Further Difficulties with the Klein-Gordon Equation
| Scientific Paper | |
|---|---|
| Title | Further Difficulties with the Klein-Gordon Equation |
| Read in full | Link to paper |
| Author(s) | E Comay |
| Keywords | Relativistic Quantum Mechanics, Dirac Equation, Klein-Gordon Equation, Hamiltonian |
| Published | 2005 |
| Journal | Apeiron |
| Volume | 12 |
| Number | 1 |
| No. of pages | 21 |
Read the full paper here
Abstract
Herein, the Dirac equation is compared with the Klein-Gordon equation. In contrast to the Dirac case, it is proved that the Klein-Gordon equation has difficulties with the Hamiltonian differential operator of relativistic quantum mechanics and with the definition of an inner product of wave functions, which is a requirement for a construction of a Hilbert space. An added discussion of the Pauli-Weisskopf article and that of Feshbach-Villars proves that their theories lack a self-consistent expression for the Hamiltonian. Related difficulties are pointed out.
Overview
E Comay, of the School of Physics and Astronomy at Tel Aviv University, wrote this Apeiron paper in 2005 as a continuation of an earlier one in the same journal. Its method is comparative: rather than attacking the Klein-Gordon (KG) equation in isolation, Comay works through the Dirac theory of a charged particle in an electromagnetic field step by step — Lagrangian density, Euler–Lagrange equation, conserved 4-current, Hamiltonian density, Hamiltonian differential operator — establishes that every step closes consistently, and then performs the same sequence on the KG equation to show where it breaks.
The verdict is uncompromising. Comay argues that the KG Hamiltonian differential operator required by the Schrödinger picture cannot be extracted from the Pauli–Weisskopf Hamiltonian density at all; that the Feshbach–Villars two-component Hamiltonian, constructed without a Lagrangian density, violates relativistic covariance; and that the KG charge density, because it depends on the external electric potential, cannot define the inner product a Hilbert space requires. He notes that Paul Dirac "maintained his negative opinion on this equation throughout his life," and takes his own results as confirming an earlier conclusion that "a KG particle cannot carry an electric charge."
The argument
The Dirac case as a standard of self-consistency
Comay begins from the Dirac matter Lagrangian density L = ψ̄[γμ(i∂μ − eAμ) − m]ψ, whose variation with respect to ψ̄ gives the Dirac equation, and from which the 4-current jμ = ψ̄γμψ follows with the conservation law jμ;μ = 0. He stresses that this conservation holds independently of the external electromagnetic field, so the density ρ = ψ†ψ is positive definite and unaffected by external quantities. The Hamiltonian density then yields the differential operator H = α·(−i∇ − eA) + βm + eV by the simple expedient of stripping off the density factor ψ†ψ, and substituting it into Hψ = i∂ψ/∂t reproduces exactly the Euler–Lagrange equation. "The complete agreement," he writes, "indicates the self-consistence of the theory."
He also makes a covariance observation that becomes his weapon in Section 4. Because the Lagrangian density is a Lorentz scalar, the Hamiltonian density is the T00 component of the second-rank tensor Tμν, and because ρ is the 0-component of a 4-vector, the Hamiltonian operator H must itself be a 0-component of a 4-vector — a property "essential for satisfying covariance of the fundamental quantum mechanical relation." Five properties of the Dirac theory are listed as the benchmark: field-independent conserved 4-current, a Lagrangian linear in ∂ψ/∂t so the Hamiltonian density contains no time derivatives, a Hamiltonian operator free of ψ and ψ̄ (as linearity and superposition demand), exact agreement between the Hamiltonian equation and the Euler–Lagrange equation, and a correct eAμ interaction term.
The Pauli–Weisskopf theory: no Hamiltonian operator
Turning to the 1934 Pauli–Weisskopf treatment, Comay writes down their Lagrangian density, the resulting second-order KG equation of motion, and their conserved density ρ = i(φ*φ;0 − φ*;0φ) − 2eVφ*φ. Unlike the Dirac current, this depends both on time derivatives of φ and on the external electromagnetic potential, and is not positive definite — which is why it is conventionally read as charge rather than probability density. He also notes that the canonical Hamiltonian obtained from the first line of the standard construction is not gauge invariant.
His central proof is short. Take the highest-order time-derivative terms in the Hamiltonian density and the charge density: φ*;0φ;0 and i(φ*φ;0 − φ*;0φ). Superposition requires the sought operator Ĥ to be independent of φ, φ* and their derivatives. But the first expression is symmetric under exchange of φ and φ*, while the second is antisymmetric, so no operator can convert one into the other. The proof uses no term containing the charge e, so it applies equally to an uncharged KG particle described by a complex field.
He then asks whether one might at least build a Hamiltonian matrix by integrating the Hamiltonian density against a basis. Two obstructions follow. First, since the KG equation is homogeneous, rescaling φ by λ rescales Hij by λ2, so the eigenvalues — which represent energies — are only well defined if an orthonormal basis fixes the normalisation; but the dimension of φ*φ is [L−2] where a normalisable density needs [L−3], and the conserved density is not positive definite, while a Hilbert space requires an inner product with (φ,φ) ≥ 0. Second, and more physically, he poses a scattering experiment: let an electron approach a charged KG particle as in Rutherford scattering. As the electron nears, V varies, the KG density (14) varies with it, and hence "for two given wave functions φi, φj, this inner product has no unique value." An inner product that depends on where an external particle happens to be is not an inner product, so "there is no Hilbert space for the Hamiltonian." He adds that the same failure propagates: expectation values cannot be computed (−i∫φ*∇φ d3x does not give momentum, given the wrong dimension of φ*φ); the Heisenberg picture inherits the problem because it borrows the Schrödinger basis at t = 0; and since Fock space is built on the Hilbert space of solutions, the quantum field theory of KG particles inherits it too.
The Feshbach–Villars Hamiltonian: a covariance failure
Feshbach and Villars (1958) evade the first problem by rewriting the theory with a two-component wave function whose entries are linear combinations of φ and φ;0, obtaining H = (τ3 + iτ2)(1/2m)(p − eA)2 + mτ3 + eV, with τ2, τ3 Pauli matrices. Comay observes that their analysis does not rest on a Lagrangian density, so covariance is not guaranteed, and no proof of it appears in their paper. He then explains why none can be given: the fundamental relation Hψ = i∂ψ/∂t requires H to be the 0-component of a 4-vector. The term eV is; the term mτ3 is not, since a Pauli matrix "certainly cannot transform like a 0-component of a 4-vector"; and the first term is a component of an even-rank tensor, since (p − eA)2 expands into products of two energy-like quantities behaving as W00. Because the metric, the Kronecker delta and the fourth-rank antisymmetric tensor all have even rank, no manipulation can put an even-rank component and an odd-rank one in the same equation. The Hamiltonian therefore violates covariance, and does so through charge-independent terms, so it fails for uncharged KG particles as well. Comay notes in passing that a standard textbook relegates this Hamiltonian to a section on the non-relativistic limit.
Diagnosis: energy-momentum operators doing double duty
The concluding section offers an explanation rather than another objection. Working in units where ℏ = c = 1 and counting everything in powers of length, the action is dimensionless, so the Lagrangian density has dimension [L−4]. In the Dirac case operators appear to the first power in energy, momentum and mass, so ψ̄ψ has dimension [L−3] — exactly the dimension of a density — and the operator part cleanly separates from the state-specific part. In the KG case operators appear squared, so φ*φ has dimension [L−2], and a density must be built by borrowing one energy-like operator: hence i(φ*φ;0 − φ*;0φ). The energy-momentum operators (i∂/∂t, −i∇) thus "play two different roles": in the equation of motion they represent energy balance, and in the current they form part of the description of a state. When minimal substitution Pμ → Pμ − eAμ is applied, the external potential is dragged along into the density. This ambiguity is offered as "probably the underlying reason" for everything that goes wrong. Six numbered difficulties close the paper, and Comay is careful to limit their scope: they apply to the KG equation "that takes a fundamental dynamical role and is derived from a Lagrangian density," not to its use as a mathematical relation — components of Dirac solutions satisfy it, and that is unobjectionable.
Assessment
The paper is careful, narrowly scoped and honest about its own limits, which is more than can be said for much dissenting literature. Comay does not assert that the KG equation is "wrong" in some vague sense; he identifies specific structural requirements — a positive-definite density, an operator independent of the wave function, covariance of the Hamiltonian, agreement between the Hamiltonian and Euler–Lagrange routes — establishes them for the Dirac case, and then shows precisely which fail for the KG case. The symmetry/antisymmetry argument against extracting a Hamiltonian operator is a genuine observation and is stated in a form anyone can check in a few lines. The Rutherford-scattering objection to the inner product is a sharp and physical way of making the point that a norm depending on a time-varying external potential cannot serve as a probability measure. And the closing dimensional diagnosis — that [L−2] for φ*φ forces the density to borrow an energy operator, mixing two distinct functions of ∂μ — is genuinely illuminating regardless of what one concludes from it. The final caveat, that the KG equation remains unobjectionable as a mathematical relation satisfied by Dirac components, shows Comay knows exactly how far his argument reaches.
The central difficulty is that the paper measures the KG equation against the standards of single-particle relativistic quantum mechanics — and the mainstream abandoned that framework, for exactly the reasons Comay rehearses, seventy years before he wrote. The non-positive-definite density, the second-order time derivative, the absence of a one-particle probability interpretation: these are not discoveries but the standard textbook motivations for passing to quantum field theory, where φ is an operator field, the conserved quantity is charge (which may legitimately be negative), and the inner product lives on Fock space built from the field's mode expansion rather than on a first-quantised Hilbert space of φ. Comay's remark that "since the Fock space is related to the Hilbert space of solutions of the Hamiltonian, one concludes that problems also exist with the quantum field theory of KG particles" is the paper's most consequential claim and its least supported one; it is asserted in a single sentence with a page reference, and it does not engage the actual construction, in which the Fock space is generated by creation operators from a vacuum and its positive-definite inner product owes nothing to the KG density. Similarly, the covariance objection to the Feshbach–Villars Hamiltonian is correct as far as it goes, but a Hamiltonian is never Lorentz-covariant on its own: it is a component of the energy-momentum 4-vector, and singling out a time direction is what Hamiltonian formulations do — the same criticism, pressed uniformly, would indict the Dirac Hamiltonian's own dependence on a choice of time slicing.
Against established measurement, the conclusion that "a KG particle cannot carry an electric charge" is the hardest thing in the paper to sustain. Charged spin-0 particles exist and are described quantitatively by scalar QED built on precisely the Lagrangian Comay dissects: charged pion decay rates, the pion form factor measured in electron–pion scattering, and the pionic-atom energy levels calculated from the KG equation with a Coulomb potential all agree with experiment. The 2012 discovery of a spin-0 boson at 125 GeV, whose production and decay are computed from a scalar field with gauge couplings of exactly this form, is a further datum the 2005 paper could not have addressed but which any current assessment must weigh. Comay's own line of escape — that his objections bind only where the KG equation "takes a fundamental dynamical role and is derived from a Lagrangian density" — unfortunately does not help here, because that is the role it plays in scalar QED. The paper is best read as a rigorous demonstration of what the KG equation cannot do as a first-quantised single-particle theory, a demonstration whose steps are sound and whose scope is narrower than its concluding sentence claims.