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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.

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

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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