The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable

Slot convention. For $\mathrm{\ell }\ge 1$, $S_{\mathrm{\ell }}$ acts on $\{0,\dots ,\mathrm{\ell }-1\}$ as in Permutation sector projections on the n-fold tensor power. On one-based tuple positions use $\widehat{\pi }(k):=1+\pi (k-1)$ for $1\le k\le \mathrm{\ell }$. Conjugation by $k\mapsto k-1$ gives $\widehat{\pi \sigma }=\widehat{\pi }\circ \widehat{\sigma }$ and $\widehat{{\pi }^{-1}}={\widehat{\pi }}^{-1}$; thus the supplier's tensor action becomes $U_{\pi }(v_1\otimes \cdots \otimes v_{\mathrm{\ell }})=v_{{\widehat{\pi }}^{-1}(1)}\otimes \cdots \otimes v_{{\widehat{\pi }}^{-1}(\mathrm{\ell })}$. The parameters of $U_{\pi }$ and $\mathrm{sgn}(\pi )$ remain canonical zero-based permutations.

(a) Adopted readings, units and choice hypotheses. Adopt the state and composition postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1), the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2), the autonomous-dynamics postulate Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution (P4), the non-relativistic kinematics postulate Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators (P5) and the identical-particle and spin postulate Identical-particle and spin assignments P6: permutation sectors and finite-spin representations (P6), in the state reading of The state space of the framework: preparations, pure rays and density operators, the observable reading of Assigned sharp observables and the outcome law of the framework, the evolution reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution and the sector vocabulary of Declared species sectors and their preparations: the framework's reading of P6. P5 and the massive spin-$s$ reading below apply to the full spatial space $H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^{2s+1})$. In the finite-mode branch, clause (b)(ii) explicitly replaces P5's full-space state and Hamiltonian assignments (iii),(iv) by retained-space assignments: P1, P2 and P4 are read on that retained space, P6(i) gives its permutation sectors, and P6(ii) still specifies spin on $H_s$, with no automatic full spin representation on the retained space. This is an additional model idealization, not a consequence of P5. Let the species parameters mass $m>0$ (kg), spin label $s\in \{0,\frac{1}{2},1,\dots \}$ and statistics choice $\sigma \in \{+,-\}$ be given in the vocabulary of that definition and let $H$ be the one-particle space of the chosen branch below. Use the sector and preparation constructions of that definition (c)-(e) on $H$; fix an integer $N\ge 1$ and let ${\mathfrak{H}}_N^{\sigma }=\mathrm{ran}{P}_{\sigma }^{(N)}\subseteq H^{\otimes N}$ be the declared $N$-particle sector. All quantities below are read in the SI conventions and units of Dimensions, SI unit conventions and quantity types of the framework (energies in joules). Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) and the Axiom of Choice (The Axiom of Choice). AC is inherited from the adopted general spectral observable and evolution interfaces of P2 and P4 and the operator suppliers of P5; it is also required by the Coulomb realization in clause (d)(ii). The bounded spatial operator construction in clause (d)(i) itself uses only Countable Choice. Adopt Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions (P7) for the qualified apparatus and parameter identifications below.

(b) Declared one-particle data. The model declares a one-particle energy operator in one of the two following realizations.

(i) Spatial realization. The one-particle space is $H=L^2({\mathbb{R}}^3;{\mathbb{C}}^{2s+1})$ and, with $\mathrm{\hslash }>0$ the reduced Planck constant, the model declares a kinetic operator $K_1:=-\frac{{\mathrm{\hslash }}^2}{2m}\mathrm{\Delta }$ on $L^2({\mathbb{R}}^3;{\mathbb{C}}^{2s+1})$ and a real, measurable, bounded external potential $V_{\mathrm{e}\mathrm{x}\mathrm{t}}:{\mathbb{R}}^3\to \mathbb{R}$ of essential supremum $\mathrm{\parallel }{V}_{\mathrm{e}\mathrm{x}\mathrm{t}}{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$; the declared one-particle operator is $h:=K_1+V_{\mathrm{e}\mathrm{x}\mathrm{t}}(Q)$, self-adjoint by CC realization of the bounded spatial operator on a finite-component H² domain with $N=1$, $M=2s+1$ and $a_1={\mathrm{\hslash }}^2\mathrm{/}(2m)$, with domain $W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^{2s+1})$. The particular singular external potential $V_{\mathrm{e}\mathrm{x}\mathrm{t}}^b-g\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ is included in the Coulomb branch of clause (c)(ii), whose one-particle $h$ is specified there using the $N=1$ instance of clause (d)(ii). Any other more singular declared $V_{\mathrm{e}\mathrm{x}\mathrm{t}}$ requires a separately supplied self-adjoint realization before its operator is read as an observable.

(ii) Finite-mode realization. A finite-dimensional complex Hilbert space $K$ of dimension $d\ge 1$ with a declared orthonormal basis $(f_1,\dots ,f_d)$ (Hilbert space) is declared together with a self-adjoint one-particle operator $h\in \mathcal{B}(K)$ (A bounded linear operator between normed spaces, Symmetric, self-adjoint and essentially self-adjoint operators). Also declare an orthonormal family $(g_1,\dots ,g_d)$ in the full spatial spin space $H_s$ of The massive spin-s particle model of the framework: state space, operators, units and frame readings clause (a). Define the retained-mode isometry, its adjoint, and its orthogonal projection by $J:K\to H_s,J\left({\sum }_{j=1}^d{c}_j{f}_j\right)={\sum }_{j=1}^d{c}_j{g}_j,J^{\ast }\psi ={\sum }_{j=1}^d⟨\psi ,g_j⟩f_j,R:=J{J}^{\ast }\mathrm{.}$ Thus $J^{\ast }J=I_K$, $R^2=R=R^{\ast }$ and $\mathrm{ran}R=\mathrm{span}\{g_1,\dots ,g_d\}$; inner products are linear in the first variable. Here $H=K$ is the effective one-particle space, and $\psi \in K$ represents the retained full-space vector $J\psi$. The tensor map $J_N:=J^{\otimes N}$ of Bounded tensor maps: extension, norm estimate, adjoint and commutation is an isometry and satisfies $J_N{U}_{\pi }^K=U_{\pi }^{H_s}{J}_N$: the two sides agree on elementary tensors by the slot convention and hence on the completion. The finite projection sums consequently give $J_N{P}_{\sigma }^{(N),K}=P_{\sigma }^{(N),H_s}{J}_N$. The retained sector is therefore isometrically embedded in the full species sector, preserving its statistics choice.

The internal spin matrices and rotations of P6(ii) act on $H_s$. Their restrictions are assigned on $K$ only when they preserve $\mathrm{ran}J$; for example a full internal rotation action on $K$ requires $(I\otimes D_s(A))\mathrm{ran}J=\mathrm{ran}J$ for every $A\in SU(2)$, with $D_s$ exactly as in P6(ii), and is then read as $J^{\ast }(I\otimes D_s(A))J$. No such invariance is imposed here, so $d$ need not be a multiple of $2s+1$: one retained spinor mode for $s=1\mathrm{/}2$ is permitted without identifying $K$ with the full two-dimensional spin factor. P5's full-space position, momentum and commutation relations are not asserted on $K$. The finite operator $h$ and pair operator $W$ of clause (c)(i) are independently adopted effective energy data; no compression of a spatial Hamiltonian, equality with its evolution, or approximation error bound is asserted. Identifying the retained modes and these energy data with a laboratory preparation is a tested identification with uncertainty under P7, not part of the mathematical embedding.

(c) Declared interaction. The model declares a pair interaction in the matching realization.

(i) Bounded interaction. In the spatial realization let $W:{\mathbb{R}}^3\to \mathbb{R}$ be real, measurable, bounded with $W(-z)=W(z)$ for almost every $z$ and $\mathrm{\parallel }W{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$. In the finite-mode realization let $W\in \mathcal{B}(K\widehat{\otimes }K)$ be self-adjoint with $U_{(01)}W{U}_{(01)}=W$, where $U_{(01)}$ is the exchange unitary of Permutation sector projections on the n-fold tensor power clause (a) in the tensor square $\widehat{K\otimes K}$ of Hilbert tensor construction: inner product, completion and product bases. The interaction strength constants entering $W$ are empirical model parameters with their uncertainties in the sense of P6 clause (iii).

(ii) Coulomb interaction. In the spatial realization the model may declare $W(z):=\kappa \mathrm{/}\mathrm{\mid }z\mathrm{\mid }$ with a real coupling $\kappa \in \mathbb{R}$ in $\mathrm{J}\mathrm{m}$ (repulsive for $\kappa >0$). Retain a real bounded measurable external term $V_{\mathrm{e}\mathrm{x}\mathrm{t}}^b$ as in clause (b)(i), and permit the additional singular term $-g\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$, where $g\in \mathbb{R}$ is also in $\mathrm{J}\mathrm{m}$, so each distance denominator in metres gives an energy in joules; the total external potential is $V_{\mathrm{e}\mathrm{x}\mathrm{t}}^b(x)-g\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$. In this branch the declared one-particle operator is $h=K_1+M_{V_{\mathrm{e}\mathrm{x}\mathrm{t}}^b-g\mathrm{/}\mathrm{\mid }x\mathrm{\mid }}$ on $W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^{2s+1})$, including the singular external term. Its self-adjointness and lower bound are the $N=1$ finite-component instance of clause (d)(ii), with no pair term. Thus the bounded one-particle reading in clause (b)(i) is replaced by this supplied Coulomb reading, rather than leaving the singular external term solely in $H_{\mathrm{l}\mathrm{a}\mathrm{b}}$. The pair interaction still consists only of $\kappa \mathrm{/}\mathrm{\mid }{x}_i-x_j\mathrm{\mid }$. The declared scalar potential on ${\mathbb{R}}^{3N}$ is $V(X)=V_b(X)+V_C(X),V_b(X)={\sum }_{i=1}^N{V}_{\mathrm{e}\mathrm{x}\mathrm{t}}^b(x_i),V_C(X)=-g{\sum }_{i=1}^N\frac{1}{\mathrm{\mid }{x}_i\mathrm{\mid }}+\kappa {\sum }_{1\le i<j\le N}\frac{1}{\mathrm{\mid }{x}_i-x_j\mathrm{\mid }}\mathrm{.}$ The cases $g=0$, $\kappa =0$, $V_{\mathrm{e}\mathrm{x}\mathrm{t}}^b=0$ and either sign of either coupling are allowed. Singular values on their null collision sets are irrelevant to the multiplier classes. The singular part is supplied by Fixed-N Coulomb self-adjointness; the bounded external addition is included by the exact-domain argument in clause (d)(ii), rather than attributed to a source statement which contains only the singular sum.

(d) The labelled energy operator and its sector restriction. Fix the particle masses $m_1=\cdots =m_N=m$ of the declared species.

(i) Spatial, bounded case. Put $r_s:=2s+1$, ${\mathcal{S}}_N:=({\mathbb{C}}^{r_s}{)}^{\otimes N}$ and $M:=r_s^N$. The finite product spin basis identifies ${\mathcal{S}}_N$ with ${\mathbb{C}}^M$. Use the full finite-component product unitary ${\mathrm{\Theta }}_N:H^{\otimes N}⟶L^2({\mathbb{R}}^{3N};{\mathbb{C}}^M),[{\mathrm{\Theta }}_N(f_1\otimes \cdots \otimes f_N){]}_a(X)={\prod }_{j=1}^N[f_j(x_j){]}_{a_j},$ where $a\in \{1,\dots ,r_s{\}}^N$. Its construction and combined spatial/spin permutation action are proved in Finite spin components of an N-fold spatial tensor product, from the finite-component extension of Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}). For $V(X)={\sum }_i{V}_{\mathrm{e}\mathrm{x}\mathrm{t}}(x_i)+{\sum }_{i<j}W(x_i-x_j)$ define $H_{\mathrm{l}\mathrm{a}\mathrm{b}}=-{\mathrm{\hslash }}^2{\mathrm{\Delta }}_X\mathrm{/}(2m)+M_V$ componentwise. The CC-only CC realization of the bounded spatial operator on a finite-component H² domain, with $a_i={\mathrm{\hslash }}^2\mathrm{/}(2m)$ and component count $M$, supplies self-adjointness on $D(H_{\mathrm{l}\mathrm{a}\mathrm{b}})=W^{2,2}({\mathbb{R}}^{3N};{\mathbb{C}}^M)=W^{2,2}({\mathbb{R}}^{3N};\mathbb{C})\otimes {\mathcal{S}}_N,$ and the lower bound $-\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$. Equal masses, identical external potentials and the even pair potential make the scalar domain and operator invariant under spatial block permutations. The finite-component lemma shows that the combined permutations ${\mathcal{U}}_{\pi }={\mathrm{\Theta }}_N{U}_{\pi }{\mathrm{\Theta }}_N^{-1}$ also permute the spin slots and commute with $H_{\mathrm{l}\mathrm{a}\mathrm{b}}$ on its domain. Thus ${\mathcal{P}}_{\sigma }:={\mathrm{\Theta }}_N{P}_{\sigma }^{(N)}{\mathrm{\Theta }}_N^{-1}$ reduces the operator by Permutation-restricted self-adjointness (a)-(d), and the transported sector energy is $H_N^{\sigma ,\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}}:=H_{\mathrm{l}\mathrm{a}\mathrm{b}}{\mid }_{D(H_{\mathrm{l}\mathrm{a}\mathrm{b}})\cap \mathrm{ran}{\mathcal{P}}_{\sigma }}\mathrm{.}$ It is self-adjoint on $\mathrm{ran}{\mathcal{P}}_{\sigma }={\mathrm{\Theta }}_N{\mathfrak{H}}_N^{\sigma }$, with the zero operator on the zero-sector branch. The model's tensor reading is $H_N^{\sigma }:={\mathrm{\Theta }}_N^{-1}{H}_N^{\sigma ,\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}}{\mathrm{\Theta }}_N$ on ${\mathfrak{H}}_N^{\sigma }$. Every constant spin action commutes with the spin-independent labelled operator; a common spin unitary $A^{\otimes N}$ preserves each identical-particle sector by the finite-component lemma. For $s=0$, $M=1$ and this is the scalar product reading; for $N=1$, $M=2s+1$ and it is the one-particle component reading.

(ii) Spatial Coulomb case. Put $r=2s+1$, ${\mathcal{S}}_N=({\mathbb{C}}^r{)}^{\otimes N}$ and $M=r^N$. Use the full spin-product unitary ${\mathrm{\Theta }}_N:(L^2({\mathbb{R}}^3;{\mathbb{C}}^r){)}^{\otimes N}\to L^2({\mathbb{R}}^{3N};{\mathbb{C}}^M)$ of Finite spin components of an N-fold spatial tensor product (a),(b), including its combined spatial/internal permutation action. To apply the literal Fourier normalization of Fixed-N Coulomb self-adjointness, choose in that mathematical theorem the masses ${\tilde{m}}_i=m\mathrm{/}(4{\pi }^2)$, its parameter $\mathrm{\hslash }$ unchanged, one fixed center at zero, coefficients $z_{i1}=-g$ and $z_{ij}=\kappa$. The mathematical masses here are substitution parameters used to realize the already declared kinetic operator; the physical species mass remains $m$. The source kinetic symbol becomes $4{\pi }^2{\mathrm{\hslash }}^2\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{/}(2m)$, so its scalar operator is exactly $T_C=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_X+V_C(X),D(T_C)=H^2({\mathbb{R}}^{3N};\mathbb{C})\mathrm{.}$ By the source's clauses (b),(c), $T_C$ and its finite-component extension are self-adjoint on these exact domains and bounded below for all stated real $g,\kappa$. The real bounded multiplier $M_{V_b}$ has norm at most $N\mathrm{\parallel }{V}_{\mathrm{e}\mathrm{x}\mathrm{t}}^b{\mathrm{\parallel }}_{\mathrm{\infty }}$, hence relative bound zero with respect to $T_C$ and to its finite-component extension. The canonical Kato-Rellich theorem makes $H_{\mathrm{l}\mathrm{a}\mathrm{b}}=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\Delta }}_X+V_C+V_b,D(H_{\mathrm{l}\mathrm{a}\mathrm{b}})=H^2({\mathbb{R}}^{3N};{\mathbb{C}}^M),$ self-adjoint on exactly this domain and bounded below; if the singular operator has lower bound $c_C$, the sum has lower bound $c_C-N\mathrm{\parallel }{V}_{\mathrm{e}\mathrm{x}\mathrm{t}}^b{\mathrm{\parallel }}_{\mathrm{\infty }}$. Equal masses, identical external slots and pair-independent signed coefficient $\kappa$ make the scalar operator invariant under spatial permutations. These permutations preserve $H^2$ by permuting the weak-derivative coordinates; the constant internal tensor permutations preserve the finite-component domain too. Consequently the combined projections ${\mathcal{P}}_{\sigma }={\mathrm{\Theta }}_N{P}_{\sigma }^{(N)}{\mathrm{\Theta }}_N^{-1}$ reduce $H_{\mathrm{l}\mathrm{a}\mathrm{b}}$, and Finite spin components of an N-fold spatial tensor product (c) gives self-adjointness of its restriction on the exact domain $H^2\cap \mathrm{ran}{\mathcal{P}}_{\sigma }$. The declared sector energy is this restriction transported back by ${\mathrm{\Theta }}_N^{-1}$ to ${\mathfrak{H}}_N^{\sigma }$, with the unique zero-space operator when that sector is zero. This Coulomb construction uses AC in addition to the inherited framework choice hypotheses of clause (a).

(iii) Finite-mode case. On $K^{\otimes N}$ put $H_{\mathrm{l}\mathrm{a}\mathrm{b}}:={\sum }_{i=1}^N{h}^{(i)}+{\sum }_{1\le i<j\le N}{W}^{(ij)},$ where $h^{(i)}$ is the slot operator of Bounded tensor maps: extension, norm estimate, adjoint and commutation (the identity on all factors but the $i$-th, $h$ on the $i$-th) and $W^{(ij)}$ is the transport of $W\in \mathcal{B}(K\widehat{\otimes }K)$ to the pair of slots $(i,j)$ by any canonical permutation $\pi$ whose transported map $\widehat{\pi }$ moves that pair to the first two slots; the symmetry $U_{(01)}W{U}_{(01)}=W$ makes the order of the pair immaterial and $W\otimes I$ commutes with permutations fixing the first two slots, so the transported operator is independent of the permutation chosen. Then $H_{\mathrm{l}\mathrm{a}\mathrm{b}}$ is bounded and self-adjoint, it commutes with every permutation unitary, and the model's sector energy operator is $H_N^{\sigma }:=H_{\mathrm{l}\mathrm{a}\mathrm{b}}{\mathrm{\mid }}_{{\mathfrak{H}}_N^{\sigma }}=P_{\sigma }^{(N)}{H}_{\mathrm{l}\mathrm{a}\mathrm{b}}{P}_{\sigma }^{(N)}{\mathrm{\mid }}_{{\mathfrak{H}}_N^{\sigma }}$.

(e) The model's energy reading and evolution. In every realization with ${\mathfrak{H}}_N^{\sigma }\mathrm{\ne }\{0\}$, the pair $(H_N^{\sigma },\mathrm{J})$ is declared to be the assigned energy observable of the $N$-particle sector in the sense of Assigned sharp observables and the outcome law of the framework clause (a): its outcome law in a declared sector preparation $\rho$ is the law ${\mu }_{\rho }^{H_N^{\sigma }}(B)=\mathrm{tr}(\rho E_{H_N^{\sigma }}(B))$ of that definition clause (b), and its spectral data are those of the self-adjoint operator constructed in clause (d). The model evolution is the unitary flow $U(t)=\exp (-it{H}_N^{\sigma }\mathrm{/}\mathrm{\hslash })$ of Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (b) with this generator. If ${\mathfrak{H}}_N^{\sigma }=\{0\}$, clause (d) retains the unique zero-space operator only as mathematical bookkeeping: every operator on this space has trace zero, so there is no trace-one preparation, no assigned physical energy observable and no preparation evolution under P1-P4. The identification of the operator with the energy of a concrete sample, and the assignment of numerical masses, couplings and potentials to a laboratory species, are tested identifications with uncertainty under P7; they are not consequences of P1-P6.

(f) The non-interacting reading. The model is called non-interacting when $W=0$ (equivalently $\kappa =0$ only in the Coulomb branch of clause (c)(ii)), so that $H_{\mathrm{l}\mathrm{a}\mathrm{b}}={\sum }_{i=1}^N{h}^{(i)}$; in the spatial realization this is the case of the external potential summed over slots of the stated bounded or Coulomb realization.

(g) Scope. The declared model is a kinematic and dynamical input of this pair: on a nonzero sector it fixes a state space, a sector, an energy operator and an evolution; a zero sector carries only the mathematical convention of clause (e). It asserts no stability of matter, no Hunziker-van Winter-Zhislin threshold statement, no statement about excited threshold structure, no thermodynamic limit, no equilibration or thermalization, no measured energy and no identification of its parameters with a particular laboratory sample beyond the tested identifications of P7. The finite-mode realization is the retained-space idealization of clause (b)(ii), with independently assigned finite energy data and no claimed accuracy relative to spatial evolution; the spatial realization is an idealization of an $N$-particle non-relativistic sample with the declared external field and pair interaction.

• Why the sector restriction is legitimate. In the spatial realizations the permutation unitaries preserve the domain $W^{2,2}$ and commute with $H_{\mathrm{l}\mathrm{a}\mathrm{b}}$, which is exactly the hypothesis of the permutation-restricted self-adjointness theorem; in the finite-mode realization the bounded operator $H_{\mathrm{l}\mathrm{a}\mathrm{b}}$ commutes with the $U_{\pi }$ by construction, and a bounded operator commuting with a projection restricts to a self-adjoint operator on its range.
• Units. $m$ is in kg, $\mathrm{\hslash }$ in J s, $V_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and $\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$ in J, $W$ in J and $g,\kappa$ in J m; in the finite-mode realization $h$ and $W$ are declared directly in joules. No magnetic coupling, spin-orbit term or velocity-dependent interaction is part of this model.