The fixed-mode polarisation model of the framework

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2, the ideal conditional-measurement postulate P3, 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, Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions) and the radiation extension postulate Radiation extension of the framework: transverse mode and photon assignments in its clause (i). Read the state vocabulary in The state space of the framework: preparations, pure rays and density operators, the observable vocabulary in Assigned sharp observables and the outcome law of the framework, the measurement vocabulary in Declared ideal measurements and instruments of the framework, and the units of angles in Dimensions, SI unit conventions and quantity types of the framework: angles and phases are dimensionless under the radian convention. Assume Countable Choice, Dependent Choice and Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain, The Axiom of Choice), including the hypotheses inherited from the adopted radiation postulate and its Fock-space suppliers.

(b) Declared fixed-mode data. A declared fixed-mode polarisation model consists of:

1. a nonzero propagation label $k_0\in {\mathbb{R}}^3\setminus \{0\}$, read in ${\mathrm{m}}^{-1}$ in a declared inertial chart, which is a parameter of the model and not a state;
2. a declared transverse frame, that is a pair $v_j=\zeta u_j$ for $j\in \{1,2\}$, where $(u_1,u_2)$ is a real orthonormal basis of $k_0^{\perp }:=\{u\in {\mathbb{R}}^3:k_0\cdot u=0\}$ and $\zeta \in \mathbb{C}$ is one common phase factor, $\mathrm{\mid }\zeta \mathrm{\mid }=1$. Here $k_0\cdot u$ is the real Euclidean dot product, and the Hermitian form on ${\mathbb{C}}^3$ is $⟨x,y⟩={\sum }_{\mathrm{\ell }=1}^3{x}_{\mathrm{\ell }}\overline{y_{\mathrm{\ell }}}$, linear in its first argument; thus $⟨v_i,v_j⟩={\delta }_{ij}$ and $k_0\cdot v_j:={\sum }_{\mathrm{\ell }=1}^3(k_0{)}_{\mathrm{\ell }}(v_j{)}_{\mathrm{\ell }}=0$;
3. the polarisation space $V:=\mathbb{C}{v}_1\oplus \mathbb{C}{v}_2$ with the restricted Hermitian inner product and the orthonormal basis $(e_1,e_2):=(v_1,v_2)$.

The model's state space is $D(V)$, the set of density operators of $V$; the pure preparations of the model are the unit rays $[\psi ]$ of $V$, that is the preparations assigned the pure state $\widehat{\rho }(\psi )$ of The state space: density operators, pure and mixed states, and purity, and a mixed preparation is a statistical mixture of preparations in the sense of P1. A pure ray is linearly polarised when it has a nonzero real representative in $k_0^{\perp }$. The map $\theta \mapsto {\psi }_{\theta }:=\cos \theta e_1+\sin \theta e_2$ assigns to each angle $\theta \in \mathbb{R}$ a distinguished pure preparation, the linear polarisation preparation at angle $\theta$ in the declared transverse frame. In particular, ${\psi }_{\theta }=\zeta (\cos \theta u_1+\sin \theta u_2)$ has that ray, whose real direction is the unoriented line $\mathbb{R}(\cos \theta u_1+\sin \theta u_2)$; hence the physical angle label is modulo $\pi$. General complex unit vectors of $V$ remain allowed as pure preparations and need not be linearly polarised.

(c) Declared polarisation analyser. For each angle $\theta \in \mathbb{R}$ the declared polarisation analyser at angle $\theta$ is the declared ideal sharp measurement of clause (a) of Declared ideal measurements and instruments of the framework with two outcomes labelled $+1$ and $-1$ and outcome structure $(P_{\theta },P_{\theta }^{\perp }),P_{\theta }x=⟨x,e_{\theta }⟩e_{\theta },e_{\theta }:=\cos \theta e_1+\sin \theta e_2,P_{\theta }^{\perp }:=I-P_{\theta },$ whose projections $P_{\theta },P_{\theta }^{\perp }$ are those of Analyzer projections of a two-dimensional polarisation space and their overlaps. The outcome law of the analyser in a preparation $\rho$ is the reading of P2 for the projections of the outcome structure, and the labels are dimensionless with the reading "$+1$: transmitted", "$-1$: blocked". Where a conditional post-measurement state is used, it is the Lueders update of clause (b) of Declared ideal measurements and instruments of the framework; the framework does not assert that a laboratory absorber leaves an outgoing one-photon polarisation state, and the absorption outcome is used only through its discarded-outcome law unless a separate instrument is declared (clause (v) of P3).

(d) Frame covariance and angle labels. For another complex orthonormal transverse pair $(e_1^{\mathrm{\prime }},e_2^{\mathrm{\prime }})$ (not necessarily a transverse frame of clause (b)), let $U:V\to V$ be the unitary map with $U{e}_i=e_i^{\mathrm{\prime }}$, explicitly $Ux={\sum }_i⟨x,e_i⟩e_i^{\mathrm{\prime }}$. Its transported sharp-measurement family is $P_{\theta }^{\mathrm{\prime }}=U{P}_{\theta }{U}^{\ast }$, because $U{e}_{\theta }=\cos \theta e_1^{\mathrm{\prime }}+\sin \theta e_2^{\mathrm{\prime }}$. Under the simultaneous transport of a preparation to ${\rho }^{\mathrm{\prime }}=U\rho U^{\ast }$ the Born probabilities agree: $\mathrm{tr}({\rho }^{\mathrm{\prime }}{P}_{\theta }^{\mathrm{\prime }})=\mathrm{tr}(U\rho P_{\theta }{U}^{\ast })=\mathrm{tr}(\rho P_{\theta })$. The last equality follows by evaluating the finite trace in a transported orthonormal basis; the same argument preserves trace one, and conjugation preserves positivity and self-adjointness of $\rho$. Thus covariance concerns the transported states and projectors together. A general complex basis change need not leave the original linear-analyser family fixed: replacing $e_2$ by $i{e}_2$ gives rays of $\cos \theta e_1+i\sin \theta e_2$, generally absent from the original family, and does not satisfy clause (b). For two transverse frames of clause (b), the bases differ by a common phase times a real orthogonal two-by-two matrix. Write its first column as $(\cos \chi ,\sin \chi )$; its second column is $(-\sin \chi ,\cos \chi )$ or $(\sin \chi ,-\cos \chi )$ according to whether its determinant is $+1$ or $-1$. The families are equal with the angle reparametrisation $\theta \mapsto \chi +\theta$ or $\chi -\theta$, respectively, by the sine and cosine addition formulas. Arbitrary complex basis transport gives a valid sharp measurement but need not give a linear polariser. The value of $\mathrm{\mid }{k}_0\mathrm{\mid }$ enters no polarisation probability: it enters the separate mode frequency and energy assignments of the radiation postulate.

(e) Scope. The model is an idealisation of a transverse propagation mode with a definite mode label. It is a finite-dimensional internal-factor model in the sense of P1 and P6: its state space is $D(V)$, its pure preparations are rays, and its analyser is a declared ideal sharp measurement. It contains no position observable, no momentum observable, no spatial wave function and no localisation claim, and it is not the massive one-particle model of P5 with $m=0$ and is not obtained from that model by a mass-to-zero limit; no such limit is used or claimed. The mode label $k_0$ is not a normalisable state of any sector of the framework, and a plane wave is not introduced as a state. Whether a laboratory polariser and source realize this declared model, with what efficiency, loss and background, is a tested identification with uncertainty under P7, not a consequence of this definition.

(f) Status. Clauses (a)-(e) fix the model data, its outcome vocabulary and its scope; they add no assumption beyond the cited postulates and the cited finite-dimensional mathematics. The mathematical properties of the projections are the content of Analyzer projections of a two-dimensional polarisation space and their overlaps; the conditional outcome law of the declared analyser is proved on this page. No empirical claim is made here.