View source for The Significance of Density in the Structure of Quantum Theories

{{Infobox paper
| title = The Significance of Density in the Structure of Quantum
Theories
| url = [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_777.pdf Link to paper]
| author = [[E Comay]]
| keywords = [[density]], [[quantum mechanics]], [[Hamiltonian]], [[Dirac equation]], [[Klein-Gordon equation]]
| published = 2007
| journal = [[Apeiron]]
| volume = 14
| number = 2
| num_pages = 15
}}

'''Read the full paper''' [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_777.pdf here]

==Abstract==

It is proved that density plays a crucial role in the structure of quantum field theory. The Dirac and the Klein-Gordon equations are examined. The results prove that the Dirac equation is consistent with density related requirements whereas the Klein-Gordon equation fails to do that. Experimental data support these conclusions.

==Overview==

Eliyahu Comay's paper, published in ''[[Apeiron]]'' vol. 14 no. 2 (April 2007), is a narrowly focused technical argument with a broad target. Its subject is the humble notion of ''density'' — the quantity whose spatial integral normalizes a wave function — and its claim is that this quantity is treated carelessly in standard quantum field theory. Comay's thesis is that a physically self-consistent density is a precondition for the whole construction that leads from a field equation to a Hamiltonian, that the Dirac field satisfies it, and that the Klein-Gordon (KG) field does not.

The departure from the textbook position is sharp but confined. Comay does not challenge [[Special Relativity]], quantum mechanics or the formalism of Lagrangian field theory; he accepts all of them and argues within them. What he rejects is the standing consensus that the KG equation describes a physically meaningful field. He aligns himself explicitly with [[Paul Dirac]], who "maintained his opinion stating that the KG equation has no physical merits," and against the modern textbooks — Bjorken and Drell, Weinberg, Peskin and Schroeder are all cited by page — which he says take the validity of a density expression "for granted when the Hamiltonian is derived from the Hamiltonian density."

==The argument==

===Why density is unavoidable===

The paper opens with the simplest possible physical state: a massive particle at rest and free of external fields. The uncertainty relations mean its position is not sharp, so it occupies some volume, so a quantum-mechanical description of that state requires an expression for its density. And because the wave functions of Hilbert space are the elements from which Fock space is built, the same requirement carries over from quantum mechanics into quantum field theory. Comay notes in the conclusions that this argument does not apply to photons, which are massless and have no rest state.

The formal chain is then laid out. One begins with a field equation ''Ô''&psi; = 0; a Lagrangian density &#8466; is defined whose action ''I'' = &int;&#8466; ''d''<sup>4</sup>''x'' reproduces that equation under variation; a Hamiltonian density &#8459; follows; and the Hamiltonian is its spatial integral, ''H'' = &int;&#8459; ''d''<sup>3</sup>''x''. Comay's point is that this passage silently changes the status of the wave function: in the field equation &psi; is "a complex mathematical function," but in the Lagrangian density "it acquires dimensions." In units with &#8463; = ''c'' = 1 the action is dimensionless and ''d''<sup>4</sup>''x'' has dimension [''L''<sup>4</sup>], so &#8466; must have dimension [''L''<sup>&minus;4</sup>], which fixes the dimension of the field. Moreover the Euler-Lagrange equation is unchanged if &#8466; is multiplied by a number, whereas the Hamiltonian represents energy and must take a definite eigenvalue — so normalization, and therefore density, is doing indispensable work.

===The three requirements===

Since the integral of density is a Lorentz scalar (the particle is found in every frame), Comay imports the conditions satisfied by charge density in electrodynamics:

* '''A.''' The dimension of density is [''L''<sup>&minus;3</sup>].
* '''B.''' Density is the 0-component of a 4-vector ''j''<sup>&mu;</sup>.
* '''C.''' That 4-vector satisfies the continuity equation ''j''<sup>&mu;</sup><sub>;&mu;</sub> = 0.

These are old and uncontroversial. Comay's new claim is that they are '''necessary but not sufficient''' — and the KG field is his demonstration of that gap.

===The Dirac field passes===

For the Dirac Lagrangian density &#8466; = &psi;&#772;[&gamma;<sup>&mu;</sup>(''i''&part;<sub>&mu;</sub> &minus; ''eA''<sub>&mu;</sub>) &minus; ''m'']&psi;, the conserved 4-current is ''j''<sup>&mu;</sup> = &psi;&#772;&gamma;<sup>&mu;</sup>&psi; with density &rho; = &psi;<sup>&dagger;</sup>&psi;. Comay lists five consequences. The current depends only on &psi; and &psi;&#772; and not on the external field ''A''<sub>&mu;</sub>, so the positive-definite density &psi;<sup>&dagger;</sup>&psi; yields an orthonormal Hilbert-space basis unaffected by changes in external quantities. Because the Dirac Lagrangian density is linear in &part;&psi;/&part;''t'', the Hamiltonian density contains no time derivatives of &psi;, and the density factors cleanly out to leave a genuine differential operator ''H'' = &alpha;&middot;(&minus;''i''&nabla; &minus; ''eA'') + &beta;''m'' + ''eV''. That operator is free of &psi;, &psi;&#772; and their derivatives, so substituting it into ''H''&psi; = ''i''&part;&psi;/&part;''t'' respects linearity and superposition and yields an explicit first-order equation. It agrees with the Euler-Lagrange equation, imposing no extra restrictions on eigenfunctions or eigenvalues. And the ''eA''<sub>&mu;</sub> term correctly represents electromagnetic interaction.

===The complex Klein-Gordon field fails===

For the complex KG field Comay takes the Lagrangian density and Hamiltonian density from Pauli and Weisskopf (1934), whose density is &rho; = ''i''(&phi;*&phi;<sub>;0</sub> &minus; &phi;*<sub>;0</sub>&phi;) &minus; 2''eV''&phi;*&phi;. This expression can be positive or negative, which is why it is normally called a charge density; and it does satisfy requirements A&ndash;C. Two problems follow.

The first is structural. The KG Hamiltonian density contains time derivatives of the wave function, so the Hamiltonian density would be expressed in terms of the Hamiltonian while the Hamiltonian is defined as an integral of the density — which Comay calls "an undesirable situation." He then cites his own earlier proof (Comay 2005) that no covariant differential operator for the Hamiltonian exists here: the highest time derivatives in the Hamiltonian density appear as the ''symmetric'' product &phi;*<sub>;0</sub>&phi;<sub>;0</sub>, while the density contains the ''antisymmetric'' combination &phi;*&phi;<sub>;0</sub> &minus; &phi;*<sub>;0</sub>&phi;.

The second is the paper's own centrepiece, a concrete counterexample. Take two states of a positively charged particle in spherical coordinates, &phi;<sub>0</sub> = ''e''<sup>&minus;''i''&omega;<sub>0</sub>''t''</sup>''f''<sub>0</sub>(''r'')''Y''<sub>00</sub> and &phi;<sub>1</sub> = ''e''<sup>&minus;''i''&omega;<sub>1</sub>''t''</sup>''f''<sub>1</sub>(''r'')''Y''<sub>10</sub>, with radial functions belonging to the lowest energy of each angular momentum, so that ''f''<sub>''i''</sub>(''r'') &ge; 0 and does not change sign. With no external potential the inner product built from the KG density vanishes, because ''Y''<sub>00</sub> and ''Y''<sub>10</sub> are orthogonal: the two states are orthogonal in Hilbert space. Now let an external positive charge approach the origin along the ''z''-axis from ''z'' > 0. The potential ''V'' now varies over space-time, and the last term of the density contributes ''U'' = &int;&minus;2''e''&phi;<sub>0</sub>''V''&phi;<sub>1</sub>''r''<sup>2</sup>sin&theta; ''dr d''&theta; ''d''&phi;. Comparing the integrand at ''P''<sub>1</sub>(''r'',&theta;,&phi;) and ''P''<sub>2</sub>(''r'',&pi;&minus;&theta;,&phi;) with &theta; < &pi;/2: the product &phi;<sub>0</sub>&phi;<sub>1</sub> has the same magnitude but opposite sign at the two points (''Y''<sub>10</sub> carries a cos&theta; factor, ''Y''<sub>00</sub> is isotropic), while ''V'' is larger at ''P''<sub>1</sub> because the approaching charge is nearer. The two hemispheres therefore fail to cancel and ''U'' > 0 — contradicting the null result. Orthogonality is destroyed by an external field, so no self-consistent inner product exists, and no Hamiltonian matrix can be built.

The real KG field fares no better: setting &phi;* = &phi; in the density makes it vanish identically, consistent with a real field carrying no charge. Finally, a dimensional discrepancy: the operator in the Dirac Lagrangian density has dimension [''L''<sup>&minus;1</sup>], giving the Dirac field [''L''<sup>&minus;3/2</sup>], whereas the KG operator has dimension [''L''<sup>&minus;2</sup>], giving the KG field [''L''<sup>&minus;1</sup>]. Since the Schrödinger density is &psi;*&psi; with &psi; of dimension [''L''<sup>&minus;3/2</sup>], Comay concludes that the nonrelativistic limit of the KG equation disagrees with the Schrödinger equation.

===The experimental appeal===

Comay closes by arguing that the long theoretical controversy is now settled empirically. Because &phi; depends on a single set of space-time coordinates, the KG field — like the Dirac field — describes a structureless pointlike particle. But no pointlike KG particle has been established: the &pi; mesons, "regarded as the primary example of a KG particle," are quark-antiquark composites, whereas the known Dirac particles (electrons, muons, quarks) are pointlike as far as measurement goes. Nature, he suggests, has not supplied the objects the equation was written for.

==Assessment==

The paper's real strength is that it does not argue by assertion. The Pauli-Weisskopf density is taken from the literature rather than constructed for the purpose; the counterexample is explicit and checkable, resting only on the parity of ''Y''<sub>10</sub> and the monotonic distance dependence of a Coulomb potential; and the conclusion is stated as a limited technical result rather than a wholesale rejection of quantum theory. The dimensional argument is clean and, as far as it goes, correct: the KG field really does have mass dimension one where the Dirac field has three halves, and the point that this complicates a naive probability-density reading of the nonrelativistic limit is a real one. The observation that requirements A&ndash;C are necessary but not sufficient is also worth making; it is easy to check a current for covariance and conservation and infer more than that check delivers. Comay's empirical remark is likewise accurate as stated — the pion is a composite, and no elementary spin-0 particle other than the Higgs boson has been observed.

The difficulties concern the target rather than the algebra. The strongest is that the modern interpretation of the KG equation does not treat &phi; as a single-particle wave function at all, and does not read the Pauli-Weisskopf &rho; as a probability density: after second quantization &phi; is an operator-valued field, &rho; is a ''charge'' density which is expected to be indefinite, and normalization is carried by the Fock-space states rather than by the classical field. Comay's proof that no self-consistent single-particle inner product exists for the complex KG field is thus a demonstration of something quantum field theory already concedes — indeed the failure of a positive-definite single-particle density is the standard textbook motivation for abandoning the single-particle reading. The paper does not engage with that response, and its dimensional argument inherits the same gap: the KG field's dimension is fixed by the two-derivative kinetic term precisely because it is not meant to be a normalizable amplitude.

Two further steps are asserted rather than established. The claim that "a massive particle can be in a motionless state and a physical theory should be able to describe its location" is offered as a physical requirement, but in a relativistic theory localization below the Compton wavelength is precisely where single-particle language breaks down, and no argument is given for why the requirement should survive that. The experimental appeal is also weaker than presented: the absence of an elementary spin-0 particle at the time of writing is a fact about the observed spectrum, not about the consistency of the equation, and the subsequent discovery of the Higgs boson at the LHC — a scalar excitation whose propagator is governed by exactly the KG operator — removes the empirical leg of the argument as it was stated in 2007. What survives, and survives intact, is the paper's narrower methodological point: that textbooks pass from Hamiltonian density to Hamiltonian without pausing on what density means, and that the passage deserves the discussion Comay gives it.

==See also==

* [[E Comay]]
* [[Paul Dirac]]
* [[Quantum mechanics]]
* [[Quantum Electrodynamics]]
* [[Apeiron]]
* [[Uncertainty Principle]]

[[Category:Scientific Paper|significance density structure quantum theories]]

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