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It is shown that the singularity spectrum of (generalized) position-space Feynman amplitude AD (x; /I) is confined to the extended Landau variety (introduced by KashiwaraKawai [8], [9] and Kashiwara-Kawai-Stapp [11]) associated with the Feynman diagram .D(O), which is obtained by replacing all the masses associated with the internal lines of D with zero, at least under the assumption that the diagram D is external. One recent trend in the investigation of analytic properties of the functions important in physics is to investigate the singularity structure of the functions in question from the view point of micro-local analysis, namely, to analyze their singularity structure on the cotangent bundle. (See Pham [15], Sato [16], lagolnizer [4], [5], [6], Kawai-Stapp [12], [13], K-K-S [11], Sato et al. [18] and references cited there. See also Chandler-Stapp [2] and lagolnitzer-Stapp [7], whose analysis is essentially of the micro-local character.) In the articles quoted above, the quantity in question is studied in the momentum-space and the celebrated LaridauNakanishi variety is naturally the starting point of their analysis. During the coruse of the precise argument on the unitarity integrals and so on, thanks to the harmony between mathematics and physics, K-K-S [11] and Kashiwara-Kawai [8], [9] have been naturally led to the notion of the extended Landau variety which describes the geometric feature of the singularity structure of phase space integrals, the (oil-shell) 5-matrix and Feynman amplitudes (K-K-S [11], Kashiwara-Kawai [8], [9]). Now, Received September 16, 1977. * Supported in part by NSF GP 36269. Department of Mathematics, Harvard University and Research Institute for Mathematical Sciences, Kyoto University. Current address: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606 Japan. (0) Hereafter referred to K-K-S [11] for short.

Extended Landau variety and the singularity spectrum of position-space Feynman amplitudes