The massive spin-s particle model of the framework: state space, operators, units and frame readings

(a) Adopted model and state space. Adopt the kinematics and massive-particle model postulate P5 for a declared particle of mass parameter $m>0$, spin label $s\in \{0,\frac{1}{2},1,\frac{3}{2},\dots \}$ and internal factor assigned as in the identical-particle and spin postulate P6 (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). Put $r:=2s+1$ and take as the model's one-particle state space the complex Hilbert space $H_s:=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)=\{f:{\mathbb{R}}^3\to {\mathbb{C}}^r\text{ measurable}:{\int }_{{\mathbb{R}}^3}\mathrm{\parallel }f(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^{2}dx<\mathrm{\infty }\}/\text{\hspace{0.17em}⁣}\sim ,$ the quotient by equality almost everywhere, with inner product $⟨f,g⟩={\int }_{{\mathbb{R}}^3}{\sum }_{l=1}^r{f}_l\overline{g_l}dx$ linear in the first argument (The space $L^p(\mu )$ as the quotient by null functions). Thus an element of $H_s$ is a class of functions, not a function, and pointwise values of a state are not defined; the position density is read from the class through the observables of clause (b) and the observable and probability postulate P2 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, The state space of the framework: preparations, pure rays and density operators, Assigned sharp observables and the outcome law of the framework). Its dimension in the conventions of Dimensions, SI unit conventions and quantity types of the framework clause (d)1 is ${\mathrm{\ell }}^{-3\mathrm{/}2}$ per component. P5 presents the same space as the completed tensor product $L^2({\mathbb{R}}^3;\mathbb{C})\widehat{\otimes }{\mathbb{C}}^r$ with the spin factor carrying the representation of P6; the unitary identification of the two descriptions and the componentwise reading of componentwise operators are those of Unitary identifications of the component-valued wave-function space and Componentwise position and momentum operators on the component-valued wave-function space. The orthonormal basis $(e_m{)}_{m=-s,\dots ,s}$ of ${\mathbb{C}}^r$ is the basis fixed by P6 (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations clause (ii)).

(b) Position and momentum. The model's $j$-th position and $k$-th momentum are the componentwise operators $Q_j$ and $P_k$ of Componentwise position and momentum operators on the component-valued wave-function space clause (a): for $f=(f_l)$, restricted to the classes with $x_j{f}_l\in L^2$ respectively with $D_k{f}_l\in L^2$ for all $l$, $(Q_{j}f{)}_l(x):=x_j{f}_l(x),(P_{k}f{)}_l:=-i\mathrm{\hslash }{D}_k{f}_l,$ with $D_k$ the weak derivative and $\mathrm{\hslash }$ the action constant of Dimensions, SI unit conventions and quantity types of the framework. These are the scalar operators of Position and momentum operators on Euclidean L2 applied componentwise. The model assigns to $(Q_j,\mathrm{m})$ and to $(P_k,\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1})$ the status of assigned sharp observables in the sense of Assigned sharp observables and the outcome law of the framework clause (a), with the units of Dimensions, SI unit conventions and quantity types of the framework clause (d)4; both are densely defined and self-adjoint by Componentwise position and momentum operators on the component-valued wave-function space clause (a). On the Schwartz core they satisfy the componentwise canonical commutation relations, and they commute with the internal-factor operators of clause (c) in the sense of Canonical commutation relations on the component-valued core and commutation with the internal factor clauses (a),(c).

(c) Internal factor. For $i\in \{1,2,3\}$ let $S_i=\mathrm{\hslash }{J}_i$ be the self-adjoint endomorphisms of ${\mathbb{C}}^r$ of the spin assignment P6 (Identical-particle and spin assignments P6: permutation sectors and finite-spin representations clause (ii), Finite spin matrices), and let the model read them on $H_s$ as the matrix actions $(S_{i}f{)}_l:=\mathrm{\hslash }{\sum }_{m=1}^r(J_i{)}_{lm}{f}_m(l=1,\dots ,r),$ that is, as the amplifications $I\otimes S_i$ under the tensor identification of clause (a) (Canonical commutation relations on the component-valued core and commutation with the internal factor clause (d)). Their declared unit is the unit of action (Dimensions, SI unit conventions and quantity types of the framework clause (d)4). The factor is internal bookkeeping: 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) Momentum-space representative. With $U^{(s)}$ the unitary of The momentum representation of the componentwise kinematics for $r=2s+1$, the momentum-space representative of a preparation vector $\psi \in H_s$ is ${\psi }_p:=U^{(s)}\psi \in L^2({\mathbb{R}}^3,dp;{\mathbb{C}}^r)$; for a Schwartz class $\psi$ it is the componentwise Fourier transform $({\psi }_p{)}_l(p)=(2\pi \mathrm{\hslash }{)}^{-3\mathrm{/}2}{\int }_{{\mathbb{R}}^3}{e}^{-ip\cdot x\mathrm{/}\mathrm{\hslash }}{\psi }_l(x)dx,$ and its dimension is $(\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}{)}^{-3\mathrm{/}2}$ per component (Dimensions, SI unit conventions and quantity types of the framework clause (d)2). By The momentum representation of the componentwise kinematics clause (b), $P_k$ acts on the momentum representative by multiplication by $p_k$; this is the model's momentum reading and no measured momentum distribution is identified with $\mathrm{\mid }{\psi }_p{\mathrm{\mid }}^2$ by this definition.

(e) Frame reading. The model's inertial frames are the global coordinate charts of P5; frame changes are the Galilean transformations of P5, with parameters and units as in Dimensions, SI unit conventions and quantity types of the framework. This definition uses P5's frame class only. For translations by $a\in {\mathbb{R}}^3$ and boosts with parameter $u\in {\mathbb{R}}^3$, choose the componentwise representative convention $(T_a\psi )(x)=\psi (x-a),(B_u(t)\psi )(x)=\exp \text{\hspace{0.17em}⁣}\left(\frac{i}{\mathrm{\hslash }}(mu\cdot x-\frac{1}{2}m\mathrm{\mid }u{\mathrm{\mid }}^{2}t)\right)\psi (x-ut)\mathrm{.}$ Here $a$ is length-valued, $u$ velocity-valued and $t$ time-valued, so the phase is dimensionless. These formulas specify classes almost everywhere and are chosen model conventions. The covariance theorems of this page derive operator and outcome-law identities conditional on these choices; they do not derive the choice of representative convention itself. No Lorentz transformation law is claimed.

(f) Status. Clauses (a)-(e) record the readings of P1, P2, P5 and P6 in the terminology of this page and fix the objects used by its theorems. They assert no time evolution beyond what the autonomous-dynamics postulate P4 assigns (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators clause (iv)), no preparation or apparatus model, no detection model and no empirical result. Massless particles, relativistic matter, radiation, singular or unbounded potentials and boundary or confined realizations are separate model assumptions with their own domains and self-adjointness statements; none is obtained from this definition.

• Why the class reading matters. $H_s$ consists of almost-everywhere classes; the operators $Q_j$ and $P_k$ are defined on the classes of clause (b) and the core identities hold on the Schwartz classes. Phrases such as "the value of the wave function at $x$" are shorthand for a chosen representative and are not part of the model's state assignment.
• Units. The wave function, its momentum representative, the position and momentum spectral coordinates and the internal matrices are read in the units fixed by Dimensions, SI unit conventions and quantity types of the framework; no numerical value, and in particular no value of $m$, $s$ or $\mathrm{\hslash }$, is asserted beyond the exact defining constants of the current SI recorded there.