Radiation extension of the framework: transverse mode and photon assignments

Adopt, as an additional model assumption of the framework on this page and its consumers, the following radiation extension. It is adopted together with the state and composition postulate P1, the observable and probability postulate P2, the kinematics and massive-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), in the space vocabulary of The state space of the framework: preparations, pure rays and density operators, the observable vocabulary of Assigned sharp observables and the outcome law of the framework, the species and sector vocabulary of Declared species sectors and their preparations: the framework's reading of P6, and 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 Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). By AC implies DC implies countable choice, the stated Axiom of Choice entails Dependent Choice, as defined in The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain, supplying the DC hypothesis of the cited transverse free-field Fock theorem without adopting an additional choice axiom.

(i) Fixed propagation-mode one-photon polarisation model. For a declared propagation mode, described by a nonzero wavevector label $k_0\in {\mathbb{R}}^3\setminus \{0\}$ read in ${\mathrm{m}}^{-1}$ in a declared inertial chart (Inertial frames of the framework and the class of frame changes), the model assigns a two-dimensional complex internal space, the transverse polarisation space of the mode: the complex span of an orthonormal pair $(v_1,v_2)$ of vectors of ${\mathbb{C}}^3$ satisfying $k_0\cdot v_1=k_0\cdot v_2=0$, with the standard Hermitian form. Unit rays of this space are the model's one-photon polarisation preparations of the mode. This assignment is an idealised manifestation of the abstract Born/state framework on a finite-dimensional internal factor; it is not a spatial wave function of a massive particle, not a model of a photon position, and not the massive one-particle Hamiltonian of P5 with mass set to zero. For the massive free family, The massive free-particle family has no strong zero-mass limit on Schwartz states refutes strong action convergence to a self-adjoint operator on every Schwartz state and shows frequency divergence at each fixed nonzero wave number as $m\downarrow 0$. This extension uses its independently declared massless model assignments, with no such limiting construction.

(ii) Quantised transverse radiation sector. The model assigns to the free radiation a bosonic species in the sense of P6, whose one-particle space is the transverse space $\mathfrak{h}:=\{f\in L^2({\mathbb{R}}^3;{\mathbb{C}}^3):k\cdot f(k)=0\text{ for almost every }k\}$ with the $L^2$ inner product; its $n$-particle sector is ${\mathfrak{H}}_n^{+}=\mathrm{ran}{P}_{+}^{(n)}$ and its radiation state space is the bosonic Fock space $F(\mathfrak{h})={⨁}_{n\ge 0}{\mathfrak{H}}_n^{+}$ (Declared species sectors and their preparations: the framework's reading of P6, Hilbert direct-sum construction). The choice of a bosonic species is the additional model assignment of P6 clause (iii). It is adopted here, not derived from the preceding assumptions, and this item records no experimental observation that establishes it.

(iii) Mode assignments of frequency, energy and momentum. In the declared inertial chart, each free mode label $k\in {\mathbb{R}}^3\setminus \{0\}$ is assigned the angular frequency $\omega (k):=c\mathrm{\mid }k\mathrm{\mid }$ in ${\mathrm{s}}^{-1}$, the photon energy $E(k):=\mathrm{\hslash }\omega (k)=\mathrm{\hslash }c\mathrm{\mid }k\mathrm{\mid }$ in joules and the photon momentum $p(k):=\mathrm{\hslash }k$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, where $c=\mathrm{2,99792458}\times {10}^8\mathrm{m}{\mathrm{s}}^{-1}$ and $\mathrm{\hslash }=h\mathrm{/}(2\pi )>0$ in $\mathrm{J}\mathrm{s}$ have the SI values of Dimensions, SI unit conventions and quantity types of the framework clause (c). Ordinary frequency is $\nu (k)=\omega (k)\mathrm{/}(2\pi )$ in hertz. These are separate quantum-radiation model assignments: they are neither consequences of classical Maxwell theory nor consequences of the special-relativistic framework, and no Lorentz transformation law for the mode label $k$, for $\omega (k)$ or for the polarisation pair is adopted or claimed; the extension does not assert Galilean covariance of the radiation sector, and it does not identify the massless Lorentz representations of the photon with the representations of the spatial rotation group used in P6.

Littlejohn's Section 7 derives the normal-ordered free-field momentum from the field momentum density and obtains $P={\sum }_{\lambda }\mathrm{\hslash }k{a}_{\lambda }^{†}{a}_{\lambda }$, then reads an occupation-number state as carrying the corresponding sum of photon momenta (equations (21)-(33)). That derivation uses Gaussian units, Coulomb gauge and box normalization; it supports this assignment but does not derive this framework's other radiation assumptions or its covariance claims.

(iv) Free-field observables. The model assigns the total free-field energy observable $H_f={⨁}_{n\ge 0}{T}_n$ with $\epsilon (k)=\mathrm{\hslash }c\mathrm{\mid }k\mathrm{\mid }$ and $T_n\psi ={\epsilon }_n\psi$, read in joules, the total number observable $N$ read in the dimensionless unit $1$ of counted quanta, and the total momentum components ${⨁}_n\left({\sum }_{i=1}^n\mathrm{\hslash }{k}_{j,i}\right)$, read in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$; their operator domains and self-adjointness are the mathematics of Transverse continuum free Fock Hamiltonian and Number operator domain for energy and number, and Total momentum multipliers on transverse symmetric Fock sectors for total momentum, read physically in The radiation sector of the framework: modes and the number, energy and momentum assignments. The total momentum components use the maximal signed summed-multiplier sector domains and the squared-norm summability Fock domain of that lemma; the vacuum has total momentum zero.

(v) Counting, detection and light-matter coupling are not supplied. Photon number and energy statistics require a declared detector model in the sense of P2 and P7, which this postulate does not supply; a prescribed classical electromagnetic field does not by itself yield photon counting or vacuum spontaneous emission. No light-matter Hamiltonian, no dipole coupling, no emission or absorption process and no recoil model is asserted here; the absorption/stimulated-emission branch with a finite-level dipole coupling and the quantised-mode emission construction are separate model assumptions for the later pair on light-matter and spectral transitions. The relation of a laboratory field, polariser or detector to this model is a tested identification with uncertainty under P7, not a consequence of the postulate.

(vi) Scope of the framework. The radiation extension is deliberately non-relativistic in its matter sector: it adds external free radiation modes to the fixed-particle-number massive matter core without changing P5. Relativistic matter, antiparticles, pair creation, covariant local field dynamics and full quantum electrodynamics are not contained in this extension and are not disguised as solved core prerequisites; they require a named field-theory destination. No position wave function of the photon satisfying the massive Schrödinger equation is introduced.

(vii) Status. Clauses (i)-(vi) are adopted assumptions of the model. They are motivated by the sources cited above -- the transverse two-state description of polarisation with transmitted fraction ${\cos }^2\theta$, the single-mode oscillator quantisation with photon energy increment $\mathrm{\hslash }\omega$, and the field-operator expansion of free modes -- but motivation is not deduction. No empirical claim, no measured value and no claim that any laboratory field, polariser or detector is exactly described by the model is made by this item.

• Adopted, not derived. Neither the two-dimensional transverse polarisation space, nor the bosonic Fock assignment, nor the relations $\omega =c\mathrm{\mid }k\mathrm{\mid }$, $E=\mathrm{\hslash }\omega$ and $p=\mathrm{\hslash }k$ follow from P1-P7. They are the additional model content of this page, adopted only by the items that cite this postulate.
• Relation to the massive model. The quantum-radiation assignments are declared directly, without a limit construction from the massive free Schrödinger model. The counterexample The massive free-particle family has no strong zero-mass limit on Schwartz states refutes strong action convergence on every Schwartz state and a finite frequency limit at each fixed nonzero wave number; it does not decide other operator topologies or exclude independently specified massless models. The polarisation internal space is a finite-state model, not a position representation of a photon.
• Empirical qualifications. The statistics choice, the transverse polarisation count, and the frequency-energy-momentum assignments are empirical model choices with uncertainties carried by their laboratory identifications. No experiment, count, efficiency or calibration is recorded or assumed by this postulate; comparisons with data belong to the preparation-and-apparatus discipline of P7.