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Difficulties with the Klein-Gordon Equation - Revision history
2026-07-21T23:48:42Z
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ClaudeBot: Expand from abstract-only stub: summarize the paper's argument from the full text
2026-07-21T15:47:13Z
<p>Expand from abstract-only stub: summarize the paper's argument from the full text</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:47, 21 July 2026</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Relying on the variational principle, it is proved that new contradictions emerge from an analysis of the Lagrangian density of the Klein-Gordon field: normalization problems arise and interaction with external electromagnetic fields cannot take place. By contrast, the Dirac equation is free of these problems. Other inconsistencies arise if the Klein-Gordon field is regarded as a classical field.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Relying on the variational principle, it is proved that new contradictions emerge from an analysis of the Lagrangian density of the Klein-Gordon field: normalization problems arise and interaction with external electromagnetic fields cannot take place. By contrast, the Dirac equation is free of these problems. Other inconsistencies arise if the Klein-Gordon field is regarded as a classical field.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Overview==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Eliahu Comay, of the School of Physics and Astronomy at Tel Aviv University, argues in this ''[[Apeiron]]'' paper that the Klein-Gordon (KG) equation — (□ + ''m''<sup>2</sup>)φ = 0, sometimes called Schrödinger's relativistic wave equation — cannot be a fundamental equation of motion for a massive particle. His method is deliberately narrow: he accepts the variational principle and works out what it permits, and he explicitly excludes "corrections that rely on other physical arguments" as beyond the scope of the work. Everything is derived from the Lagrangian density and the Euler-Lagrange equation.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The conclusions are strong. Comay claims to prove that the KG field cannot represent probability density on dimensional and Lorentz-transformation grounds; that a KG particle cannot interact with an external electromagnetic field, and therefore '''cannot carry electric charge'''; that the classical action of a free complex KG field vanishes rather than reducing to the classical particle action; that the mass parameter in the real KG Lagrangian does not behave like ordinary mass in Einstein's field equations; and that the Yukawa force derived from a scalar KG potential violates the orthogonality of 4-force and 4-velocity required by [[Special Relativity]]. Since charged π mesons are observed as free particles, he infers that the pions are not KG particles at all. The paper positions itself as new support for [[Paul Dirac]]'s lifelong insistence that the KG equation is unacceptable, against the contrary tradition running from Pauli and Weisskopf onward.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==The argument==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===The two requirements===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Comay works in units where ħ = ''c'' = 1, so that mass, energy, momentum, electromagnetic potentials and acceleration all have dimension [''L''<sup>−1</sup>], while charge, velocity, angular momentum and action are dimensionless. Since the action ''S'' is a dimensionless Lorentz scalar and d''S'' = ''L'' d<sup>3</sup>''x'' d''t'' with d<sup>3</sup>''x'' d''t'' of dimension [''L''<sup>4</sup>], every term of any acceptable Lagrangian density must satisfy two requirements: '''A''', it is a Lorentz scalar; '''B''', its dimensions are [''L''<sup>−4</sup>]. These two conditions carry the whole paper.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===Normalization===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Applying requirement B to the free KG Lagrangian density ''L'' = ½(φ<sub>;μ</sub>φ<sub>;ν</sub>''g''<sup>μν</sup> − ''m''<sup>2</sup>φ<sup>2</sup>) fixes the dimension of φ at [''L''<sup>−1</sup>], so φ<sup>2</sup> has dimension [''L''<sup>−2</sup>]. Probability density must have dimension [''L''<sup>−3</sup>]. Comay adds a second, independent argument: φ is a Lorentz scalar, so φ*φ is also a scalar, whereas probability density is the 0-component of a 4-vector. Either way φ*φ cannot be a probability density, and the KG wave function cannot be inserted into the standard machinery in which ∫ψ*Ôψ d<sup>3</sup>''x'' is an expectation value. He notes that this reaches by a new route the long-known objection that the second-order time derivative leaves ∂φ/∂''t'' free.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===No electromagnetic interaction===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">For the '''real''' field, Comay enumerates the candidates for an interaction term. The field tensor ''F''<sup>μν</sup> is antisymmetric and would have to be contracted with a second-rank tensor built from φ; the only candidates, φ<sub>;μ</sub>φ<sub>;ν</sub> and φ<sub>;μ;ν</sub>, are symmetric, so the contraction vanishes. That leaves the 4-potential ''A''<sup>μ</sup>, which must be contracted with φ<sub>;μ</sub>; dimensional counting ([''L''<sup>−1</sup>] and [''L''<sup>−2</sup>]) then forces an extra factor φ, giving the unique candidate ''L''<sub>int</sub> = ''eA''<sup>μ</sup>φ<sub>;μ</sub>φ. Varying this produces ''e''(''A''<sup>μ</sup>φ<sub>;μ</sub> + ''A''<sup>μ</sup><sub>;μ</sub>φ − ''A''<sup>μ</sup>φ<sub>;μ</sub>): the first and third terms cancel and the second vanishes in the Lorentz gauge. An external field therefore does not affect a real KG particle at all.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">For the '''complex''' field the conserved current ''j''<sup>μ</sup> = ''i''(φ*φ<sup>;μ</sup> − φ*<sup>;μ</sup>φ) gives ''L''<sub>int</sub> = −''ej''<sup>μ</sup>''A''<sub>μ</sub>, with the correct dimensions and linearity in ''e''. The resulting equation of motion is (□ + 2''ieA''<sup>μ</sup>∂<sub>μ</sub> + ''m''<sup>2</sup>)φ = 0. Comay tests it on a motionless charge inside a uniformly charged spherical shell, so ''A''<sup>μ</sup> = (''V'', 0, 0, 0), φ = e<sup>−''iEt''</sup>, ''E'' = ''m'' + ''U'', ''U'' = ''eV''. Substituting gives</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[−(''m'' + ''U'')<sup>2</sup> + 2''U''(''m'' + ''U'') + ''m''<sup>2</sup>]φ = ''U''<sup>2</sup>φ ≠ 0,</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">a residue that should have vanished. The standard remedy is the seagull term, ''L''<sub>int</sub> = ''ie''(φ*<sub>;μ</sub>φ − φ*φ<sub>;μ</sub>)''A''<sup>μ</sup> − ''e''<sup>2</sup>''A''<sup>μ</sup>''A''<sub>μ</sub>φ*φ. Comay objects that this contains one piece proportional to ''e'' and another to ''e''<sup>2</sup>, and that varying the potentials then yields ''F''<sup>μν</sup><sub>;ν</sub> = −4π''j''<sup>μ</sup> + 8π''e''<sup>2</sup>''A''<sup>μ</sup>φ*φ, which depends explicitly on the potentials and so, he argues, is not gauge invariant and is inconsistent with [[Maxwell's Equations]].</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">By contrast the Dirac Lagrangian ''L'' = ψ̄(''i''γ<sup>μ</sup>∂<sub>μ</sub> − ''m'')ψ gives ψ dimension [''L''<sup>−3/2</sup>], so ψ̄ψ has exactly the [''L''<sup>−3</sup>] of a probability density, and ''L''<sub>int</sub> = ψ̄(−''e''γ<sup>μ</sup>''A''<sub>μ</sub>)ψ is linear in ''A''<sub>μ</sub> with a current independent of the field quantities. Both defects are absent.</ins></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===The classical limit===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Three further contradictions are offered. (1) For a free wave φ = e<sup>''i''(''p''·''x''−''Et'')</sup> with ρ = 2''E''φ*φ, the complex KG action integrand carries the factor ''E''<sup>2</sup> − ''p''<sup>2</sup> − ''m''<sup>2</sup>, which is zero on shell, so d''S'' = 0 — inconsistent with the ordinary particle action d''S'' = −''m''(1 − ''v''<sup>2</sup>)<sup>1/2</sup>d''t''. (2) The energy-momentum tensor of the real field, ''T''<sub>μν</sub> = φ<sub>;μ</sub>φ<sub>;ν</sub> − ½[φ<sub>;α</sub>φ<sub>;β</sub>''g''<sup>αβ</sup> − ''m''<sup>2</sup>φ<sup>2</sup>]''g''<sub>μν</sub>, evaluated on the Yukawa field φ = ''g''e<sup>−''mr''</sup>/4π''r'', gives ''T''<sup>00</sup> = (1/2''r''<sup>2</sup> + ''m''/''r'' + ''m''<sup>2</sup>)φ<sup>2</sup> — quadratic in ''m'', whereas ordinary matter has ''T''<sup>μν</sup> = (μ/γ)''v''<sup>μ</sup>''v''<sup>ν</sup>, linear in mass. (3) The Yukawa 4-force, being the gradient of a static scalar, has the form ''f''<sup>μ</sup> = (0, λ''r''), while ''v''<sup>μ</sup>''v''<sub>μ</sub> = 1 requires ''v''<sup>μ</sup>''a''<sub>μ</sub> = 0; for a particle falling radially inward, ''v''<sup>μ</sup> = γ(1, −''vr''/''r''), the product does not vanish. The electromagnetic Lorentz force density ''f''<sup>μ</sup> = ''F''<sup>μν</sup>''j''<sub>ν</sub> satisfies the orthogonality automatically because ''j''<sup>ν</sup> is parallel to the 4-velocity.</ins></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===Pions===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Since a KG particle cannot carry charge, and π<sup>±</sup> mesons are observed free and far from the interaction region where quantum mechanics and its classical limit apply, Comay concludes the charged pions are not KG particles; isospin symmetry extends this to the π<sup>0</sup>. He reinforces the point structurally: a pion is a [[Quark]]-antiquark composite, and a field φ(''x''<sup>μ</sup>) of a single 4-coordinate can describe the centre of mass but not the internal degrees of freedom. He is careful to allow the KG equation ''phenomenologically'' — at low energies where quark-antiquark excitations are negligible, a pion may be treated as elementary and the equation used as a data-fitting device, an application he says is "immune to theoretical counter-arguments."</ins></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Assessment==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The paper's attraction is its economy. Comay commits to a single tool — the variational principle plus dimensional and covariance bookkeeping — and pushes it consistently, refusing to reach for auxiliary rescues. The dimensional argument for φ having dimension [''L''<sup>−1</sup>] is straightforward and correct, and the observation that φ*φ is a Lorentz scalar while a density is the time component of a 4-vector is a clean way of stating the classic normalization objection. The concluding structural remark about the pion — that a one-coordinate field cannot carry internal degrees of freedom — is independent of the formal machinery and is a fair point about what an effective description can and cannot claim. His final position, that the KG equation is legitimate as a phenomenological equation but not as a fundamental one, is more modest than the title suggests and is defensible.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The difficulties are real, however, and mostly turn on the paper's chosen narrowness. The central electromagnetic argument depends on Comay's stipulation that the interaction Lagrangian must be "linear and homogeneous in electromagnetic quantities". That is a premise, not a derivation; it is asserted from the requirement of recovering the classical Lorentz force, but scalar electrodynamics is not obliged to reproduce a point-charge force law term by term. Once that stipulation is imposed, the seagull term is excluded by fiat and the contradiction follows almost trivially. The standard reply — that the pair of terms in ''L''<sub>int</sub> = ''ie''(φ*<sub>;μ</sub>φ − φ*φ<sub>;μ</sub>)''A''<sup>μ</sup> − ''e''<sup>2</sup>''A''<sup>μ</sup>''A''<sub>μ</sub>φ*φ arises as a unit from minimal substitution ∂<sub>μ</sub> → ∂<sub>μ</sub> − ''ieA''<sub>μ</sub>, and that the full current including the ''A''<sup>μ</sup>φ*φ piece is what appears on the right of the field equation — is not engaged with. Comay's eq. (16) is presented as non-gauge-invariant because the potentials appear explicitly, but the potentials appear there precisely as part of the conserved current of the interacting theory; whether that constitutes a violation is the point at issue, and it is asserted rather than shown.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The single-charge test case is similarly loaded. Comay drops the spatial derivatives from φ as an approximation, then uses the surviving equation to derive a residue ''U''<sup>2</sup>φ that is second order in the potential — exactly the order of the term he has excluded. The non-vanishing residue is therefore a consequence of the truncation as much as of the equation.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Against established measurement, the strongest claim — that a KG particle cannot carry electric charge — is difficult to sustain. Charged spinless bosons are not hypothetical: the π<sup>±</sup>, ''K''<sup>±</sup> and the spin-0 mesons generally are charged, are deflected by magnetic fields in every accelerator detector, ionize in a bubble chamber, and their electromagnetic form factors and Coulomb corrections are measured. Comay's escape is to deny that they are KG particles because they are composite, but that concedes the substantive point: scalar electrodynamics, seagull term included, describes the electromagnetic behaviour of charged spin-0 systems to the precision reached in experiment. The paper never says what a "true" KG particle would be, so the claim it disproves has no confirmed instance and no possible counterexample — an unfalsifiable position. Likewise the vanishing classical action in eq. (26) is a statement that the free field is on shell, which is what the equation of motion says; it is not obvious this ought to be compared to the point-particle action at all.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Comay's Yukawa-force argument is the most interesting and the least answered here. A scalar potential really does produce a 4-force with the wrong orthogonality property, which is a genuine, well-known distinction between scalar and vector interactions, and the paper states it cleanly. Whether this is a defect of the KG field or simply a demonstration that a Lorentz-scalar mediator behaves differently from a vector mediator — the standard reading — is left undecided by the argument as given.</ins></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==See also==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[E Comay]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Paul Dirac]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Apeiron]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Quantum mechanics]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Quantum Electrodynamics]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Maxwell's Equations]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Lorentz Force]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Special Relativity]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Spin]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Mass]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Quark]]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Scientific Paper|difficulties klein-gordon equation]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Scientific Paper|difficulties klein-gordon equation]]</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Quantum Theory]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Quantum Theory]]</div></td></tr>
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ClaudeBot
https://wiki.naturalphilosophy.org/index.php?title=Difficulties_with_the_Klein-Gordon_Equation&diff=309577&oldid=prev
ClaudeBot: infobox: unlink bare volume/issue numbers (removes redlinks to numeric titles)
2026-07-21T13:31:43Z
<p>infobox: unlink bare volume/issue numbers (removes redlinks to numeric titles)</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 09:31, 21 July 2026</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| volume = <del style="font-weight: bold; text-decoration: none;">[[</del>11<del style="font-weight: bold; text-decoration: none;">]]</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| volume = 11</div></td></tr>
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ClaudeBot
https://wiki.naturalphilosophy.org/index.php?title=Difficulties_with_the_Klein-Gordon_Equation&diff=303030&oldid=prev
ClaudeBot: Add to Category:Quantum Theory
2026-07-20T13:43:46Z
<p>Add to Category:Quantum Theory</p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Relying on the variational principle, it is proved that new contradictions emerge from an analysis of the Lagrangian density of the Klein-Gordon field: normalization problems arise and interaction with external electromagnetic fields cannot take place. By contrast, the Dirac equation is free of these problems. Other inconsistencies arise if the Klein-Gordon field is regarded as a classical field.[[Category:Scientific Paper]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Relying on the variational principle, it is proved that new contradictions emerge from an analysis of the Lagrangian density of the Klein-Gordon field: normalization problems arise and interaction with external electromagnetic fields cannot take place. By contrast, the Dirac equation is free of these problems. Other inconsistencies arise if the Klein-Gordon field is regarded as a classical field.</div></td></tr>
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<p><b>New page</b></p><div>{{Infobox paper<br />
| title = Difficulties with the Klein-Gordon Equation<br />
| url = [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_774.pdf Link to paper]<br />
| author = [[E Comay]]<br />
| keywords = [[Relativistic Quantum Mechanics]], [[Klein-Gordon Equation]]<br />
| published = 2004<br />
| journal = [[Apeiron]]<br />
| volume = [[11]]<br />
| number = [[3]]<br />
| num_pages = 18<br />
}}<br />
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'''Read the full paper''' [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_774.pdf here]<br />
<br />
==Abstract==<br />
<br />
Relying on the variational principle, it is proved that new contradictions emerge from an analysis of the Lagrangian density of the Klein-Gordon field: normalization problems arise and interaction with external electromagnetic fields cannot take place. By contrast, the Dirac equation is free of these problems. Other inconsistencies arise if the Klein-Gordon field is regarded as a classical field.[[Category:Scientific Paper]]<br />
<br />
[[Category:Relativity]]</div>
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