Holonomic Systems of Linear Differential Equations and Feynman Integrals

The purpose of this report is to show that the (generalized) Feynman integral should satisfy a holonomic system ' of linear differential equations. We also discuss the analyticity of the defining function of the Feynman amplitude in complex domain as a corollary of this result. Our main result, i.e. the existence of holonomic system, gives an affirmative answer to the conjecture of Regge [15], who first understood and emphasized the importance of the role of systems of differential equations in the investigation of Feynman integrals. In his report a homological approach to this problem is suggested. It is very illuminating but seems to be accompanied with many technical difficulties, as Professor Regge himself points out in the report. This important property of the Feynman integral has also been conjectured and proved in simple cases by Sato [16] independently and in a little different context See also BarucchiPonzano [1], Kawai-Stapp [11], [12] and references cited there. Note that Kawai-Stapp ([11], [12]) discusses the ^-matrix itself, not the individual Feynman integral, as Sato [16] originally proposed. We also give the diagramatic description of the characteristic variety of the system involved. It enjoys a nice physical interpretation as is shown by Kashi-