Salpeter, E.E.; Bethe, H.A.; A Relativistic Equation for Bound-State Problems
Phys. Rev. 84 (1951) 1232;
Reprinted in The Physical Review - the First Hundred Years, AIP Press (1995) CD-ROM.
The relativistic S-matrix formalism of Feynman is applied to the bound-state problem for two interacting Fermi-Dirac particles. The bound state is described by a wave function depending on separate times for each of the two particles. Two alternative integral equations for this wave function are derived with kernels in the form of an expansion in powers of g2, the dimensionless coupling constant for the interaction. Each term in these expansions gives Lorentz-invariant equations. The
validity and physical significance of these equations is discussed. In the extreme nonrelativistic approximation and to lowest order in g2 they reduce to the appropriate Schrödinger equation. One of these integral equations is applied to the deuteron ground state using scalar mesons of mass with scalar coupling. For neutral mesons the Lorentz-invariant interaction is transformed into the sum of the instantaneous Yukawa interaction and
a retarded correction term. The value obtained for g2 differs only by a fraction proportional to (/M)2 from that obtained by using a phenomenological Yukawa potential. For a purely charged meson theory a correction term is obtained by a direct solution of the relativistic integral equation using only the first term in the expansion of the kernel. This correction is due to the fact that a nucleon can emit, or absorb, positive and negative
mesons only alternately. The constant g2 is increased by a fraction of 1.1(/M) or 15 percent.
Related references See also P. A. M. Dirac, V. A. Fock, and B. Podolsky, Phys. Zh. USSR 2 (1932) 468;
F. Bloch, Physik Zeits. Sow. 5 (1934) 301;
F. J. Dyson, Phys. Rev. 75 (1949) 486;
F. J. Dyson, Phys. Rev. 75 (1949) 1736;
E. Fermi and C. N. Yang, Phys. Rev. 76 (1949) 1739;
R. P. Feynman, Phys. Rev. 76 (1949) 769;
E. E. Salpeter, Phys. Rev. 82 (1951) 60;
E. E. Salpeter, Phys. Rev. 84 (1951) 1226;
F. M. Smith et al., Phys. Rev. 78 (1950) 86;
R. P. Feynman, Phys. Rev. 76 (1949) 749;
S. M. Dancoff, Phys. Rev. 78 (1950) 382;
Bethe-Salpeter relativistic equation for two-body bound-state problems.