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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).

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources