1965
DOI: 10.1007/bf01646307
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A proof of the crossing property for two-particle amplitudes in general quantum field theory

Abstract: In the framework of the &.6f.2£. formalism, the crossing property is proved on the mass shell for amplitudes involving two incoming and two outgoing stable particles with arbitrary masses. Any couple of physical regions in the (s 9 1, u) plane corresponding to crossed processes are shown to be connected by a certain domain of analyticity. For every negative value of ί, the amplitude is analytic in the cut s-plane outside of a large circle.

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Cited by 142 publications
(135 citation statements)
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“…F (s, t) is an analytic function of the two Mandelstam variables, s and t, in a neighborhood ofs in an interval below the threshold, 4m 2 − ρ <s < 4m 2 and also in some neighborhood of t = 0, |t| < R(s). This statement hold due to the work of Bros, Epstein and Glaser [39,45].…”
Section: Remark: This Property Is True For the Case Of D-dimensional mentioning
confidence: 85%
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“…F (s, t) is an analytic function of the two Mandelstam variables, s and t, in a neighborhood ofs in an interval below the threshold, 4m 2 − ρ <s < 4m 2 and also in some neighborhood of t = 0, |t| < R(s). This statement hold due to the work of Bros, Epstein and Glaser [39,45].…”
Section: Remark: This Property Is True For the Case Of D-dimensional mentioning
confidence: 85%
“…Note that if we choose s = s 1 , there is an analyticity domain |t| < R. Moreover, the same analyticity argument goes through for s 1 < s < 4m 2 . Now fix s, 4m 2 − R + ǫ < s < 4m 2 ; following [45], for each −s < t 0 < 4m 2 there is a neighborhood |t − t 0 | < η(s, t 0 ) of analyticity in t. Note that we have analyticity in this compact region. Then invoke the Heine-Borel-Lebesgue theorem to argue that the interval −s + ǫ ≤ t ≤ 4m 2 − ǫ can be covered by finite number of (such) compact intervals.…”
Section: Determination Of Rmentioning
confidence: 96%
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“…For the first time in physics they used very sophisticated geometrical techniques of analytic completion borrowed from the theory of functions of several complex variables. In 1965, they completed this result by proving the crossing property always for the four-point function [20]. They showed that any couple of physical regions in the (s, t, it) space corresponding to scattering amplitudes involving two incoming and two outgoing stable particles with arbitrary masses, are connected by a certain domain of analyticity.…”
Section: Cern the Center Of Europementioning
confidence: 97%
“…Together with unitarity and Poincaré invariance it became known as the "S-matrix bootstrap" but it soon ended as a result of the unmanageable nonlinear problems arising from simultaneously implementing these three properties "by hand". Another problem was the insufficient understanding of the conceptual origin of particle crossing; its derivation from the locality principle for some very special scattering amplitudes did not lead to sufficient insights and the prohibitively difficult method of analytic functions [23] of several complex variables led to an early end of these attempts. Another attempt to obtain a constructive computational access to particle theory in terms of an on-shell project based on S-matrix properties was formulated by Mandelstam [29].…”
Section: The Dual Model Misunderstandings About Particle Crossingmentioning
confidence: 99%