The vacuum-preparation and quantized-field assumption for spontaneous emission in the cutoff dipole model

Adopt, as additional model assumptions of this pair, the following clauses. It is adopted together with the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 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, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), the radiation extension postulate Radiation extension of the framework: transverse mode and photon assignments, 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 and the coupled model The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector, in the readings of The radiation sector of the framework: modes and the number, energy and momentum assignments, The state space of the framework: preparations, pure rays and density operators, Assigned sharp observables and the outcome law of the framework and Dimensions, SI unit conventions and quantity types of the framework. Assume the Axioms of Countable Choice and of Choice, the choice hypotheses of the cited mathematics (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(i) Vacuum preparation. Let $(a,b)$ be a declared upward level pair of Spectral-transition readings of the coupled model: dipole coefficients, the single-vertex photon density and ideal energy-window detection clause (b), with $\mathrm{\Omega }$ the vacuum of the radiation sector. The postulate declares that a preparation described as the declared finite-level matter in level $a$ with no quanta in the declared transverse radiation sector is represented by the ray of the unit vector $v_a\otimes \mathrm{\Omega }$ of ${\mathbb{C}}^d\otimes F(\mathfrak{h})$, and that its evolution is the model's autonomous evolution $t\mapsto e^{-it{H}_{\mathrm{L}\mathrm{M}}\mathrm{/}\mathrm{\hslash }}(v_a\otimes \mathrm{\Omega })$. This representation of a physical preparation is an adopted model assumption; it is not derived from P1-P7, from the radiation extension or from the Coulomb model.

(ii) The emission question. The postulate declares that the model's vacuum-initiated emission question for the pair is read through the declared ideal sharp measurement with outcome projection $Q_b=\mathrm{\mid }{v}_b⟩⟨v_b\mathrm{\mid }\otimes {\mathrm{\Pi }}_1$ (or through any other declared measurement of The declared ideal counting, energy-window and analyser detection models of the radiation sector whose outcome structure is explicitly declared), with its Born probability read as in P2; the model's exact probability of that outcome is the quantity studied by the emission items of this page. The identification of this outcome with the registration of an emitted photon by a laboratory apparatus is a tested identification with uncertainty under P7.

(iii) The entire field and environment content. The declared field content of the model is the quantized transverse sector with its vacuum $\mathrm{\Omega }$, the mode vectors $g_1,\dots ,g_J$ with cutoff $\mathrm{\Lambda }$ and the coupling ${\eta }_0$ of The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector clauses (c),(d). The postulate adopts no thermal radiation state, no additional modes, no reservoir or environment, no measurement apparatus and no noise model. In particular the reading of $\mathrm{\Omega }$ as the field's zero-point (vacuum) state is a model reading, and the term "spontaneous" refers to the emission process initiated by this declared vacuum state, not to a derivation of the vacuum from classical electrodynamics or of the coupling from a current.

(iv) What a rate claim would require. The postulate explicitly does not declare a rate, a lifetime, a linewidth or an exponential decay law. Such a claim would require additional adopted assumptions — for instance a declared weak-coupling window, a declared mode-density or environment model, and a declared measurement and sampling model — together with the corresponding mathematics; the theorems of this page state only the limits of the model's explicit single-vertex coefficient and the exact-model control of one projection probability, and they record those limits as model quantities.

(v) Contrast with the prescribed drive. The prescribed-field dipole-drive model of The declared prescribed dipole-drive model and its long-wavelength regime is a different model class: its field is an external datum unaffected by the matter, it contains no vacuum state and no photon number, and it supplies only stimulated transitions. The present postulate is the additional assumption under which an emission process at all is represented by the model; the two classes are not interchangeable, and no derivation of one from the other is asserted.

(vi) Scope. Clauses (i)-(v) are adopted assumptions and readings. They assert no laboratory source, atom, mode content, cutoff or coupling value; no detector response, efficiency, background or calibration; no observed spectrum, count, rate or lifetime; and no statistical inference. The identification of a laboratory system and apparatus with the model and its outcome structure is a tested identification with uncertainty under P7, and the empirical support of the radiation extension or of the coupling is the nonlogical relation recorded on this page, not a proof of the postulate.