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The spatial parity action on the component-valued wave-function space

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 inner product linear in the first argument (Complex Lp classes and Euclidean test-function conventions, The space $L^{p} (μ)$ as the quotient by null functions, Hilbert space), let $Q_{j}, P_{k}$ be the componentwise position and momentum operators (Componentwise position and momentum operators on the component-valued wave-function space clause (a)), and let $S^{(r)} = S (R^{3}; C^{r})$ and $W^{m, 2} (R^{3}; C^{r})$ be the componentwise Schwartz and Sobolev classes (Schwartz space and its seminorms, Integer-order Sobolev spaces and their norms). Define the parity action $(Π f)_{l} (x) : = f_{l} (- x) (x \in R^{3}, l = 1, \dots, r),$ let $U^{(r)}$ be the unitary momentum representation of The momentum representation of the componentwise kinematics, and let $M_{- 1}$ be the reflection operator on $L^{2} (R^{3}, d p; C^{r})$ defined by $(M_{- 1} g) (p) : = g (- p)$ . This composition operator is not pointwise multiplication by a function; reflection substitution makes it unitary. Then:

(a) $Π$ is a well-defined unitary operator with $Π^{2} = i d$ and $Π^{*} = Π$ ; moreover $Π S^{(r)} = S^{(r)}$ and $Π W^{m, 2} (R^{3}; C^{r}) = W^{m, 2} (R^{3}; C^{r})$ for $m = 1, 2$ ;

(b) $Π D (Q_{j}) = D (Q_{j})$ and $Π Q_{j} Π = - Q_{j}$ ; $Π D (P_{k}) = D (P_{k})$ and $Π P_{k} Π = - P_{k}$ ;

(d) with $H_{0} = \frac{1}{2 m_{0}} \sum_{k} P_{k}^{2}$ for a parameter $m_{0} > 0$ and a real bounded measurable $V : R^{3} \to R$ , writing $V_{P} (x) : = V (- x)$ and $H_{V} = H_{0} + M_{V}$ , one has $Π H_{0} Π = H_{0}$ and $Π M_{V} Π = M_{V_{P}}$ , so $Π H_{V} Π = H_{V_{P}}$ .

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 $Q_{j}, P_{k}$ ; $f \in H^{(r)}$ ; $ψ \in S^{(r)}$ ; the momentum unitary $U^{(r)}$ and the reflection operator $M_{- 1}$ ; a real bounded measurable $V$ ; a parameter $m_{0} > 0$ .

[F1]

Componentwise kinematics. $Q_{j}, P_{k}$ are self-adjoint with the maximal componentwise domains, $S^{(r)}$ lies in both domains and is invariant under both operators, and on $S^{(r)}$ one has $Q_{j} f = x_{j} f$ and $P_{k} f = - i ℏ D_{k} f$ (Componentwise position and momentum operators on the component-valued wave-function space clauses (a),(c), Schwartz space and its seminorms).

[F2]

Momentum representation. $U^{(r)}$ is unitary with $U^{(r)} D (P_{k}) = D (M_{p_{k}})$ , $U^{(r)} P_{k} = M_{p_{k}} U^{(r)}$ , acting on Schwartz classes by the componentwise Fourier transform with normalization $(2 π ℏ)^{- 3 / 2}$ , and $U^{(r)} S^{(r)} = S^{(r)}$ (The momentum representation of the componentwise kinematics clauses (a),(b), Componentwise position and momentum operators on the component-valued wave-function space clause (b)).

[F3]

Reflection substitution. For $f \in L^{1} (R^{3}; C)$ one has $\int f (- x) d x = \int f (y) d y$ , since $x \mapsto - x$ is a linear substitution of determinant $- 1$ and absolute determinant $1$ (A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not). A class in $H^{(r)}$ is determined by its components, $∥ f ∥^{2} = \sum_{l} ∥ f_{l} ∥^{2}$ , and componentwise operations commute with the $L^{2}$ class formation (Unitary identifications of the component-valued wave-function space clause (a), The induced length is a norm).

[F4]

Schwartz invariance. If $ψ \in S^{(r)}$ , then $ψ_{l} (- \cdot)$ is smooth and $\sup_{x} ∣ x^{α} \partial^{β} ψ_{l} (- x) ∣ = \sup_{y} ∣ (- y)^{α} \partial^{β} ψ_{l} (y) ∣$ , which is finite; hence $Π ψ \in S^{(r)}$ and by symmetry $Π S^{(r)} = S^{(r)}$ (Schwartz space and its seminorms).

[F5]

Sobolev and Fourier characterization. Under the momentum variable $p = 2 π ℏ ξ$ of [F2], $W^{1, 2}$ and $W^{2, 2}$ are the sets of classes with $p_{k} U^{(r)} f \in L^{2}$ for all $k$ , respectively with $∣ p ∣^{2} U^{(r)} f \in L^{2}$ (Integer-order W^{k,2} and H^k agree with equivalent norms, Plancherel theorem, Integer-order Sobolev spaces and their norms).

[F6]

Density and extension. $S^{(r)}$ is dense in $H^{(r)}$ (Schwartz space is dense in L2); a bounded linear operator on a dense subspace extends uniquely to the whole space (A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm).

Proof

Proof technique: direct; verify unitarity and involution by the reflection substitution, transport the action to the momentum representation for the operator conjugations, and read the Hamiltonian identities off the operator conjugations.

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

1.1F3givenalgebra

[Unitarity and involution] For $f \in H^{(r)}$ the function $x \mapsto f (- x)$ is measurable with $\int ∣ f_{l} (- x) ∣^{2} d x = \int ∣ f_{l} (y) ∣^{2} d y$ by [F3], so the assignment is well defined on classes, linear and isometric; applying it twice returns the original representative, so $Π^{2} = i d$ , $Π$ is bijective and hence unitary with $Π^{- 1} = Π$ . Being unitary and an involution, $Π$ is also self-adjoint: $⟨ Π f, g ⟩ = ⟨ f, Π g ⟩$ follows by the same substitution, $Π^{*} = Π$ .

1.2F4given

[Schwartz invariance] [F4] gives $Π S^{(r)} = S^{(r)}$ directly, with the substitution $y = - x$ used only to identify the reflected exponents.

2.1step 1.1F1givenalgebra

[Conjugation of position] Let $f \in H^{(r)}$ and compute $(Π Q_{j} Π f)_{l} (x) = (Q_{j} Π f)_{l} (- x) = (- x)_{j} (Π f)_{l} (- x) = - x_{j} f_{l} (x)$ , so $Π Q_{j} Π = - Q_{j}$ as an identity of operators with domain $D (Q_{j})$ , since $Π$ maps $D (Q_{j})$ onto itself (multiplication by $- x_{j}$ is bounded exactly where $x_{j}$ is) and $Π^{- 1} = Π$ by step 1.1.

2.2step 1.2F2F3F6givenalgebra

[Momentum-space form] Let $ψ \in S^{(r)}$ and compute componentwise with the substitution $y = - x$ of [F3]: $(U^{(r)} Π ψ)_{l} (p) = (2 π ℏ)^{- 3 / 2} \int e^{- i p \cdot x / ℏ} ψ_{l} (- x) d x = (2 π ℏ)^{- 3 / 2} \int e^{i p \cdot y / ℏ} ψ_{l} (y) d y = (U^{(r)} ψ)_{l} (- p) = (M_{- 1} U^{(r)} ψ)_{l} (p)$ . Both $U^{(r)} Π U^{(r) - 1}$ and $M_{- 1}$ are bounded, and they agree on the dense subspace $U^{(r)} S^{(r)} = S^{(r)}$ by [F2]; [F6] extends the identity to all of $L^{2} (R^{3}, d p; C^{r})$ .

3.1step 2.2F2F5givenalgebra

[Conjugation of momentum] By [F2] and step 2.2 the conjugate of $P_{k}$ corresponds to $M_{- 1} M_{p_{k}} M_{- 1} = M_{p_{k} \circ (- i d)} = M_{- p_{k}}$ , because $(M_{- 1} M_{p_{k}} M_{- 1} g) (p) = p_{k} g (p) \cdot (- 1)$ ; hence $U^{(r)} Π P_{k} Π U^{(r) - 1} = U^{(r)} (- P_{k}) U^{(r) - 1}$ and, conjugating by the unitary $U^{(r)}$ , $Π P_{k} Π = - P_{k}$ including domains by [F5] and step 2.2.

3.2step 2.2F5givenalgebra

[Sobolev invariance] By [F5] the sets $W^{1, 2}, W^{2, 2}$ are described by the conditions $p_{k} g \in L^{2}$ and $∣ p ∣^{2} g \in L^{2}$ on the momentum representative; by step 2.2 this representative is transformed by $g \mapsto g (- \cdot)$ , and $∣ (- p) ∣ = ∣ p ∣$ with the coordinate conditions again unchanged; hence $Π W^{m, 2} = W^{m, 2}$ for $m = 1, 2$ .

4.1step 3.1step 3.2F1givenalgebra

[Hamiltonian conjugation] Since $Π P_{k} Π = - P_{k}$ with domain preservation by step 3.1 and $Π W^{2, 2} = W^{2, 2}$ by step 3.2, applying the identity twice gives $Π P_{k}^{2} Π = P_{k}^{2}$ on $W^{2, 2}$ , hence $Π H_{0} Π = H_{0}$ ; and for $f \in H^{(r)}$ one has $(Π M_{V} Π f)_{l} (x) = V (- x) f_{l} (x)$ by step 1.1, so $Π M_{V} Π = M_{V_{P}}$ on all of $H^{(r)}$ . Adding on the common domain gives $Π H_{V} Π = H_{V_{P}}$ .

5.1step 1.1step 1.2step 2.1step 2.2step 3.1step 3.2step 4.1∎

[Assembly] Steps 1.1 and 1.2 prove (a) with step 3.2 for the Sobolev part, steps 2.1, 2.2, 3.1 prove (b) and (c), and step 4.1 proves (d).

Remarks

Discrete transformation. The declared parity action is the two-element group generated by an involution; no continuous one-parameter group or Stone generator is assigned to it here; its commutation with the dynamics is decided directly by the conjugation identity $Π H_{V} Π = H_{V_{P}}$ .
Scope. Parity acts on the spatial argument only; no internal-factor sign and no SU(2) lift is introduced.

Depends on

Dependency tree · 2 levels

Sources

J. Binney and D. Skinner, The Physics of Quantum Mechanics, Sections 4.1.4 and 4.2, printed pages 62-68 (PDF pages 70-76): the parity operator, its eigenvalues, and the anticommutation relations with position, momentum and vector operators