Positive transverse free photon model — Examples

The scale $\kappa$ in [F2] is a positive wave number. Apply [F1] with radii $\kappa \mathrm{/}8$ and $\kappa \mathrm{/}4$, then translate the result to the center of $\mathrm{\Omega }$. Translation preserves smoothness and support, so $b$ is smooth, bounded, and compactly supported in $\mathrm{\Omega }$; it equals one on an inner ball of positive volume. Thus $0<\mathrm{\parallel }b{\mathrm{\parallel }}_2<\mathrm{\infty }$ and $g=b\mathrm{/}\mathrm{\parallel }b{\mathrm{\parallel }}_2$ is the stated normalized wavepacket. Since the timelike coordinate has metric coefficient $-1$, direct substitution gives $[{ϵ}_{\mathrm{a}\mathrm{u}\mathrm{x}},{ϵ}_{\mathrm{a}\mathrm{u}\mathrm{x}}]=\int -\mathrm{\mid }g(\mathbf{k}){\mathrm{\mid }}^2{d}^3\mathbf{k}=-1$. On the support of $g$, the subsidiary expression is $-{ϵ}_{\mathrm{a}\mathrm{u}\mathrm{x}}^0+\mathbf{n}\cdot {\mathit{ϵ}}_{\mathrm{a}\mathrm{u}\mathrm{x}}=-g$, which is nonzero on the inner ball where $b=1$, a set of positive measure. Hence ${ϵ}_{\mathrm{a}\mathrm{u}\mathrm{x}}\mathrm{\notin }{\mathcal{K}}_0$ and it has no class in the physical quotient.

Countable Choice [A2] is carried by the selected Fock model. For every $\mathbf{k}\in \mathrm{\Omega }$, $k_3\ge 3\kappa \mathrm{/}4$ and $\mathrm{\mid }{k}_1\mathrm{\mid }\le \kappa \mathrm{/}4$, so $\mathrm{\mid }{n}_1\mathrm{\mid }=\mathrm{\mid }{k}_1\mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathbf{k}\mathrm{\mid }\le 1\mathrm{/}3$. Hence $\mathrm{\parallel }{P}_T(\mathbf{k})e_1{\mathrm{\parallel }}^2=1-\mathrm{\mid }{n}_1{\mathrm{\mid }}^2\ge 8\mathrm{/}9$ on $\mathrm{\Omega }$. The vector $u(\mathbf{k})=P_T(\mathbf{k})e_{1}g(\mathbf{k})$ is transverse and satisfies $\mathrm{\parallel }u{\mathrm{\parallel }}_{{\mathcal{H}}_T}^2={\int }_{\mathrm{\Omega }}\mathrm{\parallel }{P}_T{e}_1{\mathrm{\parallel }}^2\mathrm{\mid }g{\mathrm{\mid }}^2{d}^3\mathbf{k}\ge \frac{8}{9}\mathrm{\parallel }g{\mathrm{\parallel }}_2^2=\frac{8}{9}>0.$ Thus normalizing it gives $\mathrm{\parallel }u{\mathrm{\parallel }}_{{\mathcal{H}}_T}=1$. For every $ϵ\in {\mathcal{K}}_0$, [L1] gives $[ϵ,ϵ]=\mathrm{\parallel }Tϵ{\mathrm{\parallel }}_{{\mathcal{H}}_T}^2\ge 0$, and quotienting by $\mathcal{N}$ removes precisely the zero-norm radical. The physical wavepacket therefore has positive Hilbert norm, whereas the auxiliary negative norm belongs to no physical state.

The comparison is a hypothetical calculation in the adopted free model. It supplies no observed event, measurement outcome, or probability assignment to an auxiliary vector; the Born interpretation applies only to normalized states and supplied measurements in the physical model.

By [A3], ${\rho }_q=q_{\mathrm{S}\mathrm{I}}{\psi }_0^{†}{\psi }_0$ is constant; its SI density unit is $\mathrm{C}{\mathrm{m}}^{-3}$ by [F2] and the stated charge convention. Thus $Q=q_{\mathrm{S}\mathrm{I}}{\psi }_0^{†}{\psi }_0{L}^3<0$ because $q_{\mathrm{S}\mathrm{I}}<0$, ${\psi }_0^{†}{\psi }_0>0$, and $L>0$. A smooth periodic solution of [F1] would satisfy $\mathrm{div}{\mathit{\pi }}_A=-{\rho }_q$. Integrate this equality over the fundamental cube $[0,L{]}^3$. For each coordinate $i$, the integral of ${\mathrm{\partial }}_i({\pi }_A{)}_i$ in that coordinate is the value on the face $x_i=L$ minus the value on $x_i=0$; these are equal by periodicity, so each face difference is zero. Fubini for the smooth functions on the compact cube then gives ${\int }_{{\mathbb{T}}_L^3}\mathrm{div}{\mathit{\pi }}_A{d}^3\mathbf{x}=0$. It follows that $Q={\int }_{{\mathbb{T}}_L^3}{\rho }_q{d}^3\mathbf{x}=0$, contradicting the strict inequality in this step.

Independently, [L1] records the same compatibility requirement for the Coulomb equation: a smooth periodic potential can exist only for zero-mean charge density. Thus the contradiction is a consequence of the specified periodic model and boundary conditions; adding a boundary, puncture, or compensating source changes the hypotheses and is not covered.