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Translation invariance of bounded potentials and conservation of momentum

Physical scope

A conditional consequence of the framework postulates for the componentwise massive model with a real bounded measurable potential V: the translation T_a is a symmetry of the bounded-potential operator H_V exactly when V is invariant under translation by a, the full translation group is a symmetry group exactly when V is almost everywhere constant, and for such translation-invariant models the momentum components are conserved observables by the generator-conservation theorem. The statement concerns real bounded measurable potentials only; it asserts no momentum conservation for non-invariant potentials, no statement about singular or unbounded potentials, no lattice or discrete-group conservation claim beyond the fixed-translation criterion, no apparatus realization and no empirical result.

Statement

Adopt the postulates P1, P2, P4, P5 and P6 in the readings of The massive spin-s particle model of the framework: state space, operators, units and frame readings, Assigned sharp observables and the outcome law of the framework, Assigned energy observable, unitary flow and the two pictures of autonomous evolution and Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables for a declared particle of mass parameter $m > 0$ and spin label $s$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems, 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 Choice, the Axiom of Countable Choice and the Axiom of Dependent Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain). 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 $V : R^{3} \to R$ be real, bounded and measurable, and let $H_{V} : = \frac{1}{2 m} \sum_{k = 1}^{3} P_{k}^{2} + M_{V}, D (H_{V}) = W^{2, 2} (R^{3}; C^{2 s + 1}), V_{a} (x) : = V (x - a),$ be the model's bounded-potential energy operator with $M_{V}$ multiplication by $V$ and $T_{a} : = T_{a}^{(2 s + 1)}$ the componentwise translations (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (c), Translation covariance of the massive spin-s kinematics and of its bounded-potential dynamics). Then:

(a) for every $a \in R^{3}$ : $T_{a} H_{V} T_{a}^{- 1} = H_{V_{a}},$ and $T_{a}$ is a symmetry of $H_{V}$ in the sense of Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clause (b) if and only if $V_{a} = V$ almost everywhere;

(b) if $T_{a}$ is a symmetry for every $a \in R^{3}$ , then $V$ is almost everywhere constant; in that case $H_{V} = H_{0} + c I$ on $W^{2, 2}$ for $H_{0} = \frac{1}{2 m} \sum_{k} P_{k}^{2}$ and $c$ the constant, and the translations form a strongly continuous unitary group of symmetries;

(c) Momentum conservation for translation-invariant models. Let $V$ be almost everywhere constant and let $c$ be that constant. Then for every $k \in {1, 2, 3}$ the momentum component $P_{k}$ is a conserved observable of the model with energy $H_{V}$ : for every preparation $ρ$ , every Borel $B \subseteq R$ and every $t$ , $μ_{ρ (t)}^{P_{k}} (B) = μ_{ρ (0)}^{P_{k}} (B);$ if $ψ (0) \in D (P_{k})$ is a unit vector, the mean and second moment of the law of $P_{k}$ are conserved with time-zero values $⟨ P_{k} ψ, ψ ⟩$ and $∥ P_{k} ψ ∥^{2}$ ; and the coordinate translations satisfy $T_{t e_{k}} = e^{- i t P_{k} / ℏ}$ for all $t$ , so the generators of the one-parameter symmetry groups are the momentum components;

(d) Scope. The statement concerns the componentwise massive model with real bounded measurable potentials. It asserts no momentum conservation for a potential that is not translation invariant — the translation-covariance theorem transports the position law and conjugates the dynamics, but does not by itself conserve momentum — and it makes no claim about singular, unbounded, boundary or relativistic models. Clause (a) includes bounded periodic potentials and their fixed period translations; no additional discrete-group conservation theorem, apparatus or empirical result is asserted.

Facts & Assumptions

Given: The postulates P1, P2, P4, P5, P6 in the stated readings; the Axioms of Choice, Countable Choice and Dependent Choice; the state space $H_{s}$ ; the operators $Q_{j}, P_{k}$ and translations $T_{a}$ ; a real bounded measurable $V$ ; the operator $H_{V}$ ; a preparation $ρ$ ; a unit vector $ψ$ ; a Borel set $B$ ; $a \in R^{3}$ ; $k \in {1, 2, 3}$ ; $t \in R$ .

[F1]

Translations and the bounded-potential operator. $T_{a}$ is unitary with $T_{a} D (Q_{j}) = D (Q_{j})$ , $T_{a} Q_{j} T_{a}^{- 1} = Q_{j} - a_{j} I$ , $T_{a} D (P_{k}) = D (P_{k})$ , $T_{a} P_{k} T_{a}^{- 1} = P_{k}$ , and $T_{a} W^{2, 2} = W^{2, 2}$ ; $T_{a} H_{V} T_{a}^{- 1} = H_{V_{a}}$ ; and $T_{t e_{k}} = e^{- i t P_{k} / ℏ}$ with the generator identification of the translation group (Translation covariance of the massive spin-s kinematics and of its bounded-potential dynamics clause (c), Componentwise position and momentum operators on the component-valued wave-function space clauses (c),(d)).

[F2]

Multipliers. $M_{V}$ is the bounded self-adjoint multiplication operator by the real bounded class $V$ ; for two real bounded measurable functions $W_{1}, W_{2}$ one has $M_{W_{1}} = M_{W_{2}}$ if and only if $W_{1} = W_{2}$ almost everywhere; and $H_{V} = H_{0} + M_{V}$ is self-adjoint with domain $W^{2, 2}$ and generates a strongly continuous unitary group (Real multiplication operators and free plus bounded-potential Schrödinger operators clauses (a),(c), The space $L^{p} (μ)$ as the quotient by null functions).

[F3]

Mollifiers and Lebesgue points. Let $ρ (x) = q (∣ x ∣)$ be the unit-mass radial mollifier supported in $B_{1} (0)$ and $ρ_{ε} (x) = ε^{- 3} ρ (x / ε)$ its family (The mollifier family generated by a unit-mass smooth bump, A radial mollifier family in Rn). Since $q$ is continuous with compact support, $Φ (r) : = \sup_{u \geq r} ∣ q (u) ∣$ is bounded, nonincreasing, vanishes for $r \geq 1$ , and satisfies $∣ ρ (y) ∣ \leq Φ (∣ y ∣)$ and $\int_{R^{3}} Φ (∣ y ∣) d y < \infty$ . The radial-majorant recovery lemma therefore applies: for $f \in L^{1} (R^{3})$ and a Lebesgue point $x$ of $f$ , $(f * ρ_{ε}) (x) \to f (x)$ as $ε ↓ 0$ (Lebesgue-point convergence for radial-majorized kernels, Lebesgue points and the Lebesgue set of an $L_{l o c}^{1}$ class); almost every point is a Lebesgue point of $f$ (Almost every point is a Lebesgue point of a locally integrable function). For bounded measurable $V$ , each integral against $ρ_{ε} (x - \cdot)$ is absolutely convergent because the kernel is bounded with compact support.

[F4]

Conservation theorem. For a declared isolated system with energy $H_{o p}$ and flow $U (t)$ , an assigned sharp observable $(A, u_{A})$ with $e^{- i s A / ℏ} U (t) = U (t) e^{- i s A / ℏ}$ for all $s, t$ has a time-independent outcome law, conserved moments for vectors in $D (A)$ , and a unitary symmetry $S$ of the dynamics satisfies $S U (t) = U (t) S$ (Conservation of a symmetry generator: commuting flows, energy compatibility and conserved outcome laws clauses (a)-(c)).

[F5]

Outcome laws. The outcome law of an assigned sharp observable is $μ_{ρ}^{A} (B) = tr (ρ E_{A} (B))$ , respectively $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2}$ , with the spectral PVMs of the model's self-adjoint operators (Assigned sharp observables and the outcome law of the framework clauses (b),(e), Spectral theorem for unbounded self-adjoint operators (PVM form), Density operators and scalar spectral probability clause (a)).

Proof

Proof technique: direct; read the symmetry criterion off the conjugation identity for the bounded-potential operator, prove that a potential invariant under every translation is almost everywhere constant by mollification and Lebesgue-point convergence, and instantiate the conservation theorem for the momentum components of an invariant model.

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

1.1F1F2givenalgebra

[Symmetry criterion] The identity $T_{a} H_{V} T_{a}^{- 1} = H_{V_{a}}$ is [F1], and $T_{a}$ is a symmetry exactly when $T_{a} H_{V} T_{a}^{- 1} = H_{V}$ with $T_{a} D (H_{V}) = D (H_{V}) = W^{2, 2}$ , which by the identity holds exactly when $H_{V_{a}} = H_{V}$ . Subtracting the common free part $H_{0}$ (self-adjoint with domain $W^{2, 2}$ by [F2]) gives $M_{V_{a}} = M_{V}$ as bounded operators, and by [F2] this is equivalent to $V_{a} = V$ almost everywhere. This proves (a).

1.2F3givenalgebra

[Invariance makes the mollifications constant] Assume $V_{a} = V$ almost everywhere for every $a \in R^{3}$ , fix $ε > 0$ and $x, a \in R^{3}$ . By [F3] the defining integral is absolutely convergent, and the substitution $z = y + a$ gives $(V * ρ_{ε}) (x - a) = \int V (y) ρ_{ε} (x - a - y) d y = \int V (z - a) ρ_{ε} (x - z) d z$ . Since $V (z - a) = V (z)$ for almost every $z$ and the kernel is integrable with compact support, this equals $\int V (z) ρ_{ε} (x - z) d z = (V * ρ_{ε}) (x)$ . Thus $V * ρ_{ε}$ is translation invariant pointwise and therefore is constant, equal to some $c_{ε}$ .

2.1step 1.2F1F2F3givenalgebra

[Lebesgue points and constancy] For each positive integer $n$ put $B_{n} : = B (0, n)$ and $f_{n} : = V 1_{B (0, n + 1)} \in L^{1} (R^{3})$ . By [F3] almost every $x \in B_{n}$ is a Lebesgue point of $f_{n}$ , so choose one such point $x_{n} \in B_{n}$ ; for $0 < ε < 1$ the support of $ρ_{ε} (x_{n} - \cdot)$ lies in $B (0, n + 1)$ , and the recovery lemma gives $c_{ε} = (V * ρ_{ε}) (x_{n}) = (f_{n} * ρ_{ε}) (x_{n}) \to f_{n} (x_{n}) = V (x_{n}) = : c_{n}$ . For every Lebesgue point $x \in B_{n}$ of $f_{n}$ , the same support argument and recovery lemma give $V (x) = \lim_{ε ↓ 0} c_{ε} = c_{n}$ ; hence $V = c_{n}$ almost everywhere on $B_{n}$ . The balls $B_{n}$ are nested and each has positive measure, so their constants agree and $V$ is almost everywhere constant on $R^{3}$ . Calling that constant $c$ , [F2] gives $H_{V} = H_{0} + c I$ , and the translations are symmetries by (a) and form a strongly continuous unitary group by [F1]. This proves (b).

3.1step 2.1F1F4F5givenalgebra

[Momentum conservation] Let $V = c$ almost everywhere, so $H_{V} = H_{0} + c I$ on $W^{2, 2}$ . For fixed $k$ , the group $W (s) : = e^{- i s P_{k} / ℏ} = T_{s e_{k}}$ of [F1] consists of unitaries $S$ with $S D (H_{V}) = D (H_{V})$ and $S H_{V} S^{*} = H_{V}$ : the identity for $P_{k}$ in [F1] gives $T_{s e_{k}} P_{k} T_{s e_{k}}^{- 1} = P_{k}$ and hence the same for $P_{k}^{2}$ and for $H_{V}$ with its domain. By [F4] applied to the unitary symmetry $S = T_{s e_{k}}$ , $W (s) U (t) = U (t) W (s)$ for all $s, t$ , where $U$ is the flow of $H_{V}$ ; hence the conservation conclusion of [F4] applies with $A : = P_{k}$ , giving the displayed time-independence of the momentum law for every preparation and, for $ψ (0) \in D (P_{k})$ , the conservation of the mean and second moment with the time-zero values of [F4] and [F5]. The generator statement is [F1]. This proves (c).

4.1step 1.1step 1.2step 2.1step 3.1∎

[Assembly and scope] Steps 1.1, 1.2, 2.1 and 3.1 prove (a)-(c); clause (d) records the bounded measurable class of potentials, the explicit exclusion of momentum conservation for non-invariant potentials, and the absence of discrete-symmetry, singular-potential, apparatus and empirical claims.

Remarks

Where translation invariance fails. If a bounded measurable potential is not almost everywhere constant, the contrapositive of clause (b) gives some nonzero translation $a$ for which $V_{a}$ and $V$ are not equal almost everywhere, and then $T_{a}$ is not a symmetry; the companion page exhibits such a potential. Covariance is not conservation.
The role of boundedness. The proof of the constancy criterion uses only boundedness and measurability of $V$ ; no regularity, decay or periodicity is assumed, and no conclusion is drawn for potentials outside this class.

Translation invariance of bounded potentials and conservation of momentum

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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