A two-cell marked model: the fringe visibility equals the marker overlap exactly

Adopt the declared marked two-component model of Declared marked two-component model: path components, marker states, phase family, detection weights and fringe visibility with $s=0$ (scalar particle), $m>0$, $H_0=L^2({\mathbb{R}}^3)$, and the marker space $K={\mathbb{C}}^2$, and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice), as required by the cited position and visibility laws. Declare the following two-cell model:

(a) fix the length $\mathrm{\ell }:=1\mathrm{m}$ and a unit vector $\eta \in L^2({\mathbb{R}}^2)$, whose representatives have dimension ${\mathrm{m}}^{-1}$ for transverse coordinates in metres. Define the longitudinal profiles $f_1(x_1):={\mathrm{\ell }}^{-1\mathrm{/}2}{\mathbf{1}}_{[0,\mathrm{\ell }]}(x_1),f_2(x_1):={\mathrm{\ell }}^{-1\mathrm{/}2}({\mathbf{1}}_{[0,\mathrm{\ell }\mathrm{/}2]}(x_1)-{\mathbf{1}}_{[\mathrm{\ell }\mathrm{/}2,\mathrm{\ell }]}(x_1)),$ of dimension ${\mathrm{m}}^{-1\mathrm{/}2}$, and the unit path components, as equivalence classes of measurable representatives, ${\psi }_j(x_1,x_2,x_3):=f_j(x_1)\eta (x_2,x_3)(j=1,2),$ of dimension ${\mathrm{m}}^{-3\mathrm{/}2}$. Changes at cell boundaries are null-set changes.

(b) declare the marker states $e_1:=(1,0)$, $e_2:=(\cos \vartheta ,\sin \vartheta )$ for a fixed $\vartheta \in [0,\pi \mathrm{/}2]$, so that $⟨e_1,e_2⟩=\cos \vartheta$;

(c) declare the detection bin $B:=[0,\mathrm{\ell }\mathrm{/}2]\times {\mathbb{R}}^2$ with position effect $G(B)=M_{{\mathbf{1}}_B}$.

Then:

(i) ${\psi }_1,{\psi }_2$ are unit vectors with $⟨{\psi }_1,{\psi }_2⟩=0$ and $\mathrm{\parallel }\mathrm{\Psi }(\phi )\mathrm{\parallel }=1$ for every phase $\phi$, so the model is overlap-normalized;

(ii) $p_1(B)=p_2(B)=\frac{1}{2}$, $⟨G(B){\psi }_1,{\psi }_2⟩=\frac{1}{2}$ and $p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)=\frac{1}{2}$, hence $p(\phi )(B)=\frac{1}{2}(1+\cos \vartheta \cos \phi ),V(B)=\cos \vartheta =\mathrm{\mid }⟨e_1,e_2⟩\mathrm{\mid }:$ the visibility equals the marker overlap exactly, saturating the bound of Marker overlap contracts the cross term: the detection law and the visibility bound of an overlap-normalized marked model clause (d);

(iii) at $\vartheta =0$ (identical markers) $p(\phi )(B)=\frac{1}{2}(1+\cos \phi )$ and $V(B)=1$; at $\vartheta =\pi \mathrm{/}2$ (orthogonal markers) $p(\phi )(B)=\frac{1}{2}$ for every phase and $V(B)=0$, so the marked detection law is exactly the which-path law.

The profiles, the bin and the marker states are declared idealizations of the model; the computation asserts no measured pattern, no counts, no detector response and no empirical result.