Rotational and parity invariance of bounded-potential models: symmetries and conservation of angular momentum and parity laws

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, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations), 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 (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). Let $H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$, $r=2s+1$, be the model's state space with componentwise operators $Q_j,P_k$, let $U(R)$ be the spatial rotation action and $\mathrm{\Pi }$ the spatial parity action (The spatial rotation action on the component-valued wave-function space, The spatial parity action on the component-valued wave-function space), let $U_i(t)=U(R_i(t))$ be the coordinate rotation subgroups with Stone generators $L_i$ on the Schwartz classes (Coordinate-rotation generators on the Schwartz classes and their commutators), let $V:{\mathbb{R}}^3\to \mathbb{R}$ be real, bounded and measurable, and let $H_V:=\frac{1}{2m}{\sum }_{k=1}^3{P}_k^2+M_V,D(H_V)=W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^r),V_R(x):=V(R^{-1}x),V_P(x):=V(-x),$ be the model's bounded-potential energy operator with flow $U(t)=\exp (-it{H}_V\mathrm{/}\mathrm{\hslash })$ in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (b) (Real multiplication operators and free plus bounded-potential Schrödinger operators clause (c) applied componentwise, Componentwise position and momentum operators on the component-valued wave-function space clause (a)). Then:

(a) for every $R\in SO(3)$ one has $U(R)H_{V}U(R{)}^{-1}=H_{V_R}$, and $U(R)$ 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 $V_R=V$ almost everywhere;

(b) if $V_R=V$ almost everywhere for every $R\in SO(3)$, then every $U(R)$ is a symmetry, the map $R\mapsto U(R)$ is a strongly continuous unitary symmetry group of the declared dynamics, and each generator $L_i$ is a conserved observable of the model with energy $H_V$ in the unit $\mathrm{J}\mathrm{s}$ of the model's action (Dimensions, SI unit conventions and quantity types of the framework): for every preparation $\rho$, every Borel $B\subseteq \mathbb{R}$ and every $t$, ${\mu }_{\rho (t)}^{L_i}(B)={\mu }_{\rho (0)}^{L_i}(B),$ and for a unit vector $\psi (0)\in D(L_i)$ the mean and second moment of the law of $L_i$ are finite and independent of $t$, with time-zero values $⟨L_i\psi ,\psi ⟩$ and $\mathrm{\parallel }{L}_i\psi {\mathrm{\parallel }}^2$;

(c) if $V(-x)=V(x)$ for almost every $x$, then $\mathrm{\Pi }{H}_V\mathrm{\Pi }=H_V$ on $W^{2,2}$, so $\mathrm{\Pi }$ is a symmetry of $H_V$, and $\mathrm{\Pi }$ is a symmetry if and only if $V_P=V$ almost everywhere; moreover, reading the self-adjoint unitary $\mathrm{\Pi }$ as an assigned sharp observable with its spectral coordinate in the dimensionless unit of its eigenvalues $\pm 1$, the outcome law of $\mathrm{\Pi }$ is time-independent in every preparation: ${\mu }_{\rho (t)}^{\mathrm{\Pi }}(B)={\mu }_{\rho (0)}^{\mathrm{\Pi }}(B)(B\subseteq \mathbb{R}\text{ Borel},t\in \mathbb{R});$

(d) Scope. The statement is conditional on P1, P2, P4, P5 and P6 and the cited mathematics, and it concerns the componentwise massive model with real bounded measurable potentials. Rotations and parity act on the spatial factor only: no SU(2) lift, no rotation of the internal factor and no spin angular momentum are introduced. The rotational and parity invariance hypotheses are exactly the displayed almost-everywhere identities; no concrete potential, apparatus, measured spectrum or empirical result is asserted, and no characterization of the invariant potentials beyond these identities is claimed.

• Invariance is the hypothesis, not a universal property. No meaning of “generic” is used here. The companion counterexample $V=1_{B(0,1)}$ has no nonzero translation symmetry, but is invariant under every rotation and under parity because the Euclidean unit ball is rotation invariant and centrally symmetric. It therefore illustrates failure of translation symmetry only; no corresponding claim about a generic potential is made.
• Discrete parity has no generator. $\mathrm{\Pi }$ is an involution, so its conservation is read directly from the commutation with the flow through the self-adjoint unitary clause of the conservation theorem; no one-parameter group and no Stone generator is attached to it.