Summary
The derivation of dispersion relations for meson-nucleon scattering is reexamined. So far, all proofs given require the assumption that the Fourier representation of the scattering amplitude exists in the unphysical energy region. In this note it is shown that in the limit of fixed nucleons the causality condition and the properties of the spectrum are already sufficient to derive dispersion formulae for the forward scattering amplitudes. With the same assumptions one can also derive dispersion relations for the derivative amplitudes. The complete relativistic case is discussed briefly in the appendix.
Riassunto
Si riesamina la derivazione delle relazioni di dispersione per lo scattering mesonenucleone. Finora. tutte le prove fornite richiedono l’ipotesi che la rappresentazione di Fourier dell’ampiezza di scattering esista nella regione non fisica. In questa nota si mostra come al limite dei nucleoni fissi la condizione di causalità e le proprietà della spettro sono già sufficienti per derivare delle formule di dispersione per le ampiezze di scattering in avanti. Con la stessa ipotesi si possono anche ottenere relazioni di dispersione per le ampiezze di derivazione. Il caso relativistico complete si discute brevemente in appendice.
Similar content being viewed by others
References
M. Gell-Mann, M. L. Goldberger andW. Thirring:Phys. Rev.,95, 1612 (1954);M. L. Goldberger:Phys. Rev.,97, 508 (1955).
M. L. Goldberger:Phys, Rev.,99, 979 (1955).
M. L. Goldberger, H. Miyazawa andK. Oehme:Phys. Rev.,99, 986 (1955).
R. Karplus andM. A. Ruderman:Phys. Rev.,98, 771 (1955).
R. Oehme:Phys. Rev.,100, 1503 (1955).
R. Oehme:Phys. Rev.,102, 1174 (1956); (hereafter referred to as II).
R. H. Capps andG. Takeda :UCBL 3397.
E. C. Titchmarch:Fourier Integrals (Oxford, 1937), p. 119.
G. F. Chew andP. E. Low:Phys. Rev.,101, 1570 (1956);G. C. Wick:Rev. Mod. Phys.,27, 339 (1955);F. E. Low:Phys. Rev.,97, 1392 (1955).
Author information
Authors and Affiliations
Additional information
Supported by a grant from the U.S. Atomic Energy Commission.
Rights and permissions
About this article
Cite this article
Oehme, E. Causality and dispersion relations for the scattering of mesons by fixed nucleons. Nuovo Cim 4, 1316–1328 (1956). https://doi.org/10.1007/BF02746363
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02746363