The Yukawa Lagrangian Density is Inconsistent with the Hamiltonian
| Scientific Paper | |
|---|---|
| Title | The Yukawa Lagrangian Density is Inconsistent with the Hamiltonian |
| Read in full | Link to paper |
| Author(s) | E Comay |
| Keywords | Yukawa field, Hamiltonian. |
| Published | 2007 |
| Journal | Apeiron |
| Volume | 14 |
| Number | 1 |
| No. of pages | 11 |
Read the full paper here
Abstract
It is proved that no Hamiltonian exists for the real Klein-Gordon field used in the Yukawa interaction. It is also shown that a real Klein-Gordon particle can be neither in a free isolated state nor in a bound state having an angular momentum l > 0. The experimental data support these conclusions. This outcome is in a complete agreement with Dirac's negative opinion on the Klein-Gordon equation.
Overview
Eliahu Comay's 2007 Apeiron paper is one of a series in which he attacks the Klein-Gordon (KG) equation as a candidate quantum-mechanical wave equation for a massive particle. Here the target is the Yukawa interaction — the seventy-year-old proposal, still taught as the origin of the nuclear force, that a scalar meson field couples to the nucleon Dirac field. Comay's claim is not that the Yukawa potential fits the data badly; it is the stronger claim that the Lagrangian density from which the interaction is derived admits no Hamiltonian at all, and therefore cannot be part of quantum mechanics as quantum mechanics is actually formulated.
The paper departs from the mainstream account at the point where most textbooks pass quickly. Standard treatments write down a Hamiltonian density for the real KG field and move on. Comay argues that a Hamiltonian density is not a Hamiltonian: extracting an operator from a density presupposes a normalizable density and hence a Hilbert space, and the real KG field has neither. He then adds a second, independent line of attack — that the real-valued nature of the field forbids the particle from occupying a free momentum eigenstate or a bound state with orbital quantum number l > 0 — and finally an experimental section arguing that the observed nuclear potential and the observed properties of the π0 both contradict the Yukawa picture. His conclusion is that Dirac's long-standing hostility to the KG equation was justified.
The argument
The Yukawa Lagrangian density
The starting point is the standard Yukawa Lagrangian density, a Dirac field plus a KG field plus a trilinear coupling,
- LY = LD + LKG − gφψ̅ψ
with LD = ψ̅(iγμ∂μ − m)ψ and LKG = ½(gμνφ,μφ,ν − m2φ2). Comay works in units where ℏ = c = 1 with metric signature (1,−1,−1,−1). He stresses at the outset that because the Hamiltonian must be Hermitian, the φ appearing here is real — a one-component Lorentz scalar. Everything that follows concerns that original, real version of the field; other constructions "which may be related to the Yukawa idea" he explicitly places outside his scope.
A preliminary relativistic objection
Before the main proof, Comay recalls a difficulty with the classical limit. The Yukawa potential
- u(r) = −g2e−mr/r
yields a four-force which, he says, cannot satisfy the relativistic requirement that four-acceleration be orthogonal to four-velocity, aμvμ = 0. The electromagnetic case works precisely because the field tensor is antisymmetric: maμ = eFμνvν, and the contraction Fμνvνvμ vanishes identically. "The scalar function φ cannot yield an antisymmetric tensor," and so, he concludes, the classical Yukawa force is inconsistent with special relativity.
The main proof: order mismatch
Applying the Euler-Lagrange equations to φ gives an inhomogeneous KG equation
- (□ + m2)φ = gψ̅ψ
Comay contrasts this with what he takes to be "the fundamental quantum mechanical equation", i ∂φ/∂t = Hφ. The first is second order in time and inhomogeneous; the second is first order and homogeneous. Suppose at an instant t0 some φ0 solves both. Because the field equation is second order, ∂φ/∂t at t0 remains a free parameter, so an infinite family of solutions of the field equation matches the single Hamiltonian solution at t0 and diverges from it afterwards. Conversely, because the Hamiltonian equation is homogeneous, cφ0 solves it for any constant c — the freedom used to build an orthonormal basis — whereas the inhomogeneous field equation fixes the normalisation. "Either of these results proves that the Yukawa Lagrangian density is inconsistent with the existence of a Hamiltonian."
No density, hence no Hilbert space
The second strand is that a real KG wave function has no conserved density (he cites Berestetskii, Lifshitz and Pitaevskii; even the complex KG function has only a charge density, which is not positive definite). Without a density there is no normalisation, without normalisation no basis, without a basis no matrix representation of H. This is why, in his reading, the existence of a Hamiltonian density in Bjorken–Drell or Lurie does not settle the matter.
Reality of the field forbids free and l > 0 states
An energy-momentum eigenfunction carries the complex factor ei(kx−ωt); a real φ cannot. Hence the Yukawa particle "cannot be in an isolated free state." The same argument runs on bound states: the angular factor Ylm(θ,φ) ∝ eimφΘ(θ) is complex for every m ≠ 0, so states with l > 0 are excluded. He adds that a real wave function has a real time derivative, which through i∂φ/∂t = Hφ forces purely imaginary energy eigenvalues — unacceptable for a stable particle.
Asking whether some alternative theory could handle a real wave function, he posits an energy operator Ôχ = K ∂χ/∂t = Eχ with K a real dimensionless constant (energy has dimension [L−1] in these units). For a particle at rest, E = m > 0 and the solution is χ(t) = χ(0)eEt/K — a static state whose amplitude grows or decays exponentially. He treats this as closing off the alternative.
The experimental section
Three points. First, φ(xμ) depends on a single set of coordinates and so describes a pointlike particle, but π mesons are quark-antiquark composites and are measured to have structure. Second, the π0 lifetime of about 10−16 s gives, at relativistic speed, a path longer than 107 fermi, against a nucleon radius of about 1.2 fermi — so the π0 spends essentially all its life as a free particle, which his own theorem forbids for a Yukawa quantum. Third, the phenomenological nucleon-nucleon potential has a repulsive hard core with an attractive outer region, so both the potential and its derivative change sign; neither u(r) = −g2e−mr/r nor its derivative does.
Assessment
The paper's virtue is that it is a genuinely internal critique. Comay does not propose a rival nuclear force; he takes the textbook Lagrangian density, applies textbook Euler-Lagrange machinery, and asks a question textbooks skip — where, explicitly, is the Hamiltonian operator for this field, as opposed to the Hamiltonian density? His observation that no textbook displays one is accurate, and his arithmetic where there is arithmetic is correct: the π0 mean life is close to 10−16 s and a relativistic flight path of ct ≈ 2.5 × 10−8 m is indeed some 107 fermi, thousands of nucleon diameters. The description of the empirical nucleon-nucleon potential, with its repulsive core, is also correct.
The difficulties are structural rather than numerical. The whole proof rests on treating φ as a one-component wave function obeying i∂φ/∂t = Hφ. In the theory being criticised, φ is not a wave function at all but a field, and its Hamiltonian is a functional H[φ,π] generating first-order evolution in the pair (φ, π = φ̇) through Hamilton's equations. The "free parameter" ∂φ/∂t(t0) that Comay identifies as fatal is exactly the conjugate momentum π, an independent canonical variable in every first-order (phase-space) formulation. Read that way the second-order field equation and a first-order Hamiltonian flow are not in conflict; the order mismatch is an artefact of demanding that a field satisfy a single-particle Schrödinger-type equation. The same reply disposes of equation (13): a real field's mode functions solve φ̈ = −E2φ and oscillate as cos(Et), bounded for all time; the exponential runaway appears only because Comay has imposed a first-order real equation by hypothesis.
The relativistic objection is likewise answered in the standard literature, though the answer is not obvious. For scalar coupling the correct classical equation of motion is not maμ = force with m constant, but a variable-mass equation in which the effective mass is m + gφ; the force is then automatically projected orthogonal to the four-velocity, and aμvμ = 0 is preserved without any antisymmetric tensor. Comay's demand that a scalar interaction imitate the Lorentz force's tensor structure is a demand no scalar theory makes of itself.
The experimental section is the weakest link, because it argues against a claim Yukawa theory does not make. The meson exchanged between nucleons is an internal, off-shell line in the interaction, not an asymptotic free particle; that a physical π0 propagates freely for 107 fermi before decaying to two photons says nothing about whether virtual exchange generates a potential. Nor does the hard core refute the exponential tail: one-boson-exchange models reproduce the sign change by adding vector-meson exchange with opposite-sign coupling, keeping the single-pion-exchange form only in the long-range region, where the range ℏ/mπc ≈ 1.4 fermi does match the observed reach of the nuclear force. Finally, the composite nature of the pion is not in dispute, but it is a statement about which field is fundamental, not about whether a scalar field can be quantised.
What survives, and is worth taking seriously, is the narrower point about a real massive field in first-quantized language. There genuinely is no positive-definite single-particle probability density for the KG field, and the attempt to read φ as a one-particle wave function does fail — which is the historical reason the equation was abandoned as a wave equation and re-adopted as a field equation. Comay's paper is an unusually sharp statement of that failure; where it overreaches is in supposing the failure of the first-quantized reading is a failure of the field theory that replaced it. Dirac's scepticism, which he invokes, was of the same first-quantized kind.