Homological Study of the Generalized Singular Functions. I: -- A Preliminary Formulation Based Mainly upon the Three-Point Case --

We propose to regard Källén's generalized singular functions Δ<inf>n</inf>+(x</it><inf>1</inf>, x</it><inf>2</inf>, …x</it><inf>n</inf>; a</it>) as the inner products of suitably defined fundamental differential form and certain cocycles, hoping that various properties of Δ<inf>n</inf>+(x</it><inf>1</inf>, x</it><inf>2</inf>, …x</it><inf>n</inf>; a</it>) may be reduced to those of the cocycles. Some elementary considerations will be given for the three- and four-point cases. This viewpoint prompts us to consider other distributions than Δ<inf>n</inf>+ through appropriate choices of cycles, and to study a duality between the position space and the momentum space.