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The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice), the choice hypotheses of the cited spectral suppliers. Fix $ℏ > 0$ and $s \in {0, \frac{1}{2}, 1, \frac{3}{2}, \dots}$ , put $r : = 2 s + 1$ , and let $H_{s} = L^{2} (R^{3}; C^{r})$ 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 $S O (3)$ be the special orthogonal group (The orthogonal group $O (3)$ and the rotation group $S O (3)$ ), let $U (R)$ for $R \in S O (3)$ be the componentwise spatial rotation action $(U (R) f)_{l} (x) : = f_{l} (R^{- 1} x) (l = 1, \dots, r)$ of The spatial rotation action on the component-valued wave-function space, and let $ρ_{0} : S U (2) \to S O (3)$ be the quaternion conjugation cover of SU(2) to SO(3) as a covering homomorphism, using the matrix identification $a + b i + c j + d k ⟼ (\begin{array}{cc} a + b i & c + d i \\ - c + d i & a - b i \end{array})$ and the imaginary basis $(i, j, k)$ of The quaternion rotation map $ρ$ : conjugation on the imaginary quaternions, the unit sphere $S^{3}$ , and quaternion lifts. For the physical coordinate axes used by the Pauli matrices fix $C : = (\begin{array}{ccc} 0 & 0 & - 1 \\ 0 & - 1 & 0 \\ - 1 & 0 & 0 \end{array}) \in S O (3), ρ (W) : = C^{- 1} ρ_{0} (W) C .$ Thus $ρ$ is the same two-sheeted cover transported to the axes $(C e_{1}, C e_{2}, C e_{3}) = (- e_{3}, - e_{2}, - e_{1})$ ; it is not the literal matrix $ρ_{0}$ in the imaginary basis. It is smooth by Covering homomorphisms of Lie groups. Let $F_{s} : S_{2 s} \to V_{s} = C^{r}$ be the unitary identification $v_{k} \mapsto e_{s - k}$ of Symmetric-power SU2 representation clause (d) (for $s = 0$ , $S_{0} = C v_{0}$ and $F_{0} v_{0} : = 1$ ), define $D_{s} : S U (2) \to U (V_{s})$ by $D_{s} (W) : = F_{s} U_{2 s} (W) F_{s}^{- 1}$ for $s > 0$ and $D_{0} (W) : = I_{C}$ , and let $S_{i}^{f i n} = ℏ J_{i}$ be the self-adjoint spin endomorphisms of $V_{s}$ of Finite spin matrices clause (a). By clauses (b),(d),(f) of the theorem the maps $D_{s}$ are strongly continuous unitary representations of $S U (2)$ . For $W \in S U (2)$ define the matrix action $M_{s} (W) \in B (H_{s})$ and the lifted operator $U_{s} (W)$ by $(M_{s} (W) f) (x) : = D_{s} (W) (f (x)), U_{s} (W) : = U (ρ (W)) M_{s} (W) (f \in H_{s}, x \in R^{3}),$ where the matrix $D_{s} (W)$ acts on the column vector $f (x) \in C^{r}$ . For any $T \in End (V_{s})$ write $M_{T} f (x) : = T f (x)$ almost everywhere, and define the spin operators on $H_{s}$ by $S_{i} : = M_{S_{i}^{f i n}}, D (S_{i}) : = H_{s} .$ These bounded self-adjoint operators, rather than the endomorphisms $S_{i}^{f i n}$ on $V_{s}$ , occur in all sums, spectral commutation claims and conjugations on $H_{s}$ below. Finally let $R_{i} (t)$ , $i = 1, 2, 3$ , be the coordinate rotation matrices and $L_{i}$ the self-adjoint coordinate-rotation generators on $H_{s}$ of Coordinate-rotation generators on the Schwartz classes and their commutators (so that $U (R_{i} (t)) = e^{- i t L_{i} / ℏ}$ ), and put $A_{i} (t) : = \cos (t / 2) I_{2} - i \sin (t / 2) σ_{i} \in S U (2)$ with the Pauli matrices $σ_{i}$ of Finite spin matrices clause (e). Then:

(a) for every $W \in S U (2)$ the operator $U_{s} (W)$ is unitary, with $U_{s} (I) = i d$ and $U_{s} (W_{1} W_{2}) = U_{s} (W_{1}) U_{s} (W_{2})$ for all $W_{1}, W_{2} \in S U (2)$ , and $W \mapsto U_{s} (W)$ is strongly continuous (Strongly continuous unitary representations, invariant linear subspaces and intertwiners);

(b) $ρ (A_{i} (t)) = R_{i} (t)$ for every $t \in R$ and every $i$ , and $t \mapsto U_{s} (A_{i} (t))$ is a strongly continuous one-parameter unitary group (Strongly continuous one-parameter unitary group);

(c) the operators $L_{i}$ and $S_{i}$ strongly commute (their spectral projection-valued measures commute; Self-adjoint, positive, unitary and normal operators, Bounded borel pvm integral), the sum $J_{i} : = L_{i} + S_{i}$ with domain $D (J_{i}) : = D (L_{i})$ is self-adjoint, and $U_{s} (A_{i} (t)) = e^{- i t J_{i} / ℏ} (t \in R);$ moreover for $W \in S U (2)$ one has the conjugation identities including domains $U_{s} (W) Q_{j} U_{s} (W)^{- 1} = \sum_{l = 1}^{3} ρ (W)_{l j} Q_{l}, U_{s} (W) P_{k} U_{s} (W)^{- 1} = \sum_{l = 1}^{3} ρ (W)_{l k} P_{l},$ where $Q_{j}, P_{k}$ are the componentwise position and momentum operators of Componentwise position and momentum operators on the component-valued wave-function space clause (a) and $ρ (W)_{l j}$ are the matrix entries of the transported cover in the standard basis; the sums mean the maximal directional realizations $Q (ρ (W) e_{j})$ and $P (ρ (W) e_{k})$ of The spatial rotation action on the component-valued wave-function space clause (b), with domains $U_{s} (W) D (Q_{j})$ and $U_{s} (W) D (P_{k})$ ;

(d) Scope. The statement is a mathematical statement about the finite-dimensional spin representation and the componentwise rotation action; it asserts no physical interpretation, no measurement, no dynamics and no value of $ℏ$ , $s$ or $W$ . The spatial action enters only through $U (R) = U (ρ (W))$ and the internal action only through $M_{s} (W)$ ; no action on a second spatial or internal factor is introduced.

Facts & Assumptions

Given: the Axiom of Countable Choice and the Axiom of Choice; the spin label $s$ , $r = 2 s + 1$ , the space $H_{s}$ ; the rotation action $U (R)$ and the covering homomorphism $ρ$ ; the identification $F_{s}$ and the representation $D_{s} = F_{s} U_{2 s} F_{s}^{- 1}$ with spin matrices $S_{i}^{f i n} = ℏ J_{i}$ and operators $S_{i} = M_{S_{i}^{f i n}}$ on $H_{s}$ ; the operators $M_{s} (W)$ and $U_{s} (W)$ ; the coordinate rotations $R_{i} (t)$ , $A_{i} (t)$ and the generators $L_{i}$ ; $W, W_{1}, W_{2} \in S U (2)$ , real $s_{0}, t$ (use $u$ for the group parameter where it is not the spin label).

[F1]

The covering and its axes. The quaternion cover $ρ_{0}$ is a smooth surjective two-sheeted covering homomorphism with kernel ${I_{2}, - I_{2}}$ . The displayed $C$ satisfies $C^{T} C = I$ , $C^{- 1} = C$ and $\det C = 1$ . Conjugation $R \mapsto C^{- 1} R C$ is a smooth automorphism of $S O (3)$ with smooth inverse, so $ρ = C^{- 1} ρ_{0} C$ has the same covering, continuity, surjectivity, multiplicativity and kernel properties. The canonical matrix identification and quaternion conjugation formula are those of SU(2) to SO(3) as a covering homomorphism, The quaternion rotation map $ρ$ : conjugation on the imaginary quaternions, the unit sphere $S^{3}$ , and quaternion lifts and The quaternions $H$ : real quadruples with componentwise addition and an explicit multiplication formula matching the table on $1, i, j, k$ .

[F2]

Spatial rotation action. $U (R)$ is a unitary operator for every $R$ with $U (I) = i d$ , $U (R S) = U (R) U (S)$ and $U (R)^{- 1} = U (R^{- 1})$ ; the map $R \mapsto U (R)$ is strongly continuous on $S O (3)$ ; and $U (R) Q_{j} U (R)^{- 1} = Q (R e_{j}) = \sum_{l} R_{l j} Q_{l}$ , $U (R) P_{k} U (R)^{- 1} = P (R e_{k}) = \sum_{l} R_{l k} P_{l}$ as operator identities on the maximal directional domains defined by The spatial rotation action on the component-valued wave-function space clause (b). Explicitly, $Q (a)$ is multiplication by $a \cdot x$ on its maximal domain $D (Q (a)) = {f \in H_{s} : (a \cdot x) f \in H_{s}}$ , and $P (a) = (U^{(r)})^{- 1} M_{a \cdot p} U^{(r)}$ on its maximal domain $D (P (a)) = {f \in H_{s} : (a \cdot p) U^{(r)} f \in L^{2} (R^{3}; C^{r})}$ , where $U^{(r)}$ is the normalized Fourier unitary and $M_{a \cdot p}$ is maximal multiplication by $a \cdot p$ from that clause. In particular the conjugate domains are $U (R) D (Q_{j}) = D (Q (R e_{j}))$ and $U (R) D (P_{k}) = D (P (R e_{k}))$ , not necessarily the original individual coordinate domains (clauses (a),(b)).

[F3]

Coordinate rotation generators. For each $i$ , $U_{i} (t) : = U (R_{i} (t))$ is a strongly continuous one-parameter unitary group with $U_{i} (t) = e^{- i t L_{i} / ℏ}$ for the self-adjoint generator $L_{i}$ , and for $ψ \in S^{(r)}$ one has $L_{i} ψ = - i ℏ \sum_{j} (e_{i} \times x)_{j} D_{j} ψ = \sum_{j, k} ε_{i j k} Q_{j} P_{k} ψ$ (Coordinate-rotation generators on the Schwartz classes and their commutators clauses (a),(b), Componentwise position and momentum operators on the component-valued wave-function space).

[F4]

Internal representation. For $s > 0$ , $n = 2 s$ and $F_{s} : S_{n} \to V_{s}$ the unitary identification $v_{k} \mapsto e_{s - k}$ of Symmetric-power SU2 representation clause (d), $D_{s} (W) = F_{s} U_{n} (W) F_{s}^{- 1}$ with $U_{n} (W) = W^{\otimes n} ∣_{S_{n}}$ defines a group homomorphism $S U (2) \to U (V_{s})$ that is continuous in operator norm; for $s = 0$ , $D_{0} (W) = I_{C}$ ; and $S_{i}^{f i n} = ℏ J_{i}$ are the self-adjoint matrices of Finite spin matrices with $[S_{i}^{f i n}, S_{j}^{f i n}] = i ℏ \sum_{k} ε_{i j k} S_{k}^{f i n}$ and $\sum_{i} (S_{i}^{f i n})^{2} = ℏ^{2} s (s + 1) I$ (Symmetric-power SU2 representation clauses (b),(c),(d),(f), Finite spin matrices clauses (a),(b),(c)).

[F5]

Internal coordinate generator. For every $i$ the curve $t \mapsto A_{i} (t)$ lies in $S U (2)$ , satisfies $A_{i} (s_{0}) A_{i} (t) = A_{i} (s_{0} + t)$ and $A_{i} (0) = I_{2}$ , and the family $t \mapsto D_{s} (A_{i} (t))$ has infinitesimal generator $- i J_{i}$ : for every $v \in V_{s}$ , $\frac{1}{t} (D_{s} (A_{i} (t)) v - v) \to - i J_{i} v$ (Symmetric-power SU2 representation clauses (d),(e), Stone's theorem: unitary groups and self-adjoint generators, Infinitesimal generator of a unitary group, A self-adjoint operator generates a strongly continuous unitary group).

[F6]

Matrix actions commute with componentwise operators. For a fixed matrix $T \in End (C^{r})$ , define $(M_{T} f) (x) : = T f (x)$ . This is a bounded operator on $H_{s}$ because $∥ T f (x) ∥ \leq ∥ T ∥ ∥ f (x) ∥$ pointwise; it commutes with $U (R)$ because $T f (R^{- 1} x) = (T f) (R^{- 1} x)$ , and it commutes with $Q_{j}$ and $P_{k}$ and preserves their domains because those operators act componentwise by scalar multiplication and weak differentiation, respectively. For $W \in S U (2)$ , $M_{s} (W) = M_{D_{s} (W)}$ is unitary since $D_{s} (W)$ is unitary. These identities follow from the componentwise definitions (Componentwise position and momentum operators on the component-valued wave-function space clause (a), A bounded linear operator between normed spaces, Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces).

[F7]

Commutation with a flow. If a bounded $S \in B (H)$ commutes with every element $e^{i u T}$ of the flow of a self-adjoint $T$ , then $S$ commutes with every spectral projection $E_{T} (Λ)$ (Unitary conjugation, generator transport and strong commutation of self-adjoint observables clause (c)).

[F8]

Calculus of a strongly commuting pair. If the self-adjoint operators $A$ and $B$ have commuting spectral projection-valued measures, then $f (B) g (A) = g (A) f (B)$ for all bounded Borel $f, g$ (Bounded Borel calculus of strongly commuting self-adjoint operators clause (a), Strong spectral commutation of self-adjoint operators clause (b)).

[F9]

Product flow. If $A$ is self-adjoint and $B$ is bounded self-adjoint with $B e^{i u A} = e^{i u A} B$ for all real $u$ , then $A + B$ with domain $D (A)$ is self-adjoint and $e^{i u (A + B)} = e^{i u A} e^{i u B}$ for all $u$ (Self-adjointness and the flow of a self-adjoint operator perturbed by a bounded operator commuting with its flow clauses (a),(c)).

[F10]

Finite spectral decomposition. A normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely). Each self-adjoint $S_{i}^{f i n}$ is normal and has real eigenvalues: for a unit eigenvector $v$ , $λ = ⟨ S_{i}^{f i n} v, v ⟩$ is real. Grouping the eigenbasis by eigenvalue gives mutually orthogonal eigenprojections $P_{λ}$ with $\sum_{λ} P_{λ} = I_{V_{s}}$ and $S_{i}^{f i n} = \sum_{λ} λ P_{λ}$ ; its spectral projection on a Borel set $Σ$ is $\sum_{λ \in Σ} P_{λ}$ .

Proof

Proof technique: direct; verify unitarity and multiplicativity from the matrix action, compute the quaternion image of the coordinate subgroups, transport the commutation of the internal matrices with the spatial flow into strong spectral commutation, and identify the coordinate generators through the product-flow lemma and Stone's theorem.

Given: The data of the statement and [F1]-[F10].

1.1F1F2F4F6algebra

[Unitary and representation law] Let $W \in S U (2)$ ; the fibrewise matrix action $M_{s} (W)$ is well defined on almost-everywhere classes, linear, and isometric because $∥ D_{s} (W) f (x) ∥_{C^{r}} = ∥ f (x) ∥_{C^{r}}$ for almost every $x$ with $D_{s} (W)$ unitary, so $∥ M_{s} (W) f ∥ = ∥ f ∥$ and $M_{s} (W)$ extends to a unitary operator with inverse $M_{s} (W^{- 1})$ , and $M_{s} (W_{1}) M_{s} (W_{2}) = M_{s} (W_{1} W_{2})$ again componentwise; $U (ρ (W))$ is unitary by [F2], so $U_{s} (W)$ is unitary with $U_{s} (I) = U (I) M_{s} (I) = i d$ . Using [F6] to move $M_{s} (W_{1})$ across $U (ρ (W_{2}))$ and the multiplicativity $ρ (W_{1} W_{2}) = ρ (W_{1}) ρ (W_{2})$ of [F1] together with $D_{s} (W_{1} W_{2}) = D_{s} (W_{1}) D_{s} (W_{2})$ , one gets $U_{s} (W_{1}) U_{s} (W_{2}) = U (ρ (W_{1})) U (ρ (W_{2})) M_{s} (W_{1}) M_{s} (W_{2}) = U_{s} (W_{1} W_{2})$ .

1.2F1F3givenalgebra

[Coordinate rotations under the transported cover] Put $c = \cos (t / 2)$ and $d = \sin (t / 2)$ . In the canonical matrix identification of [F1], the matrices $A_{1} (t), A_{2} (t), A_{3} (t)$ correspond respectively to $c - d k, c - d j, c - d i$ , hence uniformly to $q = c + d φ (C e_{i})$ . For a unit axis $r$ and $v = α r + w$ with $w ⊥ r$ , quaternion multiplication $u v = - u \cdot v + u \times v$ gives $(c + d r) v (c - d r) = (c^{2} + d^{2}) α r + (c^{2} - d^{2}) w + 2 c d (r \times w) = α r + \cos t w + \sin t (r \times w) .$ Use $r = φ (C e_{i})$ and identify vectors through $φ$ . Since $C \in S O (3)$ , it preserves inner products and satisfies $(C u) \times (C v) = C (u \times v)$ (the determinant definition of the cross product); conjugating this Rodrigues formula by $C$ gives precisely the rotation about $e_{i}$ through $t$ . Therefore $ρ_{0} (A_{i} (t)) = C R_{i} (t) C^{- 1}$ and $ρ (A_{i} (t)) = R_{i} (t)$ , with exactly the coordinate convention of [F3].

1.3F4F6F7F8F10algebra

[Strong commutation of the spatial generators with internal projections] The finite spectral decomposition $S_{i}^{f i n} = \sum_{λ} λ P_{λ}$ gives $S_{i} = \sum_{λ} λ M_{P_{λ}}$ ; by [F6] the $M_{P_{λ}}$ are mutually orthogonal projections summing to $I_{H_{s}}$ . Thus $S_{i}$ is bounded self-adjoint and its spectral projections are $E_{S_{i}} (Σ) = \sum_{λ \in Σ} M_{P_{λ}} = M_{E_{S_{i}^{f i n}} (Σ)}$ . Each of these operators commutes with $U (R_{i} (u)) = e^{- i u L_{i} / ℏ}$ by [F6], hence with every $E_{L_{i}} (Λ)$ by [F7]. Both spectral measures now act on $H_{s}$ , and $E_{L_{i}} (Λ) E_{S_{i}} (Σ) = E_{S_{i}} (Σ) E_{L_{i}} (Λ)$ for all Borel $Λ, Σ$ , proving strong commutation in the sense of [F8].

1.4F1F2F4algebra

[Strong continuity of the lift] Let $W_{m} \to W$ in $S U (2)$ and $f \in H_{s}$ ; since $ρ$ is continuous by [F1], $ρ (W_{m}) \to ρ (W)$ and [F2] gives $∥ U (ρ (W_{m})) g - U (ρ (W)) g ∥ \to 0$ for every fixed $g$ ; since $∥ D_{s} (W_{m}) - D_{s} (W) ∥ \to 0$ in operator norm by [F4], $∥ M_{s} (W_{m}) f - M_{s} (W) f ∥ \leq ∥ D_{s} (W_{m}) - D_{s} (W) ∥ ∥ f ∥$ tends to $0$ ; combining, $∥ U_{s} (W_{m}) f - U_{s} (W) f ∥ \leq ∥ U (ρ (W_{m})) (M_{s} (W_{m}) f - M_{s} (W) f) ∥ + ∥ U (ρ (W_{m})) M_{s} (W) f - U (ρ (W)) M_{s} (W) f ∥ \to 0$ , so $W \mapsto U_{s} (W)$ is strongly continuous.

2.1step 1.3F4F5F6algebra

[Internal coordinate generator] By [F5] the family $t \mapsto D_{s} (A_{i} (t))$ on $V_{s}$ has infinitesimal generator $- i J_{i}$ ; the finite spectral exponential $e^{- i t S_{i}^{f i n} / ℏ}$ has the same generator $- i S_{i}^{f i n} / ℏ = - i J_{i}$ , so Stone's theorem gives $D_{s} (A_{i} (t)) = e^{- i t S_{i}^{f i n} / ℏ}$ . Lifting the finite spectral sum of step 1.3 then gives $M_{s} (A_{i} (t)) = M_{e^{- i t S_{i}^{f i n} / ℏ}} = \sum_{λ} e^{- i t λ / ℏ} M_{P_{λ}} = e^{- i t S_{i} / ℏ}$ on $H_{s}$ . This includes $s = 0$ , when both spin generators vanish.

2.2step 1.1F2F6algebra

[Conjugations of position and momentum] By [F6] the matrix action $M_{s} (W)$ commutes with $Q_{j}$ and $P_{k}$ and preserves their maximal coordinate domains, so the operator domains of the conjugates by $U_{s} (W) = U (ρ (W)) M_{s} (W)$ are $U_{s} (W) D (Q_{j}) = U (ρ (W)) D (Q_{j}) = D (Q (ρ (W) e_{j}))$ and $U_{s} (W) D (P_{k}) = U (ρ (W)) D (P_{k}) = D (P (ρ (W) e_{k}))$ , where the last equalities are [F2]. On these domains [F2] gives $U_{s} (W) Q_{j} U_{s} (W)^{- 1} = Q (ρ (W) e_{j}) = \sum_{l} ρ (W)_{l j} Q_{l}$ and $U_{s} (W) P_{k} U_{s} (W)^{- 1} = P (ρ (W) e_{k}) = \sum_{l} ρ (W)_{l k} P_{l}$ , with both sums understood as the displayed maximal directional realizations. This proves the conjugation identities of (c).

2.3step 1.3F6F8F9algebra

[Self-adjoint sum and product flow] By step 1.3 the spectral measures of $L_{i}$ and $S_{i}$ commute, so [F8] with $f (μ) = e^{- i μ t / ℏ}$ , $g (λ) = e^{- i λ u / ℏ}$ gives $e^{- i t S_{i} / ℏ} e^{- i u L_{i} / ℏ} = e^{- i u L_{i} / ℏ} e^{- i t S_{i} / ℏ}$ for all $t, u$ ; by [F6] $S_{i}$ itself also commutes with every spatial flow, so [F9] applies with $A : = L_{i}$ and the bounded self-adjoint $B : = S_{i}$ , so the sum $J_{i} = L_{i} + S_{i}$ with domain $D (L_{i})$ is self-adjoint and $e^{- i t J_{i} / ℏ} = e^{- i t L_{i} / ℏ} e^{- i t S_{i} / ℏ}$ for every $t$ .

3.1step 1.2step 2.1step 2.3F3F5algebra

[Coordinate subgroup generator] Let $t$ be real; by step 1.2, $ρ (A_{i} (t)) = R_{i} (t)$ , so by [F3] and step 2.1 $U_{s} (A_{i} (t)) = U (R_{i} (t)) M_{s} (A_{i} (t)) = e^{- i t L_{i} / ℏ} e^{- i t S_{i} / ℏ} = e^{- i t J_{i} / ℏ}$ by step 2.3; since the right-hand side is a strongly continuous one-parameter unitary group in $t$ and $A_{i}$ is multiplicative by [F5], $t \mapsto U_{s} (A_{i} (t))$ is a strongly continuous one-parameter unitary group with the displayed generator, which is (b) and the generator identity of (c).

4.1step 1.1step 1.3step 1.4step 2.2step 2.3step 3.1∎

[Assembly] Step 1.1 proves unitarity, the identity and the representation law, and step 1.4 proves strong continuity, which is (a); step 3.1 proves (b) and the generator identity of (c); step 1.3 proves the strong commutation of $L_{i}$ and $S_{i}$ , step 2.3 the self-adjointness of $J_{i}$ , and step 2.2 the conjugation identities of (c); (d) records only the scope of these mathematical assertions.

Remarks

Kernel and the two lifts. Since $D_{s} (- I_{2}) = (- 1)^{2 s} I$ by the diagonal action and Symmetric-power SU2 representation clauses (a),(b) (for $s = 0$ also $D_{0} (- I_{2}) = I$ ), one has $U_{s} (- I_{2}) = (- 1)^{2 s} i d$ . For half-integer $s$ this is $- i d \neq i d$ , so $W \mapsto U_{s} (W)$ does not factor through $ρ$ : no map $V$ with $V (ρ (W)) = U_{s} (W)$ exists, since $ρ (- I_{2}) = ρ (I_{2}) = I$ would force $V (I) = - i d$ . For integer $s$ , $- I_{2}$ acts as the identity and the action passes to the rotation group $S U (2) / {I_{2}, - I_{2}} ≅ S O (3)$ of SU(2) to SO(3) as a covering homomorphism. On a half-integer spin sector the two lifts $\pm U_{s} (W)$ of one rotation $ρ (W)$ differ by the central sign.
Units. $ℏ > 0$ is a fixed positive parameter of the statement; $L_{i}$ , $S_{i} = M_{S_{i}^{f i n}}$ and the total generator $J_{i} = L_{i} + S_{i}$ carry the unit of action. The dimensionless finite spin matrices are $S_{i}^{f i n} / ℏ$ on $V_{s}$ ; their fibrewise actions on $H_{s}$ are $S_{i} / ℏ$ . No numerical value is asserted.

The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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