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A SIMPLIFIED PROOF OF DISPERSIVE RELATIONS

CHANG TSUNG-SUI

A SIMPLIFIED PROOF OF DISPERSIVE RELATIONS

CHANG TSUNG-SUI
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  • Received Date:  07 September 1959
  • Published Online:  20 November 1959

A SIMPLIFIED PROOF OF DISPERSIVE RELATIONS

  • 1. 中国科学院数学研究所

Abstract: In this note, a simple but not rigorous proof for dispersive relations is given. The proof proceeds by expanding the causal amplitude with respect to the intermediate states and considering the analyticity of the energy denominator and the corresponding numerator separately. While the analyticity of the energy denominator is more or less obvious, the proof of the analyticity of the numerator, say N, is achieved by considering the analyticity of the corresponding numerator where the mass μ2 has been replaced by -p2 (μ = mass of meson, p = momentum of nucleon in Breit's system, the scattering of mesons by nuc-leons being considered for definiteness) and passing to the analyticity of N with the help of a suitable transformation. The idea of replacing μ2 by another quantity is due Bokolubof (Боголюбов), but here analyticity with respect to this new quantity is not considered. In the present method, p2 is allowed to be as great as M2 -μ2(M = mass of nucleon).The analyticity of phase shifts η(k) in potential scattering is also considered and it is pointed out that if the potential V→ e-αr as r→∞(α > 0), then η(k) may be extended to where |Im k| <1/2a.

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