Abstract
Coupling the mass-energy relationδE≧mc2 to the uncertainty relationδE δt ≧ ħ produces fluctuations in the number of particles at short distances and scatterings of particle pairs independent of any specific “interaction” mechanism. This observation allows the construction of a scattering theory in which there are only particles and the void, but particle number can change. We consider a system of three massive particles (hadrons) in the energy region below the first production threshold for a fourth hadron and above the first anomalous threshold for the presence of a fourth “virtual” hadron. The on-shell Faddeev equations, containing only two-particle scattering phases for positive two particle energies, provide a convergent, unitary, and readily soluble dynamics for this system. If any of the pairs can coalesce into a different particle with a rest energy less than the sum of the rest energies of the pair, the equations can be readily extended to describe 3-2 and 2–3 transitions involving this particle (coalescence, breakup) elastic scattering from it, and if there is more than one such particle 2-2 rearrangements. The three-body “bound state” requires a well defined analytic continuation. Features of more conventional calculations of three-nucleon problems which provide examples of this structure are discussed. Since only free particles occur in the theory, and the only failure of energy conservation is that required by the uncertainty principle for (free-particle) intermediate states, these one-variable equations might be extended to particles with the relativistic connection between mass, energy and momentum, and transitions in which the full rest energy of the particle which appears or disappears must be provided. The non-linear “crossed” theory for such particles has not been written down, but if the relativistic boundary condition model of Brayshaw is viewed as representing these crossed processes by a phenomenological core, then a crossed theory requiring the π to be a bound state of three π's might predict the π-π S-wave scattering length in theI=0 state in terms of the pion Compton wavelength (and hence the position and the width of theϱ) and will then show that theϱ in turn generates asingle ω resonance at about the right place. Implications are discussed.
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Invited talk at the symposium “Theory of lightest nuclei”, Liblice, Czechoslovakia, May 1974.
Supported in part by the U.S. Atomic Energy Commission.
I wish particularly to thank the organizers of this conference for their hospitality and support. Many useful conversations withLambrecht Kok, Carlo Alabiso, Enrique Zeiger, Werner Sandhas andThomas Osborn contributed to the clarification of this work, though I fear they remain skeptical. Partial support for my trip came from the Free University of Amsterdam, the University of Bonn, the University of Rome, the Institute of Theoretical Physics of Naples and the Cavendish Laboratory.
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Noyes, H.P. On-shell faddeev equations: A democritean approach to the three-body problem. Czech J Phys 24, 1205–1215 (1974). https://doi.org/10.1007/BF01587206
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DOI: https://doi.org/10.1007/BF01587206