The radiation sector of the framework: modes and the number, energy and momentum assignments

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2 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, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations) and the radiation extension postulate Radiation extension of the framework: transverse mode and photon assignments in its clauses (i)-(iv), read in the state 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 sector vocabulary of Declared species sectors and their preparations: the framework's reading of P6 and the units of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axioms of Countable Choice, Dependent Choice and Choice, the choice hypotheses of the cited Fock space mathematics (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).

(b) Radiation one-particle space and sectors. The transverse one-particle space of the radiation sector is $\mathfrak{h}:=\{f\in L^2({\mathbb{R}}^3;{\mathbb{C}}^3):k\cdot f(k)=0\text{ for almost every }k\}$ with the restricted $L^2$ inner product, a separable complex Hilbert space (Hilbert space); its elements are read as one-photon wave-packet vectors. The wave-vector coordinate is $k$ in ${\mathrm{m}}^{-1}$ and the normalization is against $d^{3}k$. With normalized state vectors dimensionless, a unit wave-packet vector $f$ therefore has coordinate representatives $f(k)$ with units ${\mathrm{m}}^{3\mathrm{/}2}$, so that $\int \mathrm{\parallel }f(k){\mathrm{\parallel }}^2{d}^{3}k$ is dimensionless. A unit mode vector has the same coordinate units. For $n\ge 1$ the $n$-photon sector is ${\mathfrak{H}}_n^{+}=\mathrm{ran}{P}_{+}^{(n)}\subseteq {\mathfrak{h}}^{\otimes n}$ and the radiation state space is the bosonic Fock space $F(\mathfrak{h})={⨁}_{n\ge 0}{\mathfrak{H}}_n^{+}$, whose sector projections are ${\mathrm{\Pi }}_n$ and whose vacuum is $\mathrm{\Omega }={\iota }_0(1)$ (Hilbert direct-sum construction, Fock sector projections). A declared radiation preparation is a state $\rho \in D(F(\mathfrak{h}))$ in the sense of The state space of the framework: preparations, pure rays and density operators; a declared one-photon preparation is a preparation whose state is supported in the one-photon sector, and for a unit $f\in \mathfrak{h}$ the ray of ${\iota }_1(f)$ is the one-photon preparation of wave-packet vector $f$.

(c) Modes. A mode of the radiation sector is a ray $[\phi ]=\{z\phi :z\in \mathbb{C}\setminus \{0\}\}$ of a nonzero class $\phi \in \mathfrak{h}$, equivalently the rank-one orthogonal projection onto $\mathbb{C}\phi$. Its unit representatives differ only by a complex factor of modulus one. A declared mode family is a finite or countable orthonormal family $({\phi }_j)$ of $\mathfrak{h}$ (Orthonormal families, complete orthonormal systems and Hilbert bases). For a unit mode vector $\phi$ and a dimensionless label $z\in \mathbb{C}$ the vector $U_{\phi }{\psi }_z,{\psi }_z=e^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}2}{\sum }_{n\in \mathbb{N}}\frac{z^n}{\sqrt{n!}}{e}_n,$ is the mode-coherent preparation vector of label $z$, and $U_{\phi }{e}_n$ is the $n$-quanta mode vector; these are the readings of the transferred vectors of The fixed-mode isometry of Fock space and the transfer of number and ladder operators. A mode vector is a normalisable wave packet; the sharp-mode label $k$ of the propagation-mode idealisation is a spectral parameter and is not a state of $\mathfrak{h}$ (Restricted momentum and energy multipliers of the transverse one-photon space clause (c)).

(d) Assigned observables. The framework assigns to the radiation sector:

1. the total number observable $(N,1)$, the self-adjoint number operator of Number operator domain, with domain $D(N)=\{\psi :{\sum }_n{n}^2\mathrm{\parallel }{\psi }_n{\mathrm{\parallel }}^2<\mathrm{\infty }\}$ and read in the dimensionless unit $1$ of counted quanta;
2. the free-field energy observable $(H_f,\mathrm{J})$, the operator $H_f={⨁}_{n\ge 0}{T}_n$ with $\epsilon (k)=\mathrm{\hslash }c\mathrm{\mid }k\mathrm{\mid }$ and $T_n\psi ={\epsilon }_n\psi$, of Transverse continuum free Fock Hamiltonian, read in joules; its one-particle restriction is the energy multiplier $M_{\mathrm{\hslash }c\mathrm{\mid }k\mathrm{\mid }}^{\mathfrak{h}}$ of Restricted momentum and energy multipliers of the transverse one-photon space;
3. for $j\in \{1,2,3\}$ the total momentum component $(P_j,\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1})$ is the self-adjoint direct sum of the signed summed multipliers $P_{j,n}$ of Total momentum multipliers on transverse symmetric Fock sectors, with $P_{j,0}=0$. In the lemma's exact Euclidean embedding $I_n$ of each symmetric transverse sector, $P_{j,n}$ multiplies by $p_{j,n}(k)=\mathrm{\hslash }{\sum }_i(k_i{)}_j$ on the maximal domain $\{u:p_{j,n}{I}_{n}u\in L^2\}$. Its Fock domain is $D(P_j)=\{\psi :{\psi }_n\in D(P_{j,n})\text{ for every }n,{\sum }_n\mathrm{\parallel }{P}_{j,n}{\psi }_n{\mathrm{\parallel }}^2<\mathrm{\infty }\}\mathrm{.}$ This is the domain of the sum, without an extra requirement that each individual momentum summand be square integrable. Its spectral PVM is the sectorwise direct sum of the inverse-image indicator multipliers, including the atom at zero on the vacuum. The one-photon momentum component is its restriction to sector one, identified with $M_{\mathrm{\hslash }{k}_j}^{\mathfrak{h}}$ of Restricted momentum and energy multipliers of the transverse one-photon space. The postulate assigns these mathematical operators their physical momentum meaning and units.

(e) Reading of the mode assignments. In the declared inertial chart the assignments $\omega (k)=c\mathrm{\mid }k\mathrm{\mid }$ in ${\mathrm{s}}^{-1}$, $E(k)=\mathrm{\hslash }\omega (k)$ in $\mathrm{J}$ and $p(k)=\mathrm{\hslash }k$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$ of clause (iii) of the radiation postulate are read through the spectral measures of clause (d): for a one-photon preparation of unit wave-packet vector $f\in \mathfrak{h}$, the assigned frequency, energy and momentum component laws are the push-forwards of the measure $\mathrm{\parallel }f(k){\mathrm{\parallel }}_{{\mathbb{C}}^3}^2{d}^{3}k$ under $k\mapsto c\mathrm{\mid }k\mathrm{\mid }$, $k\mapsto \mathrm{\hslash }c\mathrm{\mid }k\mathrm{\mid }$ and $k\mapsto \mathrm{\hslash }{k}_j$; these laws, and their finite means under the corresponding first-moment conditions, are proved on this page. Sharp frequency, sharp energy and sharp momentum-component states are not vectors of $\mathfrak{h}$; every unit one-photon vector in the corresponding multiplier domain has finite strictly positive variance in that quantity (clause (d) of Restricted momentum and energy multipliers of the transverse one-photon space). The push-forward laws themselves exist for every unit $f\in \mathfrak{h}$, without a moment-domain assumption; all their singleton probabilities vanish by the null level sets in the cited lemma, so they are non-Dirac laws even when finite means or variances do not exist. Thus "photon of energy $\mathrm{\hslash }\omega$" names a spectral assignment of the model, not a normalisable state with an exact energy.

(f) Scope. Clauses (b)-(e) fix the sector assignment and the units. They assert no detector model, no sampling model, no dynamics, no interaction with matter, no spontaneous emission and no empirical result; the counting and detection vocabulary is fixed separately on this page. The identification of a laboratory radiation field with a preparation of $F(\mathfrak{h})$ is a tested identification with uncertainty under P7, and no Lorentz transformation law for modes or their labels is adopted or claimed.