Scope and empirical status of the symmetry and conservation items

The mathematical lemmas of this page follow from their stated mathematical hypotheses; its physics theorems follow conditionally from their stated model postulates and mathematical hypotheses. Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables fixes the vocabulary: a symmetry of the declared dynamics is a unitary preserving the energy domain and commuting with the energy operator, a unitary symmetry group is a group of such symmetries, a conserved assigned sharp observable $(A,u_A)$ is one whose outcome law is time-independent in every preparation, and the translations, spatial rotations, parity and gauge multipliers are recorded as the model's candidate actions, not as symmetries. Unitary conjugation, generator transport and strong commutation of self-adjoint observables proves the equivalence between flow commutation, strong spectral commutation and domain invariance together with invariance of the observable under conjugation: for self-adjoint $K,A$ on a nonzero complex Hilbert space and $\mathrm{\hslash }>0$, clause (b)(iii) requires both $U_K(t)D(A)=D(A)$ and $U_K(t)A{U}_K(t{)}^{\ast }=A$ for every $t\in \mathbb{R}$, where $U_K(t)=e^{-itK\mathrm{/}\mathrm{\hslash }}$. Domain invariance alone is insufficient; The spatial rotation action on the component-valued wave-function space, The spatial parity action on the component-valued wave-function space, Coordinate-rotation generators on the Schwartz classes and their commutators and The free flow as a unimodular multiplier in the momentum representation provide the concrete actions and their operator identities on the component-valued model. Conservation of a symmetry generator: commuting flows, energy compatibility and conserved outcome laws proves the equivalence of its three commutation conditions, deduces conservation of the outcome law and of domain-qualified moments, and covers self-adjoint unitary symmetries; Translation invariance of bounded potentials and conservation of momentum decides exactly which bounded potentials have the translations as symmetries and derives momentum conservation for the translation-invariant case; Rotational and parity invariance of bounded-potential models: symmetries and conservation of angular momentum and parity laws does the same for rotations (with conservation of the coordinate-rotation generators) and for parity; and Gauge covariance of the prescribed-coupling model: unitary equivalence, the symmetry criterion and the transported propagator transports the prescribed-coupling operator under gauge transformations of Prescribed external electromagnetic coupling P5a: the minimal-coupling model and its gauge transformations, exhibits the exact criterion under which a gauge multiplier is a symmetry of the fixed dynamics and identifies the global-phase case. The mathematical lemmas are conditional on their displayed mathematical hypotheses, including the stated choice assumptions; they do not depend on physical postulates. The physics theorems and examples are conditional on the postulates and model hypotheses stated in each item: the general conservation result uses P1, P2 and P4; the bounded- potential translation, rotation and parity results use P1, P2, P4, P5 and P6; and gauge covariance uses P1, P2, P4 and the prescribed-coupling assumption P5a. P5a is adopted and is not derived by the gauge theorem. None of these conditional results establishes that its assumptions describe a laboratory system.

The conservation laws proved here are model consequences: they hold for the declared massive model with the displayed potential class and follow from the invariance hypotheses as identities about that model. They are not reports of measurements and do not establish that any laboratory preparation is exactly described by the model. In particular, the mass parameter $m>0$, the charge parameter $q$, the spin label $s$ and the potentials $({\mathbf{A}}_{\mathrm{e}\mathrm{x}\mathrm{t}},{\mathrm{\Phi }}_{\mathrm{e}\mathrm{x}\mathrm{t}})$ are declared model data in the units of Dimensions, SI unit conventions and quantity types of the framework; the statement that a concrete species, field configuration or apparatus realizes them is a tested identification with uncertainty, external to these items. A selection rule or a conservation law deduced here becomes an empirical prediction only after a preparation map, a detection model, a sampling model and an uncertainty budget are supplied, none of which this page provides. The agreement of such an identified model with data is evidence about the combined model and apparatus assumptions; it is not a proof of the postulates.

The page asserts no symmetry of any concrete potential outside the displayed almost-everywhere invariance hypotheses, no classification of centrally symmetric or parity-even systems beyond those hypotheses, no definite value of a conserved observable, and no converse from the constancy of outcome laws to the commutation conditions for unbounded observables. The rotation and parity actions act on the spatial factor only: no SU(2) lift, no rotation of the internal factor and no spin angular momentum appear here; antiunitary and time-reversal symmetries, superselection sectors, gauge freedom beyond the prescribed bounded-coefficient class, singular or unbounded potentials, confined and boundary realizations, arbitrary driven external fields, relativistic matter and full field theory are outside the scope of these items. Bounded periodic potentials remain within the bounded-potential class; no periodic-band classification is claimed. The stated time-dependent gauge transports are covered by Gauge covariance of the prescribed-coupling model: unitary equivalence, the symmetry criterion and the transported propagator clause (c), with its prescribed coefficient class and gauge regularity; they supply no general driven-field model.

The gauge items of this page consume the magnetic-operator development of an earlier batch of this run. The batch-53 notes preserve historical reports of role-metadata and source-locator defects in earlier versions of those suppliers; those notes are not a description of their current text. The current files inspected for this review declare the mathematics dependencies with role mathematical-premise and have these source locators:

• def-nrqm-magnetic-and-detection-magnetic-schrodinger-operator cites Teschl, Sections 6.1 and 7.2, and Binney and Skinner, Sections 3.3 and 3.3.1.
• thm-nrqm-magnetic-and-detection-gauge-h2-domain-preservation cites Teschl, Section 7.2, equations (7.22)-(7.24) and Theorem 7.8, and Binney and Skinner, Section 3.3.1, equations (3.40)-(3.45).
• thm-nrqm-magnetic-and-detection-gauge-transported-propagator cites Teschl, Section 5.1, Theorem 5.1, and Binney and Skinner, Section 3.3.1, equations (3.40)-(3.45).

The current Teschl text has the cited free-operator material in Section 7.2 (printed pages 167-168), the Kato-Rellich theorem in Section 6.1 (printed page 135), and Stone's theorem in Section 5.1 (printed pages 123-124). The current Binney text has the static gauge phase and covariant-momentum intertwining in Section 3.3.1 (printed pages 46-47). I checked these current locators against the hosted full texts. They provide background for the magnetic results; the bounded-coefficient operator and time-dependent propagator arguments are supplied locally in the magnetic items. The earlier false locators recorded in the batch notes are absent from the current files.