Primitive, adopted and derived quantities of the framework

This remark records the status of the framework's quantities under the conventions of Dimensions, SI unit conventions and quantity types of the framework and the postulates of this page. The classification is internal to the framework: a quantity called primitive here may be derived in another framework, and the same name in another framework need not denote the same object.

Primitive and adopted model inputs. The following enter the model as inputs rather than being derived within it: the arena objects and their readings (events, absolute time, the spatial metric and the frame parameters of the adopted frames); the state space assigned to each system, with its linear structure State and composition postulate P1: Hilbert spaces, preparations and composite systems; the energy operator of an autonomous system Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution; the action constant $\mathrm{\hslash }$, the particle mass $m>0$, the spin label and the potential of the one-particle model Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators; the componentwise position and momentum assignments with their maximal domains (P5 clause (iii)); and the species parameters of Identical-particle and spin assignments P6: permutation sectors and finite-spin representations (mass, charge, spin, statistics, magnetic couplings), which are empirical model inputs.

Adopted structures. Separable complex Hilbert spaces with their inner products, the completed tensor product and the space of trace-class operators are mathematical structures; the framework adopts them as the description of states, and their mathematical properties are not physical findings. Similarly the projective measurement family of the conditional-measurement postulate is an adopted apparatus model, and the permutation sector of a species is an adopted assignment.

Derived quantities and statements. Within the stated assumptions the following are derived, and each is accompanied by the hypotheses of its derivation: the outcome probability measures ${\mu }_{\rho }^A$ and their finite-resolution restrictions (from P2 and the spectral calculus); the updated preparation of the conditional-measurement rule and the discarded-outcome ensemble (from P3); the evolved vector state and density operator, with the Schroedinger equation on the operator domain (from P4); the invariance and common-core property of the Schwartz space, the canonical commutator there and the self-adjointness of the bounded-potential Hamiltonian (mathematical consequences of the operators and domains assigned in P5); the probability current $j=(\mathrm{\hslash }\mathrm{/}m)\mathrm{Im}(\bar{\psi }\mathrm{\nabla }\psi )$ and pointwise continuity equation only under the hypotheses of Smooth local continuity equation and its integrated forms clause (a): $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ and $I\subseteq \mathbb{R}$ are nonempty open sets, $m,\mathrm{\hslash }>0$, $V:\mathrm{\Omega }\to \mathbb{R}$ is continuous, and $\psi$, ${\mathrm{\partial }}_t\psi$, ${\mathrm{\partial }}_j\psi$, and ${\mathrm{\partial }}_k{\mathrm{\partial }}_j\psi$ are continuous and solve the displayed Schroedinger equation pointwise. The integrated balance additionally requires a bounded open $C^1$ region $D$ with $\bar{D}\subseteq \mathrm{\Omega }$; global norm conservation requires the domination and vanishing-flux hypotheses of clause (c). Means, variances and higher moments exist only where their corresponding integrals are finite; energies are spectral quantities of the specified model Hamiltonian, while angular momenta and other spectral quantities belong to their corresponding specified self-adjoint observables, with the units declared under P2. No derived statement is asserted outside its hypotheses.

Empirical status of measured inputs. The numerical values of $m$, of charges and of magnetic parameters are not derived by the framework; they are qualified empirical premises in the sense of P7, with uncertainty and measurement conditions. The exact defining constants of the current SI recorded in the unit conventions are conventions of the system of units, not measured findings of this framework; the measured status of ${\mu }_0$ in the same conventions is that of a qualified empirical constant wherever it is used.

No interpretation claimed. The classification above records where each object enters the model. It asserts nothing about the ontological status of states, about the completeness of the description or about the empirical truth of the framework; those questions are outside the framework's deductive content.

• A quantity's name and unit do not establish its identity with a similarly named quantity of another framework: the momentum of the one-particle model is the operator assigned in P5, not the classical momentum of a trajectory, and the identification of classical and quantum descriptions in a limit is a separate derived statement where it is claimed.
• The spin label and the internal factor are model inputs of P6; the spin operators are derived from the representation assignment, and no classical rotating-body picture is part of the framework.