CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Nonzero plane waves lie outside the finite-energy angular-momentum domain

Physical scope

Ideal nonzero monochromatic plane waves in source-free classical vacuum electromagnetism on all of R^3, in the SI laboratory chart. The claim concerns only the domain of finite-energy total angular-momentum functionals; it does not reinterpret a plane wave by box normalization or claim a finite-beam approximation.

Statement refuted

Every nonzero monochromatic plane wave in vacuum belongs to the Schwartz Fourier-packet domain on which the all-space angular-momentum functionals are defined.

Statement references

Coulomb-gauge canonical and kinetic integrals

Real plane-wave fields encoded by complex amplitudes

Dispersion, transversality and the two polarization conditions for plane-wave solutions

Period-averaged energy density, Poynting vector and momentum flux of a plane wave

Facts & Assumptions

Given: The positive dimensional scales $κ$ and $E_{0}$ , the coordinate unit vectors, $k = κ e_{3}$ , $ω = c κ$ , $e = E_{0} e_{1}$ , $b = (E_{0} / c) e_{2}$ , and the corresponding real plane wave.

[F1]

The plane-wave encoding is $E = Re (e \exp (i (k \cdot x - ω t)))$ and $B = Re (b \exp (i (k \cdot x - ω t)))$ ; its fields are identically zero exactly when both amplitudes vanish. (Real plane-wave fields encoded by complex amplitudes)

[F2]

For these transverse vacuum modes, the exact period average is $⟨ u ⟩ = ε_{0} ∣ e ∣^{2} / 2$ , and every nonzero amplitude has infinite all-space box-exhaustion energy. (Period-averaged energy density, Poynting vector and momentum flux of a plane wave)

[F3]

With $ω = c ∣ k ∣$ , $k \cdot e = 0$ , and $b = k \times e / ω$ , these plane waves satisfy the smooth source-free Maxwell equations; the plane-wave amplitudes determine the fields, and the pair is nonzero when $e \neq 0$ . (Dispersion, transversality and the two polarization conditions for plane-wave solutions) The SI conventions give $c > 0$ and $ε_{0} > 0$ . (Framework-local SI units, the vacuum constants and the exact/measured status of their values)

[F4]

The angular-momentum functionals in question are defined on the real Schwartz vacuum packet fields of the Coulomb-gauge construction; their integrands are globally integrable on that domain, and every field in the Fourier-packet class has finite total energy. (Coulomb-gauge canonical and kinetic integrals, Classical transverse Fourier amplitudes)

Counterexample

Choose $κ > 0$ in $m^{- 1}$ and $E_{0} > 0$ in $V m^{- 1}$ . Set $k = κ e_{3}$ , $ω = c κ$ , $e = E_{0} e_{1}$ , and $b = (E_{0} / c) e_{2}$ , where $e_{1} = (1, 0, 0)$ , $e_{2} = (0, 1, 0)$ and $e_{3} = (0, 0, 1)$ are the positively oriented coordinate unit vectors. Let $E, B$ be the real monochromatic plane-wave fields with phase $k \cdot x - ω t$ .

1.1givenF1F2F3algebra

Since $E_{0} > 0$ , $e \neq 0$ , and direct calculation gives $k \cdot e = 0$ , $k \times e / ω = (E_{0} / c) e_{2} = b$ , and $ω = c ∣ k ∣$ . Therefore [F3] gives a smooth source-free Maxwell solution and [F1] makes it nonzero. By [F2], its period-averaged energy density is $ε_{0} E_{0}^{2} / 2 > 0$ and its total energy over $R^{3}$ is infinite.

2.1step 1.1F4algebra∎

Every member of the Fourier-packet domain in [F4] has finite energy. Thus the nonzero wave of step 1.1 does not belong to that domain, refuting the universal statement. Componentwise formal cancellations do not make these all-space functionals defined on this field.

Example references