Let U(a, A) be a representation of the Poincard group ~ with mass and helicity zero, realized in the space of C ~-functions with compact support on IR 3 , without the origin. Let U(2)(a, A) denote the tensorial product of U(a, A) by itself. We explicitly determine the cocycles of extension of U(a, A) by U(2)(a, A) and we prove that the nontrivial cohomology is indexed by (u(X), u(X) iO, l, r 0. INTRODUCTION Recently, Taflin [1] has proved the triviality of the cohomology of extension of a representation with mass m ~ 0, spin s, of the Poincare'group ~ by the tensorial product of any number of similar representations. As a matter of fact, this implies the formal linearizability of a nonlinear representation .of ~, the linear part of which is some direct sum of unitary representations with mass m q: 0 (cf.
[2] ). In his work, Taflin gives indications that in a two-dimensional space-time, a similar result is not valid for mass-zero representations.
It is one purpose of this letter to confirm and specify these indications for a four-dimensional space-time. Indeed, we prove that the cohomology of extension of a representation with mass and helicity zero by its tensorial product alone is not trivial, each equivalence class being indexed by a pair (u(X), a), where a E tF and u(X) is a distribution in ~1o,1[. Another motivation underlying our work is tightly bound to quantum theories with an indefinite metric, a particular case of which are the local covariant gauge theories. In this customary situation, the representation of ~ is the extension of three (or more) Fock representations built on unitary mass zero representations. Obviously, theories with an indefinite metric deserve some attention only if to prove that nontrivial extensions can exist. The present letter provides indications that a positive answer will be given to this question. It is also the first small but necessary step towards the tremendous task of classifying all the physically meaningful extensions.