The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the identical-particle and spin postulate P6 (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, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations), the radiation extension postulate Radiation extension of the framework: transverse mode and photon assignments and the static finite-level dipole postulate Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field, in the state and outcome readings of The state space of the framework: preparations, pure rays and density operators, The state space: density operators, pure and mixed states, and purity and Assigned sharp observables and the outcome law of the framework, in the radiation sector vocabulary of The radiation sector of the framework: modes and the number, energy and momentum assignments and Declared species sectors and their preparations: the framework's reading of P6, and in the units and dimensions of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice, the choice hypotheses of the cited Fock space mathematics (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) Finite-level matter factor. Take the declared charged pair, the relative-coordinate factor $L^2({\mathbb{R}}^3;\mathbb{C})$, the coordinate multipliers $Q_j$ and the dipole operator components $D_j=-q{Q}_j$ of Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field clauses (i)-(ii), with the Coulomb energy operator $H_{\kappa }$ of clause (iv) of that postulate. Fix, as model data, a declared finite-dimensional subspace $W\subseteq D(H_{\kappa })\cap D(Q_1)\cap D(Q_2)\cap D(Q_3),\dim W=d<\mathrm{\infty },$ with a declared orthonormal basis $(w_1,\dots ,w_d)$ of $W$ (Orthonormal families, complete orthonormal systems and Hilbert bases), and read $W$ as ${\mathbb{C}}^d$ in that basis. The model's compressed matter energy is the Hermitian $d\times d$ matrix $H_a:={\mathrm{\Pi }}_W{H}_{\kappa }{\mid }_W,$ read in joules, where ${\mathrm{\Pi }}_W$ is the orthogonal projection onto $W$ (The Hilbert orthogonal projection onto a closed subspace); the model's compressed dipole matrices are the Hermitian $d\times d$ matrices $D_j^W:={\mathrm{\Pi }}_W{D}_j{\mid }_W(j=1,2,3),$ read in $\mathrm{C}\mathrm{m}$; and the model's levels are the pairs $(E,v)$ with $E\in \mathbb{R}$ (joules) an eigenvalue of $H_a$ and $v\in {\mathbb{C}}^d$ a nonzero eigenvector for $E$. A declared level pair is an ordered pair $(a,b)$ of orthonormal eigenvectors $v_a,v_b$ of $H_a$; its transition energy is $E_a-E_b$. The subspace, its basis and its dimension are model data: they are not derived from the Coulomb model and they replace the unbounded operators $H_{\kappa },D_j$ by their compressions throughout this pair.

(c) Radiation factor and cutoff dipole data. Take the transverse one-particle space $\mathfrak{h}$, the bosonic Fock space $F(\mathfrak{h})$, the vacuum $\mathrm{\Omega }$, the sector projections ${\mathrm{\Pi }}_n$ and the free-field energy observable $H_f={⨁}_{n\ge 0}{T}_n$ with $\epsilon (k)=\mathrm{\hslash }c\mathrm{\mid }k\mathrm{\mid }$ of The radiation sector of the framework: modes and the number, energy and momentum assignments clauses (b),(d), with the domains and self-adjointness of Transverse continuum free Fock Hamiltonian and Hilbert direct-sum construction, read in joules. Fix, as further model data, a cutoff $\mathrm{\Lambda }>0$ (joules), an integer $J\ge 1$ and vectors $g_1,\dots ,g_J\in \mathfrak{h}$ with $\mathrm{supp}{g}_j\subseteq \{k:0<\epsilon (k)\le \mathrm{\Lambda }\},\frac{g_j}{\sqrt{\epsilon }}\in \mathfrak{h},$ For each radiation channel $j=1,\dots ,J$, declare real dimensionless weights $e_{j,i}$ ($i=1,2,3$) and define the Hermitian channel dipole matrix $D_j^{\mathrm{c}\mathrm{h}}:={\sum }_{i=1}^3{e}_{j,i}{D}_i^W\mathrm{.}$ These are compressions of the actual real Cartesian dipole combination ${\sum }_i{e}_{j,i}{D}_i$ on $W$, so they have dipole units. This explicitly defines every channel even when $J>3$; the Cartesian case $J=3$, $e_{j,i}={\delta }_{ji}$ is included. The weights are declared model data, with no assumed normalization or measured direction. Fix also a real coupling number ${\eta }_0$ declared so that the interaction term of clause (d) is read in joules. The cutoff and the mode vectors are model data; no laboratory mode content, no Maxwell field, no mode density and no value of ${\eta }_0$ is asserted.

(d) The coupled model. Write $\varphi (g):=A_{-}(g)+A_{+}(g)$ for the maximal creation-annihilation sum of Closed adjoint sector domains on $F(\mathfrak{h})$. The declared finite-level cutoff dipole model is the pair of the Hilbert space ${\mathbb{C}}^d\otimes F(\mathfrak{h})$ (Hilbert space) and the operator $H_{\mathrm{L}\mathrm{M}}:=H_a\otimes I+I\otimes H_f+{\eta }_0{\sum }_{j=1}^J{D}_j^{\mathrm{c}\mathrm{h}}\otimes \varphi (g_j),D(H_{\mathrm{L}\mathrm{M}}):={\mathbb{C}}^d\otimes D(H_f)\mathrm{.}$ For $d\ge 1$, by Cutoff dipole relative bound clause (c), read with the Hermitian matrices $H_a$, $D_1^{\mathrm{c}\mathrm{h}},\dots ,D_J^{\mathrm{c}\mathrm{h}}$ and the coupling ${\eta }_0$ (its $d$ is the present $d$), the operator $H_{\mathrm{L}\mathrm{M}}$ is self-adjoint on the displayed domain and bounded below (Symmetric, self-adjoint and essentially self-adjoint operators, A bounded linear operator between normed spaces). If $d=0$, both the tensor-product Hilbert space and the displayed domain are $\{0\}$; $H_{\mathrm{L}\mathrm{M}}$ is the unique operator on that space. Its adjoint is the same operator, and $⟨H_{\mathrm{L}\mathrm{M}}0,0⟩=0$, so it is self-adjoint and bounded below without invoking the positive-dimensional supplier. This degenerate case has no levels, level pairs or normalized preparations. For $d\ge 1$, the model declares $H_{\mathrm{L}\mathrm{M}}$ its energy observable, in the reading of Assigned sharp observables and the outcome law of the framework, with energies, mode energies and the coupling read in joules.

(e) The model's evolution and states. The model's evolution is the autonomous unitary family $U(t):=e^{-it{H}_{\mathrm{L}\mathrm{M}}\mathrm{/}\mathrm{\hslash }}$, $t\in \mathbb{R}$, in the reading of clause (ii) of P4 and of Assigned energy observable, unitary flow and the two pictures of autonomous evolution; for $d\ge 1$, a preparation $\rho$ of the model is a state of ${\mathbb{C}}^d\otimes F(\mathfrak{h})$ in the sense of The state space of the framework: preparations, pure rays and density operators, with $\rho (t):=U(t)\rho U(t{)}^{\ast }$, and the outcome law of an assigned sharp observable $(A,u_A)$ is the reading of P2 (Assigned sharp observables and the outcome law of the framework). No measurement, no detector, no environment, no external field, no rate and no record is part of this definition.

(f) Scope. Clauses (b)-(e) are adopted model data and readings. They assert nothing about a laboratory atom, sample, mode content, cutoff or coupling; no Maxwell equation, source model, back-reaction or propagation is assumed; the truncation, the cutoff and the mode vectors are not derived; and no identification of ${\eta }_0$ or of the $g_j$ with laboratory quantities is made. The identification of a physical preparation with a state of this model is a tested identification with uncertainty under P7. The model contains no photon detector, no sampling model, no environment beyond the declared transverse modes, and no rate, decay or linewidth statement.