A perturbation treatment is presented for the operator u(t) giving the evolution of the state vector of a system of N particles coupled by 2-body forces. The total hamiltonian is divided into a diagonal part H/sub 0/ and offdiagonal pant H/sub 1/ in the representation with N-particle plane-wave states as basis. An effective pair-interaction operator H bar/sub 1/ is defined which includes vertex corrections of all orders, and the off-diagonal part of u is expressed exactly by an expansion which involves only H bar/sub 1/ and the exact diagonal part of u. A closed set of equations determining H bar/sub 1/ and the diagonal part of u are obtained by retaining only the leading terms in the expansion. These equations include all the corrections included in the Brueckner approximation and, in addition, they contain 3-body effects, iterated to all orders, which are omitted in that approximation. No one-to-one correspondence between the eigenstates of H/sub 0/ and those of the total hamiltonian is appealed to, and the ground state plays no special mole. The connection with statistical mechanics follows the fact that the partition function is TrSTAu(-i/ kT)!. The theory simplifies greatly for very large N. (auth)