The free model: translation symmetries and conserved momentum in the momentum representation

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 $s$, with state space $H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$, $r=2s+1$, componentwise operators $Q_j,P_k$ and translations $T_a$ (Componentwise position and momentum operators on the component-valued wave-function space clause (c)); 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 (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). Consider the free model, the bounded-potential model with $V\equiv 0$: $H_0:=\frac{1}{2m}{\sum }_{k=1}^3{P}_k^2,D(H_0)=W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^r),$ with flow $U(t)=\exp (-it{H}_0\mathrm{/}\mathrm{\hslash })$, momentum unitary $U^{(r)}$ (The momentum representation of the componentwise kinematics) and $f_t(p):=e^{-it\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m\mathrm{\hslash })}$. Then:

(a) for every $a\in {\mathbb{R}}^3$ the translation $T_a$ is a symmetry of $H_0$ in the sense of Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clause (b), and $T_a{e}^{-it{H}_0\mathrm{/}\mathrm{\hslash }}{T}_a^{-1}=e^{-it{H}_0\mathrm{/}\mathrm{\hslash }}(a\in {\mathbb{R}}^3,t\in \mathbb{R});$

(b) in the momentum representation the two actions are the multipliers $U^{(r)}{T}_a{U}^{(r)-1}=M_{e^{-ia\cdot p\mathrm{/}\mathrm{\hslash }}},U^{(r)}{e}^{-it{H}_0\mathrm{/}\mathrm{\hslash }}{U}^{(r)-1}=M_{f_t},$ which commute; consequently $T_{a}U(t)=U(t)T_a$ for all $a,t$;

(c) Momentum conservation. For every $k$ the momentum component $P_k$ is a conserved observable of the free model: for every preparation $\rho$, Borel $B\subseteq \mathbb{R}$ and $t$, ${\mu }_{\rho (t)}^{P_k}(B)={\mu }_{\rho (0)}^{P_k}(B),$ and for a unit vector $\psi (0)\in D(P_k)$ the mean and second moment of the law of $P_k$ are finite and independent of $t$, with time-zero values $⟨P_k\psi ,\psi ⟩$ and $\mathrm{\parallel }{P}_k\psi {\mathrm{\parallel }}^2$; the coordinate translations satisfy $T_{t{e}_k}=e^{-it{P}_k\mathrm{/}\mathrm{\hslash }}$, so the momentum components are the generators of the translational symmetry groups;

(d) the modulus of the momentum representative is invariant in time: $\mid \left(U^{(r)}{e}^{-it{H}_0\mathrm{/}\mathrm{\hslash }}\psi \right)(p)\mid =\mid \left(U^{(r)}\psi \right)(p)\mid \text{for almost every }p\text{ and every }\psi \in H_s;$ consequently the squared modulus of the momentum representative is the same almost-everywhere function of $p$ at every time;

(e) Scope. The statement concerns the free model $V\equiv 0$ only. For a nonconstant bounded potential the translations need not be symmetries and momentum need not be conserved; that failure is exhibited on this page. No trajectory in space, no classical limit, no measurement model and no empirical result is asserted.

• Momentum conservation is translation invariance. The conservation of $P_k$ in (c) is the special case $V\equiv 0$ of the translation-invariance theorem; the point of the example is the explicit multiplier form: both the translations and the free flow act in the momentum representation by unimodular multipliers, and multipliers of the same variable commute.
• No trajectory. The invariance of $\mathrm{\mid }{U}^{(r)}\psi \mathrm{\mid }$ in the momentum variable is a statement about the momentum representative's class, not a statement that the particle moves along a classical straight line; no position-space trajectory is asserted.