N (M + 1)$ if $M \geq 0$ , and this number equals $1$ when $M - (s + t) \in Z$ and $∣ s - t ∣ \leq M \leq s + t$ , and equals $0$ otherwise; hence the multiset of parameters in (b) is exactly ${∣ s - t ∣, ∣ s - t ∣ + 1, \dots, s + t}$ , each occurring once;

(e) for each $j \in {∣ s - t ∣, \dots, s + t}$ the space of vectors of $J_{z}^{t o t}$ -eigenvalue $j$ orthogonal to all summands $W_{j^{'}}$ with $j^{'} > j$ is one-dimensional; for a unit vector $u_{j}$ in it the vectors $v_{j, k} : = (k! (2 j) (2 j - 1) \dots (2 j - k + 1))^{- 1 / 2} (J_{-}^{t o t})^{k} u_{j} (0 \leq k \leq 2 j)$ form an orthonormal basis of $W_{j}$ with $J_{z}^{t o t} v_{j, k} = (j - k) v_{j, k}$ , $J_{+}^{t o t} v_{j, k} = \sqrt{k (2 j - k + 1)} v_{j, k - 1}$ and $J_{-}^{t o t} v_{j, k} = \sqrt{(k + 1) (2 j - k)} v_{j, k + 1}$ ; the coefficients of $v_{j, k}$ in the product basis are therefore determined by the phase of $u_{j}$ , and choosing that phase fixes the Clebsch-Gordan numbers explicitly; in particular $u_{s + t} = e_{s}^{(s)} \otimes e_{t}^{(t)}$ is a permissible choice.

The theorem is a purely mathematical statement about tensor products of the explicit matrices of Finite spin matrices; it asserts no physical coupling, conservation law or measurement fact.

Facts & Assumptions

Given: $s, t \in \frac{1}{2} N$ ; the spin models $V_{s}, V_{t}$ with their operators and orthonormal bases; the tensor product $V_{s} \otimes V_{t}$ ; the operators $J_{i}^{t o t}$ , $J_{\pm}^{t o t}$ , $J^{t o t 2}$ .

[F1]

By The elementary tensors of two bases form the product basis of the tensor product, the algebraic tensor product $V_{s} \otimes V_{t}$ has the finite basis $e_{m}^{(s)} \otimes e_{n}^{(t)}$ . Give its coordinate vectors the inner product $⟨ \sum c_{m n} e_{m} \otimes e_{n}, \sum d_{m n} e_{m} \otimes e_{n} ⟩ = \sum c_{m n} \overline{d_{m n}}$ . This is positive definite, makes that basis orthonormal, and expansion of two elementary tensors gives $⟨ u \otimes v, u^{'} \otimes v^{'} ⟩ = ⟨ u, u^{'} ⟩ ⟨ v, v^{'} ⟩$ . This is a finite construction with explicitly supplied bases; no completion or countable choice is invoked.

[F2]

Define $S \otimes T$ by its values $S e_{m} \otimes T e_{n}$ on the finite product basis. Bilinear expansion gives $(S \otimes T) (u \otimes v) = S u \otimes T v$ , and hence $(S \otimes T) (S^{'} \otimes T^{'}) = (S S^{'}) \otimes (T T^{'})$ . The inner product of [F1] gives $⟨ (S \otimes T) (u \otimes v), u^{'} \otimes v^{'} ⟩ = ⟨ u \otimes v, (S^{*} \otimes T^{*}) (u^{'} \otimes v^{'}) ⟩$ , proving $(S \otimes T)^{*} = S^{*} \otimes T^{*}$ on the spanning product basis. Also $I \otimes I = I$ , so $[S \otimes I, S^{'} \otimes I] = [S, S^{'}] \otimes I$ , $[I \otimes T, I \otimes T^{'}] = I \otimes [T, T^{'}]$ , and operators acting on different factors commute.

[F3]

On $V_{s}$ and $V_{t}$ the operators are self-adjoint, satisfy $[J_{x}, J_{y}] = i J_{z}$ and its cyclic companions, and have $J^{2} = s (s + 1) I_{V_{s}}$ , respectively $t (t + 1) I_{V_{t}}$ ; the ladder actions on the bases are $J_{\pm}^{(s)} e_{m}^{(s)} = \sqrt{(s \mp m) (s \pm m + 1)} e_{m \pm 1}^{(s)}$ and $J_{z}^{(s)} e_{m}^{(s)} = m e_{m}^{(s)}$ (Finite spin matrices).

[F4]

Every finite-dimensional triple of self-adjoint operators satisfying the commutation relations decomposes orthogonally into spin blocks with $\sum_{ν} (2 j_{ν} + 1) = \dim V$ , the block actions being the spin- $j_{ν}$ actions; the number of blocks with parameter $M \geq 0$ equals $\dim \ker (J_{z} - M I) - \dim \ker (J_{z} - (M + 1) I)$ , and $M$ must lie in $\frac{1}{2} N$ (Finite spin-block decomposition (b), (d)).

[F5]

$\dim (V_{s} \otimes V_{t}) = \dim V_{s} \dim V_{t} = (2 s + 1) (2 t + 1)$ , and dimensions add over orthogonal direct sums (Hilbert tensor construction: inner product, completion and product bases, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis, Orthogonality and the orthogonal complement).

[F6]

For a self-adjoint operator the eigenspaces for distinct eigenvalues are orthogonal, and for a unit vector $w$ and self-adjoint $T$ , $⟨ T w, w ⟩ = ⟨ w, T w ⟩$ is real; $∥ w ∥^{2} = ⟨ w, w ⟩$ (Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space, Orthogonality and the orthogonal complement, The induced length is a norm).

[F7]

Factorials and products: $k! = \prod_{i = 1}^{k} i \geq 1$ for $k \geq 0$ , and a finite product of positive numbers is positive (The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ , The natural numbers $N$ (von Neumann)).

Proof

1.1F2F3algebra

[Self-adjointness and the algebra] For $i = x, y, z$ , [F2] and [F3] give $(J_{i}^{(s)} \otimes I)^{*} = J_{i}^{(s)} \otimes I$ and $(I \otimes J_{i}^{(t)})^{*} = I \otimes J_{i}^{(t)}$ , so $J_{i}^{t o t}$ is self-adjoint. Adding the two-sided commutators, $[J_{x}^{t o t}, J_{y}^{t o t}] = [J_{x}^{(s)}, J_{y}^{(s)}] \otimes I + I \otimes [J_{x}^{(t)}, J_{y}^{(t)}] = i J_{z}^{(s)} \otimes I + i I \otimes J_{z}^{(t)} = i J_{z}^{t o t}$ , and cyclically; so the triple satisfies the commutation relations.

1.2F1F3algebra

[Weights] By [F1] and [F3], $J_{z}^{t o t} (e_{m}^{(s)} \otimes e_{n}^{(t)}) = (m + n) e_{m}^{(s)} \otimes e_{n}^{(t)}$ , so the elementary tensors form an orthonormal eigenbasis of $J_{z}^{t o t}$ with eigenvalues $m + n$ , and the number of basis vectors of eigenvalue $M$ is the number of pairs $(m, n) \in L_{s} \times L_{t}$ with $m + n = M$ , which is $# {m \in L_{s} : M - m \in L_{t}} = N (M)$ ; all sums $m + n$ lie in $s + t + Z$ , so $N (M) = 0$ off that lattice. On the lattice, the interval endpoints lie in the same unit-spaced lattice $L_{s}$ ; the displayed closed form follows by counting the interval $[\max (- s, M - t), \min (s, M + t)]$ , which is empty exactly when the maximum exceeds the minimum, and otherwise contributes one element per unit lattice step.

2.1step 1.1F2F3algebra

[The Casimir] Expanding $J^{t o t 2} = \sum_{i} (J_{i}^{(s)} \otimes I + I \otimes J_{i}^{(t)})^{2}$ gives $J^{(s) 2} \otimes I + I \otimes J^{(t) 2} + 2 C$ with $C : = \sum_{i} J_{i}^{(s)} \otimes J_{i}^{(t)}$ , which is self-adjoint by [F2] and [F3]; moreover $[J^{t o t 2}, J_{i}^{t o t}] = [J^{(s) 2}, J_{i}^{(s)}] \otimes I + I \otimes [J^{(t) 2}, J_{i}^{(t)}] + 2 [C, J_{i}^{(s)} \otimes I + I \otimes J_{i}^{(t)}]$ vanishes because the first two commutators are $0$ by [F3] and $2 [C, J_{i}^{(s)} \otimes I + I \otimes J_{i}^{(t)}] = 2 i \sum_{k, l} ε_{k i l} (J_{l}^{(s)} \otimes J_{k}^{(t)} + J_{k}^{(s)} \otimes J_{l}^{(t)}) = 0$ , the parenthesised tensor being symmetric in $k, l$ while $ε_{k i l}$ is antisymmetric.

2.2step 1.2F1algebra

[The parity-qualified difference] Restrict now to $M \geq 0$ . If $M - (s + t) \notin Z$ , neither $M$ nor $M + 1$ is a weight and both counts are zero by step 1.2. On the compatible lattice, exchange the factors if necessary to suppose $s \geq t$ . Counting the interval in step 1.2 gives $N (M) = \min (2 t + 1, s + t - M + 1)$ for $0 \leq M \leq s + t$ , and $N (M) = 0$ for $M > s + t$ . Therefore $N (M) - N (M + 1) = 0$ on the plateau $0 \leq M < s - t$ , equals one for $s - t \leq M \leq s + t$ , and is zero above $s + t$ , including the upper endpoint where $N (s + t) = 1$ and $N (s + t + 1) = 0$ . Restoring factor symmetry gives precisely the admissible lattice points in $[∣ s - t ∣, s + t]$ .

3.1step 1.1step 2.1step 2.2F4F5algebra

[Existence and parameter list] By steps 1.1, 2.1 and [F4] the triple $(J_{x}^{t o t}, J_{y}^{t o t}, J_{z}^{t o t})$ decomposes $V_{s} \otimes V_{t}$ into an orthogonal direct sum of spin blocks with parameters $j_{ν} \in \frac{1}{2} N$ , and by F4 the number of blocks with parameter $M$ equals $\dim \ker (J_{z}^{t o t} - M I) - \dim \ker (J_{z}^{t o t} - (M + 1) I) = N (M) - N (M + 1)$ for $M \geq 0$ , which by step 2.2 is $1$ precisely for $M \in {∣ s - t ∣, ∣ s - t ∣ + 1, \dots, s + t}$ and $0$ otherwise. Hence the block parameters are exactly the numbers $∣ s - t ∣, ∣ s - t ∣ + 1, \dots, s + t$ , each once; each block has dimension $2 j + 1$ and its operators are unitarily equivalent to the spin- $j$ model by F4. The dimension identity $\sum_{j = ∣ s - t ∣}^{s + t} (2 j + 1) = (2 s + 1) (2 t + 1)$ follows from $\sum_{ν} (2 j_{ν} + 1) = \dim (V_{s} \otimes V_{t})$ and [F5]. This proves (b) and (d).

4.1step 3.1F1F3F4F6F7algebra∎

[The constructive bases and Clebsch-Gordan phase] Fix $j$ in the parameter list and let $E_{j}$ be the space of $J_{z}^{t o t}$ -eigenvectors of eigenvalue $j$ orthogonal to all blocks $W_{j^{'}}$ with $j^{'} > j$ . Counting as in step 3.1, $\dim \ker (J_{z}^{t o t} - j I) = # {ν : j_{ν} - j \in Z, j_{ν} \geq j}$ (a block with parameter $j_{ν} \geq j \geq 0$ and $j_{ν} - j \in Z$ contributes exactly one eigenvector of eigenvalue $j$ ), and the higher blocks with parameters $j^{'} > j$ contribute one each, so $\dim E_{j} = # {ν : j_{ν} - j \in Z, j_{ν} \geq j} - # {ν : j_{ν} - j \in Z, j_{ν} > j} = 1$ . Pick a unit $u_{j} \in E_{j}$ , which is determined up to multiplication by a phase; applying the ladder identities of F4 in the block normal form, the vectors $v_{j, k}$ of the Statement are well defined and form the orthonormal basis of $W_{j}$ , with the stated actions, because $∥ (J_{-}^{t o t})^{k} u_{j} ∥^{2} = k! (2 j) (2 j - 1) \dots (2 j - k + 1)$ by [F7] and the recursion of F4; expanding $v_{j, k}$ in the product basis [F1] exhibits the Clebsch-Gordan coefficients, and replacing $u_{j}$ by $e^{i θ} u_{j}$ multiplies them all by the same phase. For $j = s + t$ , $e_{s}^{(s)} \otimes e_{t}^{(t)}$ has weight $s + t$ and is annihilated by $J_{+}^{t o t}$ because $J_{+}^{(s)} e_{s}^{(s)} = J_{+}^{(t)} e_{t}^{(t)} = 0$ by [F3]; this is the top vector of the unique block of parameter $s + t$ , so it is a permissible choice. This proves (e) and completes the proof.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Tsirelson bound

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $H, K$ be complex Hilbert spaces (Hilbert space) and let $A_{0}, A_{1} \in B (H)$ , $B_{0}, B_{1} \in B (K)$ be bounded self-adjoint operators (A bounded linear operator between normed spaces, Self-adjoint, positive, unitary and normal operators) with $A_{i}^{2} = I_{H}$ and $B_{j}^{2} = I_{K}$ (The Hilbert-space adjoint of a bounded operator). Let $H \hat{\otimes} K$ be the completed tensor product of Hilbert tensor construction: inner product, completion and product bases and put

$C : = A_{0} \otimes (B_{0} + B_{1}) + A_{1} \otimes (B_{0} - B_{1}) \in B (H \hat{\otimes} K) .$

Then:

(a) $C$ is self-adjoint and $C^{2} = 4 I_{H \hat{\otimes} K} - [A_{0}, A_{1}] \otimes [B_{0}, B_{1}];$

(b) $∥ C^{2} ∥ \leq 8$ and $∥ C ∥ \leq 2 \sqrt{2}$ in the operator norm (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Hilbert-adjoint identities);

(c) for all $ψ, ψ^{'} \in H \hat{\otimes} K$ , $∣ ⟨ C ψ, ψ^{'} ⟩ ∣ \leq 2 \sqrt{2} ∥ ψ ∥ ∥ ψ^{'} ∥$ (this is the Tsirelson bound for the stated pair of binary sharp models); consequently the absolute value of the CHSH combination of the four expectations $⟨ (A_{i} \otimes B_{j}) ψ, ψ ⟩$ of a unit vector $ψ$ is at most $2 \sqrt{2}$ ;

(d) the bound is attained on $H = K = C^{2}$ : let $σ_{x}, σ_{y}, σ_{z}$ be the Pauli matrices of Finite spin matrices, let $Ψ : = \frac{1}{\sqrt{2}} (e_{+} \otimes e_{-} - e_{-} \otimes e_{+})$ with the orthonormal basis $(e_{+}, e_{-})$ of $C^{2}$ , put $σ_{a} : = a_{x} σ_{x} + a_{y} σ_{y} + a_{z} σ_{z}$ for a unit vector $a \in R^{3}$ , and take $A_{0} : = σ_{z}$ , $A_{1} : = σ_{x}$ , $B_{0} : = σ_{(z + x) / \sqrt{2}}$ , $B_{1} : = σ_{(z - x) / \sqrt{2}}$ ; then $⟨ σ_{a} \otimes σ_{b} Ψ, Ψ ⟩ = - a \cdot b$ for all unit $a, b$ , and $⟨ C Ψ, Ψ ⟩ = - 2 \sqrt{2}$ , so $∣ ⟨ C Ψ, Ψ ⟩ ∣ = 2 \sqrt{2}$ ;

(e) a comparison with the stated local model: if in addition $A_{0}, A_{1}, B_{0}, B_{1}$ are four $\pm 1$ -valued random variables on one probability space, then the absolute value of their CHSH combination is at most $2$ (The CHSH bound and the finite-sampling law for independent trials (a)); this is a statement about that explicitly specified common-outcome model, not about all theories, and it is not compared here with any reported experiment.

The theorem is a mathematical statement about bounded operators; no measurement, preparation or empirical claim is asserted.

Facts & Assumptions

Given: Countable Choice; complex Hilbert spaces $H, K$ ; self-adjoint involutions $A_{0}, A_{1} \in B (H)$ , $B_{0}, B_{1} \in B (K)$ ; the completed tensor product $H \hat{\otimes} K$ ; the operator $C$ .

[F1]

The completed tensor product is a complex Hilbert space containing the algebraic tensor product densely, and the elementary tensors $u \otimes v$ have inner products $⟨ u \otimes v, u^{'} \otimes v^{'} ⟩ = ⟨ u, u^{'} ⟩ ⟨ v, v^{'} ⟩$ ; $∥ u \otimes v ∥ = ∥ u ∥ ∥ v ∥$ (Hilbert tensor construction: inner product, completion and product bases, Hilbert space).

[F2]

For bounded $S, S^{'} \in B (H)$ , $T, T^{'} \in B (K)$ : $(S \otimes T) (S^{'} \otimes T^{'}) = (S S^{'}) \otimes (T T^{'})$ , $S \otimes I_{K}$ and $I_{H} \otimes T$ commute, $(S \otimes T)^{*} = S^{*} \otimes T^{*}$ , and $∥ S \otimes T ∥ \leq ∥ S ∥ ∥ T ∥$ ; the identity operator on $H \hat{\otimes} K$ is $I_{H} \otimes I_{K}$ (Bounded tensor maps: extension, norm estimate, adjoint and commutation).

[F3]

For bounded operators on Hilbert spaces, the adjoint is additive and conjugate-linear, $(T S)^{*} = S^{*} T^{*}$ , $T^{* *} = T$ , $∥ T^{*} ∥ = ∥ T ∥$ and $∥ T^{*} T ∥ = ∥ T ∥^{2}$ under Countable Choice (Hilbert-adjoint identities). From The operator norm as the least bound and as the unit-sphere or unit-ball supremum, $∥ T S x ∥ \leq ∥ T ∥ ∥ S x ∥ \leq ∥ T ∥ ∥ S ∥ ∥ x ∥$ ; taking the supremum over the unit ball gives $∥ T S ∥ \leq ∥ T ∥ ∥ S ∥$ . Likewise the vector triangle inequality gives $∥ T + S ∥ \leq ∥ T ∥ + ∥ S ∥$ . These estimates include zero spaces.

[F4]

For a bounded operator $T$ and all $ψ, ψ^{'}$ , $∣ ⟨ T ψ, ψ^{'} ⟩ ∣ \leq ∥ T ∥ ∥ ψ ∥ ∥ ψ^{'} ∥$ by Cauchy–Schwarz and the norm definition (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors, The operator norm as the least bound and as the unit-sphere or unit-ball supremum). A self-adjoint involution satisfies $T^{*} T = T T^{*} = I$ , so it is unitary in the Hilbert-space sense (Self-adjoint, positive, unitary and normal operators).

[F5]

The Pauli matrices are self-adjoint, $σ_{i}^{2} = I_{2}$ , $σ_{x} σ_{y} = i σ_{z}$ and cyclically, and $σ_{a}$ for a unit vector $a$ is self-adjoint with $σ_{a}^{2} = I_{2}$ , because $σ_{a}^{2} = \sum_{i, j} a_{i} a_{j} σ_{i} σ_{j} = \sum_{i} a_{i}^{2} I_{2} + \sum_{i \neq j} a_{i} a_{j} σ_{i} σ_{j} = ∥ a ∥^{2} I_{2}$ by the anticommutation relations $σ_{i} σ_{j} + σ_{j} σ_{i} = 2 δ_{i j} I_{2}$ (Finite spin matrices).

[F6]

If $A_{0}, A_{1}, B_{0}, B_{1} : Ω \to {- 1, 1}$ are random variables on a probability space, then $∣ E [A_{0} B_{0}] + E [A_{0} B_{1}] + E [A_{1} B_{0}] - E [A_{1} B_{1}] ∣ \leq 2$ , with no choice principle used (The CHSH bound and the finite-sampling law for independent trials (a)).

[F7]

A bounded self-adjoint involution satisfies $∥ T x ∥^{2} = ⟨ T x, T x ⟩ = ⟨ x, T^{*} T x ⟩ = ∥ x ∥^{2}$ . Thus $∥ A_{i} ∥, ∥ B_{j} ∥ \leq 1$ (with equality on nonzero factor spaces), and the estimate proved in [F3] gives $∥ A_{0} A_{1} ∥ \leq ∥ A_{0} ∥ ∥ A_{1} ∥ \leq 1$ (The Hilbert-space adjoint of a bounded operator, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

Proof

1.1F1F2F3F7algebra

[Self-adjointness and the square identity] By [F2] and [F3], $(S \otimes T)^{*} = S^{*} \otimes T^{*}$ for the four tensor products entering $C$ , so every summand is self-adjoint and $C^{*} = C$ . Expanding the square with [F2] and [F7] (all operators below are bounded, so products may be reassociated), $C^{2} = (A_{0} \otimes (B_{0} + B_{1}))^{2} + (A_{1} \otimes (B_{0} - B_{1}))^{2} + (A_{0} \otimes (B_{0} + B_{1})) (A_{1} \otimes (B_{0} - B_{1})) + (A_{1} \otimes (B_{0} - B_{1})) (A_{0} \otimes (B_{0} + B_{1}))$ ; using $A_{i}^{2} = I$ and $B_{j}^{2} = I$ together with $(B_{0} + B_{1})^{2} = 2 I + {B_{0}, B_{1}}$ , $(B_{0} - B_{1})^{2} = 2 I - {B_{0}, B_{1}}$ , $(B_{0} + B_{1}) (B_{0} - B_{1}) = [B_{1}, B_{0}]$ and $(B_{0} - B_{1}) (B_{0} + B_{1}) = [B_{0}, B_{1}]$ , the first two summands total $I_{H} \otimes (2 I_{K} + {B_{0}, B_{1}}) + I_{H} \otimes (2 I_{K} - {B_{0}, B_{1}}) = 4 I_{H \hat{\otimes} K}$ , while the cross terms total $- A_{0} A_{1} \otimes [B_{0}, B_{1}] + A_{1} A_{0} \otimes [B_{0}, B_{1}] = - [A_{0}, A_{1}] \otimes [B_{0}, B_{1}]$ ; hence $C^{2} = 4 I - [A_{0}, A_{1}] \otimes [B_{0}, B_{1}]$ .

1.2F1F5algebra

[The singlet correlations] On $C^{2}$ with orthonormal basis $(e_{+}, e_{-})$ define $Ψ : = \frac{1}{\sqrt{2}} (e_{+} \otimes e_{-} - e_{-} \otimes e_{+})$ ; then $∥ Ψ ∥ = 1$ by [F1]. Direct computation with the displayed Pauli matrices gives $σ_{x} e_{+} = e_{-}$ , $σ_{x} e_{-} = e_{+}$ , $σ_{y} e_{+} = i e_{-}$ , $σ_{y} e_{-} = - i e_{+}$ , $σ_{z} e_{+} = e_{+}$ and $σ_{z} e_{-} = - e_{-}$ . Hence $σ_{i} \otimes σ_{i} Ψ = - Ψ$ for $i = x, y, z$ (the two elementary tensors of the difference receive the same sign change), For clarity put $q = (e_{+} \otimes e_{-} + e_{-} \otimes e_{+}) / \sqrt{2}$ and $d_{\pm} = (e_{+} \otimes e_{+} \pm e_{-} \otimes e_{-}) / \sqrt{2}$ . Direct Pauli actions give $(σ_{x} \otimes σ_{y}) Ψ = - i q$ , $(σ_{y} \otimes σ_{x}) Ψ = i q$ , $(σ_{x} \otimes σ_{z}) Ψ = - d_{+}$ , $(σ_{z} \otimes σ_{x}) Ψ = d_{+}$ , $(σ_{y} \otimes σ_{z}) Ψ = i d_{-}$ and $(σ_{z} \otimes σ_{y}) Ψ = - i d_{-}$ . Each of $q, d_{+}, d_{-}$ is orthogonal to $Ψ$ by the elementary product-basis pairings, so every off-diagonal correlation is zero. therefore $⟨ σ_{i} \otimes σ_{j} Ψ, Ψ ⟩ = - δ_{i j}$ , and for unit vectors $a, b$ , $⟨ σ_{a} \otimes σ_{b} Ψ, Ψ ⟩ = \sum_{i, j} a_{i} b_{j} (- δ_{i j}) = - a \cdot b$ , with $σ_{a}, σ_{b}$ self-adjoint involutions by [F5].

2.1step 1.1F2F3F7algebra

[The norm bound] By [F3] and [F7], $∥ [A_{0}, A_{1}] ∥ \leq ∥ A_{0} A_{1} ∥ + ∥ A_{1} A_{0} ∥ \leq 2$ and $∥ [B_{0}, B_{1}] ∥ \leq 2$ , so [F2] gives $∥ [A_{0}, A_{1}] \otimes [B_{0}, B_{1}] ∥ \leq 4$ ; hence $∥ C^{2} ∥ \leq 4 + 4 = 8$ from step 1.1 and the triangle inequality. Since $C$ is self-adjoint by step 1.1, [F3] gives $∥ C ∥^{2} = ∥ C^{*} C ∥ = ∥ C^{2} ∥ \leq 8$ , so $∥ C ∥ \leq 2 \sqrt{2}$ .

2.2step 1.2F5algebra

[Attainment] Take $A_{0} = σ_{z}$ , $A_{1} = σ_{x}$ , $B_{0} = σ_{(z + x) / \sqrt{2}}$ , $B_{1} = σ_{(z - x) / \sqrt{2}}$ , all self-adjoint involutions by [F5]. By step 1.2 the four expectations are $E_{00} = - z \cdot \frac{z + x}{\sqrt{2}} = - \frac{1}{\sqrt{2}}$ , $E_{01} = - z \cdot \frac{z - x}{\sqrt{2}} = - \frac{1}{\sqrt{2}}$ , $E_{10} = - x \cdot \frac{z + x}{\sqrt{2}} = - \frac{1}{\sqrt{2}}$ and $E_{11} = - x \cdot \frac{z - x}{\sqrt{2}} = + \frac{1}{\sqrt{2}}$ , because $x \cdot z = 0$ and $x \cdot x = z \cdot z = 1$ . The CHSH combination of the operator $C$ is $E_{00} + E_{01} + E_{10} - E_{11} = - \frac{4}{\sqrt{2}} = - 2 \sqrt{2}$ , so $∣ ⟨ C Ψ, Ψ ⟩ ∣ = 2 \sqrt{2}$ and the bound (c) is attained.

3.1step 2.1F4algebra

[The quadratic-form bound] For all $ψ, ψ^{'}$ , [F4] and step 2.1 give $∣ ⟨ C ψ, ψ^{'} ⟩ ∣ \leq ∥ C ∥ ∥ ψ ∥ ∥ ψ^{'} ∥ \leq 2 \sqrt{2} ∥ ψ ∥ ∥ ψ^{'} ∥$ ; for a unit vector this is clause (c), and writing $A_{i} \otimes B_{j}$ as self-adjoint operators with the CHSH combination equal to $C$ expresses the four-term combination as $⟨ C ψ, ψ ⟩$ .

4.1F6∎

[Local comparison] Clause (e) is exactly [F6] applied to the four $\pm 1$ -valued random variables of the stated common-outcome model; no relation between that model and the operators of this theorem is asserted.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Legendre Rodrigues equation

Statement

For $l \in N = {0, 1, 2, \dots}$ define the function $P_{l} : R \to R$ by the Rodrigues formula (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set)

$P_{l} (t) : = \frac{1}{2^{l} l!} \frac{d^{l}}{d t^{l}} [(t^{2} - 1)^{l}] .$

Then:

(a) each $P_{l}$ is the polynomial function of the polynomial $p_{l} (t) = \frac{1}{2^{l} l!} \sum_{k : 2 k \geq l}^{l} (- 1)^{l - k} (\binom{l}{k}) \frac{(2 k)!}{(2 k - l)!} t^{2 k - l} \in R [t]$ (so the two expressions agree at every real $t$ , Evaluation and roots of a polynomial in a commutative target ring), $p_{l}$ has degree $l$ with leading coefficient $\frac{(2 l)!}{2^{l} (l!)^{2}} \neq 0$ (Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree);

(b) $P_{l} (1) = 1$ and $P_{l} (- 1) = (- 1)^{l}$ ;

(c) $P_{l}$ satisfies Legendre's differential equation on $R$ : $(1 - t^{2}) P_{l}^{''} (t) - 2 t P_{l}^{'} (t) + l (l + 1) P_{l} (t) = 0 (t \in R);$

(d) $P_{0} = 1$ , $P_{1} (t) = t$ and $P_{2} (t) = \frac{1}{2} (3 t^{2} - 1)$ .

No probability, measure or physical interpretation is attached to $P_{l}$ ; this item constructs the polynomials that later items use as angular functions.

Facts & Assumptions

Given: $l \in N$ ; the function $F_{l} (t) : = (t^{2} - 1)^{l}$ ; the function $P_{l}$ defined by the Rodrigues formula; the polynomial $p_{l}$ displayed in (a).

[F1]

The derivative of $t^{0} = 1$ is $0$ ; for $n \geq 1$ , $\frac{d}{d t} t^{n} = n t^{n - 1}$ . Iterating this formula and the factorial recursion gives $\frac{d^{m}}{d t^{m}} t^{n} = \frac{n!}{(n - m)!} t^{n - m}$ for $0 \leq m \leq n$ while $\frac{d^{m}}{d t^{m}} t^{n} = 0$ for $m > n$ ; derivatives are linear and satisfy the product and power rules (For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set).

[F2]

The general Leibniz rule: for $m$ -times differentiable $f, g$ , $(f g)^{(m)} = \sum_{j = 0}^{m} (\binom{m}{j}) f^{(j)} g^{(m - j)}$ (The general Leibniz rule for the $n$ -th derivative of a product).

[F3]

The binomial coefficients satisfy $(\binom{l}{k}) = \frac{l!}{k! (l - k)!}$ , $(\binom{l}{0}) = (\binom{l}{l}) = 1$ , and $n! = (n - 1)! n$ for $n \geq 1$ , with $0! = 1$ ( $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ , The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ , The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ , The natural numbers $N$ (von Neumann)).

[F4]

Polynomial evaluation is a ring homomorphism $R [t] \to R$ , $p \mapsto p (t)$ , compatible with addition and multiplication, and a nonzero polynomial of degree $d$ has leading coefficient the coefficient of $t^{d}$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, Evaluation and roots of a polynomial in a commutative target ring, Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree).

[F5]

The binomial theorem over a commutative ring: $(x + y)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) x^{k} y^{n - k}$ in every commutative ring and for all elements $x, y$ (The binomial theorem over an arbitrary commutative ring).

Proof

1.1givenF1F3F4F5algebra

[(a), expansion] By [F5] applied in the polynomial ring to $x = t^{2}$ and $y = - 1$ , $F_{l} (t) = \sum_{k = 0}^{l} (\binom{l}{k}) (- 1)^{l - k} t^{2 k}$ as polynomials; differentiating $l$ times termwise with [F1] and [F4] (the sum is finite, so linearity applies), every term with $2 k < l$ differentiates to $0$ and every term with $2 k \geq l$ contributes $(\binom{l}{k}) (- 1)^{l - k} \frac{(2 k)!}{(2 k - l)!} t^{2 k - l}$ ; hence $P_{l}$ agrees with the polynomial $p_{l}$ of the Statement. All exponents $2 k - l$ with $2 k \geq l$ are nonnegative, and the largest is $2 l - l = l$ , attained only at $k = l$ with coefficient $(2 l)! / l! \neq 0$ by [F3]; so $p_{l}$ has degree $l$ and leading coefficient $(2 l)! / (l! \cdot 2^{l} l!) = (2 l)! / (2^{l} (l!)^{2})$ , which is nonzero.

1.2givenF1F2F3algebra

[(b), endpoint values] By the factorisation $F_{l} = (t - 1)^{l} (t + 1)^{l}$ and [F2], the $l$ -th derivative at $t = 1$ is $\sum_{j = 0}^{l} (\binom{l}{j}) \frac{d^{j}}{d t^{j}} (t - 1)^{l} ∣_{t = 1} \cdot \frac{d^{l - j}}{d t^{l - j}} (t + 1)^{l} ∣_{t = 1}$ , and every term with $j < l$ vanishes because $\frac{d^{j}}{d t^{j}} (t - 1)^{l}$ still contains a factor $(t - 1)$ ; the surviving term is $l! \cdot (1 + 1)^{l} = l! 2^{l}$ by [F1], [F3], so $P_{l} (1) = \frac{l! 2^{l}}{2^{l} l!} = 1$ . The same computation at $t = - 1$ gives $l! \cdot (- 1 - 1)^{l} = (- 1)^{l} l! 2^{l}$ , so $P_{l} (- 1) = \frac{(- 1)^{l} l! 2^{l}}{2^{l} l!} = (- 1)^{l}$ .

1.3givenF1F2F3F4algebra

[(c), Legendre's equation] If $l = 0$ , then $P_{0} = 1$ and the equation is immediate. Suppose $l \geq 1$ . Differentiating $F_{l} = (t^{2} - 1)^{l}$ with [F1] gives $F_{l}^{'} = 2 l t (t^{2} - 1)^{l - 1}$ , hence $(t^{2} - 1) F_{l}^{'} = 2 l t F_{l}$ . Apply $\frac{d^{l + 1}}{d t^{l + 1}}$ to both sides using [F2]; on the left the third and higher derivatives of $t^{2} - 1$ vanish, so $(t^{2} - 1) F_{l}^{(l + 2)} + (l + 1) (2 t) F_{l}^{(l + 1)} + (\binom{l + 1}{2}) (2) F_{l}^{(l)}$ , and on the right $2 l (t F_{l}^{(l + 1)} + (l + 1) F_{l}^{(l)})$ , where $(\binom{l + 1}{2}) = \frac{l (l + 1)}{2}$ by [F3]; subtracting gives $(t^{2} - 1) F_{l}^{(l + 2)} + 2 t F_{l}^{(l + 1)} - l (l + 1) F_{l}^{(l)} = 0$ . Since $P_{l} = \frac{1}{2^{l} l!} F_{l}^{(l)}$ is exactly the $l$ -fold derivative, dividing by the nonzero constant and multiplying by $- 1$ yields $(1 - t^{2}) P_{l}^{''} - 2 t P_{l}^{'} + l (l + 1) P_{l} = 0$ at every real $t$ : the displayed polynomial identity holds for the polynomial $p_{l}$ by [F4].

2.1givenF1algebra∎

[(d)] For $l = 0$ , $P_{0} = 1$ ; for $l = 1$ , $P_{1} (t) = \frac{1}{2} \frac{d}{d t} (t^{2} - 1) = t$ ; for $l = 2$ , $P_{2} (t) = \frac{1}{8} \frac{d^{2}}{d t^{2}} (t^{4} - 2 t^{2} + 1) = \frac{1}{8} (12 t^{2} - 4) = \frac{3 t^{2} - 1}{2}$ by [F1].

The identities proved here are polynomial identities in one real variable; the item makes no angular, measure or spectral claim.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

Sphere integrals in angle coordinates

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $S^{2} : = {x \in R^{3} : ∣ x ∣ = 1}$ and let $σ$ be the polar surface measure on $S^{2}$ of The polar surface set function on the unit sphere, a finite Borel measure (Measures on sigma-algebras, The Borel sigma-algebra of a topological space). Let

$Φ : (0, π) \times (0, 2 π) \to S^{2}, Φ (θ, φ) : = (\sin θ \cos φ, \sin θ \sin φ, \cos θ),$

and let $H : = {(x, y, 0) \in R^{3} : x \geq 0}$ denote the closed half-plane, a Lebesgue-null Borel set (Every affine hyperplane of $R^{n}$ , and hence every proper linear subspace, is Lebesgue null). Then:

(a) for every Borel function $F : S^{2} \to [0, + \infty]$ , $\int_{S^{2}} F d σ = \int_{(0, π) \times (0, 2 π)} F (Φ (θ, φ)) \sin θ d (θ, φ),$ the right-hand integral being the Lebesgue integral over the open rectangle;

(b) $σ (S^{2}) = 4 π$ ;

(c) the identity of (a) holds with $F \in L^{1} (S^{2}, σ; C)$ in the sense that $F \circ Φ$ is Lebesgue integrable on the rectangle against $\sin θ d (θ, φ)$ and the two integrals agree (real and imaginary parts, positive and negative parts).

Facts & Assumptions

Given: Countable Choice; the sphere $S^{2}$ ; the polar surface measure $σ$ ; the parametrisation $Φ$ ; the half-plane $H$ ; the map $T (r, θ, φ) : = r Φ (θ, φ)$ on $(0, \infty) \times (0, π) \times (0, 2 π)$ .

[F1]

The polar formula: for every Borel $f : R^{3} \to [0, \infty]$ , $\int_{R^{3}} f d λ_{3} = \int_{0}^{\infty} r^{2} \int_{S^{2}} f (r ω) d σ (ω) d r$ , and $σ$ is the unique Borel measure on $S^{2}$ with this property (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, The polar surface set function on the unit sphere).

[F2]

Change of variables for $C^{1}$ diffeomorphisms of open sets: if $U, V \subseteq R^{m}$ are open and $T : U \to V$ is a $C^{1}$ bijection with $C^{1}$ inverse, then for every nonnegative Borel $h$ , $\int_{V} h (y) d y = \int_{U} h (T (x)) ∣ \det D T (x) ∣ d x$ (Borel change of variables from the compact-support formula and Radon uniqueness).

[F3]

Inverse function theorem: if $T$ is $C^{1}$ near $a$ and $D T (a)$ is invertible, then $T$ restricts to a $C^{1}$ bijection of a neighbourhood of $a$ onto an open set, with $C^{1}$ inverse; and $C^{1}$ means total differentiability with continuous derivative (The Euclidean inverse function theorem, Continuously differentiable maps, local inverses, and local diffeomorphisms, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder).

[F4]

The slit $K : = {(x, 0, z) : x \geq 0}$ is closed and contained in the coordinate hyperplane ${y = 0}$ , which is Lebesgue null; complements of null sets have full measure, and $λ_{3} (K) = 0$ (Every affine hyperplane of $R^{n}$ , and hence every proper linear subspace, is Lebesgue null, Measures on sigma-algebras).

[F5]

Tonelli's theorem for nonnegative Borel functions and Fubini's theorem for integrable functions identify multiple Lebesgue integrals with iterated integrals over $σ$ -finite products; on a rectangle with product measure, integrals of continuous functions may be computed by iterated integrals, and Jordan–Fubini computes Riemann integrals over rectangles as iterated integrals (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product, Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable).

[F6]

Sine and cosine are differentiable with $\sin^{'} = \cos$ , $\cos^{'} = - \sin$ , satisfy the addition formulas and Pythagoras' identity $\sin^{2} + \cos^{2} = 1$ , $\sin > 0$ on $(0, π)$ , and $\cos$ is strictly decreasing on $[0, π]$ with $\cos 0 = 1$ , $\cos π = - 1$ ; the derivative of sine integrates by the fundamental theorem (The derivatives of sine and cosine are cosine and minus sine, The addition formulas for sine and cosine, $t \mapsto (\cos t, \sin t)$ is a bijection from $[0, 2 π)$ onto the real unit circle, Signs, monotonicity intervals, and ranges of sine and cosine, Parity and the Pythagorean identity for sine and cosine, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[F7]

Continuous maps pull Borel sets back to Borel sets, and indicators of Borel sets are measurable; the monotone convergence theorem computes integrals of sums of nonnegative measurable functions (A continuous map has Borel preimages of Borel sets, Monotone convergence for the integral, The Borel sigma-algebra of a topological space).

Proof

1.1F5F6F7algebra

[The angular measure $τ$ ] For Borel $E \subseteq S^{2}$ put $τ (E) : = \int_{(0, π) \times (0, 2 π)} 1_{E} (Φ (θ, φ)) \sin θ d (θ, φ)$ , the integral existing in $[0, \infty]$ because the integrand is nonnegative and measurable: $Φ$ is continuous, hence Borel measurable, and $1_{E} \circ Φ$ is measurable by [F7]. The set function $τ$ is countably additive by the monotone convergence theorem applied to the partial sums of $\sum_{k} 1_{E_{k}}$ for pairwise disjoint Borel $E_{k}$ , and $τ (S^{2}) = \int_{0}^{π} \int_{0}^{2 π} \sin θ d φ d θ = 2 π \cdot 2 = 4 π$ by [F5] and [F6]; hence $τ$ is a finite Borel measure.

1.2F6algebra

[ $T$ is a bijection onto the slit space] $Ω : = R^{3} ∖ K = {y \neq 0} \cup {y = 0, x < 0}$ is open. The image of $T$ is contained in $Ω$ : for $φ \in (0, π)$ the second coordinate $r \sin θ \sin φ$ is positive, for $φ \in (π, 2 π)$ it is negative, and for $φ = π$ one has $y = 0$ with first coordinate $r \sin θ \cos π = - r \sin θ < 0$ . Conversely, let $x \in Ω$ and put $r : = ∣ x ∣ > 0$ and $u : = x / r \in S^{2}$ . Since $u \notin K$ , the case $u = (0, 0, \pm 1)$ is excluded (it has $y = 0$ , $x$ -coordinate $0 \geq 0$ ), so $∣ u_{3} ∣ < 1$ ; by [F6] there is a unique $θ \in (0, π)$ with $\cos θ = u_{3}$ , and then $\sin θ > 0$ and $v : = (u_{1}, u_{2}) / \sin θ$ satisfies $v_{1}^{2} + v_{2}^{2} = 1$ by Pythagoras [F6]. By the parametrisation of the unit circle [F6] there is a unique $φ \in [0, 2 π)$ with $v = (\cos φ, \sin φ)$ ; $φ = 0$ would give $u = (\sin θ, 0, u_{3})$ with $y = 0$ and $x$ -coordinate $\sin θ > 0$ , i.e. $u \in K$ , which is excluded, so $φ \in (0, 2 π)$ . Then $x = T (r, θ, φ)$ with $(r, θ, φ)$ uniquely determined, so $T$ is a bijection of $(0, \infty) \times (0, π) \times (0, 2 π)$ onto $Ω$ .

1.3F6algebra

[ $T$ is $C^{1}$ with positive Jacobian] The components of $T$ are products and composites of the differentiable functions $\sin, \cos$ and the coordinate functions by [F6], so $T$ is $C^{1}$ with total derivative the matrix of partial derivatives (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, Total differentiability gives a local $O (∥ h ∥_{2})$ increment bound and therefore continuity), namely the columns $\partial_{r} T = (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ , $\partial_{θ} T = (r \cos θ \cos φ, r \cos θ \sin φ, - r \sin θ)$ and $\partial_{φ} T = (- r \sin θ \sin φ, r \sin θ \cos φ, 0)$ ; expanding the determinant along the third column and using $\cos^{2} φ + \sin^{2} φ = 1$ and $\sin^{2} θ + \cos^{2} θ = 1$ [F6] gives $\det D T = r^{2} \sin θ (\cos^{2} θ + \sin^{2} θ) = r^{2} \sin θ > 0$ (The Jacobian determinant of a square-dimensional $C^{1}$ map is the determinant of its Jacobian matrix).

2.1step 1.2step 1.3F2F3

[ $T$ is a $C^{1}$ diffeomorphism onto $Ω$ ] At each point of the domain $D T$ is invertible by step 1.3, so by the inverse function theorem [F3] $T$ restricts to a $C^{1}$ bijection of a neighbourhood of that point onto an open subset of $Ω$ with $C^{1}$ inverse. Since $T$ is injective with image $Ω$ by step 1.2, these local inverses agree on overlaps with the unique set-theoretic inverse $T^{- 1} : Ω \to (0, \infty) \times (0, π) \times (0, 2 π)$ , which is therefore $C^{1}$ : near any point of $Ω$ it coincides with a local $C^{1}$ inverse. Hence $T$ is a $C^{1}$ diffeomorphism of open sets, and [F2] applies to it.

3.1step 2.1step 1.1F2F4F5F7algebra

[The polar formula for $τ$ ] Let $f : R^{3} \to [0, \infty]$ be Borel. By [F4], $λ_{3} (K) = 0$ and $\int_{R^{3}} f = \int_{Ω} f$ . Applying the change of variables formula [F2] to the diffeomorphism $T$ of step 2.1 and the nonnegative Borel function $f ∣_{Ω}$ gives $\int_{Ω} f = \int_{(0, \infty) \times (0, π) \times (0, 2 π)} f (T (r, θ, φ)) r^{2} \sin θ d (r, θ, φ)$ . By Tonelli's theorem [F5] the triple integral factors as $\int_{0}^{\infty} r^{2} (\int_{(0, π) \times (0, 2 π)} f (r Φ (θ, φ)) \sin θ d (θ, φ)) d r = \int_{0}^{\infty} r^{2} \int_{S^{2}} f (r ω) d τ (ω) d r$ : for each fixed $r$ the function $ω \mapsto f (r ω)$ is nonnegative and Borel, and its $τ$ -integral equals the rectangle integral of $f (r Φ (\cdot)) \sin (\cdot)$ by the defining formula for $τ$ , extended from indicators to nonnegative Borel integrands by the monotone convergence theorem [F7]. Hence $τ$ satisfies the polar formula of [F1].

4.1step 1.1step 3.1F1F5F7algebra∎

[Identification and consequences] By the uniqueness clause of [F1] the measure satisfying the polar formula is $σ$ ; hence $τ = σ$ , which is exactly the identity of (a) for indicators and, by the monotone convergence theorem, for all nonnegative Borel $F$ [F7]; clause (b) is the computation $τ (S^{2}) = 4 π$ of step 1.1. For (c), if $F \in L^{1} (S^{2}, σ; C)$ then applying (a) to $∣ F ∣$ shows $F \circ Φ$ is integrable against $\sin θ d (θ, φ)$ , because the integral of $∣ F ∣$ over $S^{2}$ is finite; splitting $F$ into real and imaginary parts and each into positive and negative parts, and applying (a) to the four nonnegative functions (or Fubini, [F5]) gives the identity for $F$ .

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Associated Legendre function normalization

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $P_{l}$ be the Legendre polynomials of Legendre Rodrigues equation, and for $l \in N$ and $0 \leq m \leq l$ define the function $w_{l m} : (- 1, 1) \to R$ by

$w_{l m} (t) : = (- 1)^{m} (1 - t^{2})^{m / 2} \frac{d^{m}}{d t^{m}} P_{l} (t),$

where $(1 - t^{2})^{1 / 2} > 0$ denotes the nonnegative square root. Let $σ$ be the polar surface measure on $S^{2}$ and let the angle coordinates be $Φ (θ, φ) = (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ (Sphere integrals in angle coordinates, The polar surface set function on the unit sphere). The image of $Φ$ is $S^{2} ∖ C$ , where $C : = {(x, 0, z) \in S^{2} : x \geq 0}$ ; $C$ is closed and $σ (C) = 0$ by applying the integral identity of Sphere integrals in angle coordinates to its indicator. For $l \geq m \geq 0$ , define the representative ${\tilde{Y}}_{l m} : S^{2} \to C$ on $S^{2} ∖ C$ by ${\tilde{Y}}_{l m} (Φ (θ, φ)) : = b_{l m} w_{l m} (\cos θ) \frac{e^{i m φ}}{\sqrt{2 π}},$ and on the cut $C$ by ${\tilde{Y}}_{l m} (\sin θ, 0, \cos θ) : = b_{l m} w_{l m} (\cos θ) / \sqrt{2 π} (0 \leq θ \leq π) .$ The limits at $φ = 0$ and $2 π$ agree because $m$ is an integer, and the endpoint values of $w_{l m}$ from (a) give the limits at the poles; hence ${\tilde{Y}}_{l m}$ is continuous. For $1 \leq m \leq l$ set ${\tilde{Y}}_{l, - m} : = (- 1)^{m} \overline{{\tilde{Y}}_{l m}}$ , and write $Y_{l m}$ for the class of ${\tilde{Y}}_{l m}$ in $L^{2} (S^{2}, σ; C)$ . Then:

(a) each $w_{l m}$ extends continuously to $[- 1, 1]$ , with $w_{l m} (\pm 1) = 0$ when $m \geq 1$ , and on $(- 1, 1)$ it satisfies the associated Legendre equation $- ((1 - t^{2}) w_{l m}^{'})^{'} + \frac{m^{2}}{1 - t^{2}} w_{l m} = l (l + 1) w_{l m} (- 1 < t < 1),$ equivalently $(1 - t^{2}) w_{l m}^{''} - 2 t w_{l m}^{'} + [l (l + 1) - \frac{m^{2}}{1 - t^{2}}] w_{l m} = 0$ ;

(b) for fixed $m \geq 0$ the functions $w_{l m}$ , $l \geq m$ , are pairwise orthogonal in $L^{2} ((- 1, 1), d t)$ : $\int_{- 1}^{1} w_{l m} w_{k m} d t = 0$ for $k \neq l$ ; moreover $0 < \int_{- 1}^{1} w_{l m}^{2} d t < + \infty$ ;

(c) with $b_{l m} : = (\int_{- 1}^{1} w_{l m}^{2} d t)^{- 1 / 2} > 0$ and the representatives just defined, one has $∣ Y_{l, - m} ∣ = ∣ Y_{l m} ∣$ and $\overline{Y_{l m}} = (- 1)^{m} Y_{l, - m}$ for all $l$ and all $m$ with $∣ m ∣ \leq l$ ;

(d) the family ${Y_{l m} : l \geq 0, - l \leq m \leq l}$ is orthonormal in $L^{2} (S^{2}, σ; C)$ (Real and complex inner product spaces, with the inner product linear in the first argument, The space $L^{p} (μ)$ as the quotient by null functions): for all $(l, m) \neq (l^{'}, m^{'})$ with $∣ m ∣ \leq l$ , $∣ m^{'} ∣ \leq l^{'}$ , the integral $\int_{S^{2}} Y_{l m} \overline{Y_{l^{'} m^{'}}} d σ$ equals $0$ , and it equals $1$ when $(l, m) = (l^{'}, m^{'})$ ;

(e) $w_{l 0} = P_{l}$ , and $w_{l l} (t) = (- 1)^{l} \frac{(2 l)!}{2^{l} l!} (1 - t^{2})^{l / 2}$ .

The functions $Y_{l m}$ are complex-valued $L^{2}$ -functions on the sphere; no physical interpretation as angular momentum eigenfunctions or spherical harmonics of an atom is asserted.

Facts & Assumptions

Given: the Legendre polynomials $P_{l}$ ; integers $l \geq 0$ , $0 \leq m \leq l$ ; the functions $w_{l m}$ ; the measure $σ$ and the parametrisation $Φ$ ; the constants $b_{l m}$ and functions $Y_{l m}$ .

[F1]

$P_{l}$ satisfies $(1 - t^{2}) P_{l}^{''} - 2 t P_{l}^{'} + l (l + 1) P_{l} = 0$ on $R$ , is a polynomial of degree $l$ , and $P_{l}^{(m)}$ is therefore the polynomial function of a degree- $(l - m)$ polynomial (Legendre Rodrigues equation).

[F2]

The general Leibniz rule and the algebra of derivatives compute the derivatives of products; the chain rule computes derivatives of composites (The general Leibniz rule for the $n$ -th derivative of a product, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[F3]

Sphere integrals in angle coordinates: for Borel $F \geq 0$ , $\int_{S^{2}} F d σ = \int_{(0, π) \times (0, 2 π)} F (Φ (θ, φ)) \sin θ d (θ, φ)$ , and the complex $L^{1}$ version; $σ (S^{2}) = 4 π$ (Sphere integrals in angle coordinates).

[F4]

Characters of the circle: $\int_{0}^{2 π} e^{i k φ} d φ = 2 π$ if $k = 0$ and $= 0$ for every nonzero $k \in Z$ (The trigonometric characters are orthonormal in $L^{2}$ of the torus, The complex exponential by its power series).

[F5]

A continuous real function on the compact interval $[- 1, 1]$ is (Riemann and Lebesgue) integrable; products of continuous functions are continuous, and $\int_{- 1}^{1} g > 0$ for a continuous $g \geq 0$ that is not identically zero (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, Complex Lp classes and Euclidean test-function conventions).

[F6]

If $G : [a, b] \to R$ is differentiable at every point of $[a, b]$ as a function on $[a, b]$ (with one-sided derivatives at $a, b$ ) and $G^{'}$ is integrable on $[a, b]$ , then $\int_{a}^{b} G^{'} = G (b) - G (a)$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[F7]

For $f, g \in L^{2}$ the product $f \overline{g}$ is in $L^{1}$ with $\int ∣ f \overline{g} ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ ; the inner product in $L^{2} (S^{2}, σ; C)$ is linear in the first argument; conjugation satisfies $\overline{z w} = \overline{z} \overline{w}$ and $∣ z ∣^{2} = z \overline{z}$ (Cauchy-Schwarz inequality for $L^{2}$ , Real and complex inner product spaces, with the inner product linear in the first argument, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive, Complex Lp classes and Euclidean test-function conventions).

[F8]

For the given $0 \leq m \leq l$ , $d^{m} P_{l} / d t^{m}$ is a polynomial of degree $l - m$ with nonzero leading coefficient, so $w_{l m}$ is not the zero function on $(- 1, 1)$ , because $(1 - t^{2})^{m / 2} > 0$ there (Legendre Rodrigues equation, The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ ).

[F9]

If $G : [a, b] \to R$ is continuous on $[a, b]$ and differentiable at every point of $(a, b)$ as a function on $[a, b]$ , the mean value theorem gives a $c \in (a, b)$ with $G (b) - G (a) = G^{'} (c) (b - a)$ (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

Proof

1.1F1F2algebra

[Differentiated equation] Let $v : = P_{l}^{(m)}$ . Differentiating the Legendre equation of [F1] $m$ times with the general Leibniz rule [F2] gives, since the third and higher derivatives of $1 - t^{2}$ vanish, $(1 - t^{2}) v^{''} - 2 t (m + 1) v^{'} + [l (l + 1) - m (m + 1)] v = 0$ : the coefficient of $v^{''}$ is $1 - t^{2}$ ; that of $v^{'}$ collects $- 2 t$ from differentiating $1 - t^{2}$ once times the derivative count $(\binom{m}{1})$ and $- 2 t$ from the term $- 2 t P_{l}^{'}$ , total $- 2 t (m + 1)$ ; and that of $v$ collects $(\binom{m}{2}) (- 2) + m (- 2) + l (l + 1) = - m (m - 1) - 2 m + l (l + 1) = l (l + 1) - m (m + 1)$ .

1.2F1F2algebra

[Continuity at the endpoints] For $m = 0$ , $w_{l 0} = P_{l}$ is a polynomial; for $m \geq 1$ , $w_{l m} = (- 1)^{m} (1 - t^{2})^{m / 2} Q (t)$ with $Q = P_{l}^{(m)}$ a polynomial, so $∣ w_{l m} (t) ∣ \leq (1 - t^{2})^{m / 2} \sup_{[- 1, 1]} ∣ Q ∣ \to 0$ as $t \to \pm 1$ , and the function extended by $w_{l m} (\pm 1) : = 0$ is continuous on $[- 1, 1]$ .

2.1step 1.1F2algebra

[Substitution] Put $f (t) : = (1 - t^{2})^{m / 2} > 0$ on $(- 1, 1)$ , so that $w_{l m} = (- 1)^{m} f v$ ; the sign is constant and will be carried along. Put $u : = (- 1)^{m} v$ , so $w = f u$ and $u$ satisfies the same homogeneous equation as $v$ in step 1.1. From $u = f^{- 1} w$ and [F2], $u^{'} = f^{- 1} w^{'} - f^{- 2} f^{'} w$ and $u^{''} = f^{- 1} w^{''} - 2 f^{- 2} f^{'} w^{'} + (2 f^{- 3} (f^{'})^{2} - f^{- 2} f^{''}) w$ . Substituting into step 1.1 and multiplying by $f$ gives $(1 - t^{2}) w^{''} + [- 2 \frac{f^{'}}{f} (1 - t^{2}) - 2 t (m + 1)] w^{'} + [- \frac{f^{''}}{f} (1 - t^{2}) + 2 (\frac{f^{'}}{f})^{2} (1 - t^{2}) + 2 t (m + 1) \frac{f^{'}}{f} - m (m + 1) + l (l + 1)] w = 0$ . Direct differentiation gives $\frac{f^{'}}{f} = - \frac{m t}{1 - t^{2}}$ and therefore $\frac{f^{''}}{f} = (\frac{f^{'}}{f})^{'} + (\frac{f^{'}}{f})^{2} = \frac{- m - m t^{2}}{(1 - t^{2})^{2}} + \frac{m^{2} t^{2}}{(1 - t^{2})^{2}} = \frac{m (- 1 + (m - 1) t^{2})}{(1 - t^{2})^{2}}$ . Hence the coefficient of $w^{'}$ is $2 m t - 2 t (m + 1) = - 2 t$ , and the coefficient of $w$ equals $\frac{m - m (m + 1) t^{2}}{1 - t^{2}} - m (m + 1) + l (l + 1) = l (l + 1) - \frac{m^{2}}{1 - t^{2}}$ , using $\frac{m - m (m + 1) t^{2}}{1 - t^{2}} - m (m + 1) = - \frac{m^{2}}{1 - t^{2}}$ (expand the left side over the common denominator $1 - t^{2}$ ). This is the associated equation.

3.1step 2.1step 1.2F2F5F6F9algebra

[Orthogonality] Fix $m$ and let $k \neq l$ , $k \geq m$ . Multiply the associated equation for $w_{l m}$ by $w_{k m}$ and that for $w_{k m}$ by $w_{l m}$ , subtract, and use $- ((1 - t^{2}) w_{l m}^{'})^{'} w_{k m} + ((1 - t^{2}) w_{k m}^{'})^{'} w_{l m} = - \frac{d}{d t} [(1 - t^{2}) (w_{l m}^{'} w_{k m} - w_{l m} w_{k m}^{'})]$ : $(l (l + 1) - k (k + 1)) w_{l m} w_{k m} = - \frac{d}{d t} G (t)$ with $G (t) : = (1 - t^{2}) (w_{l m}^{'} (t) w_{k m} (t) - w_{l m} (t) w_{k m}^{'} (t))$ on $(- 1, 1)$ . By step 1.2, put $Q_{l} = P_{l}^{(m)}$ and $Q_{k} = P_{k}^{(m)}$ . If $m = 0$ , then both $w$ functions are polynomials and the factor $1 - t^{2}$ gives $G (t) \to 0$ at both endpoints. If $m \geq 1$ , put $f (t) = (1 - t^{2})^{m / 2}$ , so $f^{'} = - m t (1 - t^{2})^{m / 2 - 1}$ and $w_{l m} = (- 1)^{m} f Q_{l}$ , $w_{k m} = (- 1)^{m} f Q_{k}$ ; since $Q_{l}, Q_{k}$ and their derivatives are bounded on $[- 1, 1]$ , each term in $(1 - t^{2}) w_{l m}^{'} w_{k m}$ and $(1 - t^{2}) w_{l m} w_{k m}^{'}$ , after the product rule, is bounded by a constant times $(1 - t^{2})^{m}$ or $(1 - t^{2})^{m + 1}$ , so again $G (t) \to 0$ at both endpoints. Extend $G$ by $G (\pm 1) = 0$ ; these limits make it continuous on $[- 1, 1]$ . The subtracted equations give $G^{'} (t) = q (t) : = - (l (l + 1) - k (k + 1)) w_{l m} (t) w_{k m} (t)$ on $(- 1, 1)$ , and step 1.2 makes $q$ continuous on $[- 1, 1]$ . For each $x \in (- 1, 1)$ , [F9] on $[x, 1]$ gives $(G (1) - G (x)) / (1 - x) = q (ξ_{x})$ for some $ξ_{x} \in (x, 1)$ , so as $x ↑ 1$ this quotient tends to $q (1)$ . On $[- 1, x]$ , [F9] gives $(G (x) - G (- 1)) / (x + 1) = q (η_{x})$ for some $η_{x} \in (- 1, x)$ ; as $x ↓ - 1$ this quotient tends to $q (- 1)$ . Thus $G$ is differentiable at every point of $[- 1, 1]$ with the endpoint convention of [F6], and $G^{'} = q$ is continuous, hence integrable by [F5]. Now [F6] gives $\int_{- 1}^{1} G^{'} = G (1) - G (- 1) = 0$ , so $(l (l + 1) - k (k + 1)) \int_{- 1}^{1} w_{l m} w_{k m} d t = 0$ and, since $l \neq k$ , $\int_{- 1}^{1} w_{l m} w_{k m} d t = 0$ .

3.2step 2.1step 1.2F5F8algebra

[Positivity and finiteness of the norms] By steps 2.1 and 1.2, $w_{l m}^{2}$ is continuous on $[- 1, 1]$ , so its integral is finite and nonnegative by [F5]; it is not zero because $w_{l m}$ is not the zero function on $(- 1, 1)$ : $P_{l}^{(m)}$ is a nonzero polynomial by [F8] and $(1 - t^{2})^{m / 2} > 0$ in the interior, so $w_{l m}$ is nonzero at every point where the polynomial does not vanish, and a continuous nonzero function has positive integral by [F5]. Hence $0 < \int_{- 1}^{1} w_{l m}^{2} d t < + \infty$ and $b_{l m}$ of the Statement is a well-defined positive real.

3.3step 2.1F1F7algebra

[The functions $Y_{l m}$ and their conjugation] The formula in the Statement defines continuous representatives on $S^{2}$ , so each class $Y_{l m}$ belongs to $L^{2} (S^{2}, σ)$ because $σ$ is finite; their chart moduli satisfy $∣ Y_{l m} (Φ (θ, φ)) ∣ = b_{l m} ∣ w_{l m} (\cos θ) ∣ / \sqrt{2 π}$ . By definition $Y_{l, - m} = (- 1)^{m} \overline{Y_{l m}}$ , so $∣ Y_{l, - m} ∣ = ∣ Y_{l m} ∣$ and, conjugating twice, $\overline{Y_{l, - m}} = (- 1)^{m} Y_{l m}$ , which is the displayed identity for every $m \geq 0$ . This establishes (c), and the identities in (e) follow from $P_{l}^{(0)} = P_{l}$ and $P_{l}^{(l)} = \frac{(2 l)!}{2^{l} l!}$ (the leading coefficient of $P_{l}$ times $l!$ , by [F1]).

4.1step 3.1step 3.2step 3.3F3F4F7algebra∎

[Orthonormality] Let $m, m^{'} \geq 0$ . The extensions on $C$ do not affect integrals because $σ (C) = 0$ . Thus [F3] applies to the $L^{1}$ function $Y_{l m} \overline{Y_{l^{'} m^{'}}}$ , whose integral against $σ$ is the inner product; with [F4], this gives $⟨ Y_{l m}, Y_{l^{'} m^{'}} ⟩ = b_{l m} b_{l^{'} m^{'}} \frac{1}{2 π} (\int_{0}^{2 π} e^{i (m - m^{'}) φ} d φ) (\int_{- 1}^{1} w_{l m} (t) w_{l^{'} m^{'}} (t) d t) = δ_{m m^{'}} b_{l m} b_{l^{'} m} \int_{- 1}^{1} w_{l m} w_{l^{'} m} d t$ , which by steps 3.1 and 3.2 equals $0$ for $l \neq l^{'}$ and $1$ for $l = l^{'}$ . For entries with negative lower index, write the inner product as a product of conjugation operations using [F7]: $⟨ Y_{l, - m}, Y_{l^{'}, - m^{'}} ⟩ = (- 1)^{m + m^{'}} ⟨ \overline{Y_{l m}}, \overline{Y_{l^{'} m^{'}}} ⟩ = (- 1)^{m + m^{'}} \overline{⟨ Y_{l m}, Y_{l^{'} m^{'}} ⟩}$ , and $⟨ Y_{l m}, Y_{l^{'}, - m^{'}} ⟩ = (- 1)^{m^{'}} \int_{S^{2}} Y_{l m} Y_{l^{'} m^{'}} d σ$ , whose rectangle form has $φ$ -factor $\int_{0}^{2 π} e^{i (m + m^{'}) φ} d φ$ , vanishing unless $m = m^{'} = 0$ by [F4], in which case it reduces to the already treated case; collecting the cases gives orthonormality for all pairs. This proves (d), and (a), (b) were proved in steps 2.1, 3.1 and 3.2.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Weighted angular completeness

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $P_{l}$ be the Legendre polynomials of Legendre Rodrigues equation and $w_{l m}$ the associated functions of Associated Legendre function normalization, and let $L^{2} = L^{2} ((- 1, 1), d t; C)$ with inner product linear in the first argument (The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions). Then, for each fixed $m \in N$ :

(a) for every $l \geq m$ the polynomial $P_{l}^{(m)}$ has degree $l - m$ and nonzero leading coefficient, and for every $d \in N$ the polynomials $P_{m}^{(m)}, P_{m + 1}^{(m)}, \dots, P_{m + d}^{(m)}$ form a basis of the space of polynomial functions of degree at most $d$ on $[- 1, 1]$ ;

(b) the family $(w_{l m})_{l \geq m}$ has dense linear span in $L^{2}$ : if $f \in L^{2}$ satisfies $\int_{- 1}^{1} f (t) w_{l m} (t) d t = 0$ for every $l \geq m$ , then $f = 0$ almost everywhere; consequently the orthogonal family $(w_{l m})_{l \geq m}$ is complete in $L^{2}$ and its normalization $(w_{l m} / ∥ w_{l m} ∥_{2})_{l \geq m}$ is a Hilbert basis of $L^{2}$ , so every $f \in L^{2}$ has $f = \sum_{l \geq m} (\int_{- 1}^{1} f w_{l m} d t) ∥ w_{l m} ∥_{2}^{- 2} w_{l m}$ with the finite-subset-net convergence of Orthonormal families, complete orthonormal systems and Hilbert bases;

(c) for every fixed $m$ the orthogonal projection of $L^{2}$ onto the one-dimensional span of $w_{m m}$ is $f \mapsto ∥ w_{m m} ∥_{2}^{- 2} (\int_{- 1}^{1} f w_{m m} d t) w_{m m}$ , and $w_{m m} (t) = (- 1)^{m} \frac{(2 m)!}{2^{m} m!} (1 - t^{2})^{m / 2}$ .

Facts & Assumptions

Given: $m \in N$ ; the Legendre polynomials $P_{l}$ ; the functions $w_{l m}$ ; the complex Hilbert space $L^{2} ((- 1, 1), d t)$ .

[F1]

$P_{l}$ is a polynomial of degree $l$ with nonzero leading coefficient $a_{l} = \frac{(2 l)!}{2^{l} (l!)^{2}}$ (Legendre Rodrigues equation). Iterating the monomial derivative rule (For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term) and factorial recursion (The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ ) gives, for $l \geq m$ , leading term $a_{l} l! / (l - m)! t^{l - m}$ in $P_{l}^{(m)}$ ; lower-degree terms either vanish or have smaller degree. Thus its degree is $l - m$ , and $P_{m}^{(m)} = a_{m} m!$ .

[F2]

$w_{l m} = (- 1)^{m} (1 - t^{2})^{m / 2} P_{l}^{(m)}$ is continuous on $[- 1, 1]$ , the family $(w_{l m})_{l \geq m}$ is orthogonal in $L^{2}$ , and $∥ w_{l m} ∥_{2}^{2} = \int_{- 1}^{1} w_{l m}^{2} d t \in (0, \infty)$ (Associated Legendre function normalization).

[F3]

The real polynomial functions are uniformly dense in the real continuous functions on $[- 1, 1]$ : for every continuous $g : [- 1, 1] \to R$ and every $ε > 0$ there is a polynomial $p$ with $\sup_{[- 1, 1]} ∣ g - p ∣ < ε$ (Real Stone–Weierstrass theorem for compact Hausdorff spaces).

[F4]

Under Countable Choice, for every bounded interval $(a, b)$ the complex continuous functions on $[a, b]$ are dense in complex $L^{p} ((a, b))$ for $1 \leq p < + \infty$ , in particular in $L^{2}$ (Continuous functions are dense in $L^{p}$ of finite tori and of bounded intervals (2)). For continuous $g$ , integrating $∣ g (t) ∣^{2} \leq ∥ g ∥_{\infty}^{2}$ over $(a, b)$ gives $∥ g ∥_{2}^{2} \leq (b - a) ∥ g ∥_{\infty}^{2}$ , hence $∥ g ∥_{2} \leq \sqrt{b - a} ∥ g ∥_{\infty}$ by the defining $L^{2}$ norm (The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions).

[F5]

$∥ f g ∥_{1} \leq ∥ f ∥_{2} ∥ g ∥_{2}$ and $∣ \int f \overline{g} ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ in $L^{2}$ (Cauchy-Schwarz inequality for $L^{2}$ , Complex Lp classes and Euclidean test-function conventions).

[F6]

For a family that is orthonormal and complete in a Hilbert space, the expansion, Parseval identity and coefficient formulas hold with finite-subset-net convergence (Orthonormal families, complete orthonormal systems and Hilbert bases, Parseval equivalences for an orthonormal family).

Proof

1.1F1algebra

[(a)] By [F1], $P_{l}^{(m)}$ has degree $l - m$ and nonzero leading coefficient for each $l \geq m$ . Hence in the ordered list $P_{m}^{(m)}, P_{m + 1}^{(m)}, \dots, P_{m + d}^{(m)}$ the $j$ -th polynomial has degree $j$ , $0 \leq j \leq d$ : the leading-degree matrix is triangular with nonzero diagonal, so the polynomials are linearly independent; they are $d + 1$ vectors in the $(d + 1)$ -dimensional space of polynomials of degree at most $d$ , hence a basis.

2.1step 1.1F2algebra

[(b), first reduction] Let $f \in L^{2}$ satisfy $\int_{- 1}^{1} f w_{l m} d t = 0$ for all $l \geq m$ , and put $g (t) : = f (t) (1 - t^{2})^{m / 2}$ . Since $t \mapsto (1 - t^{2})^{m / 2}$ is continuous and bounded on $[- 1, 1]$ , $g \in L^{2}$ and $∥ g ∥_{2} \leq ∥ f ∥_{2}$ ; moreover for every $l \geq m$ , $\int_{- 1}^{1} g P_{l}^{(m)} d t = (- 1)^{m} \int_{- 1}^{1} f w_{l m} d t = 0$ by [F2]. By step 1.1, every polynomial $q$ of degree at most $d$ is a linear combination of $P_{m}^{(m)}, \dots, P_{m + d}^{(m)}$ , so $\int_{- 1}^{1} g q d t = 0$ for every polynomial $q$ .

3.1step 2.1F3F4F5algebra

[(b), density step] Polynomials are dense in $L^{2} ((- 1, 1))$ : given $φ \in L^{2}$ and $ε > 0$ , [F4] supplies a continuous $h$ on $[- 1, 1]$ with $∥ φ - h ∥_{2} < ε / 2$ , and apply [F3] separately to the real and imaginary parts of $h$ , obtaining real polynomials $p_{1}, p_{2}$ whose respective uniform errors are below $ε / (4 \sqrt{2})$ , and put $p = p_{1} + i p_{2}$ ; then $\sup_{[- 1, 1]} ∣ h - p ∣ < ε / (2 \sqrt{2})$ , so $∥ h - p ∥_{2} < ε / 2$ by [F4] (applied to the interval length $2$ ), whence $∥ φ - p ∥_{2} < ε$ . Applying this to the complex-valued $g$ componentwise (real and imaginary parts) gives polynomials $q_{j}$ with $∥ g - q_{j} ∥_{2} \to 0$ ; since $\int g q = 0$ for all polynomials, Cauchy–Schwarz [F5] gives $∥ g ∥_{2}^{2} = \int g \overline{g} = \lim_{j} \int g \overline{q_{j}} = 0$ , so $g = 0$ almost everywhere. On $(- 1, 1)$ the factor $(1 - t^{2})^{m / 2}$ is strictly positive, so $f = g / (1 - t^{2})^{m / 2} = 0$ almost everywhere.

4.1step 3.1F2F6algebra

[(b), completeness and expansion] By [F2] the family $(w_{l m})_{l \geq m}$ is orthogonal with all norms nonzero, so $(w_{l m} / ∥ w_{l m} ∥_{2})_{l \geq m}$ is orthonormal; if a vector of $L^{2}$ is orthogonal to the whole family, it is orthogonal to every $w_{l m}$ and vanishes by step 3.1, so the orthonormal family is complete in the sense of [F6]; the expansion, normalization and coefficient formulas follow from [F6].

5.1step 4.1F1F2F5algebra∎

[(c)] The orthogonal projection onto the span of a unit vector $u$ is $x \mapsto ⟨ x, u ⟩ u$ , and for an unnormalized nonzero vector $w$ it is $x \mapsto ∥ w ∥_{2}^{- 2} ⟨ x, w ⟩ w$ ; with $w = w_{m m}$ and the inner product linear in the first argument this is the displayed formula. Indeed its value lies in the span of $w$ , and its residual has zero inner product with $w$ , uniquely determining the orthogonal projection; Cauchy–Schwarz bounds its norm by $∥ x ∥_{2}$ . The explicit value of $w_{m m}$ follows from $P_{m}^{(m)} = \frac{(2 m)!}{2^{m} m!}$ (the leading coefficient of $P_{m}$ times $m!$ ) inserted into the definition of $w_{l m}$ .

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Sphere ONB

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $σ$ be the polar surface measure on $S^{2}$ (The polar surface set function on the unit sphere) and let $(Y_{l m})_{l \geq 0, - l \leq m \leq l}$ be the functions of Associated Legendre function normalization (The complex exponential by its power series). Then $(Y_{l m})$ is a complete orthonormal family of the complex Hilbert space $L^{2} (S^{2}, σ)$ (Orthonormal families, complete orthonormal systems and Hilbert bases, The space $L^{p} (μ)$ as the quotient by null functions): the family is orthonormal, and if $F \in L^{2} (S^{2}, σ)$ satisfies $\int_{S^{2}} F \overline{Y_{l m}} d σ = 0$ for all $l \geq 0$ and $- l \leq m \leq l$ , then $F = 0$ almost everywhere. Consequently every $F$ has the expansion $F = \sum_{l = 0}^{\infty} \sum_{m = - l}^{l} ⟨ F, Y_{l m} ⟩ Y_{l m}, ∥ F ∥^{2} = \sum_{l, m} ∣ ⟨ F, Y_{l m} ⟩ ∣^{2},$ with $⟨ F, Y_{l m} ⟩ = \int_{S^{2}} F \overline{Y_{l m}} d σ$ and finite-subset-net convergence (Parseval equivalences for an orthonormal family).

Facts & Assumptions

Given: the measure $σ$ on $S^{2}$ ; the functions $Y_{l m} = b_{l m} w_{l ∣ m ∣} (\cos θ) e^{i m φ} / \sqrt{2 π}$ with the convention $Y_{l, - m} = (- 1)^{m} \overline{Y_{l m}}$ ; a function $F \in L^{2} (S^{2}, σ)$ .

[F1]

Orthonormality of the family $(Y_{l m})$ in $L^{2} (S^{2}, σ)$ (Associated Legendre function normalization (d)).

[F2]

Angle coordinates: $\int_{S^{2}} G d σ = \int_{(0, π) \times (0, 2 π)} G (Φ (θ, φ)) \sin θ d (θ, φ)$ for $G \in L^{1}$ , with $Φ (θ, φ) = (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ (Sphere integrals in angle coordinates).

[F3]

Weighted completeness: for each $k \in N$ , if $h \in L^{2} ((- 1, 1), d t)$ satisfies $\int_{- 1}^{1} h (t) w_{l k} (t) d t = 0$ for all $l \geq k$ , then $h = 0$ a.e. (Weighted angular completeness (b)).

[F4]

The normalized characters $e_{k} (φ) = e^{i k φ} / \sqrt{2 π}$ , $k \in Z$ , are orthonormal in $L^{2} ((0, 2 π), d φ)$ , and their linear span is dense: trigonometric polynomials are uniformly dense in the continuous periodic functions (Trigonometric polynomials are uniformly dense in continuous functions on the torus), and continuous functions on $[0, 2 π]$ with equal endpoint values are dense in $L^{2} ((0, 2 π))$ (Continuous functions are dense in $L^{p}$ of finite tori and of bounded intervals, The trigonometric characters are orthonormal in $L^{2}$ of the torus); hence if $h \in L^{2} ((0, 2 π))$ has $\int_{0}^{2 π} h (φ) e^{- i k φ} d φ = 0$ for every $k \in Z$ , then $h = 0$ a.e.

[F5]

Tonelli's and Fubini's theorems identify the rectangle integral in [F2] with iterated integrals, for nonnegative and for $L^{1}$ integrands (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product).

[F6]

If $\int_{S^{2}} ∣ F ∣^{2} d σ < \infty$ , then by [F2] and [F5] the iterated integral $\int_{0}^{π} \int_{0}^{2 π} ∣ F (Φ (θ, φ)) ∣^{2} \sin θ d φ d θ$ is finite, so for almost every $θ$ the function $φ \mapsto F (Φ (θ, φ))$ lies in $L^{2} ((0, 2 π))$ (The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions).

[F7]

$∣ \int g \overline{h} ∣ \leq ∥ g ∥_{2} ∥ h ∥_{2}$ and vanishing of all pairings against a dense family forces vanishing (Cauchy-Schwarz inequality for $L^{2}$ ).

Proof

1.1givenF2F3F5F6F7algebra

[Completeness: Fourier coefficients along the circles] Suppose $F \in L^{2} (S^{2}, σ)$ is orthogonal to every $Y_{l m}$ , and for $k \in Z$ and $θ \in (0, π)$ define $G_{k} (θ) : = \frac{1}{\sqrt{2 π}} \int_{0}^{2 π} F (Φ (θ, φ)) e^{- i k φ} d φ$ ; by [F6] and Cauchy–Schwarz [F7] each $G_{k}$ is defined for almost every $θ$ , and $\int_{0}^{π} ∣ G_{k} (θ) ∣^{2} \sin θ d θ \leq \int_{0}^{π} \int_{0}^{2 π} ∣ F (Φ (θ, φ)) ∣^{2} d φ \sin θ d θ < \infty$ , so $G_{k} \in L^{2} (\sin θ d θ)$ and, after the reparametrisation $t = \cos θ$ , $G_{k} \in L^{2} ((- 1, 1), d t)$ . For $l \geq k \geq 0$ , [F2] and [F5] compute $⟨ F, Y_{l k} ⟩ = b_{l k} \int_{0}^{π} G_{k} (θ) w_{l k} (\cos θ) \sin θ d θ = b_{l k} \int_{- 1}^{1} G_{k} (t) w_{l k} (t) d t$ ; hence $\int_{- 1}^{1} G_{k} (t) w_{l k} (t) d t = 0$ for every $l \geq k$ , and by [F3] $G_{k} = 0$ almost everywhere for every $k \geq 0$ . For $k < 0$ write $m : = - k > 0$ ; the convention gives $\overline{Y_{l, - m}} = (- 1)^{m} Y_{l m}$ , so orthogonality of $F$ to $Y_{l, - m}$ gives $0 = (- 1)^{m} \int_{S^{2}} F Y_{l m} d σ$ , whose rectangle form is $b_{l m} \int_{0}^{π} G_{- m} (θ) w_{l m} (\cos θ) \sin θ d θ$ with $0 \leq m \leq l$ ; by [F3] again, $G_{- m} = G_{k} = 0$ almost everywhere. So all circle Fourier coefficients of $F$ vanish for almost every $θ$ .

2.1step 1.1F2F4F6algebra

[Completeness: vanishing on the spheres] By [F4] applied to the $L^{2} ((0, 2 π))$ -function $φ \mapsto F (Φ (θ, φ))$ , which exists for almost every $θ$ by [F6] and has all Fourier coefficients zero by step 1.1, one has $F (Φ (θ, φ)) = 0$ for almost every $φ$ and almost every $θ$ ; hence $F \circ Φ = 0$ almost everywhere on the rectangle, and by [F2] (which for the nonnegative function $∣ F ∣$ identifies $\int_{S^{2}} ∣ F ∣ d σ$ with the rectangle integral) $F = 0$ almost everywhere on $S^{2}$ .

3.1step 2.1F1F7algebra∎

[The expansion] By [F1] the family is orthonormal, and by steps 1.1 and 2.1 it is complete; the expansion, Parseval identity and coefficient formula are the Hilbert-basis statements of Orthonormal families, complete orthonormal systems and Hilbert bases and Parseval equivalences for an orthonormal family, with finite-subset-net convergence.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Sphere Laplacian diagonal domain

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix a real parameter $ℏ > 0$ . Let $(Y_{l m})_{l \geq 0, - l \leq m \leq l}$ be the complete orthonormal family of $L^{2} (S^{2}, σ)$ constructed in Associated Legendre function normalization and Sphere ONB, and let $H : = span {Y_{l m} : - l \leq m \leq l, l \in N}$ be its algebraic (finite) span. Define operators $A$ and $L_{z}$ on $L^{2} (S^{2}, σ)$ by the diagonal lemma Diagonal adjoint and graph-core lemma:

$D (A) : = {F : \sum_{l, m} l^{2} (l + 1)^{2} ∣ ⟨ F, Y_{l m} ⟩ ∣^{2} < \infty}, A F : = \sum_{l, m} l (l + 1) ⟨ F, Y_{l m} ⟩ Y_{l m},$ $D (L_{z}) : = {F : \sum_{l, m} m^{2} ∣ ⟨ F, Y_{l m} ⟩ ∣^{2} < \infty}, L_{z} F : = \sum_{l, m} ℏ m ⟨ F, Y_{l m} ⟩ Y_{l m},$

the sums being in the Hilbert-space norm (Hilbert space, Orthonormal families, complete orthonormal systems and Hilbert bases). Finally, for $F$ in the finite span $H$ , use the canonical continuous representatives ${\tilde{Y}}_{l m}$ from Associated Legendre function normalization: set $\tilde{F} : = \sum_{l, m} ⟨ F, Y_{l m} ⟩ {\tilde{Y}}_{l m}$ , a finite sum uniquely determined by the orthonormal coefficients. The same convention applies to $A F, L_{z} F \in H$ . On the chart $Φ (θ, φ) = (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ with $(θ, φ) \in (0, π) \times (0, 2 π)$ , write $F (θ, φ) : = \tilde{F} (Φ (θ, φ))$ and put

$(S F) (θ, φ) : = - \frac{1}{\sin θ} \partial_{θ} (\sin θ \partial_{θ} F) (θ, φ) - \frac{1}{\sin^{2} θ} \partial_{φ}^{2} F (θ, φ), (D F) (θ, φ) : = - i ℏ \partial_{φ} F (θ, φ),$

where the partial derivatives are those of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ . Then:

(a) $A$ and $L_{z}$ are self-adjoint operators on $L^{2} (S^{2}, σ)$ with the displayed domains; on $Y_{l m}$ they act by $A Y_{l m} = l (l + 1) Y_{l m}$ and $L_{z} Y_{l m} = ℏ m Y_{l m}$ (Symmetric, self-adjoint and essentially self-adjoint operators); $A$ is nonnegative, $⟨ A F, F ⟩ \geq 0$ , and $A 1 = 0$ ;

(b) $H$ is a core of $A$ and of $L_{z}$ : the finite truncations $F_{N} : = \sum_{l \leq N} \sum_{m} ⟨ F, Y_{l m} ⟩ Y_{l m}$ of an element $F$ of $D (A)$ satisfy $F_{N} \to F$ and $A F_{N} \to A F$ in $L^{2}$ , and analogously with $L_{z}$ ;

(c) for every $F \in H$ and every chart point with $\sin θ \neq 0$ , $S F (θ, φ) = (A F) (θ, φ), D F (θ, φ) = (L_{z} F) (θ, φ);$

(d) for every $l$ the orthogonal projection $P_{l}$ of $L^{2} (S^{2}, σ)$ onto the $(2 l + 1)$ -dimensional span of $Y_{l, - l}, \dots, Y_{l l}$ , defined by $P_{l} F : = \sum_{m = - l}^{l} ⟨ F, Y_{l m} ⟩ Y_{l m}$ (the unique vector in that span with orthogonal residual), reduces both $A$ and $L_{z}$ , i.e. $P_{l} A \subseteq A P_{l}$ and $P_{l} L_{z} \subseteq L_{z} P_{l}$ ; consequently $A$ and $L_{z}$ commute on the finite span $H$ , and for every $F \in D (A)$ with $\sum_{l, m} (l (l + 1) ℏ m)^{2} ∣ ⟨ F, Y_{l m} ⟩ ∣^{2} < \infty$ one has $A L_{z} F = L_{z} A F = \sum_{l, m} l (l + 1) ℏ m ⟨ F, Y_{l m} ⟩ Y_{l m}$ . Writing $c_{l m} = ⟨ F, Y_{l m} ⟩$ and $λ_{l} = l (l + 1)$ , the maximal product domains are $D (A L_{z}) = {F \in L^{2} : \sum_{l, m} (λ_{l} ℏ m)^{2} ∣ c_{l m} ∣^{2} < \infty}$ and $D (L_{z} A) = D (A) \cap D (A L_{z})$ . Thus this is exactly the common domain $D (A L_{z}) \cap D (L_{z} A) = D (L_{z} A)$ , while the two maximal product domains need not coincide. The product-weight condition alone does not imply $F \in D (A)$ : the vector with coefficients $c_{l 0} = 1 / l$ for $l \geq 1$ and all other coefficients zero belongs to $L^{2}$ , has zero product weight, but does not belong to $D (A)$ . The product-weight condition is not automatic for all $F \in D (A) \cap D (L_{z})$ .

The differential expression $S$ is evaluated only away from the poles, where the coordinate formulas are defined; no claim is made about a maximal differential operator on all of $S^{2}$ .

Facts & Assumptions

Given: the complete orthonormal family $(Y_{l m})$ ; the finite span $H$ ; the diagonal operators $A, L_{z}$ ; the differential expressions $S, D$ ; the canonical representatives ${\tilde{Y}}_{l m} = b_{l m} w_{l m} (\cos θ) e^{i m φ} / \sqrt{2 π}$ for $m \geq 0$ , with ${\tilde{Y}}_{l, - m} = (- 1)^{m} \overline{{\tilde{Y}}_{l m}}$ for $m > 0$ , and their finite coefficient sums $\tilde{F}$ .

[F1]

Diagonal lemma: for a complete orthonormal family $(e_{j})$ and real eigenvalues $λ_{j}$ the diagonal operator $S$ with $S e_{j} = λ_{j} e_{j}$ and $D (S) = {x : \sum_{j} λ_{j}^{2} ∣ ⟨ x, e_{j} ⟩ ∣^{2} < \infty}$ is self-adjoint and is the unique self-adjoint extension of its finite span; the finite span is a core, the finite truncations converge in the graph norm, and $⟨ S x, x ⟩ = \sum_{j} λ_{j} ∣ ⟨ x, e_{j} ⟩ ∣^{2}$ (Diagonal adjoint and graph-core lemma).

[F2]

$(Y_{l m})$ is a complete orthonormal family of $L^{2} (S^{2}, σ)$ (Sphere ONB).

[F3]

$w_{l m}$ is continuous on $[- 1, 1]$ , differentiable on $(- 1, 1)$ , and satisfies the associated Legendre equation $- ((1 - t^{2}) w_{l m}^{'})^{'} + \frac{m^{2}}{1 - t^{2}} w_{l m} = l (l + 1) w_{l m}$ (Associated Legendre function normalization (a)).

[F4]

${\tilde{Y}}_{l m} (Φ (θ, φ)) = b_{l m} w_{l m} (\cos θ) e^{i m φ} / \sqrt{2 π}$ for $m \geq 0$ and ${\tilde{Y}}_{l, - m} = (- 1)^{m} \overline{{\tilde{Y}}_{l m}}$ ; their $L^{2}$ classes are $Y_{l m}$ (Associated Legendre function normalization (c)).

[F5]

Sine and cosine are differentiable with $\sin^{'} = \cos$ , $\cos^{'} = - \sin$ ; partial derivatives are computed by the usual rules, including the chain rule for $t = \cos θ$ and the product rule (The derivatives of sine and cosine are cosine and minus sine, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ). The complex exponential has complex derivative itself (The complex exponential is entire and its complex derivative is itself). For $m \neq 0$ , its defining difference quotient along increments $i m h$ gives $\frac{d}{d φ} e^{i m φ} = i m e^{i m φ}$ ; for $m = 0$ this is the constant derivative $0$ . Repeating gives $\frac{d^{2}}{d φ^{2}} e^{i m φ} = - m^{2} e^{i m φ}$ .

[F6]

For a finite orthonormal family $(u_{j})$ in a Hilbert space, set $P x : = \sum_{j} ⟨ x, u_{j} ⟩ u_{j}$ . Its value lies in their span and $⟨ x - P x, u_{j} ⟩ = 0$ for each $j$ , so it is uniquely determined by orthogonal residual. The finite pairings give $P^{2} = P$ and $⟨ P x, y ⟩ = ⟨ x, P y ⟩$ ; also $∥ x ∥^{2} = ∥ P x ∥^{2} + ∥ x - P x ∥^{2}$ , so $∥ P ∥ \leq 1$ . Thus it is the bounded self-adjoint orthogonal projection onto that span; no finite-dimensional ambient hypothesis is needed (Real and complex inner product spaces, with the inner product linear in the first argument, The induced length is a norm).

Proof

1.1F1F2F4algebra

[(a) and (b)] Enumerate the indices by $j = l^{2} + l + m$ , which bijects the block $- l \leq m \leq l$ to $l^{2} \leq j < (l + 1)^{2}$ and all blocks to $N$ . Apply [F1] to this enumeration of the complete orthonormal family $(Y_{l m})$ of [F2] with the real eigenvalues $λ_{l m} = l (l + 1)$ : the resulting operator is exactly $A$ as defined, is self-adjoint, has the displayed domain, satisfies $⟨ A F, F ⟩ = \sum_{l m} l (l + 1) ∣ ⟨ F, Y_{l m} ⟩ ∣^{2} \geq 0$ , and has the finite span $H$ as a core with the stated truncation property; the eigenvalues are nonnegative, so $A 1 = 0$ because $1 = \sqrt{4 π} Y_{00}$ by [F4] (with $P_{0} = 1$ , $b_{00} = 1 / \sqrt{2}$ ), whose eigenvalue is zero. The same application with the real eigenvalues $ℏ m$ , $m \in Z$ (whose square summability domain is the displayed one because $ℏ > 0$ ), gives clause (a) and the second half of (b) for $L_{z}$ .

1.2F3F4F5algebra

[(c) for $A$ ] For $F = Y_{l m}$ with $m \geq 0$ , evaluate its canonical representative; by [F4], $F (θ, φ) = c w (\cos θ) e^{i m φ}$ with $w = w_{l m}$ and $c = b_{l m} / \sqrt{2 π}$ . Computing the partial derivatives with [F5]: $\partial_{φ}^{2} F = - m^{2} F$ and, writing $t = \cos θ$ , $\partial_{θ} F = - c \sin θ w^{'} (t) e^{i m φ}$ , so $\sin θ \partial_{θ} F = - c \sin^{2} θ w^{'} (t) e^{i m φ}$ and $\partial_{θ} (\sin θ \partial_{θ} F) = c (- 2 \sin θ \cos θ w^{'} + \sin^{3} θ w^{''}) e^{i m φ}$ ; dividing by $- \sin θ$ gives $c (2 t w^{'} - (1 - t^{2}) w^{''}) e^{i m φ} = - c ((1 - t^{2}) w^{'})^{'} e^{i m φ}$ , where in the last step $((1 - t^{2}) w^{'})^{'} = - 2 t w^{'} + (1 - t^{2}) w^{''}$ . Also $- \sin^{- 2} θ \partial_{φ}^{2} F = + m^{2} (1 - t^{2})^{- 1} F$ . Adding, $S F = c [- ((1 - t^{2}) w^{'})^{'} + \frac{m^{2}}{1 - t^{2}} w] e^{i m φ}$ , which by [F3] equals $c l (l + 1) w e^{i m φ} = l (l + 1) F = A F$ . For negative indices, $Y_{l, - m} = (- 1)^{m} \overline{Y_{l m}}$ and the real-coefficient differential expression $S$ commutes with conjugation, so the same eigenvalue identity holds. Both sides are linear over finite sums, so the identity extends to all of $H$ .

2.1step 1.2F4F5algebra

[(c) for $L_{z}$ ] With $F = Y_{l m}$ , $\partial_{φ} F = i m F$ , so $D F = - i ℏ (i m) F = ℏ m F = L_{z} F$ by the definition of $L_{z}$ . For negative indices [F4] gives $Y_{l, - m} = (- 1)^{m} \overline{Y_{l m}}$ and $\partial_{φ} Y_{l, - m} = (- 1)^{m} \overline{\partial_{φ} Y_{l m}} = - i m Y_{l, - m}$ , so again $D Y_{l, - m} = ℏ (- m) Y_{l, - m} = L_{z} Y_{l, - m}$ . Extending by linearity to $H$ completes the proof of (c).

3.1step 1.1step 2.1F1F6algebra∎

[(d)] On the basis $A L_{z} Y_{l m} = l (l + 1) ℏ m Y_{l m} = L_{z} A Y_{l m}$ , and both operators are linear, so $A$ and $L_{z}$ commute on the finite span $H$ . For each $l$ , the span $V_{l}$ of $Y_{l, - l}, \dots, Y_{l l}$ consists of eigenvectors of $A$ with eigenvalue $l (l + 1)$ and of $L_{z}$ with eigenvalues $ℏ (- l), \dots, ℏ l$ ; by [F6] the projection is $P_{l} x = \sum_{m} ⟨ x, Y_{l m} ⟩ Y_{l m}$ , and grouping the diagonal expansions by $l$ shows $A P_{l} x = P_{l} A x$ for $x \in D (A)$ and $L_{z} P_{l} x = P_{l} L_{z} x$ for $x \in D (L_{z})$ . Finally, let $F \in D (A)$ satisfy the displayed product-weight condition. Since $∣ m ∣ \leq l$ and $l^{2} \leq l^{2} (l + 1)^{2}$ , $F \in D (L_{z})$ . The product-weight condition gives $L_{z} F \in D (A)$ and $A F \in D (L_{z})$ , so both compositions are defined: $A L_{z} F = \sum_{l, m} l (l + 1) ℏ m c_{l m} Y_{l m}$ with $\sum ∣ l (l + 1) ℏ m ∣^{2} ∣ c_{l m} ∣^{2} < \infty$ by hypothesis, and symmetrically for $L_{z} A F$ , so the two series converge to the same element. For the maximal domains, the product-weight condition implies $F \in D (L_{z})$ , since $λ_{l} \geq 2$ whenever $m \neq 0$ ; it is then exactly the condition $L_{z} F \in D (A)$ . Hence it describes $D (A L_{z})$ alone. The other composition additionally requires $F \in D (A)$ , giving $D (L_{z} A) = D (A) \cap D (A L_{z})$ . The stated $m = 0$ counterexample shows these maximal domains can differ. To verify that the product condition is not automatic on $D (A) \cap D (L_{z})$ , take $c_{l l} = l^{- 3}$ for $l \geq 1$ and every other coefficient zero: the separate domain sums are bounded by constants times $\sum l^{- 2}$ and $\sum l^{- 4}$ , respectively, but the product sum has terms $ℏ^{2} (l + 1)^{2} / l^{2} \geq ℏ^{2}$ and diverges.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Harmonic polynomial dimension

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For $k \in N$ let $P_{k}$ be the real vector space of homogeneous polynomial functions of degree $k$ on $R^{3}$ , with $P_{k} = {0}$ for $k < 0$ . Put $r^{2} = x_{1}^{2} + x_{2}^{2} + x_{3}^{2}$ , $Δ = \partial_{1}^{2} + \partial_{2}^{2} + \partial_{3}^{2}$ , and $H_{k} = \ker (Δ ∣_{P_{k}})$ . Define their explicit complexifications $P_{k}^{C} : = {p + i q : p, q \in P_{k}}$ and $H_{k}^{C} : = {p + i q : p, q \in H_{k}}$ . The Laplacian and real rotation pullbacks extend complex-linearly to these spaces. Then:

(a) for every $l \geq 2$ , $Δ : P_{l} \to P_{l - 2}$ is onto and $P_{l} = H_{l} \oplus r^{2} P_{l - 2}$ as real vector spaces; the same decomposition holds over $C$ after complexification;

(b) for every $l \geq 0$ , $\dim_{R} P_{l} = (\binom{l + 2}{2})$ and $\dim_{R} H_{l} = 2 l + 1$ , and these equal $\dim_{C} P_{l}^{C}$ and $\dim_{C} H_{l}^{C}$ , respectively;

(c) for $l \geq 0$ and $0 \leq m \leq l$ , using the exact positive constants $b_{l m}$ and Legendre polynomials of Associated Legendre function normalization, set, initially for $x \neq 0$ , $h_{l m} (x) : = (- 1)^{m} b_{l m} (2 π)^{- 1 / 2} (x_{1} + i x_{2})^{m} r^{l - m} P_{l}^{(m)} (x_{3} / r) .$ This expression has a unique polynomial extension to $R^{3}$ , with value $0$ at the origin for $l > 0$ and the nonzero constant $b_{00} / \sqrt{2 π} = 1 / \sqrt{4 π}$ for $l = 0$ . It is a nonzero element of $H_{l}^{C}$ , and $h_{l m} ∣_{S^{2}} = Y_{l m}$ . For $1 \leq m \leq l$ define $h_{l, - m} = (- 1)^{m} \overline{h_{l m}}$ . The $2 l + 1$ functions $(h_{l m})_{- l \leq m \leq l}$ form a complex basis of $H_{l}^{C}$ , and their restrictions form a complex basis of its restriction space. The restriction maps on $H_{l}$ and $H_{l}^{C}$ are injective; a corresponding real basis of $H_{l}$ is $h_{l 0}$ together with $\sqrt{2} Re h_{l m}$ and $\sqrt{2} Im h_{l m}$ for $1 \leq m \leq l$ .

(d) let $ω, m_{0}, ℏ > 0$ , $ℓ = \sqrt{ℏ / (m_{0} ω)}$ , and let $(ψ_{N})_{N \in N^{3}}$ be the complex Hilbert-space eigenbasis of the realised isotropic oscillator in Multidimensional oscillator completeness, with $H ψ_{N} = ℏ ω (∣ N ∣ + \frac{3}{2}) ψ_{N}$ , $∣ N ∣ = N_{1} + N_{2} + N_{3}$ . Put $E_{M} : = {span}_{C} {ψ_{N} : ∣ N ∣ = M}$ . Then:

(d1) each $f \in E_{M}$ has a unique representation $f = P e^{- r^{2} / (2 ℓ^{2})}$ with a complex polynomial $P$ of degree at most $M$ , and the map $Φ_{M} : E_{M} \to P_{M}^{C}$ assigning its degree- $M$ homogeneous part (zero if its degree is lower) is a complex-linear isomorphism. It restricts to a real-linear isomorphism from the real span of the same $ψ_{N}$ onto $P_{M}$ ;

(d2) every rotation $R \in S O (3)$ preserves $E_{M}$ . With $π (R) f = f (R^{- 1} \cdot)$ on $E_{M}$ and $ρ (R) p = p (R^{- 1} \cdot)$ on $P_{M}^{C}$ , one has $Φ_{M} π (R) = ρ (R) Φ_{M}$ ; the same identity holds on the real subspaces;

(d3) one has the decomposition into rotation-invariant complex subspaces $E_{M} = ⨁_{j = 0}^{⌊ M / 2 ⌋} Φ_{M}^{- 1} (r^{2 j} H_{M - 2 j}^{C}),$ and the analogous real decomposition. The harmonic orbital degrees in this level are exactly $l = M, M - 2, M - 4, \dots \geq 0$ , each occurring once as a harmonic-degree summand. In particular not every $l \leq M$ occurs, and $\sum_{j = 0}^{⌊ M / 2 ⌋} (2 (M - 2 j) + 1) = (\binom{M + 2}{2})$ is the oscillator-level multiplicity.

All assertions concern polynomials and the realised oscillator, with no physical interpretation asserted.

Facts & Assumptions

Given: The real and complex polynomial spaces, $r^{2}$ , $Δ$ , the exact $Y_{l m}$ convention, and the oscillator parameters and basis.

[F1]

Product differentiation gives $Δ (f g) = f Δ g + 2 \nabla f \cdot \nabla g + g Δ f$ , $Δ r^{2} = 6$ , and Euler's identity $x \cdot \nabla q = d q$ for a degree- $d$ homogeneous polynomial (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , homogeneous polynomial and homogeneous ideal). Sine and cosine satisfy $\sin^{'} = \cos$ and $\cos^{'} = - \sin$ (published result The derivatives of sine and cosine are cosine and minus sine). For totally differentiable maps $f : U \to V$ and $g : V \to R^{p}$ , the total derivative satisfies $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ when the composite is defined (published result The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ). The Euclidean inverse function theorem states that a $C^{1}$ map with invertible total derivative at a point has a local $C^{1}$ inverse $g$ satisfying $D g (y) = D f (g (y))^{- 1}$ (published result The Euclidean inverse function theorem). The polar-chart map $T (r, θ, φ) = r Φ (θ, φ)$ from $(0, \infty) \times (0, π) \times (0, 2 π)$ onto $R^{3} ∖ K$ , where $K : = {(x, 0, z) : x \geq 0}$ , is a $C^{1}$ diffeomorphism: this is proved in step 2.1 of Sphere integrals in angle coordinates; its determinant $r^{2} \sin θ$ is nonzero on that open domain. Thus its inverse coordinates $r, θ, φ$ are $C^{1}$ there.

[F2]

The degree- $l$ monomials in three variables are a basis of $P_{l}$ and of its complexification; their number is $(\binom{l + 2}{2})$ (Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ , For $m \geq 1$ the number of weak compositions of $n$ into $m$ parts is $(\binom{n + m - 1}{m - 1})$ , and the number of compositions is $(\binom{n - 1}{m - 1})$ for $n \geq 1$ , Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis). Finite-dimensional surjective endomorphisms are injective, by finite basis elimination.

[F3]

$P_{l} (t) = (2^{l} l!)^{- 1} (d / d t)^{l} (t^{2} - 1)^{l}$ has degree $l$ and nonzero leading coefficient (Legendre Rodrigues equation). The supplied associated function is $w_{l m} = (- 1)^{m} (1 - t^{2})^{m / 2} P_{l}^{(m)}$ , and $Y_{l m} = b_{l m} w_{l m} (\cos θ) e^{i m φ} / \sqrt{2 π}$ , with $Y_{l, - m} = (- 1)^{m} \overline{Y_{l m}}$ and the full family orthonormal in complex $L^{2} (S^{2})$ (Associated Legendre function normalization, Sphere ONB). Its angular differential equation gives $Δ_{S^{2}} Y_{l m} = - l (l + 1) Y_{l m}$ (Sphere Laplacian diagonal domain, Sphere integrals in angle coordinates, The derivatives of sine and cosine are cosine and minus sine, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The Euclidean inverse function theorem).

[F4]

$ψ_{N} = Q_{N} e^{- r^{2} / (2 ℓ^{2})}$ with $Q_{N}$ a real product of scaled Hermite polynomials, degree $∣ N ∣$ , and degree- $∣ N ∣$ part a nonzero real multiple of $x^{N}$ . The oscillator is self-adjoint and diagonal in the complete product basis, with eigenvalue $λ_{∣ N ∣} = ℏ ω (∣ N ∣ + 3 / 2)$ ; the energies differ exactly when the total degrees differ, not whenever the multi-indices differ. On the finite product span its action is $H = - ℏ^{2} Δ / (2 m_{0}) + m_{0} ω^{2} r^{2} / 2$ (Multidimensional oscillator completeness, Oscillator scaling and exact realization, Hermite recurrence, Hermite orthogonality).

Proof

1.1F1givenalgebra

[An explicit Laplacian right inverse with the correct degree] Fix $l \geq 2$ and $q \in P_{l - 2}$ . Let $J = ⌊ (l - 2) / 2 ⌋$ and $A_{j} = 2 (j + 1) (2 l - 2 j - 1)$ for $0 \leq j \leq J$ . Every $A_{j} > 0$ . Define $c_{0} = (4 l - 2)^{- 1}$ and $c_{j} = - c_{j - 1} / A_{j}$ for $1 \leq j \leq J$ , and put $S_{l} q = \sum_{j = 0}^{J} c_{j} r^{2 j + 2} Δ^{j} q$ . Every term has degree $2 j + 2 + (l - 2 - 2 j) = l$ , and the whole polynomial lies in $r^{2} P_{l - 2}$ . More generally [F1] gives $Δ (r^{2 t} p) = 2 t (2 d + 2 t + 1) r^{2 t - 2} p + r^{2 t} Δ p$ for $p \in P_{d}$ : compute $\nabla r^{2 t} = 2 t r^{2 t - 2} x$ and $Δ r^{2 t} = 2 t (2 t + 1) r^{2 t - 2}$ and substitute Euler's identity. For $t = j + 1$ , $p = Δ^{j} q$ this is $Δ (r^{2 j + 2} Δ^{j} q) = A_{j} r^{2 j} Δ^{j} q + r^{2 j + 2} Δ^{j + 1} q$ . In $Δ S_{l} q$ , the coefficient of $q$ is $c_{0} A_{0} = 1$ , each coefficient of $r^{2 j} Δ^{j} q$ for $j \geq 1$ is $c_{j} A_{j} + c_{j - 1} = 0$ , and the last leftover vanishes since $Δ^{J + 1} q = 0$ . Hence $Δ S_{l} q = q$ .

1.2F3algebra

[Legendre parity makes the solid expression a polynomial] Rodrigues' formula in [F3] gives $P_{l} (- t) = (- 1)^{l} P_{l} (t)$ by differentiating an even polynomial $l$ times. After $m$ more derivatives, $P_{l}^{(m)} (- t) = (- 1)^{l - m} P_{l}^{(m)} (t)$ , and its degree is $l - m$ with nonzero leading coefficient for $0 \leq m \leq l$ . Thus if $P_{l}^{(m)} (t) = \sum_{k} d_{k} t^{k}$ , every nonzero term has $0 \leq k \leq l - m$ and $k \equiv l - m (m o d 2)$ . Consequently $r^{l - m} P_{l}^{(m)} (x_{3} / r) = \sum_{k} d_{k} x_{3}^{k} (r^{2})^{(l - m - k) / 2}$ is a polynomial: every exponent of $r^{2}$ is a nonnegative integer. Multiplication by $(- 1)^{m} b_{l m} (x_{1} + i x_{2})^{m} / \sqrt{2 π}$ makes the displayed $h_{l m}$ homogeneous of degree $l$ . This polynomial determines the unique extension from $x \neq 0$ , taking value zero for $l > 0$ and the nonzero constant $b_{00} / \sqrt{2 π}$ for $l = 0$ ; since $P_{0} = 1$ and $\int_{- 1}^{1} P_{0}^{2} = 2$ , this constant is $1 / \sqrt{4 π}$ . In angle coordinates $x_{1} + i x_{2} = r \sin θ e^{i φ}$ , so the factor $(- 1)^{m}$ gives exactly $h_{l m} = r^{l} Y_{l m}$ , including the required convention.

1.3F2F4algebra

[Polynomial Gaussian core and the top-degree isomorphism] Write $G (x) = e^{- r^{2} / (2 ℓ^{2})}$ . The leading terms in [F4] show by induction on degree that every complex polynomial $P$ of degree at most $D$ is a finite linear combination of the $Q_{N}$ with $∣ N ∣ \leq D$ : subtract the matching nonzero multiples of all leading monomials of degree $D$ and apply the induction to the polynomial of lower degree. The same argument uses real coefficients when $P$ is real. Hence every $P G$ belongs to the oscillator's finite product core. An element of $E_{M}$ is uniquely $P G$ with degree at most $M$ , because $G$ never vanishes; equality of the continuous functions as $L^{2}$ classes implies equality everywhere and thus polynomial uniqueness. On the basis $(ψ_{N})_{∣ N ∣ = M}$ , the degree- $M$ map $Φ_{M}$ takes each vector to a nonzero real multiple of its distinct monomial $x^{N}$ . These monomials are a basis of $P_{M}^{C}$ , so $Φ_{M}$ is directly an isomorphism, and its restriction to real spans is the stated real isomorphism. This avoids the incorrect assertion that different multi-indices necessarily have different energies.

2.1step 1.1F2algebra

[Direct sum and all dimensions] Multiplication by $r^{2}$ is injective on polynomials, so $r^{2} P_{l - 2}$ and $P_{l - 2}$ have the same finite dimension. Step 1.1 proves that $Δ : r^{2} P_{l - 2} \to P_{l - 2}$ is onto, hence bijective by [F2]. For $p \in P_{l}$ , $p - S_{l} (Δ p) \in H_{l}$ and $S_{l} (Δ p) \in r^{2} P_{l - 2}$ , proving the sum. If a vector in the intersection is $r^{2} q$ and harmonic, bijectivity makes $q = 0$ , proving directness. Thus $\dim H_{l} = (\binom{l + 2}{2}) - (\binom{l}{2}) = 2 l + 1$ for $l \geq 2$ ; for $l = 0, 1$ , every polynomial is harmonic and the dimensions are $1, 3$ . Applying the real decomposition separately to real and imaginary parts proves the complex decomposition. A real basis remains a complex basis of its explicit complexification, since taking real and imaginary parts verifies both spanning and independence, so the complex dimensions agree.

2.2step 1.3F1F4algebra

[Rotations preserve oscillator levels before intertwining] Let $f = P G \in E_{M}$ and $R \in S O (3)$ . Since $∣ R^{- 1} x ∣ = ∣ x ∣$ , its pullback is $(P \circ R^{- 1}) G$ , a polynomial Gaussian of degree at most $M$ ; step 1.3 places it in the finite product core, hence in $D (H)$ . The chain rule and $R^{- 1} (R^{- 1})^{T} = I$ give $Δ (f \circ R^{- 1}) = (Δ f) \circ R^{- 1}$ , and multiplication by $r^{2}$ also commutes with the pullback. The exact differential expression of [F4] on the core therefore gives $H π (R) f = π (R) H f = λ_{M} π (R) f$ . Expand the rotated vector in its finite product basis. Since the energies are $λ_{K} = ℏ ω (K + 3 / 2)$ for total degree $K$ and are distinct when $K \neq M$ , the eigen-equation kills every coefficient outside total degree $M$ . Thus $π (R) f \in E_{M}$ , and applying $R^{- 1}$ proves equality of the preserved space. Only after establishing this invariance define $π (R)$ on $E_{M}$ . An invertible linear substitution preserves each homogeneous degree, so the degree- $M$ part of $P \circ R^{- 1}$ is $(Φ_{M} f) \circ R^{- 1}$ . Hence $Φ_{M} π (R) = ρ (R) Φ_{M}$ , also on the real subspaces.

3.1step 2.1step 1.2F1F3algebra

[Harmonicity, complex basis and its real counterpart] On the open polar-coordinate set $r > 0$ , $0 < θ < π$ , [F1] gives that the inverse coordinates are $C^{1}$ and that their derivative is the inverse Jacobian of $T$ . Differentiating the coordinate map with the sine and cosine derivatives in [F1], the Jacobian of $T$ has columns $\partial_{r} T = (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ , $\partial_{θ} T = (r \cos θ \cos φ, r \cos θ \sin φ, - r \sin θ)$ , and $\partial_{φ} T = (- r \sin θ \sin φ, r \sin θ \cos φ, 0)$ ; its determinant is $r^{2} \sin θ \neq 0$ . Matrix inversion gives the inverse Jacobian rows, hence $\partial_{x_{1}} = \sin θ \cos φ \partial_{r} + \frac{\cos θ \cos φ}{r} \partial_{θ} - \frac{\sin φ}{r \sin θ} \partial_{φ}$ , $\partial_{x_{2}} = \sin θ \sin φ \partial_{r} + \frac{\cos θ \sin φ}{r} \partial_{θ} + \frac{\cos φ}{r \sin θ} \partial_{φ}$ , and $\partial_{x_{3}} = \cos θ \partial_{r} - \frac{\sin θ}{r} \partial_{θ}$ . Squaring these operators and adding with the product rule, the mixed terms cancel, the second-derivative coefficients are $1$ , $r^{- 2}$ , and $(r^{2} \sin^{2} θ)^{- 1}$ , and the first-derivative coefficients are $2 r^{- 1}$ , $r^{- 2} \cot θ$ , and $0$ . Thus $Δ = \partial_{r}^{2} + \frac{2}{r} \partial_{r} + \frac{1}{r^{2}} (\partial_{θ}^{2} + \cot θ \partial_{θ} + \frac{1}{\sin^{2} θ} \partial_{φ}^{2})$ . Applying this identity to $h_{l m} (x) = r^{l} Y_{l m} (θ, φ)$ gives $Δ h_{l m} = r^{l - 2} (l (l + 1) Y_{l m} - S Y_{l m}) = 0$ by [F3], where $S$ is the angular differential expression from Sphere Laplacian diagonal domain, Sphere integrals in angle coordinates, The derivatives of sine and cosine are cosine and minus sine, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The Euclidean inverse function theorem and $S Y_{l m} = l (l + 1) Y_{l m}$ . The polynomial $Δ h_{l m}$ is continuous and vanishes on this dense set, hence vanishes everywhere. Its real and imaginary parts are therefore real harmonic polynomials. Negative indices follow by complex conjugation, and nonvanishing follows from the norm-one restriction $Y_{l m}$ . Restriction is injective: if a homogeneous polynomial vanishes on $S^{2}$ , then $p (x) = ∣ x ∣^{l} p (x / ∣ x ∣) = 0$ for $x \neq 0$ , and continuity also gives its value at zero. Thus the $2 l + 1$ orthonormal restrictions are linearly independent in a complex space of dimension $2 l + 1$ by step 2.1 and hence form a basis. The invertible pairwise transformation from $h_{l m}, h_{l, - m}$ to $\sqrt{2} Re h_{l m}, \sqrt{2} Im h_{l m}$ , together with real $h_{l 0}$ , produces $2 l + 1$ real harmonic polynomials which are complex-linearly independent, hence real-linearly independent and a real basis.

3.2step 2.1step 1.3step 2.2F1F2F4algebra

[Full oscillator-level decomposition and multiplicities] Iterating the proved direct decomposition of step 2.1 yields $P_{M} = ⨁_{j = 0}^{⌊ M / 2 ⌋} r^{2 j} H_{M - 2 j}$ : at each substitution directness is preserved because multiplication by $r^{2}$ is injective. Complexifying gives the same result for $P_{M}^{C}$ . A rotation commutes with the Laplacian, preserves degree, and preserves $r^{2}$ , so each displayed harmonic-degree summand is invariant. Transfer it through the intertwining isomorphism of steps 1.3 and 2.2 to obtain exactly the stated decomposition of $E_{M}$ , with one copy of each degree $M, M - 2, \dots$ . All listed degrees occur because their dimensions $2 (M - 2 j) + 1$ are positive; no other degree is part of the exhaustive direct decomposition. Their sum is $(\binom{M + 2}{2})$ by the polynomial-space dimension, and equals the number of oscillator basis vectors with $∣ N ∣ = M$ by [F2] and [F4]. The same transfer over $R$ gives the claimed real decomposition.

4.1step 1.1step 2.1step 1.2step 3.1step 1.3step 2.2step 3.2given∎

[Conclusion] Steps 1.1 and 2.1 prove (a) and (b), steps 1.2 and 3.1 prove (c), and steps 1.3, 2.2 and 3.2 prove (d1)–(d3) with the full real and complex spaces explicit. Countable Choice is retained for the invoked sphere and realised-oscillator suppliers; the finite polynomial recursions, finite basis changes and fixed top-degree maps require no additional choice principle.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Strongly continuous rotation action

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $S^{2} : = {x \in R^{3} : ∣ x ∣ = 1}$ , let $σ$ be the polar surface measure on $S^{2}$ (The polar surface set function on the unit sphere), a finite Borel measure with $σ (S^{2}) = 4 π$ (Sphere integrals in angle coordinates), and let $L^{2} (S^{2}, σ)$ be the complex Hilbert space of equivalence classes of square-integrable functions with inner product linear in the first argument (The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions, Hilbert space, Real and complex inner product spaces, with the inner product linear in the first argument). Let $(Y_{l m})_{l \geq 0, - l \leq m \leq l}$ be the complete orthonormal family of Associated Legendre function normalization and Sphere ONB, put $V_{l} : = span {Y_{l, - l}, \dots, Y_{l l}}$ and let $H$ be the algebraic span of all $Y_{l m}$ . For $l \geq 0$ let $H_{l}^{p o l} : = {p : R^{3} \to C : p is a homogeneous polynomial of degree l with Δ p = 0}$ be the complex vector space of harmonic homogeneous polynomials of degree $l$ (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), let ${r e s}_{l} : H_{l}^{p o l} \to L^{2} (S^{2}, σ)$ send $p$ to the class of the restriction $p ∣_{S^{2}}$ , and let $R \in S O (3)$ range over the special orthogonal group, a subgroup of the general linear group with $R^{- 1} = R^{T}$ and unit operator norm, the set $M_{3} (R)$ being identified with $R^{9}$ and given the Euclidean metric (The orthogonal group $O (3)$ and the rotation group $S O (3)$ , $S O (3)$ is a subgroup of the general linear group, with $R^{- 1} = R^{T}$ ). For $F \in L^{2} (S^{2}, σ)$ define $U (R) F$ to be the class of the function $Ω \mapsto \tilde{F} (R^{- 1} Ω)$ , where $\tilde{F}$ is any representative of $F$ . Then:

(a) $U (R)$ is a well-defined $C$ -linear isometry of $L^{2} (S^{2}, σ)$ onto itself with inverse $U (R^{- 1})$ , hence a unitary operator (Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces); $U (I_{3}) = i d$ and $U (R S) = U (R) U (S)$ for all $R, S \in S O (3)$ , so $R \mapsto U (R)$ is a group homomorphism from $S O (3)$ into the group of unitary operators on $L^{2} (S^{2}, σ)$ ;

(b) for every $R \in S O (3)$ and $p \in H_{l}^{p o l}$ one has $p \circ R^{- 1} \in H_{l}^{p o l}$ ; the map $ρ (R) : p \mapsto p \circ R^{- 1}$ is a $C$ -linear automorphism of $H_{l}^{p o l}$ with inverse $ρ (R^{- 1})$ , and ${r e s}_{l} (ρ (R) p) = U (R) {r e s}_{l} (p)$ ; consequently $U (R) V_{l} = V_{l}$ for every $l \geq 0$ , and $U (R) H = H$ ;

(c) (strong continuity) for every $F \in L^{2} (S^{2}, σ)$ and every $R_{0} \in S O (3)$ , for every $ε > 0$ there is $δ > 0$ such that $∥ R - R_{0} ∥_{F} < δ$ for the Euclidean (Frobenius) norm on $M_{3} (R)$ implies $∥ U (R) F - U (R_{0}) F ∥_{2} < ε$ ; thus every orbit map $R \mapsto U (R) F$ is continuous (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form). More precisely, if $P_{N} F : = \sum_{l \leq N} \sum_{m} ⟨ F, Y_{l m} ⟩ Y_{l m}$ denotes the orthogonal truncation of $F$ to $⨁_{l \leq N} V_{l}$ (Orthonormal families, complete orthonormal systems and Hilbert bases), then for all $R$ $∥ U (R) F - F ∥_{2} \leq 2 ∥ F - P_{N} F ∥_{2} + ∥ U (R) P_{N} F - P_{N} F ∥_{2} .$

All assertions are mathematical statements about the action of $S O (3)$ on $L^{2} (S^{2}, σ)$ ; no physical interpretation of the rotation action is asserted.

Facts & Assumptions

Given: the sphere $S^{2}$ , the polar measure $σ$ , the complete orthonormal family $(Y_{l m})$ , the blocks $V_{l}$ , the finite span $H$ , the spaces $H_{l}^{p o l}$ , the restriction maps ${r e s}_{l}$ , and the group $S O (3)$ .

[F1]

The polar surface measure equals the chart surface measure on $S^{2}$ , is finite with $σ (S^{2}) = 4 π$ , and is preserved by orthogonal transformations: $σ (O E) = σ (E)$ for every orthogonal $O$ and every Borel $E \subseteq S^{2}$ (Agreement with the existing polar sphere measure, The polar surface set function on the unit sphere, Sphere integrals in angle coordinates).

[F2]

$(Y_{l m})$ is a complete orthonormal family of $L^{2} (S^{2}, σ)$ ; each $V_{l}$ has the orthonormal basis $(Y_{l, - l}, \dots, Y_{l l})$ ; $V_{l} ⊥ V_{k}$ for $l \neq k$ ; $H$ is dense; and the orthogonal truncations $P_{N} F$ converge to $F$ in norm (Parseval) (Sphere ONB, Orthonormal families, complete orthonormal systems and Hilbert bases, Parseval equivalences for an orthonormal family).

[F3]

Each $Y_{l m}$ has a continuous polynomial representative on $S^{2}$ by [F4] and Associated Legendre function normalization. For $0 \leq m \leq l$ , its chart formula is $Y_{l m} (θ, φ) = b_{l m} w_{l m} (\cos θ) e^{i m φ} / \sqrt{2 π}$ ; for $1 \leq m \leq l$ , $Y_{l, - m} = (- 1)^{m} \overline{Y_{l m}}$ . Thus the negative-index formula includes the factor $(- 1)^{m}$ , not the phase-free positive-index expression. Continuity at chart boundaries follows from the polynomial representative, and conjugation preserves it. Every $F \in H$ is consequently continuous on the whole sphere.

[F4]

The restriction map ${r e s}_{l}$ is a $C$ -linear isomorphism $H_{l}^{p o l} \to V_{l}$ , so every $F \in V_{l}$ has a unique representative $p_{F} \in H_{l}^{p o l}$ with ${r e s}_{l} (p_{F}) = F$ (Harmonic polynomial dimension (b),(c)).

[F5]

Polynomial maps on $R^{3}$ are $C^{\infty}$ ; for a $C^{2}$ function $p$ and the linear map $L (x) = R^{- 1} x$ the chain rule gives $J (p \circ L) (x) = J p (L x) R^{- 1}$ and therefore $Δ (p \circ L) (x) = tr ((R^{- 1})^{T} D^{2} p (L x) R^{- 1}) = Δ p (L x)$ , because $R^{- 1} (R^{- 1})^{T} = I_{3}$ ; the top-degree part of $p \circ L$ is the composition of the top-degree part of $p$ with $L$ , so invertibility of $L$ preserves the degree of a nonzero polynomial and preserves homogeneity; sums and scalar multiples of harmonic homogeneous polynomials of one degree are again harmonic homogeneous of that degree (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , A total derivative computes every directional derivative, and its matrix is the Jacobian, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , $C^{k}$ Euclidean maps and diffeomorphisms, $C^{k}$ Euclidean maps are closed under componentwise algebra and composition, homogeneous polynomial and homogeneous ideal, Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ ).

[F6]

Let $Φ : S^{2} \to S^{2}$ be a measurable bijection with $σ (Φ^{- 1} E) = σ (E)$ for all Borel $E$ ; then for $F, G \in L^{2} (S^{2}, σ)$ the functions $F \circ Φ$ , $G \circ Φ$ lie in $L^{2} (S^{2}, σ)$ , with $⟨ F \circ Φ, G \circ Φ ⟩ = ⟨ F, G ⟩$ (substitution formula for the measure-preserving bijection), and $F = G$ $σ$ -a.e. implies $F \circ Φ = G \circ Φ$ $σ$ -a.e.; these identities are those of the change-of-variables formula applied to indicators and to nonnegative Borel functions and then to the four positive and negative parts (Agreement with the existing polar sphere measure, The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions).

[F7]

A continuous function on the compact metric space $S^{2}$ is uniformly continuous (Every continuous map from a nonempty compact Hausdorff space to a uniform space is uniformly continuous); if $g : S^{2} \to C$ is continuous and bounded then $\int_{S^{2}} ∣ g ∣^{2} d σ \leq 4 π ∥ g ∥_{\infty}^{2}$ , because the nonnegative integral is monotone and $σ (S^{2}) = 4 π$ (The nonnegative Lebesgue integral, [F1]).

[F8]

Euclidean norms on $M_{3} (R) ≅ R^{9}$ and on $R^{3}$ : for the Frobenius norm $∥ A ∥_{F} = (\sum_{i, j} A_{i j}^{2})^{1 / 2}$ , $∣ A x ∣ \leq ∥ A ∥_{F} ∣ x ∣$ for all $x \in R^{3}$ by the Cauchy-Schwarz inequality; for $R \in S O (3)$ , $∥ R ∥_{F} = \sqrt{3}$ and $∥ R ∥ = 1$ , and $R^{- 1} - R_{0}^{- 1} = R^{- 1} (R_{0} - R) R_{0}^{- 1}$ for $R, R_{0} \in S O (3)$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors, $S O (3)$ is a subgroup of the general linear group, with $R^{- 1} = R^{T}$ , The orthogonal group $O (3)$ and the rotation group $S O (3)$ ).

[F9]

$S O (3)$ is a subgroup of the general linear group with $R^{- 1} = R^{T}$ , so $(R S)^{- 1} = S^{- 1} R^{- 1}$ and $I_{3}^{- 1} = I_{3}$ for all $R, S \in S O (3)$ ( $S O (3)$ is a subgroup of the general linear group, with $R^{- 1} = R^{T}$ , The orthogonal group $O (3)$ and the rotation group $S O (3)$ ).

Proof

1.1F1F6algebra

[(a), substitution map] Let $R \in S O (3)$ and $Φ : = R^{- 1}$ . Then $Φ$ maps $S^{2}$ bijectively onto $S^{2}$ , since $∣ R^{- 1} y ∣ = ∣ y ∣$ , and by [F1] with the orthogonal matrix $R$ one has $σ (Φ^{- 1} E) = σ (R E) = σ (E)$ for every Borel $E$ . If $\tilde{F} = {\tilde{F}}^{'}$ is any representative pair of a class $F \in L^{2}$ , then ${F \neq F^{'}}$ has $σ$ -measure zero, hence so does $R {F \neq F^{'}}$ by [F1], so all representatives produce the same class: $U (R)$ is well defined; and $U (R) F \in L^{2}$ with $∥ U (R) F ∥_{2} = ∥ F ∥_{2}$ , $⟨ U (R) F, U (R) G ⟩ = ⟨ F, G ⟩$ , by [F6].

1.2F5algebra

[(b), harmonicity] Let $p \in H_{l}^{p o l}$ and $R \in S O (3)$ ; by [F5] with $L (x) = R^{- 1} x$ , $p \circ R^{- 1}$ is $C^{\infty}$ with $Δ (p \circ R^{- 1}) (x) = Δ p (R^{- 1} x) = 0$ for all $x$ , and since $R^{- 1}$ is an invertible linear substitution, $p \circ R^{- 1}$ is a homogeneous polynomial of degree $l$ (for $p = 0$ it is $0$ ); hence $p \circ R^{- 1} \in H_{l}^{p o l}$ . The assignment $ρ (R)$ is $C$ -linear because composition with a fixed map is linear, and $ρ (R) ρ (R^{- 1}) = ρ (R^{- 1}) ρ (R) = i d$ because $p \circ R^{- 1} \circ R = p$ and $p \circ R \circ R^{- 1} = p$ ; so $ρ (R)$ is a linear automorphism of $H_{l}^{p o l}$ .

1.3F3F7F8algebra

[(c), continuity on the finite span] Let $F \in H$ and $R_{0} \in S O (3)$ . By [F3] $F$ is continuous on $S^{2}$ , hence uniformly continuous by [F7]: given $ε$ there is $η > 0$ with $∣ x - y ∣ < η$ , $x, y \in S^{2}$ , implying $∣ F (x) - F (y) ∣ < ε / (2 \sqrt{4 π})$ . For $Ω \in S^{2}$ and $R \in S O (3)$ , $U (R) F (Ω) - U (R_{0}) F (Ω) = F (R^{- 1} Ω) - F (R_{0}^{- 1} Ω)$ and, by [F8], $∣ R^{- 1} Ω - R_{0}^{- 1} Ω ∣ = ∣ R^{- 1} (R_{0} - R) R_{0}^{- 1} Ω ∣ \leq ∥ R^{- 1} ∥_{o p} ∥ R_{0} - R ∥_{F} ∥ R_{0}^{- 1} Ω ∥ = ∥ R - R_{0} ∥_{F}$ . Choosing $δ : = η$ one gets $\sup_{S^{2}} ∣ U (R) F - U (R_{0}) F ∣ \leq ε / (2 \sqrt{4 π})$ whenever $∥ R - R_{0} ∥_{F} < δ$ , and then [F7] gives $∥ U (R) F - U (R_{0}) F ∥_{2} < ε / 2 < ε$ .

2.1step 1.1F9algebra

[(a), group law and unitarity] For $F \in L^{2}$ and $Ω \in S^{2}$ , $(U (R^{- 1}) U (R) F) (Ω) = (U (R) F) (R Ω) = \tilde{F} (R^{- 1} R Ω) = \tilde{F} (Ω)$ and symmetrically $U (R) U (R^{- 1}) = i d$ , so $U (R)$ is a bijective linear isometry, i.e. unitary, with inverse $U (R^{- 1})$ (Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces); $U (I_{3}) = i d$ since $R^{- 1} = R^{T} = Ω$ -fixing for $R = I_{3}$ ; $(U (R) U (S) F) (Ω) = (U (S) F) (R^{- 1} Ω) = \tilde{F} (S^{- 1} R^{- 1} Ω) = \tilde{F} ((R S)^{- 1} Ω) = (U (R S) F) (Ω)$ by [F9] the inverse identity $(R S)^{- 1} = S^{- 1} R^{- 1}$ .

3.1step 2.1step 1.2F4algebra

[(b), intertwining and blocks] Let $F \in V_{l}$ with representative $p = p_{F} \in H_{l}^{p o l}$ ([F4]) and let $R \in S O (3)$ . For $Ω \in S^{2}$ , $U (R) F (Ω) = \tilde{F} (R^{- 1} Ω) = p (R^{- 1} Ω) = (p \circ R^{- 1}) (Ω)$ : here $R^{- 1} Ω$ lies in $S^{2}$ , so the identity of classes ${r e s}_{l} (ρ (R) p) = U (R) {r e s}_{l} (p)$ follows. Hence $U (R) V_{l} \subseteq V_{l}$ for every $l$ by step 1.2 and [F4], and applying $U (R^{- 1})$ (step 2.1) gives equality; since $H$ is the span of the $V_{l}$ and $U (R)$ is linear, $U (R) H = H$ .

4.1step 2.1step 1.3F2algebra∎

[(c), the estimate and general $F$ ] For arbitrary $F \in L^{2} (S^{2}, σ)$ and any $R, R_{0}$ , the unitarity of step 2.1 gives $∥ U (R) F - U (R_{0}) F ∥_{2} = ∥ U (R) (F - P_{N} F) + U (R) P_{N} F - U (R_{0}) P_{N} F + U (R_{0}) (P_{N} F - F) ∥_{2} \leq 2 ∥ F - P_{N} F ∥_{2} + ∥ U (R) P_{N} F - U (R_{0}) P_{N} F ∥_{2}$ , with $P_{N} F \in H$ by [F2]. Given $ε > 0$ , choose $N$ with $∥ F - P_{N} F ∥_{2} < ε / 4$ by [F2], then by step 1.3 a $δ > 0$ with $∥ U (R) P_{N} F - U (R_{0}) P_{N} F ∥_{2} < ε / 2$ when $∥ R - R_{0} ∥_{F} < δ$ ; then $∥ U (R) F - U (R_{0}) F ∥_{2} < ε$ . Taking $R_{0} = I_{3}$ and keeping the terms separately yields the displayed estimate $∥ U (R) F - F ∥_{2} \leq 2 ∥ F - P_{N} F ∥_{2} + ∥ U (R) P_{N} F - P_{N} F ∥_{2}$ .

The orbit continuity proved in (c) is the strong-continuity condition of the definition of a unitary representation (Strongly continuous unitary representations, invariant linear subspaces and intertwiners) for the group $S O (3)$ equipped with the Euclidean-metric topology inherited from $M_{3} (R) ≅ R^{9}$ ; this identification of vocabulary is not used in the proof of the present batch, whose mathematical content is exactly clauses (a)-(c).
Clause (b) is the statement that rotations permute the angular-momentum blocks $V_{l}$ and act on each of them by the finite-dimensional automorphism $ρ (R)$ transported by the restriction isomorphism ${r e s}_{l}$ ; it is the algebraic input for the self-adjoint generators of the following item.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Orbital self-adjoint generators

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Use the notation of Strongly continuous rotation action: $S^{2}$ , the polar measure $σ$ with $σ (S^{2}) = 4 π$ , the Hilbert space $L^{2} (S^{2}, σ)$ , the complete orthonormal family $(Y_{l m})$ , the blocks $V_{l}$ , the finite span $H$ , the spaces $H_{l}^{p o l}$ of complex harmonic homogeneous polynomials of degree $l$ , the restriction maps ${r e s}_{l}$ , and the unitary action $U (R) F = class of \tilde{F} (R^{- 1} \cdot)$ of $S O (3)$ . Fix $ℏ > 0$ and for $i = 1, 2, 3$ define the rotation matrices $R_{1} (t) : = (\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos t & - \sin t \\ 0 & \sin t & \cos t \end{array}), R_{2} (t) : = (\begin{array}{ccc} \cos t & 0 & \sin t \\ 0 & 1 & 0 \\ - \sin t & 0 & \cos t \end{array}), R_{3} (t) : = (\begin{array}{ccc} \cos t & - \sin t & 0 \\ \sin t & \cos t & 0 \\ 0 & 0 & 1 \end{array}),$ put $U_{i} (t) : = U (R_{i} (t))$ . Let $L_{z}$ denote the diagonal operator of Sphere Laplacian diagonal domain, set $S : = R_{2} (- π / 2)$ , $W : = R_{1} (π / 2)$ , and define $L_{3} : = L_{z}, L_{1} : = U (S)^{- 1} L_{z} U (S), L_{2} : = U (W)^{- 1} L_{z} U (W),$ with domains $D (L_{1}) = U (S)^{- 1} D (L_{z})$ , $D (L_{2}) = U (W)^{- 1} D (L_{z})$ and $D (L_{3}) = D (L_{z})$ . Here $e^{- i t L_{z} / ℏ}$ is the explicit diagonal unitary sending $Y_{l m}$ to $e^{- i t m} Y_{l m}$ ; $e^{- i t L_{1} / ℏ}$ and $e^{- i t L_{2} / ℏ}$ mean its conjugates by $U (S)$ and $U (W)$ , respectively. The proof identifies these operators with the complete strong derivative generators, so this construction has the same generator meaning without an appeal to an unrestricted spectral or Stone theorem. For a polynomial $p$ put $D_{i} p : = (x \times \nabla p)_{i} = \sum_{j, k = 1}^{3} ε_{i j k} x_{j} \partial_{k} p,$ where $ε_{i j k}$ is the sign of the permutation $(1, 2, 3) \mapsto (i, j, k)$ and is $0$ when two indices coincide. Then:

(a) each $L_{i}$ is a self-adjoint operator on $L^{2} (S^{2}, σ)$ with $U (R_{i} (t)) = e^{- i t L_{i} / ℏ}$ for all real $t$ ; the finite span $H$ is contained in $D (L_{i})$ and is a core of $L_{i}$ (Densely defined, closed and closable operators, and cores, Symmetric, self-adjoint and essentially self-adjoint operators); and for $F \in V_{l}$ with polynomial representative $p_{F} \in H_{l}^{p o l}$ one has $L_{i} F = {r e s}_{l} (- i ℏ D_{i} p_{F}),$ that is, $(L_{i} F) (Ω) = - i ℏ D_{i} p_{F} (Ω)$ for $σ$ -almost every $Ω \in S^{2}$ ; in particular $L_{i} Y_{l m} \in V_{l}$ for all $l, m$ and $L_{i} H \subseteq H$ ;

(b) on $H$ the operators satisfy the commutation relations $L_{i} L_{j} - L_{j} L_{i} = i ℏ \sum_{k = 1}^{3} ε_{i j k} L_{k};$

(c) $L_{3} = L_{z}$ as self-adjoint operators (equal domains and equal action); in particular $L_{3} Y_{l m} = ℏ m Y_{l m}$ , so that $L_{3}$ is diagonal in the basis $(Y_{l m})$ , and for every $F \in H$ and chart point $(θ, φ)$ one has $(L_{3} F) (θ, φ) = - i ℏ \partial_{φ} F (θ, φ)$ ;

(d) rotational conjugation identifies the domains: with the two quarter-turn rotations $S : = R_{2} (- π / 2)$ and $W : = R_{1} (π / 2)$ one has $U (S) L_{1} U (S)^{- 1} = L_{3}, U (W) L_{2} U (W)^{- 1} = L_{3},$ as self-adjoint operators, and consequently $U (S) D (L_{1}) = D (L_{3})$ and $U (W) D (L_{2}) = D (L_{3})$ : the three domains are unitarily identified.

All assertions are mathematical statements about operators on $L^{2} (S^{2}, σ)$ ; no physical interpretation of the generators as angular momentum is asserted, and no measurement postulate is used.

Facts & Assumptions

Given: the sphere, its measure, the harmonic basis, the blocks $V_{l}$ , their finite span $H$ , the rotation action, $ℏ > 0$ , and the matrices and explicitly defined operators in the Statement. Inner products are linear in the first argument.

[F1]

Strongly continuous rotation action supplies the unitary group law, strong continuity, $U (R) V_{l} = V_{l}$ , $U (R) H = H$ and polynomial intertwining. Harmonic polynomial dimension supplies the complex restriction isomorphism ${r e s}_{l} : H_{l}^{p o l} \to V_{l}$ .

[F2]

Sphere Laplacian diagonal domain supplies $L_{z} Y_{l m} = ℏ m Y_{l m}$ with full domain ${F : \sum_{l, m} m^{2} ∣ ⟨ F, Y_{l m} ⟩ ∣^{2} < \infty}$ . Sphere ONB and Parseval give the complete countable basis, mutually orthogonal blocks and convergence of $P_{\leq N} : = \sum_{l \leq N} P_{l}$ to the identity. Associated Legendre function normalization gives the azimuthal factor $e^{i m φ}$ , also for negative $m$ by $Y_{l, - m} = (- 1)^{m} \overline{Y_{l m}}$ .

[F3]

Direct multiplication of the displayed matrices, the sine/cosine addition and derivative formulas, and the quarter-turn values give $R_{i}^{'} (s) = J_{i} R_{i} (s) = R_{i} (s) J_{i}$ , where $J_{i} x = e_{i} \times x$ ; $R_{i} (- s)^{'} = - J_{i} R_{i} (- s)$ ; $S R_{1} (t) S^{- 1} = R_{3} (t)$ and $W R_{2} (t) W^{- 1} = R_{3} (t)$ . These are finite matrix identities, not generator identities.

[F4]

Polynomial chain and product rules, the fundamental theorem of calculus, and uniform continuity on compact sets apply (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , Every continuous map from a nonempty compact Hausdorff space to a uniform space is uniformly continuous, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ). For a continuous function $f$ on the sphere, $∥ f ∥_{2} \leq \sqrt{4 π} ∥ f ∥_{\infty}$ .

Proof

1.1F2algebra

[(a), the complete diagonal domain and adjoint] Write $c_{l m} = ⟨ F, Y_{l m} ⟩$ . On the domain of [F2], Parseval defines $A F = \sum_{l, m} ℏ m c_{l m} Y_{l m}$ with convergence in norm, and finite sums prove symmetry. If $G \in D (A^{*})$ and $K = A^{*} G$ , testing $F = Y_{l m}$ in $⟨ A F, G ⟩ = ⟨ F, K ⟩$ gives $⟨ K, Y_{l m} ⟩ = ℏ m ⟨ G, Y_{l m} ⟩$ . Parseval then implies $\sum ℏ^{2} m^{2} ∣ ⟨ G, Y_{l m} ⟩ ∣^{2} = ∥ K ∥^{2} < \infty$ : $G \in D (A)$ and $K = A G$ . Conversely, for such $G$ , the equality holds first on finite sums and then for all $F \in D (A)$ by their coefficient truncations. Thus $A = A^{*}$ on exactly the full domain of [F2], and $A = L_{z}$ . For $F \in D (A)$ , both $P_{\leq N} F \to F$ and $A P_{\leq N} F \to A F$ by the two convergent coefficient square sums; hence $H$ is a graph core. All sums use the fixed countable index family; no fiber selection or uncountable basis choice occurs.

2.1step 1.1F1algebra

[(a),(d), adjoints and cores of the conjugates] If $Q$ is either $U (S)$ or $U (W)$ , define $B = Q^{- 1} A Q$ on $Q^{- 1} D (A)$ . For $G, K$ , the adjoint identity $⟨ B F, G ⟩ = ⟨ F, K ⟩$ for every $F \in Q^{- 1} D (A)$ is, on putting $X = Q F$ , exactly $⟨ A X, Q G ⟩ = ⟨ X, Q K ⟩$ for every $X \in D (A)$ . By step 1.1 this holds exactly when $Q G \in D (A)$ and $Q K = A Q G$ . Thus $B^{*} = B$ with this full domain. Since $Q H = H$ , unitary transport of the graph approximations in step 1.1 gives $H$ as a core of $B$ . With $L_{3} = A$ , this proves self-adjointness, cores and the two stated conjugation/domain identities.

2.2step 1.1F2algebra

[(a), explicit groups and the full derivative domains] Define $E (t) F = \sum e^{- i t m} c_{l m} Y_{l m}$ . Parseval proves unitarity and the group law. For each $F$ , a finite coefficient truncation controls the tail by $2 ∥ F - P_{\leq N} F ∥$ and its finite head is continuous, proving strong continuity. If $F \in D (A)$ , $∣ (e^{- i t m} - 1) / t ∣ \leq ∣ m ∣$ and convergence on finite heads with square-summable weighted tails give $(E (t) F - F) / t \to - i A F / ℏ$ . Conversely, if the quotient converges in norm to $G$ , each coefficient tends to $- i m c_{l m}$ ; Parseval for $G$ gives $\sum m^{2} ∣ c_{l m} ∣^{2} = ∥ G ∥^{2} < \infty$ and $G = - i A F / ℏ$ . Thus this is exactly the full strong derivative domain, not merely a core calculation. The same argument transported by $Q$ gives exactly $D (Q^{- 1} A Q)$ for $Q^{- 1} E (t) Q$ , with derivative $- i Q^{- 1} A Q F / ℏ$ .

3.1step 2.2F1F2F3algebra

[(a),(c), identification with the actual rotation groups] Pullback by $R_{3} (t)^{- 1}$ replaces $φ$ by $φ - t$ . Therefore $U_{3} (t) Y_{l m} = e^{- i t m} Y_{l m}$ for every integer $m$ by [F2], including negative $m$ with its conjugation phase. Equality on the dense finite span, and boundedness, imply $U_{3} (t) = E (t)$ on all $L^{2}$ . The two finite matrix identities in [F3], transported by the group law, imply $U_{1} (t) = U (S)^{- 1} E (t) U (S)$ and $U_{2} (t) = U (W)^{- 1} E (t) U (W)$ . These are precisely the explicitly defined exponentials in the Statement. Step 2.2 gives the exact derivative domains for all three groups and their actions $- i L_{i} / ℏ$ ; it also proves uniqueness if an operator is defined by the strong derivative. In particular $L_{3} = L_{z}$ as full operators, without an AC Stone premise.

4.1step 2.2step 3.1F1F3F4algebra

[(a), polynomial differentiation with the correct inverse-curve sign] For $F = {r e s}_{l} (p) \in V_{l}$ , set $g_{i} (s, Ω) = \nabla p (R_{i} (- s) Ω) \cdot J_{i} R_{i} (- s) Ω$ . The correct chain derivative is $\frac{d}{d s} p (R_{i} (- s) Ω) = - g_{i} (s, Ω)$ , directly from $R_{i} (- s)^{'} = - J_{i} R_{i} (- s)$ in [F3]. The fundamental theorem gives $(U_{i} (t) F - F) / t = - t^{- 1} \int_{0}^{t} g_{i} (s, \cdot) d s$ . Uniform continuity on $[- δ, δ] \times S^{2}$ proves convergence uniformly, and hence in $L^{2}$ , to $- g_{i} (0, \cdot) = - D_{i} p ∣_{S^{2}}$ . By step 2.2, $F \in D (L_{i})$ and $- i L_{i} F / ℏ = - r e s (D_{i} p)$ , so $L_{i} F = r e s (- i ℏ D_{i} p)$ . The curve $p \circ R_{i} (- t)$ lies in the finite-dimensional polynomial space $H_{l}^{p o l}$ by [F1]; its coefficientwise derivative at zero also lies there, because homogeneity and the harmonic equations are finite linear coefficient constraints. This derivative is $- D_{i} p$ , proving that $D_{i} p$ is harmonic homogeneous of degree $l$ , including the zero polynomial. Thus $L_{i} V_{l} \subseteq V_{l}$ and $L_{i} H \subseteq H$ .

5.1step 1.1step 2.1step 3.1step 4.1F1F2algebra

[(a), finite blocks and their ODE identification] Put $A_{i}^{(l)} = L_{i} ∣_{V_{l}}$ . This is an everywhere-defined Hermitian matrix: symmetry follows from self-adjointness on the whole space and $V_{l} \subseteq D (L_{i})$ . The quarter turns preserve $V_{l}$ , so $A_{1}^{(l)}$ and $A_{2}^{(l)}$ are unitarily conjugate to the diagonal $A_{3}^{(l)}$ with entries $ℏ m$ . Consequently $∥ A_{i}^{(l)} ∥ \leq ℏ l$ . Polynomial differentiation at time $t$ , or the finite group law and step 4.1, gives $(U_{i} (t) ∣_{V_{l}})^{'} = (- i / ℏ) A_{i}^{(l)} U_{i} (t) ∣_{V_{l}}$ with initial value the identity. The absolutely convergent matrix exponential power series solves this equation; uniqueness follows by differentiating its inverse times any other solution. Hence the finite polynomial action equals $\exp (- i t A_{i}^{(l)} / ℏ)$ . Orthogonal block truncation then identifies its all-vector extension with the groups of step 3.1. Explicitly $D (L_{i}) = {F : \sum_{l} ∥ A_{i}^{(l)} P_{l} F ∥^{2} < \infty}$ and $L_{i} F = \sum_{l} A_{i}^{(l)} P_{l} F$ : this is the diagonal domain of step 1.1 transported by a block-preserving quarter turn. Thus no hidden narrower domain or additional time-continuity hypothesis is introduced.

5.2step 4.1F4algebra

[(b), commutators] The polynomial product rule gives $[x_{a} \partial_{b}, x_{c} \partial_{d}] = x_{a} δ_{b c} \partial_{d} - x_{c} δ_{a d} \partial_{b}$ . Substituting $D_{1} = x_{2} \partial_{3} - x_{3} \partial_{2}$ , $D_{2} = x_{3} \partial_{1} - x_{1} \partial_{3}$ gives $[D_{1}, D_{2}] = - D_{3}$ , and cyclic permutation and antisymmetry give $[D_{i}, D_{j}] = - \sum_{k} ε_{i j k} D_{k}$ . All products are defined on $H$ by step 4.1. Therefore $[L_{i}, L_{j}] = (- i ℏ)^{2} [D_{i}, D_{j}] = ℏ^{2} \sum_{k} ε_{i j k} D_{k} = i ℏ \sum_{k} ε_{i j k} L_{k}$ .

6.1step 2.1step 3.1step 4.1F4algebra∎

[(c), the chart action] The chain rule gives $\partial_{φ} p (\sin θ \cos φ, \sin θ \sin φ, \cos θ) = \nabla p (Ω) \cdot (e_{3} \times Ω) = D_{3} p (Ω)$ . Apply step 4.1 and linearity to finite harmonic sums. Together with the full equality $L_{3} = L_{z}$ of step 3.1 and the domain conjugations of step 2.1, this proves every clause (a)-(d).

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Dipole angular selection rules

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Use the notation of Harmonic polynomial dimension: $S^{2} = {x \in R^{3} : ∣ x ∣ = 1}$ with polar measure $σ$ (Sphere integrals in angle coordinates, Complex Lp classes and Euclidean test-function conventions), the Hilbert space $L^{2} (S^{2}, σ)$ with inner product linear in the first argument, the complete orthonormal family $(Y_{l m})$ (Associated Legendre function normalization, Sphere ONB), the blocks $V_{l} = span {Y_{l, - l}, \dots, Y_{l l}}$ and the finite span $H$ , the spaces $H_{l}^{p o l}$ of complex harmonic homogeneous polynomials of degree $l$ , and the polynomial representatives $h_{l m} \in H_{l}^{p o l}$ of the $Y_{l m}$ , characterized by ${r e s}_{l} (h_{l m}) = Y_{l m}$ where ${r e s}_{l} (p)$ is the class of $p ∣_{S^{2}}$ (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 $H_{- 1}^{p o l} : = {0}$ , fix $l \geq 0$ and $m \in {- l, \dots, l}$ , write $H : = h_{l m}$ , 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 $c_{0, i} = δ_{i 3}$ and $c_{\pm, i} = (δ_{i 1} \pm i δ_{i 2}) / \sqrt{2}$ . Let $D_{i}$ and $L_{z}, L_{3}$ be as in Orbital self-adjoint generators, where $L_{3} = L_{z}$ and $L_{z} Y_{l m} = ℏ m Y_{l m}$ (Sphere Laplacian diagonal domain). Then, for each $q \in {- 1, 0, 1}$ :

(a) (harmonic decomposition) the polynomial $G_{q} : = \frac{\sum_{i} c_{q, i} \partial_{i} H}{2 l + 1}$ lies in $H_{l - 1}^{p o l}$ and $Q_{q} H = R_{q} + r^{2} G_{q}, R_{q} : = Q_{q} H - r^{2} G_{q} \in H_{l + 1}^{p o l},$ where $r^{2} (x) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2}$ ; for $l = 0$ one has $G_{q} = 0$ and $R_{q} = Q_{q} H$ ;

(b) (degree selection) the function $Q_{q} ∣_{S^{2}} \cdot Y_{l m}$ lies in $V_{l + 1} \oplus V_{l - 1}$ (with $V_{- 1} : = {0}$ ); consequently, for all $l^{'} \geq 0$ and $m^{'} \in {- l^{'}, \dots, l^{'}}$ , $⟨ Q_{q} ∣_{S^{2}} Y_{l m}, Y_{l^{'} m^{'}} ⟩ = 0 whenever l^{'} \notin {l - 1, l + 1};$

(c) (weight selection) as elements of $H \subset L^{2} (S^{2}, σ)$ one has $L_{z} (Q_{q} ∣_{S^{2}} Y_{l m}) = ℏ (m + q) (Q_{q} ∣_{S^{2}} Y_{l m}),$ because $\partial_{φ} Q_{q} = i q Q_{q}$ on $S^{2}$ (Sine and cosine defined by their real power series, The complex exponential by its power series); hence in addition $⟨ Q_{q} ∣_{S^{2}} Y_{l m}, Y_{l^{'} m^{'}} ⟩ = 0 whenever m^{'} \neq m + q;$

(d) (selection rules) combining (b) and (c): the angular matrix element $⟨ Y_{l^{'} m^{'}}, Q_{q} ∣_{S^{2}} Y_{l m} ⟩$ can be nonzero only if $l^{'} = l + 1 and m^{'} = m + q, or l \geq 1, l^{'} = l - 1 and m^{'} = m + q;$ in particular only $Δ l = \pm 1$ and $Δ m \in {0, \pm 1}$ occur with the component-dependent shift $Δ m = q$ 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.

Facts & Assumptions

Given: the family $(Y_{l m})$ with representatives $h_{l m}$ , the multipliers $Q_{q}$ , the fixed indices $l, m$ , and the polynomial $H = h_{l m}$ .

[F1]

The restriction map ${r e s}_{l} : H_{l}^{p o l} \to V_{l}$ is a $C$ -linear isomorphism; $(Y_{l m})_{m}$ is an orthonormal basis of $V_{l}$ ; $V_{l} ⊥ V_{k}$ for $l \neq k$ ; $(Y_{l m})$ is complete in $L^{2} (S^{2}, σ)$ and $H$ is dense (Harmonic polynomial dimension (b),(c), Sphere ONB, Orthonormal families, complete orthonormal systems and Hilbert bases, Parseval equivalences for an orthonormal family, Real and complex inner product spaces, with the inner product linear in the first argument).

[F2]

$Δ$ is linear, satisfies the product rule $Δ (f g) = f Δ g + 2 \nabla f \cdot \nabla g + g Δ f$ , commutes with each $\partial_{i}$ , and annihilates the coordinate functions $x_{i}$ ; $r^{2}$ has $Δ r^{2} = 6$ and $\nabla r^{2} = 2 x$ ; Euler's identity $x \cdot \nabla q = k q$ holds for homogeneous $q$ of degree $k$ ; hence for homogeneous $p$ of degree $l$ and $q$ of degree $l - 1$ : $Δ (x_{i} p) = 2 \partial_{i} p$ and $Δ (r^{2} q) = (4 l + 2) q + r^{2} Δ q$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , $C^{k}$ Euclidean maps are closed under componentwise algebra and composition, A total derivative computes every directional derivative, and its matrix is the Jacobian, homogeneous polynomial and homogeneous ideal).

[F3]

$Y_{l m} (θ, φ) = b_{l m} w_{l ∣ m ∣} (\cos θ) e^{i m φ} / \sqrt{2 π}$ for $m \geq 0$ and $Y_{l, - m} = (- 1)^{m} \overline{Y_{l m}}$ ; in particular $\partial_{φ} Y_{l m} = i m Y_{l m}$ in the angular chart, and $Y_{l m}$ is continuous (Associated Legendre function normalization (a),(c), The complex exponential by its power series, Sine and cosine defined by their real power series).

[F4]

The chart is $Φ (θ, φ) = (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ ; on $S^{2}$ one has $Q_{0} = \cos θ$ , $Q_{\pm} = e^{\pm i φ} \sin θ / \sqrt{2}$ as chart functions, so $\partial_{φ} Q_{q} = i q Q_{q}$ on $S^{2}$ ; and a product of chart functions is differentiated by the product rule (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , The complex exponential by its power series, Sine and cosine defined by their real power series, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F5]

$L_{z}$ is self-adjoint on $L^{2} (S^{2}, σ)$ with $L_{z} Y_{l m} = ℏ m Y_{l m}$ , and for $F \in H$ the chart values of $L_{z} F$ are $- i ℏ \partial_{φ} F$ ; eigenvectors of $L_{z}$ for distinct eigenvalues are orthogonal, because $⟨ L_{z} F, G ⟩ = ⟨ F, L_{z} G ⟩$ for $F, G$ in the domain of the self-adjoint $L_{z}$ (Sphere Laplacian diagonal domain (a),(c), Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space, Orthogonality and the orthogonal complement).

[F6]

$L_{3} = L_{z}$ (equal self-adjoint operators) by Orbital self-adjoint generators (c).

Proof

1.1F2algebra

[(a), the two harmonic pieces] By [F2] and $Δ H = 0$ , $Δ (Q_{q} H) = \sum_{i} c_{q, i} Δ (x_{i} H) = \sum_{i} c_{q, i} 2 \partial_{i} H$ . Since $\partial_{i}$ commutes with $Δ$ and $Δ H = 0$ , each $\partial_{i} H$ lies in $H_{l - 1}^{p o l}$ , so $G_{q} = \sum_{i} c_{q, i} \partial_{i} H / (2 l + 1) \in H_{l - 1}^{p o l}$ ; by [F2] and Euler's identity for the homogeneous polynomial $G_{q}$ of degree $l - 1$ , $Δ (r^{2} G_{q}) = 6 G_{q} + 4 (l - 1) G_{q} + r^{2} Δ G_{q} = (4 l + 2) G_{q} + r^{2} Δ G_{q}$ , and here $Δ G_{q} = 0$ , so $Δ (r^{2} G_{q}) = (4 l + 2) G_{q} = 2 \sum_{i} c_{q, i} \partial_{i} H$ . Hence $Δ R_{q} = Δ (Q_{q} H) - Δ (r^{2} G_{q}) = 0$ : $R_{q} \in H_{l + 1}^{p o l}$ , and $R_{q}$ is homogeneous of degree $l + 1$ as the difference of two such polynomials; for $l = 0$ the derivative terms vanish because $H$ is constant, so $G_{q} = 0$ and $R_{q} = Q_{q} H$ .

2.1step 1.1F1algebra

[(b), degree selection] Restricting the identity of step 1.1 to $S^{2}$ , where $r^{2} = 1$ , gives $Q_{q} ∣_{S^{2}} \cdot Y_{l m} = {r e s}_{l + 1} (R_{q}) + {r e s}_{l - 1} (G_{q})$ , where for $l = 0$ the second term is omitted because $G_{q} = 0$ ; for $l \geq 1$ both restriction maps are the defined maps in [F1]. This is an element of $V_{l + 1} + V_{l - 1}$ , using $H ∣_{S^{2}} = Y_{l m}$ ([F1]) and $Q_{q} H ∣_{S^{2}} = (Q_{q} ∣_{S^{2}}) Y_{l m}$ ; the sum is direct and each summand lies in the corresponding block, so orthogonality $V_{l + 1} ⊥ V_{l^{'}}$ and $V_{l - 1} ⊥ V_{l^{'}}$ for $l^{'} \notin {l - 1, l + 1}$ ([F1]) gives $⟨ Q_{q} ∣_{S^{2}} Y_{l m}, Y_{l^{'} m^{'}} ⟩ = 0$ for such $l^{'}$ .

3.1step 2.1F3F4F5algebra

[(c), weight selection] Let $F : = Q_{q} ∣_{S^{2}} \cdot Y_{l m} \in H$ by step 2.1. On the chart, by [F3] and [F4], $\partial_{φ} F = (\partial_{φ} Q_{q}) Y_{l m} + Q_{q} \partial_{φ} Y_{l m} = i q F + i m F = i (m + q) F$ ; hence by [F5] $L_{z} F = - i ℏ \partial_{φ} F = ℏ (m + q) F$ in $H$ . If now $m^{'} \neq m + q$ , then using the self-adjointness of $L_{z}$ ([F5]) and $L_{z} Y_{l^{'} m^{'}} = ℏ m^{'} Y_{l^{'} m^{'}}$ , $ℏ (m + q) ⟨ F, Y_{l^{'} m^{'}} ⟩ = ⟨ L_{z} F, Y_{l^{'} m^{'}} ⟩ = ⟨ F, L_{z} Y_{l^{'} m^{'}} ⟩ = ℏ m^{'} ⟨ F, Y_{l^{'} m^{'}} ⟩$ , so $⟨ F, Y_{l^{'} m^{'}} ⟩ = 0$ .

4.1step 2.1step 3.1F6∎

[(d), the rules] Clause (d) restates steps 2.1 and 3.1 for the multiplier $Q_{q}$ : a nonzero angular integral requires $l^{'} \in {l - 1, l + 1}$ and $m^{'} = m + q$ ; the case $l^{'} = l - 1$ presupposes $l \geq 1$ because $- l^{'} \leq m^{'} \leq l^{'}$ with $l - 1 \geq 0$ ; and by [F6] the weight $m$ is the $L_{3}$ -eigenvalue in units $ℏ$ , so the multiplier $Q_{q}$ shifts the $L_{3}$ -eigenvalue by $ℏ q$ .

5 · Examples, experiments and counterexamples

None yet.

Qm Math Orbital And Spin

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Finite spin matrices

Statement

Facts & Assumptions

Proof

Symmetric-power SU2 representation

Statement

Facts & Assumptions

Proof

Finite spin-block decomposition

Statement

Facts & Assumptions

Proof

Tensor spin composition

Statement

Facts & Assumptions

Proof

Tsirelson bound

Statement

Facts & Assumptions

Proof

Legendre Rodrigues equation

Statement

Facts & Assumptions

Proof

Sphere integrals in angle coordinates

Statement

Facts & Assumptions

Proof

Associated Legendre function normalization

Statement

Facts & Assumptions

Proof

Weighted angular completeness

Statement

Facts & Assumptions

Proof

Sphere ONB

Statement

Facts & Assumptions

Proof

Sphere Laplacian diagonal domain

Statement

Facts & Assumptions

Proof

Harmonic polynomial dimension

Statement

Facts & Assumptions

Proof

Strongly continuous rotation action

Statement

Facts & Assumptions

Proof

Orbital self-adjoint generators

Statement

Facts & Assumptions

Proof

Dipole angular selection rules

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources