Declared species sectors and their preparations: the framework's reading of P6

(a) Adopted readings. Adopt the state and composition postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1) and the identical-particle and spin postulate Identical-particle and spin assignments P6: permutation sectors and finite-spin representations (P6), read with the state space vocabulary of The state space of the framework: preparations, pure rays and density operators and the quantities and units of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the choice hypothesis of the cited tensor construction.

(b) Declared species and sector. A declared species sector of the model consists of: a nonzero separable complex Hilbert space $H$, the one-particle space of the species, as assigned in P1; a mass parameter $m>0$ in kg; a spin label $s\in \{0,\frac{1}{2},1,\frac{3}{2},\dots \}$ and the internal factor $V_s={\mathbb{C}}^{2s+1}$ of P6 clause (ii), with the orbital factor $H_{\mathrm{s}\mathrm{p}}$ the factor carrying no internal degree of freedom when the one-particle space is given as a completed product $H_{\mathrm{s}\mathrm{p}}\widehat{\otimes }{V}_s$; for the massive spin-$s$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings the one-particle space is $L^2({\mathbb{R}}^3;{\mathbb{C}}^{2s+1})$, whose spin components are the internal factor of P6 clause (ii) in the reading of that definition; a statistics choice $\sigma \in \{+,-\}$, where $+$ declares the species bosonic and $-$ declares it fermionic; and, for a charged species, a charge parameter $q\in \mathbb{R}$ in coulombs. The parameters and the statistics choice are empirical model assignments with their uncertainties, in the sense of P6 clause (iii); no value and no statistics choice is derived by this definition.

(c) The $n$-particle sector. Fix $n\ge 1$. The labelled $n$-particle space of the species is the $n$-fold completed tensor power $H^{\otimes n}$ (Hilbert tensor construction: inner product, completion and product bases), whose tensor factors are mathematical labels and not observable particle identities, equipped with the permutation unitaries $(U_{\pi }{)}_{\pi \in S_n}$ and the sector projections $P_{\sigma }^{(n)}$ (Permutation sector projections on the n-fold tensor power), where $P_{+}^{(n)}$ is the symmetrizing projection and $P_{-}^{(n)}$ the alternating projection. The $n$-particle sector space of the declared species is ${\mathfrak{H}}_n^{\sigma }:=\mathrm{ran}{P}_{\sigma }^{(n)}\subseteq H^{\otimes n},$ a closed linear subspace carrying the inner product of $H^{\otimes n}$; on it every permutation acts by the scalar $U_{\pi }{\mathrm{\mid }}_{{\mathfrak{H}}_n^{\sigma }}={\mathrm{sgn}}_{\sigma }(\pi ){\mathrm{i}\mathrm{d}}_{{\mathfrak{H}}_n^{\sigma }},{\mathrm{sgn}}_{+}(\pi )=1,{\mathrm{sgn}}_{-}(\pi )=\mathrm{sgn}(\pi ),$ by clause (c) of the cited permutation lemma. For $n=1$ one has ${\mathfrak{H}}_1^{+}={\mathfrak{H}}_1^{-}=H$.

(d) Declared sector preparations. A declared sector preparation of $n$ particles of the species is a state of the framework on the sector space, that is a density operator $\rho \in D({\mathfrak{H}}_n^{\sigma })$ in the sense of The state space: density operators, pure and mixed states, and purity and clause (a) of The state space of the framework: preparations, pure rays and density operators, read as the state assigned to the preparation by P1 clause (iii). A pure sector preparation is a pure state, i.e. the rank-one operator of a unit ray of ${\mathfrak{H}}_n^{\sigma }$; a statistical mixture of sector preparations is a convex combination. The sector is part of the declaration of the species: a model that declares one statistics choice does not switch it within the same declaration. When a sector preparation is paired with an operator or a spectral projection on the labelled space, use its canonical zero extension $\tilde{\rho }:=j\rho j^{\ast }$, where $j:{\mathfrak{H}}_n^{\sigma }\hookrightarrow H^{\otimes n}$ is the isometric inclusion and $j^{\ast }$ is the orthogonal projection viewed as a map into the sector. This extension is a density operator on the labelled space: by clause (a) of Density operators and scalar spectral probability, $\rho ={\sum }_k{w}_k{T}_{u_k,u_k}$ in trace norm for an orthonormal family in the sector and nonnegative weights summing to one. The family $(j{u}_k)$ is orthonormal in the labelled space, so the same clause gives the density ${\sum }_k{w}_k{T}_{j{u}_k,j{u}_k}$ there; its action is $j\rho j^{\ast }$, since both agree on the sector and vanish on its orthogonal complement. If the sector is zero, there is no density preparation of trace one and this convention is vacuous.

(e) Sector quantities. Let $A$ be a linear operator on the labelled space $H^{\otimes n}$ with domain $D(A)$. The model reads $A$ as a quantity of the declared sector only when $A$ preserves the sector, $P_{\sigma }^{(n)}D(A)\subseteq D(A),A(D(A)\cap {\mathfrak{H}}_n^{\sigma })\subseteq {\mathfrak{H}}_n^{\sigma },$ in which case the sector restriction $A_{\sigma }:=A{\mathrm{\mid }}_{{\mathfrak{H}}_n^{\sigma }}$ with domain $D(A)\cap {\mathfrak{H}}_n^{\sigma }$ is part of the declaration and is the quantity actually read on sector preparations; For any bounded $A$, commutation with $P_{\sigma }^{(n)}$ implies sector preservation; for bounded self-adjoint $A$ the converse also holds. Indeed, if such an $A$ preserves the sector, $y$ in its orthogonal complement and $x$ in the sector satisfy $⟨Ay,x⟩=⟨y,Ax⟩=0$, so the complement is invariant as well; splitting each vector into its two orthogonal components gives $A{P}_{\sigma }^{(n)}=P_{\sigma }^{(n)}A$. For an arbitrary bounded operator, invariance alone need not preserve the complement and is not asserted equivalent to commutation. An operator that does not preserve the sector is not read on the sector; in particular no quantity of a single labelled factor is assumed to define a quantity of the sector unless it preserves it. If $A_{\sigma }$ is self-adjoint, its outcome law in a sector preparation is the law of Assigned sharp observables and the outcome law of the framework read on ${\mathfrak{H}}_n^{\sigma }$.

(f) Status and scope. Clauses (a)-(e) fix the vocabulary with which the framework reads the adopted P6 assumptions: they add no assumption to P1 and P6, and the permutation invariance of sector preparations and of sector readouts is a conditional consequence proved on this page (Permutation invariance of sector preparations and the indistinguishability of relabelled readouts), not a content of this definition. The definition asserts no spin-statistics theorem, no dynamics, no measurement instrument and no empirical fact; whether a laboratory ensemble is described by a declared sector preparation is a tested identification with uncertainty under the preparation and apparatus postulate P7, not a consequence of the declaration.

• Labels versus particles. The tensor factors of $H^{\otimes n}$ are bookkeeping slots, not particle identities: by clause (c) the model assigns the same sector state to every relabelling, and clause (i) of P6 records that the labels carry no physical content.
• Why the sector restriction is stated for operators. The rule of clause (e) is the one recorded in the owner direction for permutation sectors: a bounded quantity restricts to a sector only when it preserves the sector. For $n=1$, both sector projections are the identity, so the assigned one-particle position and momentum operators preserve both sectors and their domains. For $n\ge 2$, an operator on a single labelled factor need not preserve either sector; clause (e), including its domain condition, must be checked before reading it as a sector quantity.