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We can assume that Φ which is a function of k'2, k"2 and k"'2 falls off very rapidly as any of these arguments becomes large. Consequently, we can consider that the domain of integration of the variables pt is practically limited to the values for which all vectors k associated with the various lines of the graph have bounded four-dimensional lengths k*. The following property will be useful: if a time-like vector k has a bounded scalar product with a fixed time-like vector k', its four components are bounded.
The functions Φ limit then the domain of integration to values of the variables pt and p\ such that the scalar products Κρι> Ρ1Ρ2, P2P3, · · · 103 * 0 p ' , p'p'n, p'p' 9 ··· Nr. 8 45 are bounded. As k0 is a fixed time-like vector and the p satisfy equations such as p2 + m2 = 0, it follows from the inequality (7,17) that the corresponding domain of integration is bounded. Thus, the corresponding terms are convergent, such as, for instance, the second-order self-energies. We consider now the case where there are other undoubled -^Ό— directed doubled line undoubled line with orientation Q" vertex with orientation Fig.
Nr. 8 29 The first order terms can be computed as in the preceding case, and are given by ψ(1) (x) = \ dx'dx"dx'" [fin ( x ) , Ητ(χ', χ", x'")], *X>X' W+V\x) = \dx'dx"dx'"[y>+il\x), Η^χ',χ", χ'")], (5,11) *'x > x'" u (1) (x) = \ dx'dx"dx'" [u in ( x ) , Ητ(χ', χ", x'")] ■ It should be noted in these formulas that the domain of integration of only one of the three variables occurring in Ητ is restricted by an inequality. This variable is different depending on the field function which is being computed; it is the variable of the field function which does not commute with the field function which is being computed.