The declared two-path preparation model and its ideal detection bins

(a) Framework data and state convention. Adopt the state and composition postulate P1, the observable and probability postulate P2, the kinematics and one-particle model postulate P5, the identical-particle and spin postulate P6, and the preparation and apparatus discipline P7 (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, 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, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). Fix a spin label $s\in \{0,\frac{1}{2},1,\frac{3}{2},\dots \}$, put $r:=2s+1$, and fix a mass parameter $m>0$. P5 assigns the one-particle state space $H_s:=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$ with almost-everywhere equivalence classes and pairing $⟨f,g⟩:={\int }_{{\mathbb{R}}^3}{\sum }_{l=1}^r{f}_l(x)\overline{g_l(x)}dx,$ linear in the first argument (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, The space $L^p(\mu )$ as the quotient by null functions, Real and complex inner-product spaces and their induced length). P1 identifies a pure preparation with the ray $[\psi ]:=\{z\psi :z\in \mathbb{C}\setminus \{0\}\}$ of a unit vector $\psi$ and a general preparation with a density operator, meaning a bounded self-adjoint positive trace-class operator $\rho$ with $\mathrm{tr}\rho =1$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Trace class operator, Self-adjoint, positive, unitary and normal operators). Write $D(H_s):=\{\rho :\rho \text{ is a bounded self-adjoint positive trace-class operator on }{H}_s\text{ and }\mathrm{tr}\rho =1\}$ for the set of general preparations.

For a unit vector $u$, define $P_{u}f:=⟨f,u⟩u$. By Cauchy--Schwarz (Cauchy–Schwarz: $\mathrm{\mid }⟨x,y⟩\mathrm{\mid }\le \mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }$, with equality exactly for dependent pairs), $\mathrm{\parallel }{P}_{u}f\mathrm{\parallel }=\mathrm{\mid }⟨f,u⟩\mathrm{\mid }\le \mathrm{\parallel }f\mathrm{\parallel }$, so $P_u$ is bounded. Then $P_u^2=P_u$ and $P_u$ is self-adjoint because for $f,g\in H_s$, $⟨P_{u}f,g⟩=⟨f,u⟩⟨u,g⟩=⟨f,P_{u}g⟩$. Moreover $⟨P_{u}f,f⟩=\mathrm{\mid }⟨f,u⟩{\mathrm{\mid }}^2\ge 0$ by the inner-product identities and $⟨u,u⟩=1$. Its range is $\mathbb{C}u$, so it has finite rank and is trace class by Trace class operator. The one-term nuclear representation $P_{u}f=⟨f,u⟩u$ gives $\mathrm{tr}{P}_u=⟨u,u⟩=1$ by Trace is absolutely convergent and basis independent. Thus $P_u$ is a density operator. For $\psi \mathrm{\ne }0$ define $\widehat{\rho }(\psi ):=P_{\psi \mathrm{/}\mathrm{\parallel }\psi \mathrm{\parallel }},\widehat{\rho }(\psi )f=\frac{⟨f,\psi ⟩\psi }{\mathrm{\parallel }\psi {\mathrm{\parallel }}^2}\mathrm{.}$ For every nonzero scalar $z$, conjugate-linearity in the second argument and $\mathrm{\parallel }z\psi {\mathrm{\parallel }}^2=\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{\parallel }\psi {\mathrm{\parallel }}^2$ give $\widehat{\rho }(z\psi )f=\frac{⟨f,z\psi ⟩z\psi }{\mathrm{\parallel }z\psi {\mathrm{\parallel }}^2}=\frac{\mathrm{\mid }z{\mathrm{\mid }}^2⟨f,\psi ⟩\psi }{\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{\parallel }\psi {\mathrm{\parallel }}^2}=\widehat{\rho }(\psi )f,$ so the pure density operator depends only on the ray.

Position PVM and Born-density calculation. P5 assigns the componentwise position operators $Q_{j}f=x_{j}f$ on $D(Q_j):=\{f\in H_s:x_{j}f\in H_s\},j=1,2,3.$ Each $Q_j$ is self-adjoint on this domain. To see density, for $f\in H_s$ put $f_n={\mathbf{1}}_{\{\mathrm{\mid }{x}_j\mathrm{\mid }\le n\}}f$; then $f_n\in D(Q_j)$ and $f_n\to f$ in $H_s$ by dominated convergence. If $f,g\in D(Q_j)$, then $⟨Q_{j}f,g⟩=⟨f,Q_{j}g⟩$ because $x_j$ is real-valued and both products are integrable by Cauchy--Schwarz. If $g\in D(Q_j^{\ast })$ with $Q_j^{\ast }g=h$, let $E_n=\{x:\mathrm{\mid }{x}_j\mathrm{\mid }\le n\}$. For every $q\in H_s$ supported in $E_n$, one has $q\in D(Q_j)$ and $⟨q,(x_{j}g-h){\mathbf{1}}_{E_n}⟩=0$ by the defining adjoint identity. The vector $(x_{j}g-h){\mathbf{1}}_{E_n}$ lies in $H_s$ because $x_j$ is bounded on $E_n$; taking it for $q$ shows $x_{j}g=h$ almost everywhere on $E_n$. Since $E_n\uparrow {\mathbb{R}}^3$, $x_{j}g=h\in H_s$ almost everywhere, so $g\in D(Q_j)$ and $Q_j^{\ast }g=Q_{j}g$. Symmetry gives the reverse domain inclusion; hence $Q_j=Q_j^{\ast }$ by Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators.

For a Borel set $S\subseteq {\mathbb{R}}^3$, define the indicator multiplication operator $M_{{\mathbf{1}}_S}:H_s\to H_s$ by $(M_{{\mathbf{1}}_S}f)(x):={\mathbf{1}}_S(x)f(x)$ almost everywhere. This is independent of the representative of the $L^2$ class and is bounded, since $\mathrm{\parallel }{M}_{{\mathbf{1}}_S}f{\mathrm{\parallel }}^2={\int }_S\mathrm{\parallel }f(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^{2}dx\le \mathrm{\parallel }f{\mathrm{\parallel }}^2\mathrm{.}$ It is linear and a bounded operator with bound one (A bounded linear operator between normed spaces). It is an orthogonal projection: ${\mathbf{1}}_S^2={\mathbf{1}}_S$ and the real indicator multiplier is self-adjoint for the declared pairing.

For Borel $C\subseteq \mathbb{R}$, define $E_j(C):=M_{{\mathbf{1}}_{\{x:x_j\in C\}}}\mathrm{.}$ Each value is a bounded orthogonal projection since $\mathrm{\parallel }{\mathbf{1}}_{\{x:x_j\in C\}}f\mathrm{\parallel }\le \mathrm{\parallel }f\mathrm{\parallel }$, and $E_j(\mathrm{\varnothing })=0$, $E_j(\mathbb{R})=I$, and $E_j(C\cap D)=E_j(C)E_j(D)$. For a disjoint sequence $(C_n)$ with union $C$, the squared norm of $E_j(C)f-{\sum }_{n\le N}{E}_j(C_n)f$ equals ${\int }_{\{x:x_j\in C\setminus {\cup }_{n\le N}{C}_n\}}\mathrm{\parallel }f(x){\mathrm{\parallel }}^{2}dx,$ which tends to zero by dominated convergence; hence $E_j$ is strongly countably additive. Its scalar measures are finite Borel measures $⟨E_j(C)f,f⟩={\int }_{\{x:x_j\in C\}}\mathrm{\parallel }f(x){\mathrm{\parallel }}^{2}dx,$ and are regular on $\mathbb{R}$ by Locally finite Borel measures on second-countable LCH spaces are regular. The push-forward integral identity ${\int }_{\mathbb{R}}g(\lambda )d⟨E_j(\lambda )f,f⟩={\int }_{{\mathbb{R}}^3}g(x_j)\mathrm{\parallel }f(x){\mathrm{\parallel }}^{2}dx$ holds first for indicators and nonnegative simple functions and then for nonnegative measurable $g$ by monotone convergence. In particular the domain of $\int \lambda d{E}_j(\lambda )$ is exactly $D(Q_j)$ by the same identity with $g(\lambda )={\lambda }^2$. For bounded Borel $g$, the bounded PVM calculus equals multiplication by $g(x_j)$: this holds for indicators by the definition of $E_j$, for finite simple functions by linearity, and for bounded Borel functions by uniform simple approximation and the norm bound of the bounded calculus (Bounded borel pvm integral). Therefore the truncations of the unbounded integral are multiplication by $x_j{\mathbf{1}}_{\{\mathrm{\mid }{x}_j\mathrm{\mid }\le n\}}$. For $f\in D(Q_j)$ these converge in $H_s$ to $x_{j}f$, since the squared tail norm is ${\int }_{\{\mathrm{\mid }{x}_j\mathrm{\mid }>n\}}{x}_j^2\mathrm{\parallel }f(x){\mathrm{\parallel }}^{2}dx\to 0$ by dominated convergence. Thus $\int \lambda d{E}_j(\lambda )=Q_j$ on exactly $D(Q_j)$, and the regular spectral theorem identifies $E_j$ as the spectral PVM of $Q_j$ (Integral of a measurable function against a projection-valued measure, Spectral theorem for unbounded self-adjoint operators (PVM form)).

Define the joint position PVM by $G(B):=M_{{\mathbf{1}}_B}(B\subseteq {\mathbb{R}}^3\text{ Borel})\mathrm{.}$ For each $f$, $\mathrm{\parallel }G(B)f\mathrm{\parallel }\le \mathrm{\parallel }f\mathrm{\parallel }$, so its values are bounded; indicator multiplication gives its projection, identity, and intersection laws. Strong countable additivity follows from dominated convergence applied to $\mathrm{\parallel }f{\mathrm{\parallel }}^2$. Its scalar measures are $B\mapsto {\int }_B\mathrm{\parallel }f(x){\mathrm{\parallel }}^{2}dx$, finite regular Borel measures on ${\mathbb{R}}^3$. On coordinate rectangles, $G(C_1\times C_2\times C_3)=E_1(C_1)E_2(C_2)E_3(C_3)\mathrm{.}$ The rectangles generate $\mathcal{B}({\mathbb{R}}^3)$; the published Borel-product and finite-measure uniqueness theorems show that these rectangle values uniquely determine the joint PVM (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, Finite measures agreeing on a generating pi-system and on the whole space are equal, Scalar and complex measures from a pvm): for any two candidate PVMs, their finite scalar measures agree on the generating rectangles and on the whole space, so they agree on every Borel set; the polarization identity in the last lemma then gives equality of the operator values. This constructs the mathematical joint position PVM; a physical joint detector reading is supplied only for the finite effect families declared below. For a density operator $\rho$ define ${\mu }_{\rho }^G(B):=\mathrm{tr}(\rho G(B))$ for each Borel $B\subseteq {\mathbb{R}}^3$. Since $G$ is a PVM, this set function is a probability measure by the general density/PVM calculation used in clause (ii) of P2. For a finite Borel partition $(B_a{)}_{a=1}^m$, the operators $G(B_a)$ are orthogonal projections and sum to $I$. For each $f\in H_s$, $⟨G(B_a)f,f⟩=\mathrm{\parallel }G(B_a)f{\mathrm{\parallel }}^2\in [0,\mathrm{\parallel }f{\mathrm{\parallel }}^2]$, so they are bounded effects with $0\le G(B_a)\le I$ in quadratic-form order. Clause (iv) of P2 therefore assigns the channel probabilities $p_a(\rho )={\mu }_{\rho }^G(B_a)$; the two-channel partition $(B,{\mathbb{R}}^3\setminus B)$ assigns ${\mu }_{\rho }^G(B)$ to bin $B$ (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes). Thus $G$ is the mathematical joint position PVM, while the detector reading used here is the declared finite-effect model of P2(iv); this does not assert that every three sharp coordinate measurement is jointly realized by a laboratory apparatus. For a unit vector $u$, the trace formula for the rank-one operator $P_u$ gives ${\mu }_{P_u}^G(B)=\mathrm{tr}(P_{u}G(B))=\mathrm{\parallel }G(B)u{\mathrm{\parallel }}^2={\int }_B\mathrm{\parallel }u(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^{2}dx,$ and P7(iii) leaves identification of this ideal detector with a concrete apparatus as a tested physical claim with uncertainty.

Assume the Axiom of Countable Choice and the Axiom of Choice as required by the cited trace and spectral suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) Declared two-path assignment. A declared two-path preparation model of the apparatus is the assignment of two nonzero classes ${\psi }_1,{\psi }_2\in H_s$, called the path components of the model with respect to the declared pair of alternative path conditions of the apparatus, together with a declared relative phase $\phi \in \mathbb{R}$ and a declared weight $w\in [0,1]$. The assignment of ${\psi }_1,{\psi }_2$, $\phi$ and $w$ is part of the preparation model in the sense of clause (i) of P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions): it is declared, not computed from the framework, and no item of this page asserts that a particle described by the model "took" one of the two paths.

(c) Coherent preparation. The declared two-path model is admissible at phase $\phi$ when ${\psi }_1+e^{i\phi }{\psi }_2\mathrm{\ne }0$ in $H_s$. When it is admissible, the coherent preparation of relative phase $\phi$ is the pure preparation ${\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}(\phi ):=\widehat{\rho }({\psi }_1+e^{i\phi }{\psi }_2),$ the normalized rank-one density operator defined in clause (a). The relative phase $\phi$ is read in radians (dimensionless). Multiplying both path components by one common nonzero scalar leaves their associated pure density operator unchanged, while the declared relative phase remains part of the preparation assignment.

(d) Incoherent preparation. For a declared weight $w\in [0,1]$ the incoherent preparation, or which-path preparation, of the model is ${\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(w):=w\widehat{\rho }({\psi }_1)+(1-w)\widehat{\rho }({\psi }_2)\mathrm{.}$ It is a density operator: both summands are positive trace-class operators of trace one; trace-class operators form a vector space, nonnegative real combinations preserve positivity and self-adjointness, and the trace is linear, so $\mathrm{tr}{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(w)=w+(1-w)=1$ (Trace class is a two sided Banach operator ideal, Trace is absolutely convergent and basis independent). The weight $w$ is a declared preparation weight, not an outcome probability.

(e) Ideal detection bins. A declared ideal detector of the model is a finite family $\mathcal{B}=(B_a{)}_{a=1}^m$ of pairwise disjoint Borel subsets of ${\mathbb{R}}^3$ with ${\bigcup }_a{B}_a={\mathbb{R}}^3$; the ideal bin probabilities of a preparation $\rho \in D(H_s)$ are the numbers $p_a(\rho ):={\mu }_{\rho }^G(B_a)=\mathrm{tr}(\rho G(B_a))(a=1,\dots ,m),$ the reading of the finite-effect clause (iv) of P2 for the declared detector and its mathematical joint position PVM (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Projection valued measure). By clause (iv) of P2 these numbers are nonnegative, are at most one, and satisfy ${\sum }_a{p}_a(\rho )=1$. A single declared bin $B$ is the two-element ideal detector $(B,{\mathbb{R}}^3\setminus B)$; its probability is ${\mu }_{\rho }^G(B)$.

For the ideal support, choose a Lebesgue-measurable representative $f$ of the unit $L^2$ class and form $S_f:=\{x:\mathrm{\parallel }f(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}>0\}$. The ideal support is the class of $S_f$ under equality up to a Lebesgue-null symmetric difference. This class is independent of the representative, and it is not asserted to be a Borel detector bin.

(f) Status and scope. Clauses (a)-(e) fix the model data of the pair: the framework reading, the declared path components and preparations, and the ideal bin reading of the position observable. They adopt no assumption beyond P1, P2, P5, P6, P7 and the cited mathematics; the declaration of ${\psi }_1,{\psi }_2,\phi ,w$ and of the bins is an apparatus-model input, and the claim that a concrete two-slit apparatus realizes a particular such declaration is a tested identification with uncertainty under P7(iii). The declared bins are subsets of ${\mathbb{R}}^3$; the model does not claim that a physical screen of lower dimension is literally a spatial region, and a marginal, coarse-graining or flux reduction of a state to a screen coordinate or surface is a separate detection model, not supplied here. No dynamics, no recorded-event response, no sampling model and no empirical result is asserted; the recorded-event and sampling models are declared separately on this page.

• Classes, not functions. The path components are almost-everywhere classes; pointwise values such as ${\psi }_j(x)$ are read only through the measurable representatives, densities and pairings defined above, and the interference identities of the page hold up to null sets.
• "Path" is a declared decomposition. The phrase "path component" names an element of the declared decomposition of the preparation model. The pair proves laws of the declared coherent and incoherent preparations; it does not assert which-path ontology, does not assign a trajectory to the particle, and does not assume that the two components are orthogonal or have disjoint ideal supports. Whether a preparation model with an orthogonal or disjoint-support decomposition describes a given apparatus is a physical identification under P7.
• Units. The outcome sets are sets of position with unit metre, the probability numbers are dimensionless, and the phase is read in radians (Dimensions, SI unit conventions and quantity types of the framework).