Three-Dimensional Formulation of the Relativistic Two-Body Problem and Infinite-Component Wave Equations

C. Itzykson, V. G. Kadyshevsky, and I. T. Todorov
Phys. Rev. D 1, 2823 – Published 15 May 1970
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Abstract

A relativistic quasipotential equation is derived from the conventional Hamiltonian formalism and old-fashioned "noncovariant" off-energy-shell perturbation theory in a similar way to that by which the four-dimensional Bathe-Salpeter equation is obtained from the off-mass-shell Feynman rules. The three-dimensional equation for the (off-energy-shell) scattering amplitude appears as a straightforward generalization of the nonrelativistic Lippmann-Schwinger equation. The corresponding homogeneous equation for the bound-state wave function and the normalization condition for its solutions are derived from the equation for the complete four-point Green's function. In order to obtain a solvable model, we consider a simplified version of the quasipotential equation which still reproduces correctly the on-shell scattering amplitude and is consistent with the elastic unitarity condition. It involves a "local" approximation to the potential V(pq) which defines the kernel of our integral equation (the integration being carried over a two-sheeted hyperboloid in the energy-momentum space). It is shown that for the scalar Coulomb potential V(pq)=α(pq)2, our model equation is equivalent to a simple infinite-component wave equation of the type considered by Nambu, Barut, and Fronsdal. The energy eigenvalues for the bound-state problem are calculated explicitly in this case and are found to be O(4) degenerate (just as in the nonrelativistic Coulomb problem and in Wick and Cutkosky's treatment of the Bethe-Salpeter equation in the same approximation).

  • Received 14 November 1969

DOI:https://doi.org/10.1103/PhysRevD.1.2823

©1970 American Physical Society

Authors & Affiliations

C. Itzykson*

  • Institute for Advanced Study, Princeton, New Jersey 08540

V. G. Kadyshevsky

  • Joint Institute for Nuclear Research, Dubna, USSR

I. T. Todorov

  • Institute for Advanced Study, Princeton, New Jersey 08540

  • *Work supported by the National Science Foundation. On leave from DPhT-CEN Saclay, BP No. 2, 91, Gif-sur-Yvette, France.
  • On leave from Joint Institute for Nuclear Research, Dubna, USSR, and from Physical Institute of the Bulgarian Academy of Sciences, Sofia, Bulgaria.

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Vol. 1, Iss. 10 — 15 May 1970

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