In a 1948 Letter to Physical Review, F. J. Dyson proved that the pseudovector and pseudoscalar couplings of a pseudoscalar meson field to a nucleon field are equivalent under a contact (canonical) transformation, both in the neutral theory and, with additional second- and higher-order terms, in the charged and symmetrical (charge-independent) theories; and he stated verbatim that the result holds when the external potential V is either an ordinary or a spin-exchange potential, but not when V is a charge-exchange potential, without analyzing the excluded case. The later literature made the equivalence rigorous for the closed meson-nucleon system (Case; Foldy; Berger-Foldy-Osborn; Wentzel; Chisholm; Bollini), proved general field-redefinition invariance of the S-matrix (Chisholm; Kamefuchi-O'Raifeartaigh-Salam), and revisited the external-potential question for isoscalar nuclear mean fields (Friar-Gibson; Hecking-Brockmann-Weise; Noble), but the case of a fixed external charge-exchange potential has not been analyzed. The problem is to determine the explicit relationship between the two coupling forms when the same charge-exchange potential is kept in both, thereby verifying or refuting Dyson's assertion and characterizing any physical processes in which the two forms differ.

Background

In the Yukawa meson theory of nuclear forces, a nucleon described by the Dirac field Ψ\Psi (with Dirac adjoint Π\Pi, i.e. the row spinor conjugate to Ψ\Psi) interacts with a meson field of spin-parity 0−0^-, called a pseudoscalar meson because its wave function changes sign under spatial inversion (the pion is the physical example). Kemmer (1938) systematized the possible couplings; for a pseudoscalar meson field ψ\psi with canonical momentum π\pi and mass μ\mu there are two canonical forms: the pseudovector (derivative) coupling

H1=g∫Π(σ⋅∇ψ−(1/c)γ5π)Ψ dτH_1=g\int \Pi(\sigma\cdot\nabla\psi-(1/c)\gamma_5\pi)\Psi\,d\tau

and the pseudoscalar (non-derivative) coupling

H2=G∫Πβγ5ψ Ψ dτ,H_2=G\int \Pi\beta\gamma_5\psi\,\Psi\,d\tau,

where gg and GG are coupling constants, MM is the nucleon mass, cc the speed of light, dτd\tau a volume element, and α\alpha, β\beta, σ\sigma, γ5\gamma_5 the Dirac matrices acting on the nucleon spinor indices (γ5\gamma_5 exchanges the large and small components of the Dirac spinor and anticommutes with β\beta). In 1948 the standard procedure was to drop the γ5\gamma_5 term of H1H_1 and all of H2H_2 for nonrelativistic nucleons; unpublished calculations by Bethe (cited in Dyson's Letter) found instead that the two interactions give identical nuclear potentials and identical scattering, and Nelson (1941) had already noticed the coincidence. F. J. Dyson's 1948 Letter supplied the precise statement. With the free Hamiltonian

H0=12∫(π2+c2∣∇ψ∣2+μ2c2ψ2) dτ−c∫Π(α⋅∇+iMβ−V)Ψ dτH_0=\tfrac{1}{2}\int(\pi^2+c^2|\nabla\psi|^2+\mu^2c^2\psi^2)\,d\tau-c\int \Pi(\alpha\cdot\nabla+iM\beta-V)\Psi\,d\tau

containing an external potential VV acting on the nucleon, the contact (canonical) transformation H′=eiSHe−iSH'=e^{iS}He^{-iS} with generator

S=−(g/ℏc)∫Πγ5ψ Ψ dτS=-(g/\hbar c)\int \Pi\gamma_5\psi\,\Psi\,d\tau

maps H=H0+H1H=H_0+H_1 onto H′=H0+H1′H'=H_0+H_1', where H1′H_1' is c∫Π(iMβ(1−e−2(g/c)γ5ψ))Ψ dτc\int \Pi(iM\beta(1-e^{-2(g/c)\gamma_5\psi}))\Psi\,d\tau plus a nucleon-nucleon contact term localized where the fields coincide (a delta-function interaction, which Dyson discarded as having no physical meaning); the exponential survives only in the β\beta term because γ5\gamma_5 commutes with α\alpha and σ\sigma but not with β\beta. Expanding the exponential yields H2H_2 with G=2iMgG=2iMg plus higher-order terms that Dyson classified mostly as mass effects. The two couplings are therefore, in his words, completely equivalent for first-order radiative processes and for many higher-order processes as well, and he recommended that H2H_2 be used exclusively in future calculations. External potentials are classified by their exchange character: ordinary (Wigner) potentials act as scalars in the nucleon's spin and isotopic-spin space (isotopic spin being the two-valued internal degree of freedom distinguishing neutron and proton); spin-exchange (Bartlett) potentials are built from the spin matrices σ\sigma; and charge-exchange potentials are off-diagonal in charge, built from the isotopic-spin matrices τ1,τ2\tau_1,\tau_2 that convert a proton into a neutron. Dyson extended the method to the charged meson theory and to the symmetrical (charge-independent) theory, in which the meson field is an isovector τ⋅ϕ\tau\cdot\phi with neutral and charged components, finding additional terms only of second and higher order in the meson field; and then stated, verbatim, that the result is true when VV is either an ordinary or a spin-exchange potential, but not when VV is a charge-exchange potential. The structural reason visible in the Letter's formulas is that in the neutral theory the generator is an isotopic-spin scalar, whereas in the charged and symmetrical theories the generator carries the charge structure of the meson field, with which a charge-exchange VV fails to commute. Dyson did not analyze the excluded case. The later literature made the equivalence rigorous but dropped the external potential. Case (1949) re-proved the theorems to second order in the coupling constants with an external electromagnetic four-potential included, finding the two couplings equivalent except for a predominantly self-energy term and, for charged mesons only, a term that modifies the electromagnetic properties and the scattering of charged mesons, and showing that the delta-function interactions found earlier are actually absent. Foldy (1951) gave an exact canonical transformation for a single nucleon with a neutral field; Berger, Foldy and Osborn (1952) established rigorous equivalence theorems for the neutral, charge-symmetrical, charged and symmetrical theories with a second-quantized nucleon field (the nucleon treated as a field with creation and annihilation operators rather than a fixed source), the isospin structure generating additional nonlinear terms, while explicitly omitting interactions with electromagnetic fields; Wentzel (1952) stressed that the transformed mass-effect terms do contribute to meson-nucleon scattering, to meson pair creation by nucleons in the presence of external nuclear fields, and to second-order nuclear forces; Drell and Henley (1952) applied the transformation to charged-meson scattering and photoproduction. Chisholm (1956, 1961), Chisholm and Dixon (1958), Singh (1959), Bollini (1962), and Kamefuchi, O'Raifeartaigh and Salam (1961) carried the program to the S-matrix level (the S-matrix being the scattering operator relating asymptotic incoming and outgoing states) and to the general theorem that the S-matrix is invariant under point transformations of the field operators, from which the ordinary equivalence theorems follow. A separate modern line revisited the external-potential question, but only for isoscalar (charge-diagonal) nuclear mean fields, the case Dyson said was safe: Friar and Gibson (1977) analyzed the roles of the equivalence theorem and of equivalence-breaking terms in pion-exchange currents and threshold pion photoproduction; Hecking, Brockmann and Weise (1978) analyzed the non-equivalence of the two couplings for a Dirac nucleon bound in a scalar-vector potential and related it to the spin-orbit splitting; Noble (1979) showed that the large discrepancies between the couplings found in threshold pion-production calculations with relativistic bound-nucleon wave functions, traceable to the Lorentz-scalar part of the nuclear potential, disappear to second order in the coupling once partial conservation of the axial-vector current (PCAC, the statement that the divergence of the axial-vector current is proportional to the pion field) and bound-state orthogonality are properly imposed; and Noble (1989) showed that in relativistic mean-field theory of nuclear structure the pion mean field can be treated as pseudovector-coupled to all orders in the coupling. Dyson's transformation also survives as the ancestor of the chiral rotations relating the two pion-nucleon couplings in modern chiral perturbation theory (the low-energy effective field theory of pions and nucleons). None of these works analyzes the case Dyson excluded: the relationship between H1H_1 and H2H_2 when the same charge-exchange potential is kept in both theories.

Problem Statement

Within the canonical Hamiltonian framework of Dyson's 1948 Letter, a nucleon field interacting with a pseudoscalar meson field through either the pseudovector interaction H1=g∫Π(σ⋅∇ψ−(1/c)γ5π)Ψ dτH_1=g\int \Pi(\sigma\cdot\nabla\psi-(1/c)\gamma_5\pi)\Psi\,d\tau or the pseudoscalar interaction H2=G∫Πβγ5ψ Ψ dτH_2=G\int \Pi\beta\gamma_5\psi\,\Psi\,d\tau with G=2iMgG=2iMg, and a free Hamiltonian H0H_0 containing an external potential VV acting on the nucleon, determine the relationship between the two coupling forms when VV is a charge-exchange potential, an external potential off-diagonal in the nucleon's charge (isotopic spin) space, the case Dyson explicitly excluded. The substantive setting is the charged and the symmetrical (charge-independent, isovector meson field) theory, where the transformation generator carries the charge structure of the meson field; in the neutral theory the generator is an isotopic-spin scalar, and the answer should make that case's status explicit. Concretely, an answer should: (i) carry out the contact transformation eiSHe−iSe^{iS}He^{-iS} with the generator appropriate to the charged and symmetrical theories in the presence of such a VV, and identify all terms beyond those Dyson obtained, in particular those induced by the non-commutativity of VV with the generator; (ii) decide whether H1H_1 and H2H_2 remain equivalent for on-shell physical processes (processes involving real, physical particles rather than virtual intermediate states) when VV is held fixed in both theories, that is, verify or refute Dyson's assertion that the equivalence fails for charge-exchange VV, to the order in the coupling constants at which any difference first appears; and (iii) if the two forms differ, characterize the physical processes, for example meson emission, absorption, or scattering in the charge-exchange background, in which they give different predictions. The problem may equivalently be posed in Dyson's canonical formulation or in an explicitly equivalent modern formulation (canonical field redefinitions), in either case with the comparison 'same VV in both coupling forms' made precise, including whether the external field couples to the mesons as well as to the nucleons.

References

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