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	Errors pertaining to the following physical theories are discussed: the Dirac magnetic monopole theory; the Klein-Gordon equation; the Yukawa theory of nuclear force; the idea of Vector Meson Dominance; the Aharonov-Bohm effects; the idea of diffraction-free electromagnetic beams and Quantum Chromodynamics. Implications of the theoretical errors are discussed briefly. In particular, relations between the Dirac monopole theory, the idea of Vector Meson Dominance and Quantum Chromodynamics cast doubt on the current interpretation of strong interactions.		Errors pertaining to the following physical theories are discussed: the Dirac magnetic monopole theory; the Klein-Gordon equation; the Yukawa theory of nuclear force; the idea of Vector Meson Dominance; the Aharonov-Bohm effects; the idea of diffraction-free electromagnetic beams and Quantum Chromodynamics. Implications of the theoretical errors are discussed briefly. In particular, relations between the Dirac monopole theory, the idea of Vector Meson Dominance and Quantum Chromodynamics cast doubt on the current interpretation of strong interactions.

			==Overview==

			This 2006 ''Apeiron'' article by Eliahu Comay of Tel Aviv University is a survey of what its author regards as outright '''theoretical errors''' — not merely unattractive or unfashionable ideas — surviving inside mainstream twentieth-century physics. Six specific targets are examined in turn: Dirac's magnetic monopole theory, the Klein-Gordon equation as a fundamental quantum equation, the Yukawa interaction, Vector Meson Dominance (VMD), the Aharonov-Bohm (AB) effects, and the claim that diffraction-free electromagnetic beams exist. A seventh section catalogues experimental results that, in Comay's reading, [[Quark]]-based strong-interaction theory has not explained.

			What distinguishes the paper from ordinary criticism is the criterion it adopts. Comay does not argue that these theories are wrong because a rival theory says otherwise; he explicitly rejects that move as illegitimate. Instead he defines a theoretical error as "a mathematical part of a theory that yields predictions which are clearly inconsistent with experimental results" within the theory's own domain of validity — or, in the indirect sense, a set of axioms whose predictions contradict an already-established part of physics. The departure from the mainstream account is therefore methodological as much as substantive: the paper insists that established lower-rank theories impose binding constraints on higher-rank ones, and it applies that rule mechanically to structures the community treats as settled. Its final pages complain openly that physics journals have all but abandoned the publication of critical debate.

			==The argument==

			===Domains of validity and the rank of theories===

			The framing section sets out the machinery used throughout. A physical theory, like a mathematical one, rests on axioms and deduction, but it must additionally account for measurement. Its validity is assessed only over a limited set of experiments — its ''domain of validity''. If the domain ''D<sub>A</sub>'' of theory A is properly contained in the domain ''D<sub>B</sub>'' of theory B, then B holds the higher hierarchical rank; but A is not thereby "wrong", and crucially A '''constrains''' B, since B must reproduce A wherever A works. Comay calls this "restrictions imposed by a lower rank theory" and uses it repeatedly — the standard illustration being that relativistic mechanics must return Newtonian formulas for ''v'' ≪ ''c''. He also warns that neatness, simplicity and "physical acceptability" are subjective and secondary; a nineteenth-century physicist would have found relativity of length and time, non-Euclidean spacetime and quantum nonlocality all unphysical.

			===The Dirac monopole===

			Monopoles are defined by the duality transformation '''E''' → '''B''', '''B''' → −'''E''', ''e'' → ''g'', ''g'' → −''e''. Applying it to Maxwellian electrodynamics yields a ''dual'' Maxwellian theory of monopoles and fields with no electric charges. Any covering theory of charges ''and'' monopoles must therefore reduce to ordinary electrodynamics when monopoles vanish '''and''' to the dual theory when charges vanish. Comay's charge is that Dirac's theory fails the second requirement, and so violates a restriction imposed by a lower-rank theory. Dirac's construction, he argues, implicitly adds an unsupported axiom — that the fields of charges and of monopoles have identical dynamical properties — which forces a single 4-potential ''A''<sup>μ</sup> and hence the Dirac "string" singularity, since for regular '''A''' one has ∇·'''B''' = ∇·(∇×'''A''') = 0 and monopoles cannot exist. He adds older objections from the literature (inconsistency with S-matrix theory, with relativistic covariance) and the non-vanishing interaction angular momentum of a charge-monopole pair even as their separation tends to infinity. He notes Dirac's own late remark: "I am inclined now to believe that monopoles do not exist. So many years have gone by without any encouragement from the experimental side." Comay's conclusion is not that monopoles are impossible but that a ''regular'' charge-monopole theory can be built without the extra axiom, and that such a theory itself predicts the failure of the Dirac-monopole searches.

			===The Klein-Gordon equation===

			Comay concedes the formula (□ + ''m''<sup>2</sup>)φ = 0 is correct — components of Dirac solutions satisfy it. What he disputes is its status as a ''fundamental'' equation derived from the Pauli-Weisskopf Lagrangian density. He lists eight difficulties: no expression for particle density (only charge density, itself dependent on external particles' coordinates); a Hamiltonian density depending on ∂φ/∂''t''; no covariant differential operator serving as Hamiltonian, and a Hamiltonian matrix that destroys the Hilbert-space inner product; a second-order equation not identical to ''i''∂φ/∂''t'' = ''H''φ; no self-consistent electromagnetic interaction; no explanation why the energy-momentum operators are redeployed to represent charge density; and a nonrelativistic limit disagreeing with the Schrödinger equation — including a dimensional mismatch, [''L''<sup>−3/2</sup>] for Ψ against [''L''<sup>−1</sup>] for φ. Experimentally, he stresses that a wave function of one set of coordinates describes a pointlike object, whereas the historical KG candidates, the 0<sup>−</sup> π mesons, have a charge radius of 0.672 ± 0.008 fm and are quark-antiquark composites.

			===Yukawa, VMD, and the beam arguments===

			The Yukawa Lagrangian contains the KG Lagrangian and so inherits its problems; worse, its interaction term depends on the scalar density ψ̄ψ rather than the actual density ψ<sup>†</sup>ψ. Comay argues the classical limit ''V''(''r'') = −''g''<sup>2</sup>e<sup>−μ''r''</sup>/''r'' is incompatible with the relativistic orthogonality ''a''<sup>μ</sup>''v''<sub>μ</sub> = 0, which electrodynamics satisfies through the antisymmetry of ''F''<sup>μν</sup> but a scalar field cannot. Empirically the Yukawa potential never changes sign, whereas the nuclear force has a hard repulsive core, a tensor component and spin-orbit dependence.

			Against VMD — the proposal that an energetic photon's state is \|γ⟩ = ''c''<sub>0</sub>\|γ<sub>0</sub>⟩ + ''c''<sub>h</sub>\|h⟩ — he offers a thought experiment: two crossing optical rays that plainly do not exchange energy or momentum in the laboratory frame would, viewed from a frame moving fast transversely, consist of energetic photons carrying hadronic components and so ought to interact at the crossing point. A frame-dependent interaction is a contradiction.

			For the AB effects, Comay writes the system wave function as Ψ(''r''<sub>s</sub>, ''r''<sub>e</sub>) = φ<sub>1</sub>ψ<sub>1</sub> + φ<sub>2</sub>ψ<sub>2</sub>, so that phase is a property of a ''term'', not of a single particle. For the magnetic effect the ferromagnetic source is inert, φ factors out, and the prediction stands — he accepts it as correct and observed. For the electric effect the source's state changes, the two contributions cancel, and the effect disappears; retaining the single-particle treatment, he says, violates energy conservation. He also denies that the effects establish topology as inherent to quantum mechanics, since the fundamental two-body interaction is a sum over individual atoms and no genuinely field-free region exists.

			Finally, "diffraction-free" Bessel beams φ = e<sup>''i''β''z''</sup>''J''<sub>0</sub>(''aρ'') are rejected as inconsistent with the [[Uncertainty Principle]]: the alternating sign of the Bessel amplitude produces destructive interference at a circle in the wave zone, so energy conservation requires part of the beam to miss it — and since ''J''<sub>1</sub>(0) = 0, the φ-invariant Maxwell solutions predict a minimum of energy current at the beam centre, whereas experiments show a strong central peak.

			===Unexplained data===

			Section 8 lists six results Comay says textbook strong-interaction theory does not account for: the undetected Higgs mesons (as of 2004 data); the near-identical cross sections of hard photons on protons and neutrons; the narrower ''x''-width of antiquark structure functions, implying antiquarks occupy a larger volume than quarks; the absence of strongly bound pentaquarks (the Θ<sup>+</sup> at 1540 MeV lies above the 1435 MeV ''nK''<sup>+</sup> threshold); the uniform density of nuclear matter and the van der Waals-like form of the nuclear force; and the EMC effect.

			==Assessment==

			The paper's real strength is its discipline about what counts as a refutation. Comay states plainly that showing the existence of a contradictory theory F never establishes that theory E is erroneous, and he keeps to that: nearly every objection is either an internal inconsistency or a clash with a lower-rank result the mainstream itself accepts. The dimensional argument against the KG equation's nonrelativistic limit, the ''a''<sup>μ</sup>''v''<sub>μ</sub> = 0 objection to a scalar force law, and the ''J''<sub>1</sub>(0) = 0 point about the on-axis energy current of φ-invariant Bessel solutions are all sharply stated and checkable. His treatment of the AB effects is notably even-handed: he ''endorses'' the magnetic effect as theoretically correct and experimentally confirmed, and only the electric variant and the topological gloss are rejected. That willingness to grant ground is rare in critical literature and lends the rest more weight.

			The difficulties are of several kinds. First, much of the load is carried by references to Comay's own earlier papers — the "regular monopole theory", the analyses of the KG Lagrangian, of VMD and of the electric AB effect are all asserted here and derived elsewhere, so the article is closer to a summary brief than a self-contained demonstration. Second, several arguments turn on a strong reading of an idealisation rather than on how practitioners use it. "Diffraction-free" beams are universally understood as approximately propagation-invariant over a finite range, not literally non-diffracting to infinity; Comay says as much ("taking the diffraction-free idea literally") but then treats the literal reading as the target. Likewise, Vector Meson Dominance is presented in the literature as a phenomenological model, a point he acknowledges — and then converts into a separate complaint that the Standard Model therefore has no theoretical account of photon-hadron interaction, which understates the role of perturbative parton-level calculations at high ''Q''<sup>2</sup>.

			Third, and most importantly, parts of the empirical section have simply been overtaken. The Higgs boson, listed in 2006 as undetected after prolonged search, was observed at the LHC in 2012 at about 125 GeV — the single item in Section 8 that Comay presents as bearing on the Standard Model as a whole. The pentaquark case has moved the other way from his framing: the Θ<sup>+</sup> claims of the early 2000s were not confirmed, but LHCb has since reported pentaquark states in the charmonium sector. Neither reversal touches his internal arguments about the KG Lagrangian or the monopole duality constraint, but both are a reminder that "no adequate explanation in textbooks" is a claim with a shelf life. Finally, the pion-radius objection to KG particles establishes only that the historical candidates are composite; it does not by itself show that a genuinely elementary spin-0 field is impossible, and the Higgs discovery bears directly on that point. Where the paper is on firmest ground is the narrower charge that the second-order Lagrangian formalism does not deliver a probability density, a Hilbert-space Hamiltonian, or a clean Schrödinger limit in the way the [[Paul Dirac\|Dirac]] equation does — a criticism with a long and respectable history.

			==See also==

			* [[E Comay]]
			* [[Apeiron]]
			* [[Standard Model]]
			* [[Quark]]
			* [[Proton]]
			* [[Neutron]]
			* [[Photon]]
			* [[Uncertainty Principle]]
			* [[Quantum Electrodynamics]]
			* [[Maxwell's Equations]]
			* [[Lorentz Force]]
			* [[Paul Dirac]]

	[[Category:Scientific Paper\|theoretical errors contemporary physics]]		[[Category:Scientific Paper\|theoretical errors contemporary physics]]
			[[Category:Particle Physics]]
			[[Category:Quantum Theory]]
			[[Category:Electrodynamics]]
			[[Category:Philosophy of Science]]

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	\| published = 2006		\| published = 2006
	\| journal = [[Apeiron]]		\| journal = [[Apeiron]]
	\| volume = [[13]]		\| volume = 13
	\| number = [[2]]		\| number = 2
	\| num_pages = 34		\| num_pages = 34
	}}		}}

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	==Abstract==		==Abstract==

	Errors pertaining to the following physical theories are discussed: the Dirac magnetic monopole theory; the Klein-Gordon equation; the Yukawa theory of nuclear force; the idea of Vector Meson Dominance; the Aharonov-Bohm effects; the idea of diffraction-free electromagnetic beams and Quantum Chromodynamics. Implications of the theoretical errors are discussed briefly. In particular, relations between the Dirac monopole theory, the idea of Vector Meson Dominance and Quantum Chromodynamics cast doubt on the current interpretation of strong interactions.[[Category:Scientific Paper]]		Errors pertaining to the following physical theories are discussed: the Dirac magnetic monopole theory; the Klein-Gordon equation; the Yukawa theory of nuclear force; the idea of Vector Meson Dominance; the Aharonov-Bohm effects; the idea of diffraction-free electromagnetic beams and Quantum Chromodynamics. Implications of the theoretical errors are discussed briefly. In particular, relations between the Dirac monopole theory, the idea of Vector Meson Dominance and Quantum Chromodynamics cast doubt on the current interpretation of strong interactions.

			[[Category:Scientific Paper\|theoretical errors contemporary physics]]

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{{Infobox paper
| title = Theoretical Errors in Contemporary Physics
| url = [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_776.pdf Link to paper]
| author = [[E Comay]]
| keywords = [[Monopoles]], [[Klein-Gordon]], [[QCD]], [[Yukawa theory]], [[VMD]], [[Aharonov-Bohm]], [[Diffraction-free beams]]
| published = 2006
| journal = [[Apeiron]]
| volume = [[13]]
| number = [[2]]
| num_pages = 34
}}

'''Read the full paper''' [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_776.pdf here]

==Abstract==

Errors pertaining to the following physical theories are discussed: the Dirac magnetic monopole theory; the Klein-Gordon equation; the Yukawa theory of nuclear force; the idea of Vector Meson Dominance; the Aharonov-Bohm effects; the idea of diffraction-free electromagnetic beams and Quantum Chromodynamics. Implications of the theoretical errors are discussed briefly. In particular, relations between the Dirac monopole theory, the idea of Vector Meson Dominance and Quantum Chromodynamics cast doubt on the current interpretation of strong interactions.[[Category:Scientific Paper]]

Theoretical Errors in Contemporary Physics - Revision history

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