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Qm Massive Particle Kinematics

1 · Prerequisites

2 · Summary

This page develops the kinematic one-particle structure of the massive spin- $s$ model that the kinematics postulate P5 assigns. The model reading is fixed in The massive spin-s particle model of the framework: state space, operators, units and frame readings: the state space is the component-valued Hilbert space $H_{s} = L^{2} (R^{3}; C^{2 s + 1})$ of almost-everywhere classes, read equivalently as the complete tensor product of the spatial factor with the finite internal factor assigned in P6; position and momentum are the componentwise operators $Q_{j}, P_{k}$ with their maximal domains and declared units; the internal spin matrices act as matrix multiplications; and the momentum representative and frame reading are fixed. The mathematics supporting the definition is local to this page: Finite orthogonal sums of self-adjoint operators and of bounded operators proves self-adjointness and norms for finite orthogonal sums, Unitary identifications of the component-valued wave-function space identifies the direct-sum and tensor descriptions, Componentwise position and momentum operators on the component-valued wave-function space proves that the componentwise operators are self-adjoint with their maximal domains, gives the Fourier representation and the translation identities, and identifies the coordinate translation groups with the spectral flows of the momentum components, Canonical commutation relations on the component-valued core and commutation with the internal factor proves the canonical commutation relations on the Schwartz core, determines the spectral projection-valued measures of $Q_{j}$ and $P_{k}$ and shows that the spatial measures commute with the internal factor, and The momentum representation of the componentwise kinematics constructs the unitary momentum representation and its operator transforms.

On this basis Position and momentum outcome laws of the massive spin-s model derives the position and momentum outcome laws of the spin- $s$ model: the joint position law with density $∥ ψ (x) ∥_{C^{2 s + 1}}^{2}$ , the momentum laws with density $∥ ψ_{p} (p) ∥^{2}$ in the momentum variable, the atomlessness of exact values and the invariance of the spatial laws under the internal factor. Translation covariance of the massive spin-s kinematics and of its bounded-potential dynamics and Galilean boost covariance of the free massive spin-s model and the moving-frame reading establish the covariance of these laws and of the bounded-potential dynamics under the Galilean translations and boosts of the model's frame class, including the free-propagator identity and the moving-frame bounded-potential model. The remaining items record the exact scope: the model is conditional on the framework postulates, the mass and spin labels are empirical parameters when a species is identified, and massless, relativistic, singular-potential, boundary and radiation models are separate assumptions. The page is the kinematic prerequisite of the interference, uncertainty, one-dimensional-potential, oscillator and symmetry pages, which consume its operators, Fourier normalization and covariance results.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Finite orthogonal sums of self-adjoint operators and of bounded operators

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), the choice hypothesis of the cited definitions of the adjoint. Fix an integer $r \geq 1$ , let $H_{1}, \dots, H_{r}$ be complex Hilbert spaces with inner products linear in the first argument (Hilbert space), and let $H : = ⨁_{l = 1}^{r} H_{l}, ⟨ (x_{l}), (y_{l}) ⟩ : = \sum_{l = 1}^{r} ⟨ x_{l}, y_{l} ⟩_{H_{l}},$ be their orthogonal sum. For each $l$ let $A_{l} : D (A_{l}) \subset H_{l} \to H_{l}$ be a densely defined self-adjoint operator (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators) and put $D (A) : = {(x_{l}) \in H : x_{l} \in D (A_{l}) for all l, \sum_{l = 1}^{r} ∥ A_{l} x_{l} ∥_{H_{l}}^{2} < \infty}, A (x_{l}) : = (A_{l} x_{l}) .$ Then:

(a) $H$ is a complex Hilbert space, $D (A)$ is a dense linear subspace of $H$ , and $A : D (A) \to H$ is a linear operator;

(b) $A$ is symmetric and self-adjoint, and $∥ A x ∥^{2} = \sum_{l = 1}^{r} ∥ A_{l} x_{l} ∥_{H_{l}}^{2}$ for every $x \in D (A)$ ;

(c) if $B_{l} \in B (H_{l})$ for every $l$ (A bounded linear operator between normed spaces), then $B : = ⨁_{l = 1}^{r} B_{l}$ , defined by $B (x_{l}) : = (B_{l} x_{l})$ for all $(x_{l}) \in H$ , is bounded with $∥ B ∥ = \max_{1 \leq l \leq r} ∥ B_{l} ∥$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum), its adjoint is $B^{*} = ⨁_{l = 1}^{r} B_{l}^{*}$ (The Hilbert-space adjoint of a bounded operator), and $B$ is unitary if every $B_{l}$ is unitary.

No assertion is made about infinite orthogonal sums: only the finite case is proved here.

Facts & Assumptions

Given: The Axiom of Countable Choice; the integer $r \geq 1$ ; complex Hilbert spaces $H_{1}, \dots, H_{r}$ with pairings linear in the first argument; densely defined self-adjoint operators $A_{l}$ on $H_{l}$ ; bounded operators $B_{l} \in B (H_{l})$ ; vectors $x = (x_{l}), y = (y_{l}) \in H$ ; unit vectors $u_{l} \in H_{l}$ ; scalars $α, β \in C$ .

[F1]

The orthogonal sum. $⟨ (x_{l}), (y_{l}) ⟩ = \sum_{l} ⟨ x_{l}, y_{l} ⟩_{l}$ is a Hermitian sesquilinear form, linear in the first argument, with $⟨ x, x ⟩ = \sum_{l} ∥ x_{l} ∥^{2} \geq 0$ and equality exactly for $x = 0$ ; the induced norm is $∥ x ∥^{2} = \sum_{l} ∥ x_{l} ∥^{2}$ (Hilbert space). A complex Hilbert space is complete for its norm.

[F2]

Adjoints. For a densely defined operator $T$ on a Hilbert space, $D (T^{*})$ consists of those $y$ for which $x \mapsto ⟨ T x, y ⟩$ is a bounded linear functional on $D (T)$ , and $T^{*} y$ is its Riesz representative; $T$ is symmetric when $⟨ T x, y ⟩ = ⟨ x, T y ⟩$ for all $x, y \in D (T)$ , and self-adjoint when $D (T^{*}) = D (T)$ and $T^{*} = T$ (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators). In particular a symmetric operator with $D (T^{*}) \subseteq D (T)$ is self-adjoint.

[F3]

Bounded operators. For $B \in B (H)$ one has $∥ B x ∥ \leq ∥ B ∥ ∥ x ∥$ , and $∥ B ∥ = \sup_{∥ x ∥ \leq 1} ∥ B x ∥$ is attained as a supremum over the unit ball (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum). The Hilbert adjoint $B^{*}$ is characterized by $⟨ B x, y ⟩ = ⟨ x, B^{*} y ⟩$ for all $x, y \in H$ ; $B$ is unitary exactly when $B^{*} B = B B^{*} = I$ (The Hilbert-space adjoint of a bounded operator).

Proof

technique · direct; verify the Hilbert-space axioms componentwise, prove density, symmetry and the adjoint computation slot by slot, and compute the norm and adjoint of a bounded orthogonal sum

Given: The data of the statement and [F1]-[F3].

1.1F1algebra

[Hilbert-space axioms] [F1] gives sesquilinearity, Hermitian symmetry and positive definiteness of the displayed form, since all of these are checked componentwise and the sum over $l = 1, \dots, r$ is finite. For completeness let $(x^{(n)})_{n}$ be a Cauchy sequence in $H$ , $x^{(n)} = (x_{l}^{(n)})_{l}$ . For each $l$ and all $m, n$ one has $∥ x_{l}^{(n)} - x_{l}^{(m)} ∥_{l} \leq ∥ x^{(n)} - x^{(m)} ∥$ , so $(x_{l}^{(n)})_{n}$ is Cauchy in $H_{l}$ and converges to some $x_{l} \in H_{l}$ ; then $\sum_{l} ∥ x_{l} ∥^{2} = \lim_{n} \sum_{l} ∥ x_{l}^{(n)} ∥^{2} \leq {lim sup}_{n} ∥ x^{(n)} ∥^{2} < \infty$ because a Cauchy sequence is bounded, so $x : = (x_{l}) \in H$ , and $∥ x^{(n)} - x ∥^{2} = \lim_{m} \sum_{l} ∥ x_{l}^{(n)} - x_{l}^{(m)} ∥^{2} = \lim_{m} ∥ x^{(n)} - x^{(m)} ∥^{2}$ , which tends to $0$ as $n \to \infty$ uniformly in $n$ because the sequence is Cauchy; hence $x^{(n)} \to x$ . This proves (a) for the space.

1.2F1F2algebra

[Density and linearity of the domain] The algebraic sum $⨁_{l} D (A_{l}) \subseteq D (A)$ : each $(x_{l})$ with $x_{l} \in D (A_{l})$ has a finite sum $\sum_{l} ∥ A_{l} x_{l} ∥^{2}$ . It is dense in $H$ : given $x \in H$ and $ε > 0$ , density of $D (A_{l})$ gives for each $l$ a vector $y_{l} \in D (A_{l})$ with $∥ x_{l} - y_{l} ∥ < ε / \sqrt{r}$ , so $y : = (y_{l}) \in ⨁_{l} D (A_{l})$ and $∥ x - y ∥^{2} = \sum_{l} ∥ x_{l} - y_{l} ∥^{2} < ε^{2}$ . The set $D (A)$ is a linear subspace because $\sum_{l} ∥ α A_{l} x_{l} + β A_{l} y_{l} ∥^{2} \leq 2 ∣ α ∣^{2} \sum_{l} ∥ A_{l} x_{l} ∥^{2} + 2 ∣ β ∣^{2} \sum_{l} ∥ A_{l} y_{l} ∥^{2} < \infty$ for $x, y \in D (A)$ , and $A$ is linear because each $A_{l}$ is.

1.3F2algebra

[Symmetry] Let $x, y \in D (A)$ . Then $⟨ A x, y ⟩ = \sum_{l} ⟨ A_{l} x_{l}, y_{l} ⟩_{l} = \sum_{l} ⟨ x_{l}, A_{l} y_{l} ⟩_{l} = ⟨ x, A y ⟩$ , because each $A_{l}$ is self-adjoint and hence symmetric; all sums are finite.

1.4F1F2algebra

[The adjoint domain] Let $y \in D (A^{*})$ and $z : = A^{*} y$ . For fixed $l$ and each $u \in D (A_{l})$ let $x^{(l, u)} \in H$ have $u$ in slot $l$ and $0$ in all other slots; then $x^{(l, u)} \in D (A)$ and $A x^{(l, u)}$ has $A_{l} u$ in slot $l$ . Using [F2] for $A$ and the componentwise pairing, $⟨ A_{l} u, y_{l} ⟩_{l} = ⟨ A x^{(l, u)}, y ⟩ = ⟨ x^{(l, u)}, z ⟩ = ⟨ u, z_{l} ⟩_{l}$ for every $u \in D (A_{l})$ . Hence $u \mapsto ⟨ A_{l} u, y_{l} ⟩_{l}$ is bounded on $D (A_{l})$ , represented by $z_{l}$ ; so $y_{l} \in D (A_{l}^{*}) = D (A_{l})$ and $A_{l}^{*} y_{l} = z_{l}$ , that is $A_{l} y_{l} = z_{l}$ . Summing, $\sum_{l} ∥ A_{l} y_{l} ∥^{2} = \sum_{l} ∥ z_{l} ∥^{2} = ∥ z ∥^{2} < \infty$ , so $y \in D (A)$ and $A y = z$ . Hence $D (A^{*}) \subseteq D (A)$ and $A^{*} y = A y$ .

2.1step 1.1step 1.2step 1.3step 1.4F1F2algebra

[Self-adjointness] Step 1.3 gives symmetry; symmetry gives $D (A) \subseteq D (A^{*})$ with $A^{*}$ agreeing with $A$ there (Adjoint of a densely defined operator), and step 1.4 gives the converse inclusion, so $A^{*} = A$ by [F2]. The norm identity is [F1] applied to $A x = (A_{l} x_{l})$ . This proves (b).

3.1F3step 1.1algebra∎

[Bounded sums] Let $B_{l} \in B (H_{l})$ and put $M : = \max_{1 \leq l \leq r} ∥ B_{l} ∥$ , which exists because the index set is finite and nonempty. For $x \in H$ , $∥ B x ∥^{2} = \sum_{l} ∥ B_{l} x_{l} ∥^{2} \leq M^{2} \sum_{l} ∥ x_{l} ∥^{2} = M^{2} ∥ x ∥^{2}$ , so $B$ is bounded with $∥ B ∥ \leq M$ by [F3]. If $M > 0$ , choose $l_{0}$ with $∥ B_{l_{0}} ∥ = M$ ; since $∥ B_{l_{0}} ∥ > 0$ , $H_{l_{0}} \neq {0}$ . For each $η > 0$ , the unit-sphere formula for the operator norm in [F3] gives a unit vector $u_{l_{0}} \in H_{l_{0}}$ with $∥ B_{l_{0}} u_{l_{0}} ∥ > M - η$ . The vector with $u_{l_{0}}$ in slot $l_{0}$ and $0$ elsewhere is a unit vector of $H$ , so $∥ B ∥ \geq ∥ B x ∥ = ∥ B_{l_{0}} u_{l_{0}} ∥ > M - η$ ; letting $η ↓ 0$ gives $∥ B ∥ \geq M$ . If $M = 0$ , the upper bound already gives $∥ B ∥ = 0 = M$ . Thus $∥ B ∥ = \max_{l} ∥ B_{l} ∥$ . For $B_{l}^{*}$ the adjoint of $B_{l}$ , the operator $⨁_{l} B_{l}^{*}$ satisfies $⟨ B x, y ⟩ = \sum_{l} ⟨ B_{l} x_{l}, y_{l} ⟩_{l} = \sum_{l} ⟨ x_{l}, B_{l}^{*} y_{l} ⟩_{l} = ⟨ x, (⨁_{l} B_{l}^{*}) y ⟩$ for all $x, y \in H$ , so $B^{*} = ⨁_{l} B_{l}^{*}$ by [F3]. If every $B_{l}$ is unitary then $B^{*} B = ⨁_{l} B_{l}^{*} B_{l} = ⨁_{l} I_{H_{l}} = I$ and likewise $B B^{*} = I$ , so $B$ is unitary. This proves (c).

Remarks

Where the finiteness is used. Both the norm identity of (b) and the norm formula of (c) are finite-sum statements; for infinite orthogonal sums the domain condition in (b) and the supremum in (c) require separate arguments, and no such statement is made here.
Choice. Only the Axiom of Countable Choice is declared, as the choice hypothesis of the cited definitions of the adjoint; the argument itself handles finitely many slots and makes no further selection.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Unitary identifications of the component-valued wave-function space

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix integers $d \geq 1$ and $r \geq 1$ , let $H : = L^{2} (R^{d}; C)$ with the Euclidean integral pairing (Complex Lp classes and Euclidean test-function conventions, The space $L^{p} (μ)$ as the quotient by null functions, $L^{2}$ with the integral pairing is a Hilbert space), let $H^{(r)} : = L^{2} (R^{d}; C^{r}), ⟨ f, g ⟩ : = \int_{R^{d}} \sum_{l = 1}^{r} f_{l} (x) \overline{g_{l} (x)} d x,$ and let $C^{r}$ carry the standard inner product with orthonormal basis $e_{1}, \dots, e_{r}$ (Hilbert space, Orthonormal families, complete orthonormal systems and Hilbert bases). Write $H \hat{\otimes} C^{r}$ for the completed tensor product of Hilbert tensor construction: inner product, completion and product bases with inclusion $κ$ . Then:

(a) the map $Ψ_{0} ((f_{l})) : = (x \mapsto (f_{1} (x), \dots, f_{r} (x)))$ is a well-defined unitary isomorphism $⨁_{l = 1}^{r} H \to H^{(r)}$ , and $∥ Ψ_{0} (f) ∥_{H^{(r)}}^{2} = \sum_{l = 1}^{r} ∥ f_{l} ∥_{H}^{2}, ⟨ Ψ_{0} (f), Ψ_{0} (g) ⟩ = \sum_{l = 1}^{r} ⟨ f_{l}, g_{l} ⟩_{H};$

(b) the assignment on elementary tensors $Θ_{0} (κ (u \otimes e_{l})) : = ι_{l} (u)$ , where $ι_{l} (u)$ is the $r$ -tuple with $u$ in slot $l$ and $0$ elsewhere, extends uniquely to a surjective linear isometry $Θ : H \hat{\otimes} C^{r} \to H^{(r)}$ (so $Θ$ is unitary), and $Θ (κ (u \otimes z)) = \sum_{l = 1}^{r} z_{l} ι_{l} (u)$ for all $u \in H$ , $z = \sum_{l} z_{l} e_{l} \in C^{r}$ ;

(c) for every $a \in L^{\infty} (R^{d})$ and every $v \in R^{d}$ one has the intertwining identities $Θ^{- 1} (a \cdot Θ (w)) = (M_{a} \otimes I_{C^{r}}) w, Θ^{- 1} (T_{v}^{(r)} Θ (w)) = (T_{v} \otimes I_{C^{r}}) w (w \in H \hat{\otimes} C^{r}),$ where $M_{a}$ is multiplication by $a$ , $T_{v}$ is the Euclidean translation $T_{v} h : = h (\cdot - v)$ of Self-adjointness, Fourier representation and translation identities of position and momentum clause (c), $T_{v}^{(r)}$ is its componentwise action on $H^{(r)}$ , and the bounded amplifications are those of Bounded tensor maps: extension, norm estimate, adjoint and commutation.

Facts & Assumptions

Given: The Axiom of Countable Choice; integers $d, r \geq 1$ ; the spaces $H$ , $H^{(r)}$ and the orthonormal basis $e_{1}, \dots, e_{r}$ of $C^{r}$ ; the completed tensor product $H \hat{\otimes} C^{r}$ with inclusion $κ$ ; tuples $f = (f_{l}), g = (g_{l}) \in ⨁_{l} H$ ; an element $w \in H \hat{\otimes} C^{r}$ ; $a \in L^{\infty} (R^{d})$ ; $v \in R^{d}$ .

[F1]

Tensor construction. $H ⊙ C^{r}$ carries the sesquilinear form $⟨ \sum_{i} u_{i} \otimes z_{i}, \sum_{j} u_{j}^{'} \otimes z_{j}^{'} ⟩ = \sum_{i, j} ⟨ u_{i}, u_{j}^{'} ⟩_{H} ⟨ z_{i}, z_{j}^{'} ⟩_{C^{r}}$ , which is an inner product; for finite-dimensional subspaces with orthonormal bases, expansions diagonalize it; the completion $H \hat{\otimes} C^{r}$ with inclusion $κ$ is a complex Hilbert space with dense image and $∥ κ (u \otimes z) ∥ = ∥ u ∥ ∥ z ∥$ (Hilbert tensor construction: inner product, completion and product bases clauses (a)-(c)). Moreover $(e_{l})_{l}$ is an orthonormal basis of $C^{r}$ , so the elementary tensors $u \otimes e_{l}$ have $⟨ \sum_{l} u_{l} \otimes e_{l}, \sum_{l} u_{l}^{'} \otimes e_{l} ⟩ = \sum_{l} ⟨ u_{l}, u_{l}^{'} ⟩$ and $∥ \sum_{l} u_{l} \otimes e_{l} ∥^{2} = \sum_{l} ∥ u_{l} ∥^{2}$ (Hilbert tensor construction: inner product, completion and product bases clause (b)).

[F2]

Bounded amplifications. For bounded $S \in B (H)$ and $T \in B (C^{r})$ there is a unique bounded $S \otimes T$ on $H \hat{\otimes} C^{r}$ with $(S \otimes T) κ (u \otimes z) = κ (S u \otimes T z)$ and $∥ S \otimes T ∥ \leq ∥ S ∥ ∥ T ∥$ ; for $S = M_{a}$ bounded and $T = I$ this is $M_{a} \otimes I$ , and for $S = T_{v}$ unitary and $T = I$ it is $T_{v} \otimes I$ (Bounded tensor maps: extension, norm estimate, adjoint and commutation clauses (a),(d)).

[F3]

Euclidean $L^{2}$ . For $f = (f_{1}, \dots, f_{r}) \in L^{2} (R^{d}; C^{r})$ each component $f_{l}$ is a well-defined class in $L^{2} (R^{d}; C)$ , two representatives that agree almost everywhere have almost everywhere equal components, and $∣ f_{l} ∣ \leq ∥ f ∥_{C^{r}}$ pointwise, so $∥ f_{l} ∥_{H} \leq ∥ f ∥_{H^{(r)}}$ ; conversely every $r$ -tuple of $L^{2}$ classes defines a class in $H^{(r)}$ (Complex Lp classes and Euclidean test-function conventions, The space $L^{p} (μ)$ as the quotient by null functions).

[F4]

Tonelli and Cauchy-Schwarz. For $σ$ -finite measure spaces, integrals of nonnegative product-measurable functions may be iterated (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product); for vectors $f, g$ in an inner product space $∣ ⟨ f, g ⟩ ∣ \leq ∥ f ∥ ∥ g ∥$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

[F5]

Dense extension. A bounded linear map on a dense subspace of a normed space extends uniquely to a bounded linear map on the whole space, with the same norm (A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm).

[F6]

Scalar completeness. The scalar complex space $H = L^{2} (R^{d}; C)$ is complete for its integral-pairing norm ( $L^{2}$ with the integral pairing is a Hilbert space).

Proof

technique · direct; check well-definedness of the component map, compute norms and pairings by Tonelli and Cauchy-Schwarz, and build the tensor identification isometrically on the algebraic tensor product before extending by density

Given: The data of the statement and [F1]-[F5].

1.1F3algebra

[Well-definedness of $Ψ_{0}$ ] Let $(f_{l}) \in ⨁_{l} H$ and let $({\tilde{f}}_{l})$ be another choice of representatives with ${\tilde{f}}_{l} = f_{l}$ almost everywhere for each $l$ ; then $({\tilde{f}}_{1}, \dots, {\tilde{f}}_{r}) = (f_{1}, \dots, f_{r})$ almost everywhere, so $Ψ_{0} (f)$ is the same class of $H^{(r)}$ and $Ψ_{0}$ is well defined [F3]. It is complex-linear because addition and scalar multiplication in $H^{(r)}$ are componentwise.

1.2F3F4algebra

[Isometry] For $f = (f_{l})$ and $g = (g_{l})$ in $⨁_{l} H$ , the function $x \mapsto \sum_{l} ∣ f_{l} (x) ∣^{2}$ is nonnegative and measurable, so [F4] gives $\int_{R^{d}} \sum_{l} ∣ f_{l} ∣^{2} = \sum_{l} \int_{R^{d}} ∣ f_{l} ∣^{2} = \sum_{l} ∥ f_{l} ∥_{H}^{2}$ , that is $∥ Ψ_{0} (f) ∥^{2} = ∥ f ∥_{\oplus}^{2}$ ; the map $Ψ_{0}$ is thus an isometry. For the inner product, Cauchy-Schwarz gives $\int_{R^{d}} \sum_{l} ∣ f_{l} ∣ ∣ g_{l} ∣ \leq \sum_{l} ∥ f_{l} ∥_{H} ∥ g_{l} ∥_{H} \leq (\sum_{l} ∥ f_{l} ∥^{2})^{1 / 2} (\sum_{l} ∥ g_{l} ∥^{2})^{1 / 2} < \infty$ . Each product $f_{l} \overline{g_{l}}$ is integrable by Cauchy-Schwarz, and finite linearity of the Lebesgue integral therefore gives $⟨ Ψ_{0} (f), Ψ_{0} (g) ⟩ = \int \sum_{l} f_{l} \overline{g_{l}} = \sum_{l} ⟨ f_{l}, g_{l} ⟩_{H}$ .

1.3F1F3algebra

[The tensor identification on the algebraic tensor product] On elementary tensors put $Θ_{0} (κ (u \otimes z)) : = \sum_{l} z_{l} ι_{l} (u)$ for $z = \sum_{l} z_{l} e_{l}$ . The map $(u, z) \mapsto \sum_{l} z_{l} ι_{l} (u)$ $H \times C^{r} \to H^{(r)}$ is $C$ -bilinear, so $Θ_{0}$ factors through the defining bilinearity relations of the algebraic tensor product $H ⊙ C^{r}$ and is a well-defined complex-linear map (Hilbert tensor construction: inner product, completion and product bases clause (a)). Every element of $H ⊙ C^{r}$ can be written as $\sum_{l} w_{l} \otimes e_{l}$ : a finite sum $\sum_{i} u_{i} \otimes z_{i}$ equals $\sum_{l} (\sum_{i} z_{i, l} u_{i}) \otimes e_{l}$ by bilinearity. For such an element, [F1] gives $Θ_{0} (\sum_{l} w_{l} \otimes e_{l}) = \sum_{l} ι_{l} (w_{l}) = (w_{1}, \dots, w_{r})$ and $∥ \sum_{l} w_{l} \otimes e_{l} ∥^{2} = \sum_{l} ∥ w_{l} ∥^{2} = ∥ Θ_{0} (\sum_{l} w_{l} \otimes e_{l}) ∥_{H^{(r)}}^{2}$ . By [F3], every element of $H^{(r)}$ is an $r$ -tuple of classes in $H$ , so $Θ_{0}$ is onto $H^{(r)}$ . Hence $Θ_{0}$ is an isometry of $H ⊙ C^{r}$ onto $H^{(r)}$ .

2.1step 1.1step 1.2F3

[Surjectivity] Let $g \in H^{(r)}$ . By [F3] each component $g_{l}$ is an $L^{2}$ class and $g = (g_{1}, \dots, g_{r}) = Ψ_{0} ((g_{l}))$ ; hence $Ψ_{0}$ is surjective, and (a) is proved.

2.2step 1.3F1F3F5F6algebra

[Extension and surjectivity] First $H^{(r)}$ is complete: if $(g^{(n)})_{n}$ is Cauchy, each component sequence $g_{l}^{(n)}$ is Cauchy in the complete scalar space $H$ by [F6] because $∥ g_{l}^{(n)} - g_{l}^{(m)} ∥_{H} \leq ∥ g^{(n)} - g^{(m)} ∥_{H^{(r)}}$ . Let $g_{l} \in H$ be its limit. By [F3] the finite tuple $g = (g_{l})$ lies in $H^{(r)}$ , and $∥ g^{(n)} - g ∥_{H^{(r)}}^{2} = \sum_{l = 1}^{r} ∥ g_{l}^{(n)} - g_{l} ∥_{H}^{2} \to 0$ , so $H^{(r)}$ is complete. By [F5] the isometry $Θ_{0}$ extends uniquely to a linear isometry $Θ : H \hat{\otimes} C^{r} \to H^{(r)}$ , because $κ (H ⊙ C^{r})$ is dense in the completion (Hilbert tensor construction: inner product, completion and product bases clause (c)); since $Θ_{0}$ is onto by step 1.3, so is $Θ$ . Thus $Θ$ is unitary. This proves (b) except for the displayed formula, which holds on elementary tensors by construction and extends to all $w$ by continuity of the bounded operators involved.

3.1step 1.3step 2.2F1F2algebra∎

[Intertwining identities] For $u \in H$ and $z = \sum_{l} z_{l} e_{l}$ , $Θ^{- 1} (a Θ (κ (u \otimes z))) = Θ^{- 1} (a \sum_{l} z_{l} ι_{l} (u)) = Θ^{- 1} (\sum_{l} z_{l} ι_{l} (a u)) = κ (a u \otimes z) = (M_{a} \otimes I) κ (u \otimes z)$ , where the last identity is [F2]; similarly $Θ^{- 1} (T_{v}^{(r)} Θ (κ (u \otimes z))) = κ (T_{v} u \otimes z) = (T_{v} \otimes I) κ (u \otimes z)$ , since $T_{v}^{(r)}$ acts componentwise and $T_{v}^{(r)} ι_{l} (u) = ι_{l} (T_{v} u)$ (Self-adjointness, Fourier representation and translation identities of position and momentum clause (c)). Both sides of each identity are bounded operators (the amplifications are bounded by [F2], and $Θ$ is unitary by step 2.2), and the elementary tensors have dense span in $H \hat{\otimes} C^{r}$ [F1]; hence the identities hold on all of $H \hat{\otimes} C^{r}$ . This proves (c).

Remarks

Two descriptions of one space. Clause (b) says that a finite internal factor $C^{r}$ may be read either as a tensor factor or as component bookkeeping; the component reading is the one used by the model definition on this page. Nothing here identifies a component index with a spatial direction or an observable.
Only bounded amplifications. The intertwining identities of clause (c) use the bounded amplifications of the cited supplier; no unbounded operator is extended or transported by this lemma.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Componentwise position and momentum operators 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), the choice hypotheses of the cited scalar operator and spectral suppliers. Fix integers $d \geq 1$ , $r \geq 1$ and $ℏ > 0$ . Let $H^{(r)} = L^{2} (R^{d}; C^{r})$ be the component-valued space of Unitary identifications of the component-valued wave-function space with its componentwise inner product, let $Q_{j}^{(1)}, P_{k}^{(1)}$ be the scalar position and momentum operators on $L^{2} (R^{d}; C)$ of Position and momentum operators on Euclidean L2, and let $F_{2}$ be the unitary Plancherel transform with $\hat{f} (ξ) = \int f (x) e^{- 2 π i x \cdot ξ} d x$ on $L^{1}$ (Fourier transform on complex L1 classes, Plancherel theorem). Write $F_{2}^{(r)} : = Ψ_{0} (⨁_{l} F_{2}) Ψ_{0}^{- 1}$ for its componentwise version, with $Ψ_{0}$ the unitary of Unitary identifications of the component-valued wave-function space clause (a). Define, componentwise, $D (Q_{j}) : = {f = (f_{l}) \in H^{(r)} : x_{j} f_{l} \in L^{2} for all l}, (Q_{j} f)_{l} : = x_{j} f_{l},$ $D (P_{k}) : = {f = (f_{l}) \in H^{(r)} : D_{k} f_{l} exists in L^{2} for all l}, (P_{k} f)_{l} : = - i ℏ D_{k} f_{l},$ where $D_{k}$ is the weak derivative of Weak derivative of a locally integrable function. Write $W^{m, 2} (R^{d}; C^{r})$ for the classes whose components lie in $W^{m, 2} (R^{d}; C)$ (Integer-order Sobolev spaces and their norms), and $S (R^{d}; C^{r})$ for the componentwise Schwartz class (Schwartz space and its seminorms). Then:

(a) under $Ψ_{0}$ the operators $Q_{j}$ and $P_{k}$ are the orthogonal sums $⨁_{l} Q_{j}^{(1)}$ and $⨁_{l} P_{k}^{(1)}$ of Finite orthogonal sums of self-adjoint operators and of bounded operators; in particular $Q_{j}$ and $P_{k}$ are densely defined and self-adjoint with the displayed domains;

(b) $F_{2}^{(r)}$ is unitary, $D (P_{k}) = {f \in H^{(r)} : ξ_{k} F_{2}^{(r)} f \in L^{2} (R^{d}; C^{r})},$ and for every $f \in D (P_{k})$ $F_{2}^{(r)} (P_{k} f) (ξ) = 2 π ℏ ξ_{k} F_{2}^{(r)} f (ξ) for almost every ξ;$

(c) the componentwise translation $T_{a}^{(r)} f : = f (\cdot - a)$ , $a \in R^{d}$ , is unitary, leaves $S (R^{d}; C^{r})$ , $W^{1, 2} (R^{d}; C^{r})$ and $W^{2, 2} (R^{d}; C^{r})$ invariant, satisfies $D_{k} (T_{a}^{(r)} f) = T_{a}^{(r)} (D_{k} f)$ for $f \in W^{1, 2} (R^{d}; C^{r})$ , and $T_{a}^{(r)} D (Q_{j}) = D (Q_{j}), T_{a}^{(r)} Q_{j} (T_{a}^{(r)})^{- 1} = Q_{j} - a_{j} I, T_{a}^{(r)} D (P_{k}) = D (P_{k}), T_{a}^{(r)} P_{k} (T_{a}^{(r)})^{- 1} = P_{k};$

(d) for every $f \in D (P_{j})$ , $\lim_{t \to 0} \frac{1}{t} (T_{t e_{j}}^{(r)} f - f) = - \frac{i}{ℏ} P_{j} f in H^{(r)},$ so the strongly continuous one-parameter unitary group $t \mapsto T_{t e_{j}}^{(r)}$ has generator $- \frac{i}{ℏ} P_{j}$ and $T_{t e_{j}}^{(r)} = \exp (- \frac{i}{ℏ} t P_{j}) (t \in R),$ the exponential being the spectral-calculus flow of the self-adjoint operator $P_{j}$ .

Facts & Assumptions

Given: The Axiom of Countable Choice and the Axiom of Choice; integers $d, r \geq 1$ ; $ℏ > 0$ ; the scalar operators $Q_{j}^{(1)}, P_{k}^{(1)}$ ; the componentwise operators $Q_{j}, P_{k}$ ; the unitaries $Ψ_{0}$ and $F_{2}, F_{2}^{(r)}$ ; the translations $T_{a}, T_{a}^{(r)}$ ; classes $f, g \in H^{(r)}$ ; a vector $a \in R^{d}$ ; a real $t \neq 0$ ; a coordinate index $j, k$ .

[F1]

Scalar position and momentum. On $L^{2} (R^{d}; C)$ the operators $Q_{j}^{(1)}$ and $P_{k}^{(1)}$ are self-adjoint with $D (Q_{j}^{(1)}) = {f : x_{j} f \in L^{2}}$ and $D (P_{k}^{(1)}) = {f : ξ_{k} F_{2} f \in L^{2}}$ , $F_{2} (P_{k}^{(1)} f) (ξ) = 2 π ℏ ξ_{k} F_{2} f (ξ)$ for $f \in D (P_{k}^{(1)})$ , the scalar translations $T_{a} g = g (\cdot - a)$ are unitary with $T_{a} D (Q_{j}^{(1)}) = D (Q_{j}^{(1)})$ , $T_{a} Q_{j}^{(1)} T_{a}^{- 1} = Q_{j}^{(1)} - a_{j} I$ , $T_{a} D (P_{k}^{(1)}) = D (P_{k}^{(1)})$ , $T_{a} P_{k}^{(1)} T_{a}^{- 1} = P_{k}^{(1)}$ , and $T_{a}$ preserves $W^{1, 2}, W^{2, 2}$ with $D_{k} (T_{a} g) = T_{a} (D_{k} g)$ (Self-adjointness, Fourier representation and translation identities of position and momentum clauses (a)-(c)). For a Schwartz class the classical derivative is the weak derivative and lies in the Schwartz class (Classical derivatives agree with weak derivatives, Schwartz space and its seminorms).

[F2]

Componentwise identifications. $Ψ_{0} : ⨁_{l} L^{2} (R^{d}; C) \to H^{(r)}$ is a unitary isomorphism, a tuple of classes is mapped to the class of its component function, and, directly from this component formula, multiplication by a bounded scalar function $b \in L^{\infty} (R^{d})$ and translation by $a \in R^{d}$ correspond under $Ψ_{0}$ to the direct sums $⨁_{l} M_{b}$ and $⨁_{l} T_{a}$ : their $l$ -th components are respectively $b f_{l}$ and $T_{a} f_{l}$ (Unitary identifications of the component-valued wave-function space clause (a)). In particular a class in $H^{(r)}$ has all its components in the Schwartz class exactly when it lies in $S (R^{d}; C^{r})$ (Schwartz space and its seminorms).

[F3]

Orthogonal sums. For finitely many densely defined self-adjoint operators $A_{l}$ the orthogonal sum $⨁_{l} A_{l}$ is self-adjoint with domain ${(x_{l}) : x_{l} \in D (A_{l}), \sum_{l} ∥ A_{l} x_{l} ∥^{2} < \infty}$ , and the orthogonal sum of unitaries is unitary with norm one (Finite orthogonal sums of self-adjoint operators and of bounded operators clauses (b),(c)).

[F4]

Plancherel and multipliers. $F_{2}$ is unitary (Plancherel theorem); for $L^{2}$ classes $u, w$ with $- i ℏ D_{k}$ defined componentwise, the Fourier multiplier identity is $F_{2} (D_{k} u) (ξ) = 2 π i ξ_{k} F_{2} u (ξ)$ for classes whose weak derivative is in $L^{2}$ (Distributional derivatives are polynomial Fourier multipliers); the scalar function $F_{2} (T_{a} u) (ξ) = e^{- 2 π i a \cdot ξ} F_{2} u (ξ)$ holds for $u \in L^{1} \cap L^{2}$ and extends by density; the exponential is entire with derivative itself, so $\frac{e^{i θ} - 1}{θ} \to i$ as $θ \to 0$ , and $∣ e^{i θ} - 1 ∣ \leq ∣ θ ∣$ (The complex exponential is entire and its complex derivative is itself; the bound follows from the mean value estimate for the curve $s \mapsto e^{i s θ}$ over $[0, 1]$ with $∣ e^{i θ} - 1 ∣ \leq ∣ θ ∣$ ). Translation depends norm-continuously on $a$ in $L^{2}$ , so $a \mapsto T_{a}^{(r)}$ is strongly continuous (Complex translation, convolution, approximate identities, and mollification). [F4]

[F5]

Generation and uniqueness. For a self-adjoint $T$ the family $t \mapsto e^{i t T}$ is a strongly continuous unitary group with generator $i T$ , and two strongly continuous unitary groups with one and the same generator agree (A self-adjoint operator generates a strongly continuous unitary group, Strongly continuous one-parameter unitary group, Infinitesimal generator of a unitary group, Stone's theorem: unitary groups and self-adjoint generators).

[F6]

Dominated convergence. If $∣ h_{t} ∣ \leq G$ with $G$ integrable and $h_{t} \to h$ pointwise as $t \to 0$ , then $\int ∣ h_{t} - h ∣ \to 0$ (Dominated convergence).

Proof

technique · direct; transport the componentwise operators to orthogonal sums of scalar operators, read off self-adjointness and the Fourier representation, verify the translation identities componentwise and prove the difference-quotient limit by Plancherel with dominated convergence before identifying the group

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

1.1F1F2algebra

[Operators as orthogonal sums] Let $f = (f_{l}) \in H^{(r)}$ . By [F2] one has $Ψ_{0}^{- 1} f = (f_{l}) \in ⨁_{l} L^{2}$ , and $f \in D (Q_{j})$ holds exactly when $x_{j} f_{l} \in L^{2}$ for every $l$ , that is when $Ψ_{0}^{- 1} f \in ⨁_{l} D (Q_{j}^{(1)})$ ; on that domain $Ψ_{0}^{- 1} (Q_{j} f) = (x_{j} f_{l})_{l} = (⨁_{l} Q_{j}^{(1)}) Ψ_{0}^{- 1} f$ . The same computation with $D_{k} f_{l}$ in place of $x_{j} f_{l}$ gives $Ψ_{0}^{- 1} (P_{k} f) = (⨁_{l} P_{k}^{(1)}) Ψ_{0}^{- 1} f$ on $Ψ_{0}^{- 1} D (P_{k}) = ⨁_{l} D (P_{k}^{(1)})$ . The finite-sum condition in the domain of an orthogonal sum is automatic here: if every $∥ Q_{j}^{(1)} f_{l} ∥$ is finite then $\sum_{l} ∥ Q_{j}^{(1)} f_{l} ∥^{2} = ∥ Q_{j} f ∥^{2} < \infty$ , and similarly for $P_{k}$ . This proves the identification in (a).

1.2F1F4F6algebra

[A scalar difference quotient] Let $g \in L^{2} (R^{d}; C)$ with $ξ_{j} F_{2} g \in L^{2}$ , that is $g \in D (P_{j}^{(1)})$ by [F1], and let $t \neq 0$ . The modulation identity $F_{2} (T_{t e_{j}} g) (ξ) = e^{- 2 π i t ξ_{j}} F_{2} g (ξ)$ of [F4] gives $F_{2} (\frac{1}{t} (T_{t e_{j}} g - g)) (ξ) = \frac{e^{- 2 π i t ξ_{j}} - 1}{t} F_{2} g (ξ) .$ Writing $θ : = - 2 π t ξ_{j}$ , the factor equals $\frac{e^{i θ} - 1}{i θ} \cdot (- 2 π i ξ_{j})$ ; by [F4] the first factor tends to $1$ and $θ \to 0$ for every fixed $ξ$ as $t \to 0$ , so the pointwise limit of the right-hand side is $- 2 π i ξ_{j} F_{2} g (ξ) = F_{2} (- D_{j} g) (ξ)$ [F4]. For domination, [F4] gives $∣ \frac{e^{i θ} - 1}{t} ∣ \leq ∣ θ / t ∣ = 2 π ∣ ξ_{j} ∣$ , so the modulus of the difference $h_{t} (ξ) : = (\frac{e^{- 2 π i t ξ_{j}} - 1}{t} + 2 π i ξ_{j}) F_{2} g (ξ)$ is at most $4 π ∣ ξ_{j} ∣ ∣ F_{2} g (ξ) ∣$ , whose square is integrable because $ξ_{j} F_{2} g \in L^{2}$ . Since $h_{t} \to 0$ pointwise, [F6] applied to $∣ h_{t} ∣^{2}$ gives $\int ∣ h_{t} ∣^{2} \to 0$ , so the Fourier-space difference converges to $F_{2} (- D_{j} g)$ in $L^{2}$ . As $F_{2}$ is a surjective isometry [F4], and $- D_{j} g = - \frac{i}{ℏ} P_{j}^{(1)} g$ , this yields $\lim_{t \to 0} \frac{1}{t} (T_{t e_{j}} g - g) = - \frac{i}{ℏ} P_{j}^{(1)} g in L^{2} .$

2.1step 1.1F1F3

[Self-adjointness] Each scalar $Q_{j}^{(1)}$ and $P_{k}^{(1)}$ is self-adjoint by [F1], so [F3] makes their finite orthogonal sums self-adjoint with the displayed domains and dense domains; by step 1.1 these sums are the conjugates of $Q_{j}$ and $P_{k}$ by the unitary $Ψ_{0}$ , hence $Q_{j}$ and $P_{k}$ are densely defined and self-adjoint. This completes (a).

2.2step 1.1F1F2F3algebra

[Fourier representation] By [F2] and [F3], $F_{2}^{(r)}$ is the orthogonal sum of the unitaries $F_{2}$ , hence unitary. For $f \in D (P_{k})$ its components satisfy $F_{2} (P_{k}^{(1)} f_{l}) (ξ) = 2 π ℏ ξ_{k} F_{2} f_{l} (ξ)$ for almost every $ξ$ by [F1]; assembling components gives $F_{2}^{(r)} (P_{k} f) (ξ) = 2 π ℏ ξ_{k} F_{2}^{(r)} f (ξ)$ for almost every $ξ$ . The domain description follows because $ξ_{k} F_{2} f_{l} \in L^{2}$ for all $l$ is exactly $ξ_{k} F_{2}^{(r)} f \in L^{2} (R^{d}; C^{r})$ . This proves (b).

2.3step 1.1F1F2F3algebra

[Translation identities] By [F2] the translation $T_{a}^{(r)}$ corresponds to $⨁_{l} T_{a}$ , which is unitary by [F3] and [F1]; it preserves the componentwise Schwartz and Sobolev classes because each scalar $T_{a}$ does [F1]. The identities $D_{k} (T_{a}^{(r)} f) = T_{a}^{(r)} (D_{k} f)$ and $T_{a}^{(r)} D (Q_{j}) = D (Q_{j})$ , $T_{a}^{(r)} Q_{j} (T_{a}^{(r)})^{- 1} = Q_{j} - a_{j} I$ , $T_{a}^{(r)} P_{k} (T_{a}^{(r)})^{- 1} = P_{k}$ are componentwise instances of [F1]; for instance, $(T_{a}^{(r)} Q_{j} (T_{a}^{(r)})^{- 1} f)_{l} = T_{a} (x_{j} T_{- a} f_{l}) = (x_{j} - a_{j}) f_{l}$ for $f_{l} \in D (Q_{j}^{(1)})$ , and the domain equality follows from the scalar one. This proves (c).

2.4step 1.2F1F2algebra

[The componentwise limit] Let $f \in D (P_{j})$ . Applying step 1.2 to each component $f_{l} \in D (P_{j}^{(1)})$ and assembling components through the unitary $Ψ_{0}$ [F2] gives $\frac{1}{t} (T_{t e_{j}}^{(r)} f - f) \to (- \frac{i}{ℏ} P_{j}^{(1)} f_{l})_{l} = - \frac{i}{ℏ} P_{j} f$ in $H^{(r)}$ as $t \to 0$ ; this is the first assertion of (d).

3.1step 2.4F1F5algebra∎

[The group of translations and its generator] Fix $j$ . The family $U_{j} (t) : = T_{t e_{j}}^{(r)}$ is a strongly continuous one-parameter unitary group: the group law and unitarity follow from the componentwise definition of translations and [F3], and strong continuity from [F4]. By step 2.4 its generator $G$ satisfies $G \supseteq - \frac{i}{ℏ} P_{j}$ ; by [F5] applied to the self-adjoint operator $T : = - P_{j} / ℏ$ , the family $V_{j} (t) : = \exp (- \frac{i}{ℏ} t P_{j}) = \exp (i t T)$ is a strongly continuous unitary group with generator $i T = - \frac{i}{ℏ} P_{j}$ and domain $D (P_{j})$ , and it preserves $D (P_{j})$ [F5]. On $D (P_{j})$ both groups have the same right derivative at $0$ , and both preserve $D (P_{j})$ ; hence $w (t) : = U_{j} (t) g - V_{j} (t) g$ for fixed $g \in D (P_{j})$ satisfies $w^{'} (t) = (- \frac{i}{ℏ} P_{j}) w (t)$ and $\frac{d}{d t} ∥ w (t) ∥^{2} = 2 Re ⟨ w (t), - \frac{i}{ℏ} P_{j} w (t) ⟩ = 0$ because $⟨ P_{j} w (t), w (t) ⟩$ is real; with $w (0) = 0$ this gives $U_{j} (t) g = V_{j} (t) g$ for all $t$ on the dense set $D (P_{j})$ , hence $U_{j} (t) = V_{j} (t)$ by continuity and unitarity. This proves (d).

Remarks

Domains are part of the statement. $Q_{j}$ and $P_{k}$ are unbounded; clause (a) exhibits them as orthogonal sums with maximal componentwise domains, and no product or commutator of two such operators is asserted here beyond the domains displayed.
The difference-quotient clause is the generator statement. Clause (d) identifies the coordinate translation group with the spectral flow of the momentum component; it is used by the physics items of this page to read the translation unitaries as generated by momentum, and no global exponential $\exp (- i a \cdot P / ℏ)$ for a vector $a$ is asserted in this lemma beyond the displayed coordinate statement.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Canonical commutation relations on the component-valued core and commutation with the internal factor

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 integers $d \geq 1$ , $r \geq 1$ and $ℏ > 0$ , let $H^{(r)} = L^{2} (R^{d}; C^{r})$ with the componentwise position and momentum operators $Q_{j}, P_{k}$ of Componentwise position and momentum operators on the component-valued wave-function space, let $S^{(r)} : = S (R^{d}; C^{r})$ be the componentwise Schwartz class (Schwartz space and its seminorms), and let $Ψ_{0}$ and $Θ$ be the unitaries of Unitary identifications of the component-valued wave-function space. For $S = (S_{l m}) \in M_{r} (C)$ write $S_{\times} f : = (\sum_{m = 1}^{r} S_{l m} f_{m})_{l} (f \in H^{(r)}) .$ Then:

(a) $S^{(r)} \subseteq D (Q_{j}) \cap D (P_{k})$ for all $j, k$ , $Q_{j} S^{(r)} \subseteq S^{(r)}$ and $P_{k} S^{(r)} \subseteq S^{(r)}$ , and on $S^{(r)}$ the canonical commutation relations hold: $(Q_{j} P_{k} - P_{k} Q_{j}) f = i ℏ δ_{j k} f (f \in S^{(r)});$

(b) $S_{\times}$ is bounded with $∥ S_{\times} ∥ = ∥ S ∥_{o p}$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum), and $S_{\times} D (Q_{j}) \subseteq D (Q_{j}), Q_{j} S_{\times} f = S_{\times} Q_{j} f (f \in D (Q_{j})), S_{\times} D (P_{k}) \subseteq D (P_{k}), P_{k} S_{\times} f = S_{\times} P_{k} f (f \in D (P_{k}));$

(c) the spectral projection-valued measures of $Q_{j}$ and $P_{k}$ are $E_{Q_{j}} (B) = M_{1_{x_{j}^{- 1} (B)}}, E_{P_{k}} (B) = F_{2}^{(r) *} M_{1_{{2 π ℏ ξ_{k} \in B}}} F_{2}^{(r)} (B \subseteq R Borel),$ where $M_{g}$ is multiplication by $g$ ; consequently $S_{\times} E_{Q_{j}} (B) = E_{Q_{j}} (B) S_{\times}$ and $S_{\times} E_{P_{k}} (B) = E_{P_{k}} (B) S_{\times}$ for every Borel $B$ , and for unitary $V \in U (r)$ and every $f \in H^{(r)}$ $∥ E_{Q_{j}} (B) V_{\times} f ∥ = ∥ E_{Q_{j}} (B) f ∥, ∥ E_{P_{k}} (B) V_{\times} f ∥ = ∥ E_{P_{k}} (B) f ∥;$

(d) under the tensor identification $Θ$ the operator $S_{\times}$ corresponds to the bounded amplification $I \otimes S$ of Bounded tensor maps: extension, norm estimate, adjoint and commutation, that is $Θ^{- 1} S_{\times} Θ = I \otimes S$ ; in particular the internal-factor operators commute with the spatial ones in the sense of (b).

Facts & Assumptions

Given: The Axiom of Countable Choice and the Axiom of Choice; integers $d, r \geq 1$ ; $ℏ > 0$ ; the componentwise operators $Q_{j}, P_{k}$ and their scalar counterparts $Q_{j}^{(1)}, P_{k}^{(1)}$ ; the unitaries $Ψ_{0}, Θ$ and the componentwise Plancherel transform $F_{2}^{(r)}$ ; a matrix $S \in M_{r} (C)$ ; classes $f \in H^{(r)}$ and $u \in S^{(r)}$ ; Borel sets $B \subseteq R$ .

[F1]

Componentwise formulas on the Schwartz class. For $u = (u_{l}) \in S^{(r)}$ , the componentwise operators satisfy $(Q_{j} u)_{l} = x_{j} u_{l}$ and $(P_{k} u)_{l} = - i ℏ \partial_{k} u_{l}$ : the Schwartz functions are smooth, their classical derivatives are their weak derivatives (Classical derivatives agree with weak derivatives), and multiplication by $x_{j}$ and differentiation preserve the Schwartz class (Schwartz space and its seminorms, Componentwise position and momentum operators on the component-valued wave-function space clauses (a),(b)).

[F2]

Componentwise operators and their identification. $Q_{j}, P_{k}$ are the orthogonal sums of the scalar operators under $Ψ_{0}$ , they are self-adjoint with their maximal componentwise domains, and $F_{2}^{(r)}$ is unitary with the Fourier representation of $P_{k}$ (Componentwise position and momentum operators on the component-valued wave-function space clauses (a),(b)); the bounded multipliers and translations correspond to the amplifications of Unitary identifications of the component-valued wave-function space clause (c).

[F3]

Weak derivatives are linear. If $f_{1}, \dots, f_{r} \in L^{2} (R^{d}; C)$ have weak derivatives $D_{k} f_{m} \in L^{2}$ and $α_{1}, \dots, α_{r} \in C$ , then $\sum_{m} α_{m} f_{m}$ has weak derivative $\sum_{m} α_{m} D_{k} f_{m}$ : the defining integration-by-parts identity is verified term by term against test functions (Weak derivative of a locally integrable function).

[F4]

Multiplier spectral measure and uniqueness. For a $σ$ -finite measure space and a real measurable finite-valued multiplier $m$ , the multiplication operator $M_{m}$ is self-adjoint with regular spectral PVM $E (B) = M_{1_{m^{- 1} (B)}}$ , and its law is $∥ E (B) ψ ∥^{2} = \int_{m^{- 1} (B)} ∣ ψ ∣^{2}$ (Spectral measure of a real L2 multiplication operator clauses (a)-(c), Projection valued measure). A regular PVM representing a self-adjoint operator is its spectral PVM (Spectral theorem for unbounded self-adjoint operators (PVM form)). Integrals of nonnegative product-measurable functions may be iterated (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product); dominated convergence for the componentwise strong additivity is Dominated convergence.

[F5]

Bounded matrix action. For $S \in M_{r} (C)$ and $f \in L^{2} (R^{d}; C^{r})$ , $∥ S f (x) ∥_{C^{r}} \leq ∥ S ∥_{o p} ∥ f (x) ∥_{C^{r}}$ pointwise (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors), so $S_{\times}$ is bounded with $∥ S_{\times} ∥ \leq ∥ S ∥_{o p}$ ; every elementary tensor $κ (u \otimes z)$ satisfies $(I \otimes S) κ (u \otimes z) = κ (u \otimes S z)$ (Bounded tensor maps: extension, norm estimate, adjoint and commutation clause (a)).

Proof

technique · direct; compute the core commutator from the componentwise differential formulas and the classical product rule, verify the matrix-multiplier commutation from linearity of weak derivatives, transport the scalar spectral measures through the unitaries, and identify the amplification on elementary tensors

Given: The data of the statement and [F1]-[F5].

1.1F1algebra

[The core and the CCR] For $u = (u_{l}) \in S^{(r)}$ , [F1] gives $Q_{j} u, P_{k} u \in S^{(r)}$ , so both products are defined there. For each component, using the ordinary product rule for the smooth function $x_{j} u_{l} (x)$ , $(Q_{j} P_{k} u - P_{k} Q_{j} u)_{l} = - i ℏ x_{j} \partial_{k} u_{l} + i ℏ \partial_{k} (x_{j} u_{l}) = i ℏ δ_{j k} u_{l} .$ Thus $(Q_{j} P_{k} - P_{k} Q_{j}) u = i ℏ δ_{j k} u$ as an equality of $L^{2}$ classes. This proves (a).

1.2F3F2algebra

[Linearity of the matrix action] For $f \in D (Q_{j})$ and every $m$ , $x_{j} f_{m} \in L^{2}$ , so the components $x_{j} (S_{\times} f)_{l} = x_{j} \sum_{m} S_{l m} f_{m} = \sum_{m} S_{l m} x_{j} f_{m}$ are $L^{2}$ classes (finite sums), and $S_{\times} f \in D (Q_{j})$ . For $f \in D (P_{k})$ , [F3] applied to the finite linear combinations $\sum_{m} S_{l m} f_{m}$ gives $D_{k} (S_{\times} f)_{l} = \sum_{m} S_{l m} D_{k} f_{m}$ , so $S_{\times} f \in D (P_{k})$ ; the actions agree with the matrix action on each component, whence $Q_{j} S_{\times} f = S_{\times} Q_{j} f$ and $P_{k} S_{\times} f = S_{\times} P_{k} f$ .

1.3F4F2algebra

[The position spectral measure] Put $E (B) : = M_{1_{x_{j}^{- 1} (B)}}$ on $H^{(r)}$ , the componentwise multiplication by the bounded indicator. Each $E (B)$ is an orthogonal projection and $E$ is a regular PVM: regularity follows by applying [F4] to the scalar multiplier $x \mapsto x_{j}$ on $L^{2} (R^{d})$ and taking the finite orthogonal sum over the $r$ components; multiplicativity and the endpoint values are pointwise, and for pairwise disjoint Borel $B_{n}$ with union $B$ the strong additivity follows from $∥ E (B) x - \sum_{n \leq N} E (B_{n}) x ∥^{2} = \int ∣ 1_{B} - \sum_{n \leq N} 1_{B_{n}} ∣ (x_{j}) ∥ x ∥^{2} \to 0$ by dominated convergence. For $x \in D (Q_{j})$ and $y \in H^{(r)}$ , the scalar pairing measure is $E_{x, y} (B) = \int_{x_{j}^{- 1} (B)} ⟨ x (z), y (z) ⟩_{C^{r}} d z$ , the pushforward under $z \mapsto z_{j}$ of the complex measure with density $⟨ x (z), y (z) ⟩$ . Its first absolute moment is finite because $\int ∣ z_{j} ∣ ∣ ⟨ x (z), y (z) ⟩ ∣ d z \leq ∥ Q_{j} x ∥ ∥ y ∥$ by Cauchy--Schwarz. The pushforward integral formula (first for indicator functions, then simple functions and integrable limits) therefore gives $\int λ d E_{x, y} (λ) = \int z_{j} ⟨ x (z), y (z) ⟩ d z = ⟨ Q_{j} x, y ⟩$ . Similarly, $E_{x}$ is the pushforward of $∥ x (z) ∥^{2} d z$ , and $\int λ^{2} d E_{x} (λ) = \int z_{j}^{2} ∥ x (z) ∥^{2} d z$ , finite exactly when $x \in D (Q_{j})$ . Thus $E$ represents $Q_{j}$ , and by uniqueness [F4] it is the spectral PVM of $Q_{j}$ .

2.1step 1.2F5algebra

[Norm of the matrix action] [F5] gives the pointwise bound $∥ S_{\times} f (x) ∥_{C^{r}} \leq ∥ S ∥_{o p} ∥ f (x) ∥_{C^{r}}$ , so $∥ S_{\times} f ∥_{H^{(r)}} \leq ∥ S ∥_{o p} ∥ f ∥_{H^{(r)}}$ and $∥ S_{\times} ∥ \leq ∥ S ∥_{o p}$ . Choose a unit vector $z_{0} \in C^{r}$ with $∥ S z_{0} ∥ = ∥ S ∥_{o p}$ , and a unit $h \in L^{2} (R^{d}; C)$ with $∥ h ∥ = 1$ ; then $f = h z_{0}$ (the class $x \mapsto h (x) z_{0}$ ) has $∥ f ∥_{H^{(r)}} = ∥ h ∥ ∥ z_{0} ∥ = 1$ and $∥ S_{\times} f ∥_{H^{(r)}} = ∥ h ∥ ∥ S z_{0} ∥ = ∥ S ∥_{o p}$ , so $∥ S_{\times} ∥ = ∥ S ∥_{o p}$ ; this proves the norm statement of (b), and step 1.2 is the commutation statement.

2.2step 1.3F2F4algebra

[The momentum spectral measure] Put $E^{'} (B) : = F_{2}^{(r) *} M_{1_{{2 π ℏ ξ_{k} \in B}}} F_{2}^{(r)}$ , the unitary conjugate of the multiplication PVM by the scalar multiplier $m (ξ) : = 2 π ℏ ξ_{k}$ ; [F4] gives the scalar multiplication PVM and its regularity, which pass to the finite component sum and unitary conjugate, so $E^{'}$ is a regular PVM. For $x \in D (P_{k})$ put $g : = F_{2}^{(r)} x$ and for $y^{'} \in H^{(r)}$ put $h^{'} : = F_{2}^{(r)} y^{'}$ ; then $E_{x, y^{'}}^{'} (B) = \int_{m^{- 1} (B)} ⟨ g (ξ), h^{'} (ξ) ⟩ d ξ$ is the pushforward under $m$ of the complex measure with density $⟨ g (ξ), h^{'} (ξ) ⟩$ . Its first absolute moment is finite since $\int ∣ m (ξ) ∣ ∣ ⟨ g, h^{'} ⟩ ∣ d ξ \leq ∥ m g ∥_{2} ∥ h^{'} ∥_{2} < \infty$ , so the pushforward integral formula (first for indicator functions, then simple functions and integrable limits) and [F2] give $\int λ d E_{x, y^{'}}^{'} (λ) = \int m (ξ) ⟨ g, h^{'} ⟩ d ξ = ⟨ m g, h^{'} ⟩ = ⟨ P_{k} x, y^{'} ⟩$ . Similarly, $E_{x}^{'}$ is the pushforward of $∥ g (ξ) ∥^{2} d ξ$ , so $\int λ^{2} d E_{x}^{'} (λ) = ∥ m g ∥_{2}^{2}$ , finite exactly when $x \in D (P_{k})$ . Thus $E^{'}$ represents $P_{k}$ , and by uniqueness [F4] it is the spectral PVM of $P_{k}$ .

3.1step 1.2step 1.3step 2.2F2algebra

[Commutation with the spectral measures] The multiplier $M_{1_{x_{j}^{- 1} (B)}}$ commutes with $S_{\times}$ because both are multiplication by functions on the finite-dimensional values: pointwise, $1_{x_{j}^{- 1} (B)} (x) S z = S (1_{x_{j}^{- 1} (B)} (x) z)$ for every $z \in C^{r}$ . For the momentum case, $S_{\times} F_{2}^{(r)} = F_{2}^{(r)} S_{\times}$ because $F_{2}^{(r)}$ acts on components and matrix multiplication acts on the finite index, so $F_{2}^{(r)} (S_{\times} g) = (\sum_{m} S_{l m} F_{2} g_{m})_{l} = S_{\times} F_{2}^{(r)} g$ ; and $S_{\times}$ commutes with the scalar multiplier by the same pointwise argument. Hence $S_{\times}$ commutes with both $E_{Q_{j}} (B)$ and $E_{P_{k}} (B)$ by steps 1.3 and 2.2. For unitary $V \in U (r)$ and $f \in H^{(r)}$ , $V_{\times}$ is unitary (it is matrix multiplication by a unitary pointwise) and commutes with the PVMs, so $∥ E (B) V_{\times} f ∥ = ∥ V_{\times} E (B) f ∥ = ∥ E (B) f ∥$ , which is the last assertion of (c).

4.1step 2.1F5algebra∎

[Amplification] Let $u \in H$ and $z = \sum_{l} z_{l} e_{l} \in C^{r}$ , and put $w : = Θ (κ (u \otimes z))$ ; by Unitary identifications of the component-valued wave-function space clause (b), $w = \sum_{l} z_{l} ι_{l} (u)$ , where $ι_{l} (u) \in H^{(r)}$ denotes the vector with $u$ in slot $l$ and $0$ elsewhere. Then $Θ^{- 1} S_{\times} Θ κ (u \otimes z) = Θ^{- 1} (S_{\times} w)$ , and $S_{\times} w = \sum_{l} z_{l} S_{\times} ι_{l} (u) = \sum_{l, m} z_{l} S_{m l} ι_{m} (u)$ ; applying $Θ^{- 1}$ with clause (b) gives $\sum_{l, m} z_{l} S_{m l} κ (u \otimes e_{m}) = κ (u \otimes S z)$ , where the last step expands $S z = \sum_{m} (\sum_{l} S_{m l} z_{l}) e_{m}$ ; by [F5] this equals $(I \otimes S) κ (u \otimes z)$ . Both sides are bounded operators (Bounded tensor maps: extension, norm estimate, adjoint and commutation clause (a) and step 2.1) that agree on the elementary tensors, whose span is dense in $H \hat{\otimes} C^{r}$ (Hilbert tensor construction: inner product, completion and product bases clause (c)); hence $Θ^{- 1} S_{\times} Θ = I \otimes S$ , which is (d).

Remarks

Only the core carries the CCR. Clause (a) is a relation on the common Schwartz core; no product of the unbounded operators is claimed to be defined on all of $H^{(r)}$ , and the companion counterexample of this page shows that no bounded-operator form with nonzero commutator can hold.
The internal factor is a bookkeeping factor. Clause (d) says that the matrix action is exactly the amplification $I \otimes S$ ; together with (c) it says that the spatial spectral measures are invariant under the action of the internal factor, so the spatial laws do not distinguish internal components. No identification of the internal factor with a spatial object is made.

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The momentum representation of the componentwise kinematics

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 $d \geq 1$ , $r \geq 1$ and $ℏ > 0$ , let $H^{(r)} = L^{2} (R^{d}; C^{r})$ with the componentwise operators $Q_{j}, P_{k}$ of Componentwise position and momentum operators on the component-valued wave-function space, let $U$ be the unitary of Momentum Fourier Jacobian: the unitary momentum representation clause (c), which on Schwartz classes acts by $(U ψ) (p) = (2 π ℏ)^{- d / 2} \int_{R^{d}} e^{- i p \cdot x / ℏ} ψ (x) d x,$ and put $U^{(r)} : = Ψ_{0} (⨁_{l} U) Ψ_{0}^{- 1}$ with $Ψ_{0}$ as in Unitary identifications of the component-valued wave-function space. Then:

(a) $U^{(r)} : L^{2} (R^{d}, d x; C^{r}) \to L^{2} (R^{d}, d p; C^{r})$ is unitary; for $f = (f_{l}) \in S (R^{d}; C^{r})$ , $(U^{(r)} f)_{l} (p) = (2 π ℏ)^{- d / 2} \int_{R^{d}} e^{- i p \cdot x / ℏ} f_{l} (x) d x (l = 1, \dots, r),$ $U^{(r)} f \in S (R^{d}; C^{r})$ , and $\int_{R^{d}} ∥ U^{(r)} f (p) ∥_{C^{r}}^{2} d p = \int_{R^{d}} ∥ f (x) ∥_{C^{r}}^{2} d x$ ;

(b) $U^{(r)} D (P_{k}) = D (M_{p_{k}})$ , where $M_{p_{k}} g (p) : = p_{k} g (p)$ on $D (M_{p_{k}}) = {g \in L^{2} (d p; C^{r}) : p_{k} g \in L^{2}}$ , and $U^{(r)} P_{k} = M_{p_{k}} U^{(r)};$ equivalently $(U^{(r)} P_{k} f) (p) = p_{k} (U^{(r)} f) (p)$ for every $f \in D (P_{k})$ and almost every $p$ ;

(c) for every $f \in S (R^{d}; C^{r})$ and every $j$ , $U^{(r)} (Q_{j} f) (p) = i ℏ \frac{\partial}{\partial p_{j}} (U^{(r)} f) (p) (p \in R^{d}),$ and both sides are Schwartz classes; no identity of unbounded operators on the full domain of $Q_{j}$ is asserted;

(d) for every $a \in R^{d}$ , with $T_{a}^{(r)}$ the componentwise translation of Componentwise position and momentum operators on the component-valued wave-function space clause (c), $U^{(r)} T_{a}^{(r)} U^{(r) - 1} = M_{e^{- i a \cdot p / ℏ}},$ the bounded multiplication operator by the unimodular function $p \mapsto e^{- i a \cdot p / ℏ}$ ;

(e) under $Ψ_{0}$ the operator $U^{(r)}$ is the orthogonal sum $⨁_{l} U$ ; no physical identification of $U^{(r)} f$ with a measured momentum distribution is made by this lemma.

Facts & Assumptions

Given: The Axiom of Countable Choice and the Axiom of Choice; integers $d, r \geq 1$ ; $ℏ > 0$ ; the scalar unitary $U$ and its componentwise version $U^{(r)}$ ; the operators $Q_{j}, P_{k}$ and their scalar counterparts; a class $f = (f_{l})$ ; a vector $a \in R^{d}$ ; indices $j, k$ .

[F1]

The scalar momentum representation. $U$ is a unitary $L^{2} (R^{d}, d x) \to L^{2} (R^{d}, d p)$ with the displayed Schwartz formula; $U = D_{ℏ} F_{2}$ on Schwartz classes, where $D_{ℏ} g (p) = (2 π ℏ)^{- d / 2} g (p / (2 π ℏ))$ is a surjective isometry of $L^{2} (R^{d}; C)$ and $F_{2}$ is the unitary Plancherel transform of Plancherel theorem; and on Schwartz classes $U$ acts as the topological automorphism $D_{ℏ} F_{2}$ (Momentum Fourier Jacobian: the unitary momentum representation clauses (a)-(c), Fourier transform is a topological automorphism of Schwartz space).

[F2]

Scalar domains, Fourier representation and translations. $D (P_{k}^{(1)}) = {f : ξ_{k} F_{2} f \in L^{2}}$ and $F_{2} (P_{k}^{(1)} f) (ξ) = 2 π ℏ ξ_{k} F_{2} f (ξ)$ ; the scalar translations are unitary and satisfy $F_{2} (T_{a} f) (ξ) = e^{- 2 π i a \cdot ξ} F_{2} f (ξ)$ on $L^{1} \cap L^{2}$ , hence on $L^{2}$ by density and continuity (Self-adjointness, Fourier representation and translation identities of position and momentum clauses (b),(c), Fourier transform on complex L1 classes).

[F3]

Componentwise operators and identifications. $Ψ_{0}$ is unitary and $Q_{j}, P_{k}$ are its conjugates of the corresponding orthogonal sums; $F_{2}^{(r)}$ is unitary with the componentwise multiplication representation of $P_{k}$ ; the componentwise translations are the conjugates of $⨁_{l} T_{a}$ (Componentwise position and momentum operators on the component-valued wave-function space clauses (a)-(c), Unitary identifications of the component-valued wave-function space clauses (a),(c)).

[F4]

Differentiation under the integral. If for a parameter $p$ the integrand is differentiable with derivative dominated by an integrable function independent of the parameter, the derivative may be taken under the integral sign (Differentiation under the integral sign). Change of variables under a translation holds for integrable functions (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). Tonelli's theorem for nonnegative measurable integrands (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F5]

Dense extension of bounded identities. If two bounded operators agree on a dense subspace, they agree everywhere (A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm).

Proof

technique · direct; reduce unitarity and the Schwartz formula to the scalar supplier through the orthogonal-sum identification, transport the momentum representation and its domain, differentiate the Schwartz transform under the integral for the core identity of position, and extend the translation conjugation by density

Given: The data of the statement and [F1]-[F5].

1.1F1F3algebra

[Unitarity, Schwartz formula and Plancherel identity] By [F3] and Finite orthogonal sums of self-adjoint operators and of bounded operators clause (c) the orthogonal sum $⨁_{l} U$ is unitary, hence so is its conjugate $U^{(r)}$ by the unitary $Ψ_{0}$ . On a Schwartz class $f = (f_{l})$ each component is a scalar Schwartz class, so [F1] gives $(U^{(r)} f)_{l} = U f_{l}$ with the displayed integral formula and $U f_{l} \in S (R^{d}; C)$ ; hence $U^{(r)} f \in S (R^{d}; C^{r})$ . The Plancherel identity is the componentwise scalar identity of [F1], summed over $l$ (finite sum) and read with the componentwise norms. This proves (a) and (e).

2.1step 1.1F1F2F3algebra

[Momentum domain and action] Let $f = (f_{l}) \in H^{(r)}$ . By [F2], for each $l$ one has $f_{l} \in D (P_{k}^{(1)})$ exactly when $ξ_{k} F_{2} f_{l} \in L^{2}$ , and then $F_{2} (P_{k}^{(1)} f_{l}) = 2 π ℏ ξ_{k} F_{2} f_{l}$ . Since $U = D_{ℏ} F_{2}$ and $p_{k} D_{ℏ} h = D_{ℏ} (2 π ℏ ξ_{k} h)$ pointwise for every $h \in L^{2}$ with $ξ_{k} h \in L^{2}$ , one gets $p_{k} U f_{l} = U (P_{k}^{(1)} f_{l})$ and, conversely, $p_{k} U f_{l} \in L^{2}$ exactly when $ξ_{k} F_{2} f_{l} \in L^{2}$ because $D_{ℏ}$ is an isometry; thus $U D (P_{k}^{(1)}) = D (M_{p_{k}})$ and $U P_{k}^{(1)} = M_{p_{k}} U$ . Assembling components through $Ψ_{0}$ [F3] gives $U^{(r)} D (P_{k}) = D (M_{p_{k}})$ and $U^{(r)} P_{k} = M_{p_{k}} U^{(r)}$ , which is (b).

2.2step 1.1F1F3F4algebra

[Position in the momentum representation] Let $f \in S (R^{d}; C^{r})$ and consider one component $f_{l} \in S (R^{d}; C)$ . For fixed $x$ , the $p_{j}$ -derivative of $e^{- i p \cdot x / ℏ} f_{l} (x)$ is $(- i / ℏ) x_{j} e^{- i p \cdot x / ℏ} f_{l} (x)$ . Its modulus is $∣ x_{j} f_{l} (x) ∣ / ℏ$ , an integrable bound independent of $p$ because $f_{l}$ is Schwartz; [F4] therefore permits differentiation under the integral, giving $\frac{\partial}{\partial p_{j}} (U f_{l}) (p) = (2 π ℏ)^{- d / 2} \int e^{- i p \cdot x / ℏ} (- \frac{i}{ℏ} x_{j}) f_{l} (x) d x = (- \frac{i}{ℏ}) (U (Q_{j}^{(1)} f_{l})) (p),$ that is $U (Q_{j}^{(1)} f_{l}) = i ℏ \partial_{p_{j}} (U f_{l})$ . Since $U f_{l} \in S$ by [F1], its partial derivative is again Schwartz (Schwartz space and its seminorms), and assembling components through [F3] proves (c).

3.1step 1.1F1F3F4F5algebra∎

[Translations] Let $f = (f_{l})$ be a Schwartz class and fix $a$ . For each component, the substitution $y = x - a$ and [F4] give $(U T_{a} f_{l}) (p) = (2 π ℏ)^{- d / 2} \int e^{- i p \cdot (x + a) / ℏ} f_{l} (x) d x = e^{- i a \cdot p / ℏ} (U f_{l}) (p)$ , so $U^{(r)} T_{a}^{(r)} = M_{e^{- i a \cdot p / ℏ}} U^{(r)}$ on the dense Schwartz subspace of $H^{(r)}$ ; both sides are bounded operators (products of unitaries, a translation and a unimodular multiplier), so [F5] extends the identity to all of $H^{(r)}$ and $U^{(r)} T_{a}^{(r)} U^{(r) - 1} = M_{e^{- i a \cdot p / ℏ}}$ , which is (d).

Remarks

What the momentum representative is. $U^{(r)} f$ is a class in $L^{2} (R^{d}, d p; C^{r})$ obtained from $f$ by the unitary transform of clause (a); the lemma makes no statement about measuring it, and the physical reading of $∥ U^{(r)} f (p) ∥^{2} d p$ as an outcome density is a separate consequence of the observable postulate, proved with the physics items of this page.
Core only for position. Clause (c) is asserted on Schwartz classes, where both sides are ordinary functions; no claim is made that the mapping $g \mapsto i ℏ \partial_{p_{j}} g$ with its maximal $L^{2}$ domain is the unitary conjugate of the full position operator.

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The massive spin-s particle model of the framework: state space, operators, units and frame readings

Physical scope

The kinematic reading of the massive-particle model postulate P5 for one particle of spin label s and mass parameter m>0: the component-valued state space L^2(R^3;C^(2s+1)) with its class reading and internal factor, the componentwise position and momentum operators with their declared units, the internal spin operators as matrix actions, the momentum-space representative, and the frame reading of the model. The item fixes terminology and readings of P1, P2, P5 and P6; it asserts no dynamics beyond what P4 assigns, no apparatus or detection model, no empirical identification of any species, and excludes massless, relativistic, singular-potential, boundary and radiation models, which require their own assumptions.

Definition

(a) Adopted model and state space. Adopt the kinematics and massive-particle model postulate P5 for a declared particle of mass parameter $m > 0$ , spin label $s \in {0, \frac{1}{2}, 1, \frac{3}{2}, \dots}$ and internal factor assigned as in the identical-particle and spin postulate P6 (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations). Put $r : = 2 s + 1$ and take as the model's one-particle state space the complex Hilbert space $H_{s} : = L^{2} (R^{3}; C^{r}) = {f : R^{3} \to C^{r} measurable : \int_{R^{3}} ∥ f (x) ∥_{C^{r}}^{2} d x < \infty} / ⁣ \sim,$ the quotient by equality almost everywhere, with inner product $⟨ f, g ⟩ = \int_{R^{3}} \sum_{l = 1}^{r} f_{l} \overline{g_{l}} d x$ linear in the first argument (The space $L^{p} (μ)$ as the quotient by null functions). Thus an element of $H_{s}$ is a class of functions, not a function, and pointwise values of a state are not defined; the position density is read from the class through the observables of clause (b) and the observable and probability postulate P2 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, The state space of the framework: preparations, pure rays and density operators, Assigned sharp observables and the outcome law of the framework). Its dimension in the conventions of Dimensions, SI unit conventions and quantity types of the framework clause (d)1 is $ℓ^{- 3 / 2}$ per component. P5 presents the same space as the completed tensor product $L^{2} (R^{3}; C) \hat{\otimes} C^{r}$ with the spin factor carrying the representation of P6; the unitary identification of the two descriptions and the componentwise reading of componentwise operators are those of Unitary identifications of the component-valued wave-function space and Componentwise position and momentum operators on the component-valued wave-function space. The orthonormal basis $(e_{m})_{m = - s, \dots, s}$ of $C^{r}$ is the basis fixed by P6 (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations clause (ii)).

(b) Position and momentum. The model's $j$ -th position and $k$ -th momentum are the componentwise operators $Q_{j}$ and $P_{k}$ of Componentwise position and momentum operators on the component-valued wave-function space clause (a): for $f = (f_{l})$ , restricted to the classes with $x_{j} f_{l} \in L^{2}$ respectively with $D_{k} f_{l} \in L^{2}$ for all $l$ , $(Q_{j} f)_{l} (x) : = x_{j} f_{l} (x), (P_{k} f)_{l} : = - i ℏ D_{k} f_{l},$ with $D_{k}$ the weak derivative and $ℏ$ the action constant of Dimensions, SI unit conventions and quantity types of the framework. These are the scalar operators of Position and momentum operators on Euclidean L2 applied componentwise. The model assigns to $(Q_{j}, m)$ and to $(P_{k}, k g m s^{- 1})$ the status of assigned sharp observables in the sense of Assigned sharp observables and the outcome law of the framework clause (a), with the units of Dimensions, SI unit conventions and quantity types of the framework clause (d)4; both are densely defined and self-adjoint by Componentwise position and momentum operators on the component-valued wave-function space clause (a). On the Schwartz core they satisfy the componentwise canonical commutation relations, and they commute with the internal-factor operators of clause (c) in the sense of Canonical commutation relations on the component-valued core and commutation with the internal factor clauses (a),(c).

(c) Internal factor. For $i \in {1, 2, 3}$ let $S_{i} = ℏ J_{i}$ be the self-adjoint endomorphisms of $C^{r}$ of the spin assignment P6 (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations clause (ii), Finite spin matrices), and let the model read them on $H_{s}$ as the matrix actions $(S_{i} f)_{l} : = ℏ \sum_{m = 1}^{r} (J_{i})_{l m} f_{m} (l = 1, \dots, r),$ that is, as the amplifications $I \otimes S_{i}$ under the tensor identification of clause (a) (Canonical commutation relations on the component-valued core and commutation with the internal factor clause (d)). Their declared unit is the unit of action (Dimensions, SI unit conventions and quantity types of the framework clause (d)4). The factor is internal bookkeeping: it is not a spatial coordinate, not a rotating body and not a relativistic spinor, and the model assigns no position or worldline to it.

(d) Momentum-space representative. With $U^{(s)}$ the unitary of The momentum representation of the componentwise kinematics for $r = 2 s + 1$ , the momentum-space representative of a preparation vector $ψ \in H_{s}$ is $ψ_{p} : = U^{(s)} ψ \in L^{2} (R^{3}, d p; C^{r})$ ; for a Schwartz class $ψ$ it is the componentwise Fourier transform $(ψ_{p})_{l} (p) = (2 π ℏ)^{- 3 / 2} \int_{R^{3}} e^{- i p \cdot x / ℏ} ψ_{l} (x) d x,$ and its dimension is $(k g m s^{- 1})^{- 3 / 2}$ per component (Dimensions, SI unit conventions and quantity types of the framework clause (d)2). By The momentum representation of the componentwise kinematics clause (b), $P_{k}$ acts on the momentum representative by multiplication by $p_{k}$ ; this is the model's momentum reading and no measured momentum distribution is identified with $∣ ψ_{p} ∣^{2}$ by this definition.

(e) Frame reading. The model's inertial frames are the global coordinate charts of P5; frame changes are the Galilean transformations of P5, with parameters and units as in Dimensions, SI unit conventions and quantity types of the framework. This definition uses P5's frame class only. For translations by $a \in R^{3}$ and boosts with parameter $u \in R^{3}$ , choose the componentwise representative convention $(T_{a} ψ) (x) = ψ (x - a), (B_{u} (t) ψ) (x) = \exp ⁣ (\frac{i}{ℏ} (m u \cdot x - \frac{1}{2} m ∣ u ∣^{2} t)) ψ (x - u t) .$ Here $a$ is length-valued, $u$ velocity-valued and $t$ time-valued, so the phase is dimensionless. These formulas specify classes almost everywhere and are chosen model conventions. The covariance theorems of this page derive operator and outcome-law identities conditional on these choices; they do not derive the choice of representative convention itself. No Lorentz transformation law is claimed.

(f) Status. Clauses (a)-(e) record the readings of P1, P2, P5 and P6 in the terminology of this page and fix the objects used by its theorems. They assert no time evolution beyond what the autonomous-dynamics postulate P4 assigns (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators clause (iv)), no preparation or apparatus model, no detection model and no empirical result. Massless particles, relativistic matter, radiation, singular or unbounded potentials and boundary or confined realizations are separate model assumptions with their own domains and self-adjointness statements; none is obtained from this definition.

Remarks

Why the class reading matters. $H_{s}$ consists of almost-everywhere classes; the operators $Q_{j}$ and $P_{k}$ are defined on the classes of clause (b) and the core identities hold on the Schwartz classes. Phrases such as "the value of the wave function at $x$ " are shorthand for a chosen representative and are not part of the model's state assignment.
Units. The wave function, its momentum representative, the position and momentum spectral coordinates and the internal matrices are read in the units fixed by Dimensions, SI unit conventions and quantity types of the framework; no numerical value, and in particular no value of $m$ , $s$ or $ℏ$ , is asserted beyond the exact defining constants of the current SI recorded there.

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Position and momentum outcome laws of the massive spin-s model

Physical scope

The position and momentum outcome laws of the spin-s massive-particle model of the framework: the spectral projection-valued measures of the componentwise position and momentum operators, the joint position law with Born density the squared component norm, the momentum laws with density the squared momentum-representative norm, the atomlessness of exact values, and the invariance of the spatial laws under the internal factor. The statement is a conditional consequence of the observable and probability postulate P2 and the model postulates P5 and P6 and the cited mathematics; it asserts no joint position-momentum statistics, no dynamics, no apparatus or detector realization, no species identification and no empirical result.

Statement

Adopt the observable and probability postulate in the reading of Assigned sharp observables and the outcome law of the framework, the kinematics and massive-particle model postulate P5, and the internal-factor assignment of the postulate P6, all read for the spin- $s$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with mass parameter $m > 0$ , spin label $s \in {0, \frac{1}{2}, 1, \dots}$ and $r = 2 s + 1$ (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations), and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $H_{s}$ be the model's state space, $Q_{j}, P_{k}$ its componentwise position and momentum operators, $U^{(s)}$ its momentum transform, and let $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ be the outcome law of an assigned sharp observable $A$ in a unit vector $ψ$ (Assigned sharp observables and the outcome law of the framework clause (b)). Then:

(a) for each $j$ the spectral projection-valued measure of $Q_{j}$ is $E_{Q_{j}} (B) = M_{1_{x_{j}^{- 1} (B)}} (B \subseteq R Borel),$ the componentwise multiplication by the indicator, and $μ_{ψ}^{Q_{j}} (B) = \int_{x_{j}^{- 1} (B)} ∥ ψ (x) ∥_{C^{r}}^{2} d x;$ the law is atomless, $μ_{ψ}^{Q_{j}} ({λ}) = 0$ for every $λ \in R$ , and $σ (Q_{j}) = R$ ;

(b) there is a unique regular projection-valued measure $G$ on the Borel sets of $R^{3}$ with $G (B_{1} \times B_{2} \times B_{3}) = E_{Q_{1}} (B_{1}) E_{Q_{2}} (B_{2}) E_{Q_{3}} (B_{3})$ for all Borel $B_{1}, B_{2}, B_{3}$ , namely $G (B) = M_{1_{B}}$ , and its scalar law is the model's Born position law, $μ_{ψ}^{G} (B) = ∥ G (B) ψ ∥^{2} = \int_{B} ∥ ψ (x) ∥_{C^{r}}^{2} d x;$

(c) for each $k$ the spectral projection-valued measure of $P_{k}$ is $E_{P_{k}} (B) = U^{(s) - 1} M_{1_{B} (p_{k})} U^{(s)},$ and with $ψ_{p} : = U^{(s)} ψ$ $μ_{ψ}^{P_{k}} (B) = \int_{{p \in R^{3} : p_{k} \in B}} ∥ ψ_{p} (p) ∥_{C^{r}}^{2} d p;$ the law is atomless and $σ (P_{k}) = R$ ;

(d) for every unitary $V \in U (r)$ , with $V_{\times}$ the matrix action of Canonical commutation relations on the component-valued core and commutation with the internal factor clause (b), $μ_{V_{\times} ψ}^{Q_{j}} = μ_{ψ}^{Q_{j}}, μ_{V_{\times} ψ}^{P_{k}} = μ_{ψ}^{P_{k}};$

(e) Scope. The statement is conditional on P2, P5 and P6 and the cited mathematics, and extends the position and momentum laws of the massive model to the spin- $s$ state space. No joint law of a position and a momentum observable is asserted, no dynamics is used or claimed, and the identification of the operators with concrete laboratory apparatus is a tested identification under the preparation and apparatus postulate, not a consequence of this theorem; no empirical result is recorded. Exact values have probability zero although both spectra equal $R$ . The real outcome spaces remain $R$ ; zero singleton mass means atomlessness and supplies no normalizable exact-value eigenstate. Finite-resolution records use the displayed Borel bin probabilities.

Facts & Assumptions

Given: The postulates P2, P5, P6 read through The massive spin-s particle model of the framework: state space, operators, units and frame readings; the Axiom of Countable Choice and the Axiom of Choice; the state space $H_{s}$ with $r = 2 s + 1$ ; the operators $Q_{j}, P_{k}$ and the unitaries $Ψ_{0}, U^{(s)}$ ; a unit vector $ψ \in H_{s}$ ; Borel sets $B, B_{1}, B_{2}, B_{3} \subseteq R$ ; a Borel set $B^{'} \subseteq R^{3}$ ; a unitary matrix $V \in U (r)$ ; a real $λ$ .

[F1]

Outcome law. For an assigned sharp observable $A$ with spectral PVM $E_{A}$ the law in a unit vector is $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ , a probability measure on the Borel sets; the spectral support of $E_{A}$ is the spectrum $σ (A)$ , and $μ_{ψ}^{A} ({λ}) = 0$ unless $λ$ is an eigenvalue of $A$ (Assigned sharp observables and the outcome law of the framework clauses (b)-(d)). If $A$ has no eigenvalues its laws are atomless, with real outcome space $R$ and no normalizable exact-value eigenstate; zero singleton mass does not exclude real point realizations (clause (c) of the same definition, Projection valued measure).

[F2]

Componentwise operators and their spectral measures. $Q_{j}$ and $P_{k}$ are the componentwise operators of the model, self-adjoint with their maximal domains; $E_{Q_{j}} (B) = M_{1_{x_{j}^{- 1} (B)}}$ and $E_{P_{k}} (B) = F_{2}^{(r) *} M_{1_{{2 π ℏ ξ_{k} \in B}}} F_{2}^{(r)}$ are their spectral PVMs (Canonical commutation relations on the component-valued core and commutation with the internal factor clauses (b),(c), Componentwise position and momentum operators on the component-valued wave-function space clause (a)); the matrix action $V_{\times}$ commutes with these PVMs and is unitary for unitary $V$ (Canonical commutation relations on the component-valued core and commutation with the internal factor clauses (b)-(d)).

[F3]

Momentum representation. $U^{(s)} = U^{(\cdot)}$ for $r = 2 s + 1$ is unitary, $U^{(s)} D (P_{k}) = D (M_{p_{k}})$ and $U^{(s)} P_{k} = M_{p_{k}} U^{(s)}$ ; on Schwartz classes the displayed integral formula holds and the Plancherel identity $\int ∥ U^{(s)} f ∥^{2} d p = \int ∥ f ∥^{2} d x$ (The momentum representation of the componentwise kinematics clauses (a),(b)).

[F4]

Multiplication spectral measure. For a $σ$ -finite measure space and a real measurable finite-valued multiplier $m$ , the multiplication operator $M_{m}$ is self-adjoint with regular spectral PVM $M_{1_{m^{- 1} (B)}}$ and $σ (M_{m})$ equal to the essential range of $m$ (Spectral measure of a real L2 multiplication operator clauses (a),(c),(d)); a regular PVM representing a self-adjoint operator is its spectral PVM (Spectral theorem for unbounded self-adjoint operators (PVM form)).

[F5]

Measure toolkit. For $σ$ -finite measure spaces, integrals of nonnegative product-measurable functions may be iterated (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product); for nonnegative measurable $h$ the set function $A \mapsto \int_{A} h d μ$ is a measure (The indefinite integral of a nonnegative measurable function is a measure); finite measures on the $σ$ -algebra generated by a $π$ -system that agree on the $π$ -system and have equal mass are equal (Finite measures agreeing on a generating pi-system and on the whole space are equal); the Euclidean Borel $σ$ -algebra of $R^{3}$ is generated by the rectangles (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}). Under Countable Choice, every Borel measure finite on compact sets on a second-countable locally compact Hausdorff space is regular (Locally finite Borel measures on second-countable LCH spaces are regular). The usual Euclidean $R^{3}$ is such a space: rational-center, positive rational-radius balls form a countable base; the metric is Hausdorff; and closed bounded balls are compact by Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, so every point has a compact neighborhood. A coordinate hyperplane in $R^{3}$ has Lebesgue measure zero: its indicator has iterated integral zero by [F5] because the inner integral over a singleton vanishes.

Proof

technique · direct; read the spectral measures of the componentwise operators from the conjugation lemmas, compute the laws by Tonelli, construct the joint position projection-valued measure and prove its uniqueness on rectangles, and transport the momentum measure through the unitary momentum representation

Given: The data of the statement and [F1]-[F5].

1.1F1F2F4F5algebra

[Position spectral measure and law] [F2] gives $E_{Q_{j}} (B) = M_{1_{x_{j}^{- 1} (B)}}$ , so for a unit $ψ$ , $μ_{ψ}^{Q_{j}} (B) = ∥ E_{Q_{j}} (B) ψ ∥^{2} = \int_{x_{j}^{- 1} (B)} ∥ ψ (x) ∥_{C^{r}}^{2} d x$ by [F1] and the definition of the componentwise norm. The measure is the indefinite integral $A \mapsto \int_{A} ∥ ψ ∥^{2} d x$ of a nonnegative integrable function [F5], hence atomless: for every $λ$ the set $x_{j}^{- 1} ({λ})$ is a coordinate hyperplane, whose indicator has iterated integral zero by [F5], so $μ_{ψ}^{Q_{j}} ({λ}) = 0$ . Finally the essential range of $x_{j}$ is $R$ because every interval slab $x_{j}^{- 1} ((λ - ε, λ + ε))$ has positive Lebesgue measure, so [F4] gives $σ (Q_{j}) = R$ . This proves (a).

1.2F1F2algebra

[Internal-factor invariance] Let $V \in U (r)$ . By [F2] the matrix action $V_{\times}$ is unitary and commutes with $E_{Q_{j}} (B)$ and $E_{P_{k}} (B)$ ; hence $μ_{V_{\times} ψ}^{Q_{j}} (B) = ∥ E_{Q_{j}} (B) V_{\times} ψ ∥^{2} = ∥ V_{\times} E_{Q_{j}} (B) ψ ∥^{2} = ∥ E_{Q_{j}} (B) ψ ∥^{2} = μ_{ψ}^{Q_{j}} (B)$ , and the same computation with $P_{k}$ proves (d).

2.1step 1.1F2F4F5algebra

[Joint position law] Define $G (B) : = M_{1_{B}}$ for Borel $B \subseteq R^{3}$ ; each $G (B)$ is a bounded self-adjoint projection because pointwise multiplication by an indicator is one, and $G$ is a PVM: $G (B) G (C) = G (B \cap C)$ , the total mass is $I$ , and strong countable additivity follows by dominated convergence for $∥ G (B) ψ - \sum_{n \leq N} G (B_{n}) ψ ∥^{2} = \int ∣ 1_{B} - \sum_{n \leq N} 1_{B_{n}} ∣ ∥ ψ ∥^{2}$ . For every $h \in H_{s}$ , the scalar measure is $G_{h} (B) = ⟨ G (B) h, h ⟩ = \int_{B} ∥ h (x) ∥_{C^{r}}^{2} d x$ , a Borel measure of finite total mass $∥ h ∥^{2}$ by [F5], hence finite on every compact set. Apply the regularity corollary in [F5] on the usual Euclidean $R^{3}$ , using the Countable Choice already assumed: each $G_{h}$ is regular. In particular the measure for a unit $ψ$ is the regular probability measure $μ_{ψ} (B) = \int_{B} ∥ ψ (x) ∥^{2} d x$ ; the zero vector gives the zero regular measure. This proves regularity of the actual component-valued PVM $G$ in Projection valued measure, including its finite matrix amplification. On rectangles, $G (B_{1} \times B_{2} \times B_{3}) = M_{1_{B_{1}} (x_{1}) 1_{B_{2}} (x_{2}) 1_{B_{3}} (x_{3})} = E_{Q_{1}} (B_{1}) E_{Q_{2}} (B_{2}) E_{Q_{3}} (B_{3})$ by [F2], since the componentwise multipliers commute. For a unit $ψ$ the scalar law is $μ_{ψ}^{G} (B) = ∥ G (B) ψ ∥^{2} = \int_{B} ∥ ψ (x) ∥^{2} d x$ . If $G^{'}$ is another regular PVM with the same rectangle values, then for every unit $φ$ the finite measures $B \mapsto ⟨ G (B) φ, φ ⟩$ and $B \mapsto ⟨ G^{'} (B) φ, φ ⟩$ agree on the rectangles, a $π$ -system generating the Borel $σ$ -algebra of $R^{3}$ [F5], and have equal total mass $∥ φ ∥^{2}$ ; [F5] makes them equal on all Borel sets, and polarization recovers all pairings, so $G^{'} = G$ . This proves (b).

2.2step 1.1F3F4algebra

[Momentum spectral measure and law] By [F3] and [F4], $U^{(s)}$ is unitary, $D (M_{p_{k}}) = U^{(s)} D (P_{k})$ and $M_{p_{k}} = U^{(s)} P_{k} U^{(s) - 1}$ ; the multiplication operator $M_{p_{k}}$ on $L^{2} (d p; C^{r})$ is self-adjoint with spectral PVM $M_{1_{{p_{k} \in B}}}$ [F4], so transporting through the unitary $U^{(s)}$ and using uniqueness of the spectral PVM [F4] gives $E_{P_{k}} (B) = U^{(s) - 1} M_{1_{{p_{k} \in B}}} U^{(s)}$ . For a unit $ψ$ , $μ_{ψ}^{P_{k}} (B) = ∥ E_{P_{k}} (B) ψ ∥^{2} = ∥ M_{1_{{p_{k} \in B}}} U^{(s)} ψ ∥^{2} = \int_{{p_{k} \in B}} ∥ ψ_{p} (p) ∥^{2} d p$ [F3]. The law is the indefinite integral of the integrable function $∥ ψ_{p} ∥^{2}$ over a slice condition, hence atomless by the hyperplane argument of step 1.1; and the essential range of $p_{k}$ is $R$ , so $σ (P_{k}) = R$ [F4]. This proves (c).

3.1givenstep 1.1step 2.1step 2.2step 1.2∎

[Scope] Clauses (a)-(d) are consequences of P2, P5 and P6 as read by The massive spin-s particle model of the framework: state space, operators, units and frame readings together with the displayed mathematics; no dynamics, joint position-momentum statistics, apparatus model or empirical claim is used, and the last sentences of the statement record the scope.

Remarks

Relation to the spinless laws. For $s = 0$ one has $r = 1$ and the statement reduces to the position and momentum outcome laws of the spinless massive model already established on the observables page; the present theorem supplies the component-valued case and the invariance of the spatial laws under the internal factor.
Nondegenerate continuum. Both spectra equal $R$ and both laws are atomless: the model admits no normalizable position or momentum eigenvector, and no exact value is an outcome with positive probability. Finite-resolution readings are the Borel bin probabilities displayed above.

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Translation covariance of the massive spin-s kinematics and of its bounded-potential dynamics

Physical scope

The translation covariance of the spin-s massive-particle model: the unitarity and operator conjugation identities of the componentwise translations, the resulting translation of the position law and invariance of the momentum laws, the conjugation of the free-plus-bounded-potential energy operator and of its flow, and the model's reading of a passive change of spatial origin. The statement is conditional on P2, P4, P5 and P6 and the cited mathematics, holds for real bounded measurable potentials only, and asserts nothing about singular or unbounded potentials, boundary realizations, relativistic models, radiation, apparatuses or empirical results.

Statement

Adopt the observable and probability postulate P2 in the reading of Assigned sharp observables and the outcome law of the framework, the autonomous-dynamics postulate P4 in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution, and the kinematics and massive-particle model postulate P5 and internal-factor postulate P6 read through The massive spin-s particle model of the framework: state space, operators, units and frame readings for a particle of mass parameter $m > 0$ and spin label $s$ (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators), and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $H_{s}$ be the model's state space with componentwise operators $Q_{j}, P_{k}$ and translations $T_{a} : = T_{a}^{(r)}$ ( $r = 2 s + 1$ ) of Componentwise position and momentum operators on the component-valued wave-function space clause (c), let $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ be the outcome law of an assigned sharp observable $A$ in a unit vector $ψ$ , and let $V : R^{3} \to R$ be real, bounded and measurable with $V_{a} (x) : = V (x - a)$ ; write $H_{V} : = \frac{1}{2 m} \sum_{k = 1}^{3} P_{k}^{2} + M_{V}, D (H_{V}) = W^{2, 2} (R^{3}; C^{r}),$ for the model's free-plus-bounded-potential energy operator (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (c) applied componentwise, assembled by Finite orthogonal sums of self-adjoint operators and of bounded operators), with flow $U_{V} (t) = \exp (- i t H_{V} / ℏ)$ in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (b). Then:

(a) for every $a \in R^{3}$ the operator $T_{a}$ is unitary, $T_{a} D (Q_{j}) = D (Q_{j})$ , $T_{a} D (P_{k}) = D (P_{k})$ , and $T_{a} Q_{j} T_{a}^{- 1} = Q_{j} - a_{j} I, T_{a} P_{k} T_{a}^{- 1} = P_{k};$

(b) Outcome-law covariance. For every unit $ψ \in H_{s}$ and Borel $B \subseteq R$ , $μ_{T_{a} ψ}^{Q_{j}} (B) = μ_{ψ}^{Q_{j}} (B - a_{j}), μ_{T_{a} ψ}^{P_{k}} (B) = μ_{ψ}^{P_{k}} (B) :$ translating the preparation by $a$ translates the position law by $a$ and leaves every momentum law unchanged;

(c) Dynamics covariance. With $H_{V}$ and $H_{V_{a}}$ the energy operators of the potentials $V$ and $V_{a}$ , $T_{a} H_{V} T_{a}^{- 1} = H_{V_{a}}, T_{a} U_{V} (t) T_{a}^{- 1} = U_{V_{a}} (t) (t \in R);$ in particular, if $V$ is invariant under translation by $a$ (equivalently constant almost everywhere when this holds for every $a$ ), then $T_{a}$ leaves $H_{V}$ and its flow invariant, and the free flow ( $V = 0$ ) commutes with every translation;

(d) Frame reading. The model reads the inertial-frame change $t^{'} = t$ , $x^{'} = x + a$ of P5 (a change of spatial origin; the parameter $a$ has the unit of length, Dimensions, SI unit conventions and quantity types of the framework) on state representatives by $ψ^{'} (x^{'}) = ψ (x^{'} - a) = (T_{a} ψ) (x^{'})$ ; this is the translation convention of the model, consistent with P5's frame class, and clauses (b) and (c) are its covariance consequences for the outcome laws and for the bounded-potential dynamics;

(e) Scope. The statement is conditional on P2, P4, P5 and P6 and the cited mathematics, and it concerns the componentwise massive model with real bounded measurable potentials only; singular or unbounded potentials, confined or boundary realizations, massless and relativistic models and radiation are separate model assumptions for which no translation covariance is asserted. No apparatus, laboratory identification or empirical result is recorded.

Facts & Assumptions

Given: The postulates P2, P4, P5, P6 read through The massive spin-s particle model of the framework: state space, operators, units and frame readings; the Axiom of Countable Choice and the Axiom of Choice; the state space $H_{s}$ with $r = 2 s + 1$ ; the operators $Q_{j}, P_{k}$ ; the translations $T_{a}$ ; a real bounded measurable $V$ ; a unit vector $ψ$ ; $a \in R^{3}$ ; $λ \in R$ ; Borel $B \subseteq R$ .

[F1]

Componentwise kinematics. $Q_{j}$ and $P_{k}$ are self-adjoint with the maximal componentwise domains; the translations $T_{a}$ are unitary with $T_{a} D (Q_{j}) = D (Q_{j})$ , $T_{a} Q_{j} T_{a}^{- 1} = Q_{j} - a_{j} I$ , $T_{a} D (P_{k}) = D (P_{k})$ , $T_{a} P_{k} T_{a}^{- 1} = P_{k}$ ; on Schwartz classes $T_{a}$ is the shift and $P_{k} f = - i ℏ D_{k} f$ (Componentwise position and momentum operators on the component-valued wave-function space clauses (a),(c), Position and momentum operators on Euclidean L2).

[F2]

Laws and spectral measures. The outcome law of an assigned sharp observable is $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ ; the spectral measures of the model are $E_{Q_{j}} (B) = M_{1_{x_{j}^{- 1} (B)}}$ and $E_{P_{k}} (B) = U^{(s) - 1} M_{1_{B} (p_{k})} U^{(s)}$ (Assigned sharp observables and the outcome law of the framework clause (b), Position and momentum outcome laws of the massive spin-s model clauses (a),(c), The momentum representation of the componentwise kinematics clause (b), Projection valued measure).

[F3]

Bounded-potential energy and flow. For a real bounded measurable $V$ the operator $(1 / 2 m) \sum_{k} P_{k}^{2} + M_{V}$ is self-adjoint on $W^{2, 2} (R^{3}; C)$ with domain invariant under the flow $e^{- i t H / ℏ}$ , which is a strongly continuous unitary group solving $i ℏ \frac{d}{d t} ψ (t) = H ψ (t)$ exactly on the domain (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (c), Finite orthogonal sums of self-adjoint operators and of bounded operators, The Schroedinger-picture flow: unitarity, domain invariance and the domain-qualified equation clauses (a)-(c), Strongly continuous one-parameter unitary group). The generator of a strongly continuous unitary group determines the group (Stone's theorem: unitary groups and self-adjoint generators, A self-adjoint operator generates a strongly continuous unitary group).

[F4]

Multipliers shift. $T_{a} M_{V} T_{a}^{- 1} = M_{V_{a}}$ with $V_{a} (x) = V (x - a)$ , and $T_{a} W^{2, 2} (R^{3}; C^{r}) = W^{2, 2} (R^{3}; C^{r})$ , because the componentwise translation preserves the weak derivative and shifts the pointwise multiplier (Componentwise position and momentum operators on the component-valued wave-function space clause (c), Unitary identifications of the component-valued wave-function space clause (c)).

[F5]

Borel representatives. Under Countable Choice, a Lebesgue-measurable function on $R^{3}$ is equal almost everywhere to a Borel-measurable function (A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra).

[F6]

Tonelli. For sigma-finite measure spaces, a nonnegative product-measurable function has measurable sections and its integral is the iterated integral in either order (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F7]

Linear change of variables. An invertible linear map $A : R^{6} \to R^{6}$ carries Lebesgue-measurable sets to measurable sets and scales their measure by $∣ \det A ∣$ (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).

[F8]

Translations preserve null sets. For a $C^{1}$ diffeomorphism $T$ the change-of-variables formula holds for nonnegative Lebesgue-measurable functions; in particular translations of Lebesgue-null sets are null (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions).

Proof

technique · direct; read clause (a) from the componentwise translation identities, compute the law transport from the explicit multiplication and momentum-multiplier forms of the spectral measures, and conjugate the bounded-potential Hamiltonian and its flow, identifying the generator of the conjugated group

Given: The data of the statement and [F1]-[F4].

1.1F1

[Operators and domains] Clause (a) is exactly [F1]: unitarity of $T_{a}$ , the domain equalities and the two conjugation identities.

1.2F1F2algebra

[Position-law transport] By [F2], the two compositions $E_{Q_{j}} (B) T_{a}$ and $T_{a} E_{Q_{j}} (B - a_{j})$ act componentwise as follows: for a class $f$ , $(E_{Q_{j}} (B) T_{a} f) (x) = 1_{B} (x_{j}) f (x - a)$ and $(T_{a} E_{Q_{j}} (B - a_{j}) f) (x) = 1_{B - a_{j}} (x_{j} - a_{j}) f (x - a) = 1_{B} (x_{j}) f (x - a)$ , so the operators agree. Taking norms and using unitarity of $T_{a}$ and the law formula [F2], $μ_{T_{a} ψ}^{Q_{j}} (B) = ∥ E_{Q_{j}} (B) T_{a} ψ ∥^{2} = ∥ E_{Q_{j}} (B - a_{j}) ψ ∥^{2} = μ_{ψ}^{Q_{j}} (B - a_{j})$ .

1.3F1F3F4algebra

[Hamiltonian covariance] By [F1], $T_{a} P_{k} T_{a}^{- 1} = P_{k}$ , hence also $T_{a} P_{k}^{2} T_{a}^{- 1} = P_{k}^{2}$ with domain invariance; and [F4] gives $T_{a} M_{V} T_{a}^{- 1} = M_{V_{a}}$ , with $T_{a} W^{2, 2} = W^{2, 2}$ . Therefore $T_{a} H_{V} T_{a}^{- 1} = H_{V_{a}}$ as operators with domain $W^{2, 2} (R^{3}; C^{r})$ .

1.4F5F6F7F8algebra

[Functions invariant under every translation] Suppose $V (\cdot - a) = V$ almost everywhere for every $a \in R^{3}$ . By [F5] choose a Borel representative $\tilde{V} = V$ almost everywhere. For each fixed $a$ , [F8] preserves the exceptional null set under translation, so $\tilde{V} (\cdot - a) = \tilde{V}$ almost everywhere. For $n \geq 1$ put $C_{n} : = (- n, n)^{3}$ , $A_{n} : = (- 2 n, 2 n)^{3}$ , and $N_{n} : = {(x, a) \in C_{n} \times A_{n} : \tilde{V} (x - a) \neq \tilde{V} (x)}$ . This is a Borel set, and each $x$ -section at fixed $a$ is null, so [F6] gives $λ_{6} (N_{n}) = 0$ . The invertible linear map $Φ (x, a) : = (x, x - a)$ has $∣ \det Φ ∣ = 1$ ; by [F7], $Φ [N_{n}]$ is null. Since every $(x, y) \in C_{n} \times C_{n}$ has $a = x - y \in A_{n}$ , the set ${(x, y) \in C_{n} \times C_{n} : \tilde{V} (y) \neq \tilde{V} (x)}$ is contained in $Φ [N_{n}]$ and is null. By [F6], for almost every $x \in C_{n}$ this equality holds for almost every $y \in C_{n}$ ; choose such an $x_{n}$ with $\tilde{V} (x_{n})$ finite. Hence $V = \tilde{V} (x_{n})$ almost everywhere on $C_{n}$ . The cubes are nested and their overlaps have positive measure, so these constants agree for all $n$ and $V$ is constant almost everywhere on $R^{3}$ . Conversely, if $V$ is constant almost everywhere, [F8] gives $V (\cdot - a) = V$ almost everywhere for every $a$ . Thus invariance under all translations is equivalent to being constant almost everywhere.

2.1step 1.2F1F2algebra

[Momentum-law invariance] By [F2], $U^{(s)} E_{P_{k}} (B) U^{(s) - 1} = M_{1_{B} (p_{k})}$ and $U^{(s)} T_{a} U^{(s) - 1} = M_{e^{- i a \cdot p / ℏ}}$ (The momentum representation of the componentwise kinematics clause (d)); the two multipliers commute, so $E_{P_{k}} (B) T_{a} = T_{a} E_{P_{k}} (B)$ and $μ_{T_{a} ψ}^{P_{k}} (B) = ∥ E_{P_{k}} (B) T_{a} ψ ∥^{2} = ∥ E_{P_{k}} (B) ψ ∥^{2} = μ_{ψ}^{P_{k}} (B)$ . Together with step 1.2 this proves (b).

2.2step 1.3F1F3F4algebra

[Flow covariance] Define $W (t) : = T_{a} U_{V} (t) T_{a}^{- 1}$ . It is a strongly continuous unitary group because $U_{V}$ is one and conjugation by the unitary $T_{a}$ preserves the group law, unitarity and strong continuity. For $x \in D (H_{V_{a}}) = T_{a} D (H_{V})$ one has $T_{a}^{- 1} x \in D (H_{V})$ and $\frac{1}{t} (W (t) x - x) = T_{a} \frac{1}{t} (U_{V} (t) T_{a}^{- 1} x - T_{a}^{- 1} x) ⟶ T_{a} (- \frac{i}{ℏ} H_{V} T_{a}^{- 1} x) = - \frac{i}{ℏ} H_{V_{a}} x,$ using the generator property of $U_{V}$ [F3] and step 1.3; the limit exists exactly on $D (H_{V_{a}})$ because the limit for $U_{V}$ exists exactly on $D (H_{V})$ [F3]. Hence $W$ is a strongly continuous unitary group with generator $- \frac{i}{ℏ} H_{V_{a}}$ ; by [F3] the group $t \mapsto U_{V_{a}} (t) = \exp (- i t H_{V_{a}} / ℏ)$ has the same generator, and uniqueness of the group with a given generator gives $T_{a} U_{V} (t) T_{a}^{- 1} = U_{V_{a}} (t)$ . If $V (\cdot - a) = V$ almost everywhere, then $V_{a} = V$ as a multiplier and $T_{a} H_{V} T_{a}^{- 1} = H_{V}$ with $T_{a} U_{V} (t) T_{a}^{- 1} = U_{V} (t)$ ; the free case $V = 0$ is translation invariant for every $a$ . This proves the fixed-translation assertion of (c).

3.1givenstep 1.1step 1.2step 2.1step 1.3step 2.2step 1.4∎

[Frame reading and scope] Clause (d) records the model's convention for the passive spatial translation of P5: the parameter $a$ is the shift of the spatial origin and the representative transforms by the shift $T_{a}$ , so steps 1.2, 2.1 and 2.2 are exactly the transformations of the law and of the bounded-potential dynamics under the frame change. Clause (e) records the hypotheses actually used: P2, P4, P5 and P6, the componentwise model, real bounded measurable $V$ and the cited mathematics; nothing was used about singular potentials, boundaries, massless or relativistic models, radiation, apparatus or measurements.

Remarks

Sign of the shift. The convention is that translating the preparation by $a$ moves the position law by $+ a$ : clause (b) says $μ_{T_{a} ψ}^{Q_{j}} (B) = μ_{ψ}^{Q_{j}} (B - a_{j})$ . Momentum laws are invariant because the translation commutes with $P_{k}$ ; this is the kinematic statement that translations are generated by momentum, and the generator identification itself is the mathematical content of Componentwise position and momentum operators on the component-valued wave-function space clause (d).
What is not claimed. No covariance of the internal factor under spatial transformations, no rotation or boost statement (those are separate items of this page), and no general covariance for potentials outside the bounded measurable class. The frame reading of clause (d) is a convention of the model and gains content only through the covariance statements (b) and (c).

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Galilean boost covariance of the free massive spin-s model and the moving-frame reading

Physical scope

The Galilean boost covariance of the free spin-s massive-particle model and the moving-frame reading of its bounded-potential dynamics: unitarity and domain preservation of the boost family, the operator identities shifting position by ut and momentum by mu together with the resulting law transports, the free-propagator covariance identity, the conjugated free Hamiltonian identity, and the moving-frame Schroedinger equation with the pulled-back potential. The statement is conditional on P2, P4, P5 and P6 and the cited mathematics, holds for the free model and for real bounded measurable potentials only, and asserts no Lorentz covariance, no massless or photon limit, no covariance for singular or unbounded potentials, and no apparatus or empirical result.

Statement

Adopt the observable and probability postulate P2 in the reading of Assigned sharp observables and the outcome law of the framework, the autonomous-dynamics postulate P4 in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution, and the kinematics and massive-particle model postulate P5 and internal-factor postulate P6 read through The massive spin-s particle model of the framework: state space, operators, units and frame readings for a particle of mass parameter $m > 0$ and spin label $s$ (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators), and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $H_{s}$ be the model's state space with componentwise operators $Q_{j}, P_{k}$ (Componentwise position and momentum operators on the component-valued wave-function space), let $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ be the outcome law of an assigned sharp observable $A$ in a unit vector $ψ$ , let $H_{0} : = \frac{1}{2 m} \sum_{k} P_{k}^{2}$ with domain $D (H_{0}) = W^{2, 2} (R^{3}; C^{r})$ and flow $U_{0} (t) = \exp (- i t H_{0} / ℏ)$ , and for real bounded measurable $V : R^{3} \to R$ let $H_{V}$ and $U_{V}$ be the operator and flow of Translation covariance of the massive spin-s kinematics and of its bounded-potential dynamics, with $D (H_{V}) = W^{2, 2} (R^{3}; C^{r})$ and $V_{u} (x, t) : = V (x - u t)$ . For $u \in R^{3}$ and $t \in R$ define the boost operator $B_{u} (t)$ componentwise by $(B_{u} (t) ψ) (x) : = \exp (\frac{i}{ℏ} (m u \cdot x - \frac{m ∣ u ∣^{2}}{2} t)) ψ (x - u t) .$ Then:

(a) for every $u, t$ the operator $B_{u} (t)$ is unitary, $B_{u} (t)^{- 1} = B_{u} (t)^{*}$ , and it maps $W^{2, 2} (R^{3}; C^{r})$ and $S (R^{3}; C^{r})$ onto themselves;

(b) $B_{u} (t) D (Q_{j}) = D (Q_{j})$ and $B_{u} (t) D (P_{k}) = D (P_{k})$ ; for $f \in D (Q_{j})$ and $g \in D (P_{k})$ , respectively, $Q_{j} B_{u} (t) f = B_{u} (t) (Q_{j} + u_{j} t I) f, P_{k} B_{u} (t) g = B_{u} (t) (P_{k} + m u_{k} I) g .$ For a Borel $B \subseteq R$ and $a \in R$ , write $B - a : = {b - a : b \in B}$ . Consequently, for every unit $ψ$ and Borel $B \subseteq R$ $μ_{B_{u} (t) ψ}^{Q_{j}} (B) = μ_{ψ}^{Q_{j}} (B - u_{j} t), μ_{B_{u} (t) ψ}^{P_{k}} (B) = μ_{ψ}^{P_{k}} (B - m u_{k});$

(c) free covariance. $B_{u} (t) U_{0} (t - s) = U_{0} (t - s) B_{u} (s)$ for all $s, t \in R$ ; in particular the freely evolved boosted preparation equals the boost at time $t$ of the initial boosted preparation, $B_{u} (t) U_{0} (t) ψ = U_{0} (t) B_{u} (0) ψ (ψ \in H_{s});$ and the conjugated free Hamiltonian identity $B_{u} (t)^{- 1} H_{0} B_{u} (t) = H_{0} + u \cdot P + \frac{m ∣ u ∣^{2}}{2} I$ holds on $W^{2, 2} (R^{3}; C^{r})$ (in the chart velocity $v = - u$ of clause (e) it reads $H_{0} - v \cdot P + \frac{m ∣ v ∣^{2}}{2} I$ ), the operator form of the frame change on the free generator;

(d) moving-frame bounded-potential model. For every real bounded measurable $V$ and every $u$ , the two-parameter family $W_{u} (t, s) : = B_{u} (t) U_{V} (t - s) B_{u} (s)^{- 1}$ consists of unitaries preserving $W^{2, 2} (R^{3}; C^{r})$ , and for every $s \in R$ and $ψ_{0} \in W^{2, 2} (R^{3}; C^{r})$ the curve $t \mapsto W_{u} (t, s) ψ_{0}$ takes values in $W^{2, 2}$ , is differentiable as an $L^{2}$ -valued curve and satisfies $i ℏ \frac{d}{d t} ψ (t) = (\frac{1}{2 m} \sum_{k} P_{k}^{2} + M_{V_{u} (\cdot, t)}) ψ (t), ψ (s) = ψ_{0};$ it is unique among maps $w : R \to W^{2, 2}$ that are differentiable into $L^{2}$ and satisfy this equation in $L^{2}$ for every $t$ with $w (s) = ψ_{0}$ ;

(e) Frame reading and scope. The model reads the inertial-frame change $t^{'} = t$ , $x^{'} = x + u t$ of P5 (the new chart moves with velocity $- u$ relative to the old one) on state representatives by $ψ^{'} (x^{'}, t) = (B_{u} (t) ψ (\cdot, t)) (x^{'})$ ; clauses (b)-(d) are the covariance consequences of that reading for the laws of the free model and for the bounded-potential dynamics in the moving frame. The statement is conditional on P2, P4, P5 and P6 and the cited mathematics, concerns the componentwise massive model with real bounded measurable potentials only, and asserts no Lorentz covariance, no photon limit, no covariance for singular or unbounded potentials and no worldline of an unmeasured particle; no apparatus or empirical result is recorded.

Facts & Assumptions

Given: The postulates P2, P4, P5 and P6 read through The massive spin-s particle model of the framework: state space, operators, units and frame readings; the Axiom of Countable Choice and the Axiom of Choice; the state space $H_{s}$ , $r = 2 s + 1$ ; the componentwise operators $Q_{j}, P_{k}$ and their scalar counterparts; the boost operators $B_{u} (t)$ ; the free energy $H_{0}$ ; a bounded real measurable $V$ and the operators $H_{V}$ , $U_{V}$ , $W_{u} (t, s)$ ; a unit vector $ψ$ ; a Borel set $B \subseteq R$ ; vectors $u \in R^{3}$ , $x \in R^{3}$ ; times $s, t$ .

[F1]

Phase derivatives. For $φ (x, t) : = m u \cdot x / ℏ - m ∣ u ∣^{2} t / (2 ℏ)$ , one has $\nabla φ = m u / ℏ$ and $\partial_{t} φ = - m ∣ u ∣^{2} / (2 ℏ)$ . The function $e^{i φ}$ is smooth and unimodular, with $\partial_{k} e^{i φ} = i (m u_{k} / ℏ) e^{i φ}$ and $\partial_{t} e^{i φ} = - i (m ∣ u ∣^{2} / (2 ℏ)) e^{i φ}$ , by the complex-exponential derivative and modulus formulas (The complex exponential is entire and its complex derivative is itself, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[F2]

Componentwise kinematics, momentum representation and laws. $Q_{j}, P_{k}$ are the componentwise operators with their maximal domains and $P_{k} f = - i ℏ D_{k} f$ on its domain (Componentwise position and momentum operators on the component-valued wave-function space clause (a)). The unitary momentum transform $U^{(s)}$ sends $P_{k}$ to multiplication by $p_{k}$ with the corresponding domain equality, and sends $T_{a}$ to multiplication by $e^{- i a \cdot p / ℏ}$ (The momentum representation of the componentwise kinematics clauses (b),(d)). The spectral measures and outcome laws are those of Position and momentum outcome laws of the massive spin-s model clauses (a),(c).

[F3]

Translation identities. $T_{a} D (Q_{j}) = D (Q_{j})$ , $T_{a} Q_{j} T_{a}^{- 1} = Q_{j} - a_{j} I$ , $T_{a} D (P_{k}) = D (P_{k})$ , $T_{a} P_{k} T_{a}^{- 1} = P_{k}$ , and $T_{a}$ preserves $W^{2, 2} (R^{3}; C^{r})$ and the Schwartz class (Translation covariance of the massive spin-s kinematics and of its bounded-potential dynamics clause (a), Componentwise position and momentum operators on the component-valued wave-function space clause (c)).

[F4]

Weak-derivative test identity. For $f \in D (P_{k})$ , $D_{k} f \in L^{2}$ and $\int f \partial_{k} η = - \int (D_{k} f) η$ for every $η \in C_{c}^{\infty} (R^{3})$ , by the definition of weak derivative (Weak derivative of a locally integrable function, Componentwise position and momentum operators on the component-valued wave-function space clause (a)).

[F5]

Flows and uniqueness. $H_{0}$ and $H_{V}$ are self-adjoint on $W^{2, 2} (R^{3}; C^{r})$ with strongly continuous unitary flows preserving the domain and satisfying the Schroedinger equation exactly on it (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (c), The Schroedinger-picture flow: unitarity, domain invariance and the domain-qualified equation clauses (a)-(c), Strongly continuous one-parameter unitary group).

[F6]

A real function continuous on a closed interval and differentiable in its interior with zero derivative is constant, by 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 $f (b) - f (a) = f^{'} (c) (b - a)$ ; restrictions of a differentiable curve to compact intervals satisfy these continuity hypotheses.

[F7]

If $g_{h} \to g$ pointwise as $h \to 0$ and $∣ g_{h} ∣ \leq G$ for a fixed integrable $G$ , then the corresponding integrals converge (Dominated convergence).

[F8]

For a differentiable Banach-valued curve with derivative norm bounded by $C$ on an interval, the endpoint difference has norm at most $C$ times the interval length (Mean value inequality for a differentiable Banach-valued curve).

Proof

technique · direct; derive the boost's domain and time-derivative identities from its phase and translation formula, compute free covariance in the momentum representation, and conjugate the bounded-potential evolution

Given: The data of the statement and [F1]-[F8].

1.1F1F3algebra

[Unitarity and domain preservation] Write $B_{u} (t) = M_{e^{i φ (\cdot, t)}} T_{u t}$ componentwise, where $T_{u t}$ is the componentwise translation and $φ (\cdot, t)$ the smooth real phase of [F1]. By [F3] the translation is unitary and preserves $W^{2, 2}$ and the Schwartz class; multiplication by the unimodular smooth function $e^{i φ}$ is unitary and preserves both classes because the phase is affine with bounded derivatives. Hence $B_{u} (t)$ is unitary with the stated inverse and preserves both classes, componentwise as in the scalar case of [F1]. This proves (a).

1.2F1F3algebra

[Position identity] By [F3], $Q_{j} T_{u t} = T_{u t} (Q_{j} + u_{j} t I)$ on $D (Q_{j})$ ; multiplication by $e^{i φ}$ commutes with $Q_{j}$ (both are componentwise multipliers). Hence, on $D (Q_{j})$ , $Q_{j} B_{u} (t) = M_{e^{i φ}} Q_{j} T_{u t} = M_{e^{i φ}} T_{u t} (Q_{j} + u_{j} t I) = B_{u} (t) (Q_{j} + u_{j} t I)$ . Since $T_{u t}$ and multiplication by the unimodular $e^{i φ}$ both preserve $D (Q_{j})$ in both directions, $B_{u} (t) D (Q_{j}) = D (Q_{j})$ .

1.3F1F3F4algebra

[Momentum identity] Put $χ = e^{i φ (\cdot, t)}$ and $c = m u_{k} / ℏ$ . For each component $f_{ℓ}$ of $f \in D (P_{k})$ and each scalar test function $η \in C_{c}^{\infty} (R^{3})$ , $χ η$ is again a test function, so [F4] and the classical product rule give [F1, F3, F4, algebra] $\int χ f_{ℓ} \partial_{k} η = \int f_{ℓ} \partial_{k} (χ η) - \int f_{ℓ} (\partial_{k} χ) η = - \int (χ D_{k} f_{ℓ} + i c χ f_{ℓ}) η .$ The proposed weak derivative is in $L^{2}$ since $∣ χ ∣ = 1$ and $c$ is constant; therefore $χ f \in D (P_{k})$ and $P_{k} (χ f) = χ (P_{k} + ℏ c I) f = χ (P_{k} + m u_{k} I) f$ . The same calculation with $χ^{- 1}$ shows that multiplication by $χ$ maps $D (P_{k})$ onto itself. By [F3], $T_{u t}$ also preserves $D (P_{k})$ in both directions and $P_{k} T_{u t} = T_{u t} P_{k}$ ; hence $B_{u} (t) D (P_{k}) = D (P_{k})$ and $P_{k} B_{u} (t) = M_{e^{i φ}} (P_{k} + m u_{k} I) T_{u t} = B_{u} (t) (P_{k} + m u_{k} I)$ with the stated domains.

1.4F1F2F7algebra

[Time derivative of the boost] Put $λ : = m ∣ u ∣^{2} / (2 ℏ)$ and $q (p) : = u \cdot p / ℏ$ . For $f \in W^{1, 2}$ and $h \neq 0$ , the momentum representation gives [F1, F2, F7, algebra] $U^{(s)} (\frac{T_{u h} f - f}{h} + u \cdot D f) = (\frac{e^{- i h q} - 1}{h} + i q) U^{(s)} f .$ The multiplier tends pointwise to $0$ by the derivative of the complex exponential. Applying [F8] to $r \mapsto e^{- i r h q (p)}$ for $r \in [0, 1]$ gives $∣ e^{- i h q (p)} - 1 ∣ \leq ∣ h q (p) ∣$ , so the squared integrand is bounded by $4 ∣ q ∣^{2} ∣ U^{(s)} f ∣^{2}$ , which is integrable because $u \cdot D f \in H_{s}$ ; [F7] proves convergence to $0$ in $L^{2}$ . From the definition, $B_{u} (t + h) = e^{- i λ h} B_{u} (t) T_{u h}$ ; the resulting difference quotient yields $\frac{d}{d t} B_{u} (t) f = - B_{u} (t) (u \cdot D f) - i λ B_{u} (t) f = - \frac{i}{ℏ} B_{u} (t) (u \cdot P + \frac{m ∣ u ∣^{2}}{2} I) f .$ For every $f \in H_{s}$ , the multiplier formula for $T_{u h}$ and $∣ e^{- i h q} - 1 ∣ \leq 2$ from [F1] give $T_{u h} f \to f$ by [F7]; the scalar phase $e^{- i λ h}$ is continuous, so $B_{u} (t + h) f \to B_{u} (t) f$ . Thus $B_{u} (t)$ is strongly continuous on $H_{s}$ .

2.1step 1.2step 1.3F2F3algebra

[Law transports] For a componentwise representative $f$ , $(E_{Q_{j}} (B) B_{u} (t) f) (x) = 1_{B} (x_{j}) e^{i φ (x, t)} f (x - u t)$ , whereas $(B_{u} (t) E_{Q_{j}} (B - u_{j} t) f) (x) = e^{i φ (x, t)} 1_{B - u_{j} t} (x_{j} - u_{j} t) f (x - u t)$ . Since the two indicators agree, $E_{Q_{j}} (B) B_{u} (t) = B_{u} (t) E_{Q_{j}} (B - u_{j} t)$ ; taking norms gives $μ_{B_{u} (t) ψ}^{Q_{j}} (B) = μ_{ψ}^{Q_{j}} (B - u_{j} t)$ . For momentum, changing variables $y = x - u t$ in the Fourier transform first on smooth compactly supported representatives yields $(U^{(s)} B_{u} (t) f) (p) = e^{- i t u \cdot p / ℏ + i m ∣ u ∣^{2} t / (2 ℏ)} (U^{(s)} f) (p - m u)$ . The equality extends to all $L^{2}$ by density and unitarity. Therefore the indicator $1_{B} (p_{k})$ on the left equals the transported indicator $1_{B - m u_{k}} (p_{k} - m u_{k})$ on the right, so $E_{P_{k}} (B) B_{u} (t) = B_{u} (t) E_{P_{k}} (B - m u_{k})$ and $μ_{B_{u} (t) ψ}^{P_{k}} (B) = μ_{ψ}^{P_{k}} (B - m u_{k})$ . These are the two law transports in (b); neither depends on the phase in the position density.

2.2step 1.1step 1.3F2algebra

[Free Hamiltonian conjugation] By step 1.3, $P_{k} B_{u} (t) = B_{u} (t) (P_{k} + m u_{k} I)$ for every $k$ . Applying this identity twice on $W^{2, 2}$ and using preservation of that domain (step 1.1) gives $P_{k}^{2} B_{u} (t) = B_{u} (t) (P_{k} + m u_{k} I)^{2}$ . Therefore [step 1.1, step 1.3, F2, algebra] $B_{u} (t)^{- 1} H_{0} B_{u} (t) = \frac{1}{2 m} \sum_{k} (P_{k} + m u_{k} I)^{2} = H_{0} + u \cdot P + \frac{m ∣ u ∣^{2}}{2} I$ on $W^{2, 2}$ , with the domain equality inherited from $B_{u} (t) W^{2, 2} = W^{2, 2}$ . This is the Hamiltonian identity of (c).

3.1step 2.1F2F5algebra

[Free covariance] Put $τ : = t - s$ . Under $U^{(s)}$ , each $P_{k}$ is multiplication by $p_{k}$ (The momentum representation of the componentwise kinematics clause (b)); hence $H_{0} = (2 m)^{- 1} \sum_{k} P_{k}^{2}$ is multiplication by $∣ p ∣^{2} / (2 m)$ on the transformed $W^{2, 2}$ domain, and its spectral-calculus flow $U_{0} (τ)$ is multiplication by $e^{- i τ ∣ p ∣^{2} / (2 m ℏ)}$ (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (b), Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (b)). By the momentum-transform formula of step 2.1, both sides of the free-covariance identity have the same momentum representative because [step 2.1, F2, F5, algebra] $e^{- i t u \cdot p / ℏ + i m ∣ u ∣^{2} t / (2 ℏ)} e^{- i τ ∣ p - m u ∣^{2} / (2 m ℏ)} = e^{- i τ ∣ p ∣^{2} / (2 m ℏ)} e^{- i s u \cdot p / ℏ + i m ∣ u ∣^{2} s / (2 ℏ)},$ as follows by expanding $∣ p - m u ∣^{2}$ and using $τ = t - s$ . The equality holds on all of $H_{s}$ because $U^{(s)}$ is unitary and all displayed multipliers are unimodular.

3.2step 1.1step 1.3step 1.4step 2.2F5F6algebra

[Moving-frame equation and uniqueness] Fix $s \in R$ and $ψ_{0} \in W^{2, 2}$ , put $y_{0} : = B_{u} (s)^{- 1} ψ_{0}$ , $y (t) : = U_{V} (t - s) y_{0}$ , and $ψ (t) : = B_{u} (t) y (t)$ . By [F5], $y (t) \in W^{2, 2}$ and $y^{'} (t) = - i H_{V} y (t) / ℏ$ in $L^{2}$ ; step 1.1 gives $ψ (t) \in W^{2, 2}$ . The difference-quotient decomposition [step 1.1, step 1.3, step 1.4, step 2.2, F5, F6, algebra] $\frac{ψ (t + h) - ψ (t)}{h} = B_{u} (t + h) \frac{y (t + h) - y (t)}{h} + \frac{B_{u} (t + h) - B_{u} (t)}{h} y (t)$ and strong continuity of $B_{u} (t)$ (step 1.4) yield $ψ^{'} = B_{u} (t) y^{'} - \frac{i}{ℏ} B_{u} (t) (u \cdot P + m ∣ u ∣^{2} I / 2) y$ . Using $H_{V} = H_{0} + M_{V}$ , step 2.2 gives $B_{u} (t) (H_{0} + u \cdot P + m ∣ u ∣^{2} I / 2) y = H_{0} ψ$ , while $B_{u} (t) M_{V} = M_{V_{u} (\cdot, t)} B_{u} (t)$ follows from $B_{u} (t) = M_{e^{i φ}} T_{u t}$ , the commutation of multiplication operators, and $T_{u t} M_{V} = M_{V_{u} (\cdot, t)} T_{u t}$ . Therefore $i ℏ ψ^{'} = (H_{0} + M_{V_{u} (\cdot, t)}) ψ$ and $ψ (s) = ψ_{0}$ , proving existence. For uniqueness, let $w_{1}, w_{2} : R \to W^{2, 2}$ be $L^{2}$ -differentiable solutions and put $z = w_{1} - w_{2}$ . For each $t$ , $A (t) : = H_{0} + M_{V_{u} (\cdot, t)}$ is symmetric on $W^{2, 2}$ , so $\frac{d}{d t} ∥ z (t) ∥^{2} = 2 Re ⟨ - i A (t) z (t) / ℏ, z (t) ⟩ = 0$ . By the real mean-value theorem, $∥ z (t) ∥^{2}$ is constant; if $z (s) = 0$ , then $z (t) = 0$ for all $t$ . This proves uniqueness in the class stated in (d).

4.1givenstep 1.2step 1.3step 1.4step 2.1step 3.1step 2.2step 3.2∎

[Frame reading and scope] Clause (e) records the model's convention for the passive boost of P5: the chart change $x^{'} = x + u t$ is read by $B_{u} (t)$ , so steps 1.2, 1.3, 2.1, 3.1, 2.2 and 3.2 are the covariance statements under that change. The hypotheses actually used are P2, P4, P5 and P6, the componentwise model and real bounded measurable potentials; no Lorentz structure, massless limit, singular potential, boundary realization, apparatus or empirical datum is used.

Remarks

Momentum shift from the phase. The phase $φ (x, t) = m u \cdot x / ℏ - m ∣ u ∣^{2} t / (2 ℏ)$ depends on $x$ and shifts the model momentum law by $m u$ , as clause (b) records. The boost is a kinematic frame change of the model, and its free-propagator identity (clause (c)) gives the covariance of the free dynamics.
What is not claimed. No boost covariance is asserted for potentials outside the real bounded measurable class, for confined or boundary realizations, or for the internal factor beyond its passive transport by the componentwise definition; no relativistic boost or Lorentz transformation law is either assumed or derived. The boost family is defined by the phase and translation formula in the Statement; the covariance identities are derived here from that formula and the componentwise model operators.

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Scope and empirical status of the massive-particle kinematics items

Physical scope

The reading of the kinematics pair as a whole: its model assumptions, the empirical status of its mass and spin parameters, the excluded realizations and the limits of its predictions. The remark asserts no new dynamics or mathematics and records no measurement.

What the pair establishes

The items of this pair are conditional consequences of the framework's postulates and of the cited mathematics. The massive spin-s particle model of the framework: state space, operators, units and frame readings founds the spin- $s$ model on the state space $H_{s} = L^{2} (R^{3}; C^{2 s + 1})$ with the componentwise position and momentum operators $Q_{j}, P_{k}$ ; Componentwise position and momentum operators on the component-valued wave-function space proves that these operators are self-adjoint with their maximal domains and provides the Fourier representation and the translation identities; Canonical commutation relations on the component-valued core and commutation with the internal factor proves the canonical commutation relations on the Schwartz core, identifies the spectral measures and proves that the internal factor commutes with the spatial measures; The momentum representation of the componentwise kinematics constructs the momentum representation; Position and momentum outcome laws of the massive spin-s model derives the position and momentum outcome laws of the spin- $s$ model, including the Born density $∥ ψ (x) ∥_{C^{2 s + 1}}^{2}$ and the momentum density $∥ ψ_{p} (p) ∥^{2}$ ; Translation covariance of the massive spin-s kinematics and of its bounded-potential dynamics and Galilean boost covariance of the free massive spin-s model and the moving-frame reading establish the covariance of these laws and of the bounded-potential dynamics under the Galilean translations and boosts of the model's frame class. All of this is conditional on P1, P2, P4, P5 and P6 and on the cited mathematical suppliers; none of it is a statement that any physical system is exactly described by the model.

Empirical status of the parameters

The mass parameter $m > 0$ and the spin label $s$ are model parameters: within the framework they are declared values, and the identical-particle and spin postulate assigns to a species its mass, spin, statistics and couplings as empirical model parameters with uncertainty. This pair derives no value of $m$ or $s$ , records no measurement and asserts no species identification; the units in which $m$ , $ℏ$ , $Q_{j}$ and $P_{k}$ are read are those of Dimensions, SI unit conventions and quantity types of the framework, and the exact defining constants of the current SI recorded there are unit conventions, not measurements of this framework.

What is not claimed

The outcome laws are sharp-law statements: an exact position or momentum value has probability zero in every preparation, finite-resolution outcomes are Borel bins, and no detector, efficiency, resolution or sampling model is introduced. The covariance theorems hold for the free model and for real bounded measurable potentials; they assert nothing for singular or unbounded potentials, confined or boundary realizations, massless particles, radiation or relativistic matter. The internal factor is bookkeeping: it is not identified with a spatial rotation, a classical angular momentum or a relativistic spinor, and no empirical consequence of the spin assignment alone is derived. The identification of the operators $Q_{j}, P_{k}$ with concrete laboratory procedures remains a tested identification under the preparation and apparatus postulate, external to this pair.

5 · Examples, experiments and counterexamples

None yet.

Qm Massive Particle Kinematics

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Finite orthogonal sums of self-adjoint operators and of bounded operators

Statement

Facts & Assumptions

Proof

Remarks

Unitary identifications of the component-valued wave-function space

Statement

Facts & Assumptions

Proof

Remarks

Componentwise position and momentum operators on the component-valued wave-function space

Statement

Facts & Assumptions

Proof

Remarks

Canonical commutation relations on the component-valued core and commutation with the internal factor

Statement

Facts & Assumptions

Proof

Remarks

The momentum representation of the componentwise kinematics

Statement

Facts & Assumptions

Proof

Remarks

The massive spin-s particle model of the framework: state space, operators, units and frame readings

Definition

Remarks

Position and momentum outcome laws of the massive spin-s model

Statement

Facts & Assumptions

Proof

Remarks

Translation covariance of the massive spin-s kinematics and of its bounded-potential dynamics

Statement

Facts & Assumptions

Proof

Remarks

Galilean boost covariance of the free massive spin-s model and the moving-frame reading

Statement

Facts & Assumptions

Proof

Remarks

Scope and empirical status of the massive-particle kinematics items

What the pair establishes

Empirical status of the parameters

What is not claimed

5 · Examples, experiments and counterexamples

Sources