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On the problem of causality. (English) Zbl 0098.20105

quantum theory
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[1] H. Lehmann, K. Symanzik andW. Zimmermann:Nuovo Cimento,6, 319 (1957). · Zbl 0077.42501 · doi:10.1007/BF02832508
[2] If, according toR. Haag (Varenna Lectures, 1958), one assumes that\(\Phi\)[g]= =(dx)4 g(x)\(\Phi\)(x) defines a “ quasi-localizable ” state\(\Phi\)[g]|0g(x) being a smooth test-function, causality implies only: <0| [\(\Phi\) 1(x 1),\(\Phi\) 2(x 2)]|00 for (x 1 2)2 .
[3] It may be shown that the knowledge of the correspondenceA 0(\(\zeta\)) [A 0, \(\zeta\)] uniquely definesS(p).
[4] Retardation ofA(x) againstA 0(x) can be characterized, in spite of the impossibility of sharp fronts, by the condition: \(\int\limits_{\alpha _0 }^\infty {dx\left| {A(x)} \right.} \left| {^2 } \right. \leqslant \int\limits_{x_0 }^\infty {dx\left| A \right._0 (x)\left| {^2 } \right.} \) for everyx 0. Arguments similar to those ofVan Kampen (Phys. Rev.,91, 1267 (1953)) show that this condition implies analiticity ofS(p) in the upper half-plane. Therefore strict retardation excludes the occurence of poles on the positive imaginary axis, due to bound states. · doi:10.1103/PhysRev.91.1267
[5] Cf. Theorem XII, p. 16 inPaley andWiener:Fourier Transforms in the Complex Domain (American Mathematical Society, 1934).
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