Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4, the kinematics and massive-particle model postulate P5 and the internal-factor assignment of the identical-particle and spin postulate P6, all read for the spin-$s$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with mass parameter $m>0$, spin label $s\in \{0,\frac{1}{2},1,\frac{3}{2},\dots \}$ and $r=2s+1$ (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, The state space of the framework: preparations, pure rays and density operators, Assigned sharp observables and the outcome law of the framework). Let $H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$ be the model's state space with componentwise operators $Q_j,P_k$, and let ${\mathcal{S}}^{(r)}$ be its componentwise Schwartz class (Componentwise position and momentum operators on the component-valued wave-function space). Units are those of Dimensions, SI unit conventions and quantity types of the framework; the unit of angular momentum is the unit of action (J$$s).

(b) Orbital angular momentum. For $i=1,2,3$ let $L_i$ be the self-adjoint operator on $H_s$ of Coordinate-rotation generators on the Schwartz classes and their commutators, the Stone generator of the coordinate rotation group $t\mapsto U(R_i(t))$, so that $U(R_i(t))=e^{-it{L}_i\mathrm{/}\mathrm{\hslash }}$ and, on the Schwartz class, $L_i\psi ={\sum }_{j,k=1}^3{\epsilon }_{ijk}{Q}_j{P}_k\psi$ (Coordinate-rotation generators on the Schwartz classes and their commutators clause (b), Strongly continuous one-parameter unitary group). The model reads $(L_i,\mathrm{J}\mathrm{s})$ as an assigned sharp observable, called the $i$-th orbital angular momentum of the model, with outcome law ${\mu }_{\rho }^{L_i}$ in a preparation $\rho$ (Assigned sharp observables and the outcome law of the framework clauses (a),(b), The state space: density operators, pure and mixed states, and purity). The operators $L_i$ act on the spatial factor only; their reading on the angular degrees of freedom is the one supplied by the sphere operators Orbital self-adjoint generators and Sphere ONB, and the finite-blocks identity ${\sum }_i{L}_i^{2}F={\mathrm{\hslash }}^{2}l(l+1)F$ on the harmonic span is the content of a separate lemma of this page (The orbital Casimir identity: the restrictions of the orbital generators to the degree-l blocks are spin-l models).

(c) Spin angular momentum. Let $S_i=\mathrm{\hslash }{J}_i$ be the internal operators of The massive spin-s particle model of the framework: state space, operators, units and frame readings clause (c), the matrix actions $(S_{i}f{)}_l=\mathrm{\hslash }{\sum }_m(J_i{)}_{lm}{f}_m$ of the self-adjoint matrices of Finite spin matrices. The model reads $(S_i,\mathrm{J}\mathrm{s})$ as an assigned sharp observable, the $i$-th spin angular momentum; on the internal factor, $[S_i,S_j]=i\mathrm{\hslash }{\sum }_{k=1}^3{\epsilon }_{ijk}{S}_k,S^2:=S_x^2+S_y^2+S_z^2={\mathrm{\hslash }}^{2}s(s+1)I$ (Finite spin matrices clauses (a),(b),(c), Identical-particle and spin assignments P6: permutation sectors and finite-spin representations clause (ii)). The ladder operators $S_{\pm }:=S_x\pm i{S}_y$ and the finite projections on the basis vectors $e_m$ are structural objects of the internal factor, not assigned sharp observables. They satisfy $S_{+}^{\ast }=S_{-}$; for $s>0$ they are not self-adjoint, since $S_{+}{e}_{-s}=\mathrm{\hslash }\sqrt{2s}{e}_{-s+1}\mathrm{\ne }0$ whereas $S_{-}{e}_{-s}=0$, while for $s=0$ both vanish and are self-adjoint. For $s=\frac{1}{2}$ one has $S_i=\frac{\mathrm{\hslash }}{2}{\sigma }_i$ in the Pauli matrices (Finite spin matrices clause (e)). The internal factor is a finite-dimensional bookkeeping factor: it is not a spatial coordinate, not a rotating body and not a relativistic spinor, and the model assigns no position or worldline to it.

(d) Total angular momentum. For each $i$ let $J_i^{\mathrm{t}\mathrm{o}\mathrm{t}}:=L_i+S_i,D(J_i^{\mathrm{t}\mathrm{o}\mathrm{t}}):=D(L_i),$ the self-adjoint sum of The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators clause (c) (well defined because $S_i$ is bounded and strongly commutes with $L_i$; Self-adjointness and the flow of a self-adjoint operator perturbed by a bounded operator commuting with its flow clause (a), Symmetric, self-adjoint and essentially self-adjoint operators). The model reads $(J_i^{\mathrm{t}\mathrm{o}\mathrm{t}},\mathrm{J}\mathrm{s})$ as an assigned sharp observable, the $i$-th total angular momentum; its flow is the coordinate subgroup $t\mapsto U_s(A_i(t))$ of the lifted rotation action, using the physical-axes cover $\rho (W)=C^{-1}{\rho }_0(W)C$ defined in that lemma (rather than the literal quaternion-coordinate cover ${\rho }_0$), $U_s(A_i(t))=e^{-it{J}_i^{\mathrm{t}\mathrm{o}\mathrm{t}}\mathrm{/}\mathrm{\hslash }}$ (The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators clauses (b),(c)).

(e) Status and scope. Clauses (a)-(d) fix the angular momentum vocabulary of this page within P1, P2, P4, P5 and P6 and the cited readings. No new physical assumption is added; in particular no magnetic coupling, no spin-orbit interaction, no SU(2) gauge structure and no anomaly is introduced. Whether a laboratory angular momentum measurement is described by one of the assigned pairs $(L_i,\mathrm{J}\mathrm{s})$, $(S_i,\mathrm{J}\mathrm{s})$ or $(J_i^{\mathrm{t}\mathrm{o}\mathrm{t}},\mathrm{J}\mathrm{s})$ is a tested identification with uncertainty under the preparation and apparatus postulate P7, not a consequence of this definition; no measurement procedure, apparatus, resolution or empirical value is supplied here. Antiunitary symmetries, superselection rules and symmetries of preparations as opposed to dynamics are outside the definition.

• Why the sum is a definition and not a theorem. The operator identity $J_i^{\mathrm{t}\mathrm{o}\mathrm{t}}=L_i+S_i$ is self-adjoint on $D(L_i)$ because $S_i$ is bounded self-adjoint; the adjoint-domain argument in Self-adjointness and the flow of a self-adjoint operator perturbed by a bounded operator commuting with its flow Proof steps 1.1, 1.3 and 2.1 uses no commutation hypothesis for this conclusion. Strong commutation identifies the flow of the sum with the product of the two flows; that commutation and product-flow identity are proved in The SU(2) lift of the spatial rotation action on the component-valued wave-function space and its coordinate generators, and this definition only names the resulting assigned observable.
• Orbital and spin are distinct. $L_i$ acts on the spatial factor and $S_i$ on the internal factor; no term of this definition identifies the spin with a rotation of a spatial body, and the page proves no spin-orbit coupling.