On infrared singularities

The traditional separation of infrared divergent part of the S-matrix from a finite remainder ([YFS], [GY]) is effective only at points where the S-matrix is non-singular, as was pointed out in [S2]. This limitation is due primarily to the approximation 1 $${e^{ikx}} \sim 1\;\;\;(|k| \ll 1),$$ which is used to replace $${\delta ^4}(p + k) = \int {{{{d^4}x} \over {{{(2\pi )}^4}}}} {e^{i(p + k)x}}\;\;by\;\;{\delta ^4}(p) = \int {{{{d^4}x} \over {{{(2\pi )}^4}}}} {e^{ipx}}$$ in the demonstration that the infrared divergent terms originating from real photons are cancelled by those originating from virtual photons (e.g. [GY] (3.20) ff). An approximation of this sort seems to be necessary, if we treat the separation of the infrared divergent parts in momentum space. (Cf. Problem A below.) However, the separation can be neatly done in coordinate space, even at singular points of the S-matrix. ([S2]) In view of the fact that a point x in the coordinate space represents the cotangential component of the singularity spectrum of a function on the momentum space (e.g. [KS1],[Sa]) the recipe of Stapp [S2] may be regarded as the microlocalization of the traditional separation of infrared divergences. The core-spirit of microlocal analysis (e.g. [K3]) is to make use of both p-variables and x-variables in the analysis. In fact, to study the infrared finiteness of the remainder terms (the Q-coupling part in the sense of [S2] and [KS4]) p-variables play a central role, while the cancellation of infrared divergent terms (the C-coupling part in the sense of [S2]) proceeds in coordinate space.