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Coordinate-rotation generators on the Schwartz classes and their commutators

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Fix an integer $r \geq 1$ and $ℏ > 0$ , let $H^{(r)} = L^{2} (R^{3}; C^{r})$ with the componentwise operators $Q_{j}, P_{k}$ and the Schwartz class $S^{(r)} = S (R^{3}; C^{r})$ (Componentwise position and momentum operators on the component-valued wave-function space, Schwartz space and its seminorms), and let $U (R)$ for $R \in S O (3)$ be the spatial rotation action of The spatial rotation action on the component-valued wave-function space (so that $(U (R) f) (x) = f (R^{- 1} x)$ componentwise and $U$ is a strongly continuous unitary representation). 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 $U_{i} (t) : = U (R_{i} (t))$ , and let $L_{i}$ be the self-adjoint operator with $U_{i} (t) = e^{- i t L_{i} / ℏ}$ furnished by Stone's theorem (Stone's theorem: unitary groups and self-adjoint generators, Infinitesimal generator of a unitary group). Then:

(a) for each $i$ the family $t \mapsto U_{i} (t)$ is a strongly continuous one-parameter unitary group with $U_{i} (s + t) = U_{i} (s) U_{i} (t)$ and $U_{i} (0) = i d$ (Strongly continuous one-parameter unitary group);

(b) for every $ψ \in S^{(r)}$ the curve $t \mapsto U_{i} (t) ψ$ is differentiable at $t = 0$ into $H^{(r)}$ with $\frac{d}{d t} ∣_{t = 0} U_{i} (t) ψ = - \sum_{j = 1}^{3} (e_{i} \times x)_{j} D_{j} ψ,$ and consequently $S^{(r)} \subseteq D (L_{i})$ and $L_{i} ψ = - i ℏ \sum_{j = 1}^{3} (e_{i} \times x)_{j} D_{j} ψ = \sum_{j, k = 1}^{3} ε_{i j k} Q_{j} P_{k} ψ (ψ \in S^{(r)});$

(c) $L_{i} S^{(r)} \subseteq S^{(r)}$ for each $i$ , and on $S^{(r)}$ the commutation relations hold: $L_{i} L_{k} ψ - L_{k} L_{i} ψ = i ℏ \sum_{j = 1}^{3} ε_{i k j} L_{j} ψ (ψ \in S^{(r)}) .$ (d) For each $i$ the Schwartz class is a core for $L_{i}$ : on $D (L_{i})$ the graph norm $∥ ψ ∥_{L_{i}} : = (∥ ψ ∥_{H^{(r)}}^{2} + ℏ^{- 2} ∥ L_{i} ψ ∥_{H^{(r)}}^{2})^{1 / 2}$ makes $S^{(r)}$ dense in $D (L_{i})$ (equivalently, the graph closure of $L_{i} ∣_{S^{(r)}}$ is $L_{i}$ ). This norm is equivalent to the ordinary graph norm because $ℏ > 0$ is fixed.

Facts & Assumptions

Given: The Axiom of Countable Choice and the Axiom of Choice; $r \geq 1$ ; $ℏ > 0$ ; the space $H^{(r)}$ ; the componentwise operators $Q_{j}, P_{k}$ ; the rotation action $U$ and the groups $U_{i}$ ; the generators $L_{i}$ ; a class $ψ \in S^{(r)}$ ; real $s, t$ and $h \neq 0$ .

[F1]

Rotation action. $(U (R) f) (x) = f (R^{- 1} x)$ componentwise defines a strongly continuous unitary representation of $S O (3)$ on $H^{(r)}$ ; for $R$ near the identity and $ψ \in S^{(r)}$ the reflected class $ψ \circ R^{- 1}$ is Schwartz, and $R \mapsto U (R) ψ$ is continuous (The spatial rotation action on the component-valued wave-function space clauses (a),(b), Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form).

[F2]

Matrices. $t \mapsto R_{i} (t)$ is a continuous group homomorphism $(R, +) \to S O (3)$ with $R_{i} (0) = I$ , $R_{i} (s) R_{i} (t) = R_{i} (s + t)$ and $R_{i} (t)^{- 1} = R_{i} (- t)$ ; and $\frac{d}{d t} ∣_{t = 0} R_{i} (t) = G_{i}$ , where $G_{1} x = (0, - x_{3}, x_{2})$ , $G_{2} x = (x_{3}, 0, - x_{1})$ and $G_{3} x = (- x_{2}, x_{1}, 0)$ for $x \in R^{3}$ , so that $G_{i} x = e_{i} \times x$ . These follow by direct computation from the addition formulas and the derivatives $\sin^{'} = \cos$ , $\cos^{'} = - \sin$ and the multiplication rules for $3 \times 3$ matrices (The addition formulas for sine and cosine, The derivatives of sine and cosine are cosine and minus sine, 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 cross product in $R^{3}$ , The orthogonal group $O (3)$ and the rotation group $S O (3)$ , $S O (3)$ is a subgroup of the general linear group, with $R^{- 1} = R^{T}$ ).

[F3]

Componentwise operators and derivatives. $Q_{j}, P_{k}$ are self-adjoint with maximal componentwise domains, they map $S^{(r)}$ into itself, and $D_{j}$ denotes the weak derivative, which on Schwartz classes agrees with the classical partial derivative (Componentwise position and momentum operators on the component-valued wave-function space clause (a), Weak derivative of a locally integrable function, Schwartz space and its seminorms).

[F4]

Stone's theorem. $T \mapsto e^{i t T}$ is a bijection from self-adjoint operators onto strongly continuous one-parameter unitary groups, and the domain $D (T)$ is exactly the set of vectors where $t \mapsto e^{i t T} x$ is differentiable at $0$ , with derivative $i T x$ (Stone's theorem: unitary groups and self-adjoint generators, Infinitesimal generator of a unitary group).

[F5]

Schwartz decay and dominated convergence. For $ψ \in S^{(r)}$ each component and each of its first partial derivatives is bounded and decays faster than any polynomial; hence for fixed $x$ the map $t \mapsto ψ_{l} (R_{i} (t)^{- 1} x)$ is differentiable with derivative $- \sum_{j} (G_{i} x)_{j} (D_{j} ψ_{l}) (x)$ at $t = 0$ , and the difference quotients are dominated by a fixed Schwartz-decaying function near $t = 0$ (Schwartz space and its seminorms, The mean value inequality: if $f : [a, b] \to R^{m}$ is continuous and differentiable on $(a, b)$ with $∥ f^{'} ∥_{2} \leq M$ , then $∥ f (b) - f (a) ∥_{2} \leq M (b - a)$ , Dominated convergence, 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 $g (c) \neq 0$ ).

[F6]

Canonical commutation relations. On $S^{(r)}$ one has $(Q_{j} P_{k} - P_{k} Q_{j}) ψ = i ℏ δ_{j k} ψ$ (Canonical commutation relations of position and momentum on the Schwartz core clause (a)).

[F7]

Schwartz density. The componentwise Schwartz classes are dense in $H^{(r)}$ (Schwartz space is dense in L2).

Proof

Proof technique: direct; read the group law and continuity from the rotation representation, differentiate the orbit in the position representation along the coordinate subgroups, identify the generator with the angular-momentum expression on Schwartz classes, and compute the commutators from the canonical commutation relations.

Given: The data of the statement and [F1]-[F6].

1.1F1F2givenalgebra

[Group law and strong continuity] By [F1], $U_{i} (s + t) = U (R_{i} (s + t)) = U (R_{i} (s) R_{i} (t)) = U (R_{i} (s)) U (R_{i} (t)) = U_{i} (s) U_{i} (t)$ using the group law of [F2] and the representation property of [F1]; $U_{i} (0) = U (I) = i d$ ; each $U_{i} (t)$ is unitary; and strong continuity of $t \mapsto U_{i} (t) ψ$ is the composition of the continuous map $t \mapsto R_{i} (t)$ of [F2] with the continuous orbit map of [F1]. This proves (a).

1.2F1F2F5givenalgebra

[Differentiation of the orbit on Schwartz classes] Fix $ψ \in S^{(r)}$ , $i$ and one component. By [F2], $R_{i} (h)^{- 1} = R_{i} (- h) = I - h G_{i} + O (h^{2})$ entrywise, so $R_{i} (h)^{- 1} x = x - h G_{i} x + O (h^{2})$ uniformly for $x$ in a compact set; hence $ψ_{l} (R_{i} (h)^{- 1} x) - ψ_{l} (x) = - h \sum_{j} (G_{i} x)_{j} (D_{j} ψ_{l}) (x) + o (h)$ pointwise as $h \to 0$ by the differentiability of $ψ_{l}$ and the chain rule [F5], with $(G_{i} x)_{j} = (e_{i} \times x)_{j}$ by [F2]. The difference quotients are dominated near $h = 0$ by a fixed Schwartz-decaying function because $ψ_{l}$ and its first derivatives decay faster than any polynomial and $∣ x - h G_{i} x + O (h^{2}) ∣ \leq 2 ∣ x ∣$ for small $h$ ; dominated convergence [F5] therefore gives convergence in $H^{(r)}$ , so the orbit is differentiable at $0$ with the displayed derivative.

2.1step 1.2F2F3F4givenalgebra

[Generator on Schwartz classes] By step 1.2 the difference quotient of $t \mapsto U_{i} (t) ψ$ at $0$ converges, so $ψ \in D (L_{i})$ and $- \frac{i}{ℏ} L_{i} ψ = - \sum_{j} (e_{i} \times x)_{j} D_{j} ψ$ by [F4]; multiplying by $i ℏ$ gives $L_{i} ψ = - i ℏ \sum_{j} (e_{i} \times x)_{j} D_{j} ψ$ . Since $(e_{i} \times x)_{j} = x_{j^{'}}$ for the two indices $j^{'}$ complementary to $i$ with the signs of [F2], this reads $L_{i} ψ = \sum_{j, k} ε_{i j k} Q_{j} P_{k} ψ$ : for example $L_{3} ψ = - i ℏ (- x_{2} D_{1} + x_{1} D_{2}) ψ = x_{1} (- i ℏ D_{2}) ψ - x_{2} (- i ℏ D_{1}) ψ = (Q_{1} P_{2} - Q_{2} P_{1}) ψ$ , and the other two cases are the cyclic permutations. This proves (b).

2.2step 1.1F1F4F7givenalgebra

[Schwartz classes are a graph core] Fix $i$ and choose a nonnegative $η \in C_{c}^{\infty} (R)$ supported in $[- 1, 1]$ with $\int_{R} η (t) d t = 1$ ; for example, normalize $e^{- 1 / (1 - t^{2})}$ on $∣ t ∣ < 1$ and extend it by zero. Its derivatives at the endpoints vanish because each is a finite sum of powers of $(1 - t^{2})^{- 1}$ times the exponential, and $u^{- N} e^{- 1 / u} \to 0$ as $u ↓ 0$ for every $N$ . Put $η_{ϵ} (t) = ϵ^{- 1} η (t / ϵ)$ and, for $f \in H^{(r)}$ , define $S_{ϵ} f : = \int_{R} η_{ϵ} (t) U_{i} (t) f d t$ , where the integral is the norm limit of Riemann sums for the continuous $H^{(r)}$ -valued integrand on its compact support; completeness of $H^{(r)}$ gives this limit. If $f \in S^{(r)}$ , then $S_{ϵ} f \in S^{(r)}$ : its representative is $\int η_{ϵ} (t) f (R_{i} (- t) x) d t$ , and on the compact $t$ -support every spatial derivative and polynomially weighted derivative is bounded uniformly by finitely many Schwartz seminorms, so differentiation under the integral gives finite seminorms of every order. By the group law and the substitution $s = t + h$ , $\frac{U_{i} (h) S_{ϵ} f - S_{ϵ} f}{h} = \int_{R} \frac{η_{ϵ} (s - h) - η_{ϵ} (s)}{h} U_{i} (s) f d s ⟶ - \int_{R} η_{ϵ}^{'} (s) U_{i} (s) f d s$ in $H^{(r)}$ : the scalar difference quotients converge uniformly to $- η_{ϵ}^{'}$ and have support in a common compact interval for small $h$ . The Stone-generator domain characterization in [F4] therefore gives $S_{ϵ} f \in D (L_{i})$ and $L_{i} S_{ϵ} f = - i ℏ \int η_{ϵ}^{'} (s) U_{i} (s) f d s$ , so $ℏ^{- 1} ∥ L_{i} S_{ϵ} f ∥ \leq ϵ^{- 1} ∥ η^{'} ∥_{1} ∥ f ∥$ . If $f \in D (L_{i})$ , instead write the quotient as $\int η_{ϵ} (t) U_{i} (t) \frac{U_{i} (h) f - f}{h} d t$ ; its integrand converges uniformly in $t$ in $H^{(r)}$ to $- \frac{i}{ℏ} η_{ϵ} (t) U_{i} (t) L_{i} f$ , so $L_{i} S_{ϵ} f = S_{ϵ} L_{i} f$ . Strong continuity and nonnegativity of $η$ now give $∥ S_{ϵ} f - f ∥ \leq \sup_{∣ t ∣ \leq ϵ} ∥ U_{i} (t) f - f ∥ \to 0$ , and the same estimate for $L_{i} f$ shows $S_{ϵ} f \to f$ in $∥ \cdot ∥_{L_{i}}$ . Finally take $ϵ_{n} = 1 / n$ and choose $φ_{n} \in S^{(r)}$ with $∥ φ_{n} - f ∥ < [n (1 + ϵ_{n}^{- 1} ∥ η^{'} ∥_{1})]^{- 1}$ , possible by [F7]. The bounds above give $∥ S_{ϵ_{n}} (φ_{n} - f) ∥_{L_{i}} \to 0$ , while $S_{ϵ_{n}} φ_{n} \in S^{(r)}$ ; hence $S^{(r)}$ is dense in $D (L_{i})$ in the stated graph norm.

3.1step 2.1F3F5givenalgebra

[Invariance of the Schwartz class] The expression $L_{i} ψ = - i ℏ \sum_{j} (e_{i} \times x)_{j} D_{j} ψ$ is a finite sum of products of a linear polynomial with a first weak derivative of a Schwartz class; hence $L_{i} ψ \in S^{(r)}$ by the closure of $S^{(r)}$ under polynomial multiplication and differentiation [F5], so $L_{i} S^{(r)} \subseteq S^{(r)}$ .

4.1step 2.1step 3.1F3F6givenalgebra

[Commutators on the Schwartz class] Let $ψ \in S^{(r)}$ and use the product rule for commutators $[A B, C D] = A [B, C] D + A C [B, D] + [A, C] D B + C [A, D] B$ , verified by expanding both sides. By step 2.1, $L_{1} = Q_{2} P_{3} - Q_{3} P_{2}$ and $L_{2} = Q_{3} P_{1} - Q_{1} P_{3}$ , and by [F3] all displayed products are defined on $S^{(r)}$ . Using [F6] and bilinearity, only the terms containing a nonvanishing commutator contribute: $[Q_{2} P_{3}, Q_{3} P_{1}] = Q_{2} [P_{3}, Q_{3}] P_{1} = - i ℏ Q_{2} P_{1}$ , and $[Q_{3} P_{2}, Q_{1} P_{3}] = Q_{1} [Q_{3}, P_{3}] P_{2} = i ℏ Q_{1} P_{2}$ , while $[Q_{2} P_{3}, Q_{1} P_{3}] = 0$ and $[Q_{3} P_{2}, Q_{3} P_{1}] = 0$ because in the remaining products every commutator is of coordinate with coordinate or of momentum with momentum, namely $[P_{3}, Q_{1}] = [Q_{2}, Q_{1}] = [Q_{2}, P_{3}] = 0$ for the first and $[P_{2}, Q_{3}] = [Q_{3}, Q_{3}] = [Q_{3}, P_{1}] = 0$ for the second. Hence $[L_{1}, L_{2}] ψ = - i ℏ Q_{2} P_{1} ψ + i ℏ Q_{1} P_{2} ψ = i ℏ (Q_{1} P_{2} - Q_{2} P_{1}) ψ = i ℏ L_{3} ψ$ . The index triple $(1, 2, 3)$ and the three coordinate directions enter the definitions of $L_{1}, L_{2}, L_{3}$ cyclically symmetrically, and the canonical commutation relations are cyclically invariant, so the two cyclic permutations of this computation give $[L_{2}, L_{3}] = i ℏ L_{1}$ and $[L_{3}, L_{1}] = i ℏ L_{2}$ on $S^{(r)}$ ; that is $[L_{i}, L_{k}] ψ = i ℏ \sum_{j} ε_{i k j} L_{j} ψ$ for all $i, k$ . This proves (c).

5.1step 1.1step 1.2step 2.1step 3.1step 2.2step 4.1∎

[Assembly] Step 1.1 proves (a), steps 1.2 and 2.1 prove (b), step 3.1 proves the Schwartz invariance in (c), steps 2.2 and 4.1 prove the graph-core and commutator assertions, and the fixed definition of the graph norm in the Statement completes (d).

Remarks

Sign and normalization. All three generators are fixed by the convention $U_{i} (t) = e^{- i t L_{i} / ℏ}$ already used for the energy flow; with it, $L_{i} = \sum_{j, k} ε_{i j k} Q_{j} P_{k}$ on $S^{(r)}$ , the classical expression $L = x \times p$ read with $P = (- i ℏ D)$ .
Core and commutators. The graph-core statement concerns each coordinate generator separately and does not assert a joint graph core for products or for a sum such as $\sum_{i} L_{i}^{2}$ .

Coordinate-rotation generators on the Schwartz classes and their commutators

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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