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A remark on the proof of dispersion relations in quantum field theory

Published online by Cambridge University Press:  24 October 2008

J. G. Taylor
J. G. Taylor
Affiliation:
Laboratoire de Physique Théorique et Hautes Energies B.P. 12, Orsay (Seine-et-Oise), France and Institut des Hautes Etudes Scienttfiques, Paris, France
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Extract

In order to complete fully the proof of the fixed momentum-transfer dispersion relations for elastic scattering in quantum field theory given by Bremermann, Oehme and Taylor (1) it is necessary (2) to discuss more fully the analyticity of the absorptive part A(t, γ, Δ) in the mass variable γ. We are using the notation of (1), with 2Δ the momentum transfer and 4t2 the total energy of the initial system. For simplicity of discussion we will only treat the case of the elastic scattering of equal mass particles of mass m, the other cases discussed in (1) can be treated similarly. The crucial point of the proof, as first pointed out in (3), is to prove that

is regular in γ in a strip . It was only proved explicitly in (1) that A(t, γ, Δ) is regular for each t, in a strip , where δ may depend on t. In order to prove the analyticity in S of the integral in equation (1) it is necessary to show that δ(t) may be chosen to be independent of t, at least above some finite value of t (possibly large).

Type
Research Notes
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 3 , July 1961 , pp. 694 - 695
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

REFERENCES

(1)Bremermann, H. J., Oehme, R. and Taylor, J. G.Phys. Rev. (2), 109 (1958), 2178–90.CrossRefGoogle Scholar
(2) Private communications from R., Streater and R., Froissart, whom the author would like to thank for raising this question.Google Scholar
(3)Bogoliubov, N. N., Medvedev, B. V. and Polivanov, M. K.Problems in the theory of dispersion relations (Moscow, 1959).Google Scholar
(4)Taylor, J. G.Proc. Camb. Phil Soc. 54, (1958), 377–82.CrossRefGoogle Scholar