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Axiomatic Perturbation Theory for Retarded Functions
1.H. Lehmann, K. Symanzik, and W. Zimmermann, Nuovo Cimento 1, 205 (1955).
2.K. Nishijima, Phys. Rev. 119, 485 (1960).
3.These relations have been shown to follow from the axioms only for the special case of decay processes, and must therefore be introduced into the formalism as a separate and essential postulate.
3.See H. M. Fried and D. L. Pursey, Phys. Rev. 124, 1281 (1961).
4.K. Symanzik, J. Math. Phys. 1, 249 (1960).
5.In momentum space, Eq. (4) takes the form of an integral over an absorptive type commutator; hence the adjective.
6.For the 3‐point operator, this follows with the aid of the Jacobi identity and the relation The general proof follows by induction on n.
7.The only remaining unspecified terms are pure connected contact terms of form discussion of this point is deferred to Sect. IV (c). Such terms first appear in Eq. (27).
8.This can always be done if the coefficients of each are symmetric in This will always be the case because the terms arise from the application of to the particular factors of Eq. (19), and the coefficients of these factors are symmetric in all the other y coordinates. Another way of seeing this last point is to consider the fully amputated λ expansion of the right‐hand side of (19), which must be symmetric in all . Here F represents the nonsingular (in x) fully retarded terms already shown to be symmetric in all hence the sum of all the singular terms must also be symmetric in all and this requires that each be independent of the superscript α and be symmetric in all y coordinates.
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