Free transverse-photon energy multiplier

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Use the transverse space ${\mathcal{H}}_T=\mathrm{Ran}{M}_{P_T}$ of Positive transverse photon one-particle space, with $P_T(\mathbf{k})=I_3-\mathbf{n}{\mathbf{n}}^T$ for $\mathbf{k}\mathrm{\ne }0$ and any assigned value at the null point $\mathbf{k}=0$. In the QFT length convention, assign $\mathbf{k}$ units m${}^{-1}$ and define the nonnegative measurable multiplier $\omega (\mathbf{k}):=\mathrm{\mid }\mathbf{k}\mathrm{\mid }\mathrm{.}$ Its maximal multiplication operator on ${\mathcal{H}}_T$ is $(M_{\omega }f)(\mathbf{k}):=\omega (\mathbf{k})f(\mathbf{k}),D(M_{\omega }):=\{f\in {\mathcal{H}}_T:\omega f\in L^2({\mathbb{R}}^3,d^3\mathbf{k};{\mathbb{C}}^3)\}\mathrm{.}$ Because $\omega$ is a scalar multiplier, $\mathbf{k}\cdot f(\mathbf{k})=0$ a.e. implies $\mathbf{k}\cdot (M_{\omega }f)(\mathbf{k})=0$ a.e.; thus the displayed multiplication maps its stated domain into ${\mathcal{H}}_T$. The value $\omega (0)=0$ is immaterial to the $L^2$ operator. For $\mathrm{\hslash },c>0$, the corresponding one-particle energy convention is $E_1:=\mathrm{\hslash }c{M}_{\omega }$, in joules.

The QFT-MG01 transverse-space draft is present and its exact projection-space definition was inspected, and its page manifest lists the draft supplier items. This local multiplier is conditional on that mathematical supplier. No photon polarization frame, interacting dynamics, or empirical photon-energy claim is part of this definition.