SEMIQUANTUM CHAOS AND THE LARGE-N EXPANSION

We consider the dynamical system consisting of a quantum degree of fre edom A interacting with N quantum oscillators described by the Lagrang ian L = 1/2 A(2) + Sigma(i=1)(N) {1/2x(i)(2) - 1/2(m(2)+e(2)A(2))x(i)( 2)}. In the limit N --> infinity , with e(2)N fixed, the quantum fluct uations in A are of order l/N. In this limit, the x oscillators behave as harmonic oscillators with a time dependent mass determined by the solution of a semiclassical equation for the expectation value (A(t)). This system can be described, when [x(t)] = 0, by a classical Hamilto nian for the variables G(t) = (x(2)(t)), G(t), A(c)(t) = (A(t)), and A (c)(t). The dynamics of this latter system turns out to be chaotic. We propose to study the nature of this large N limit by considering both the exact quantum system as well as by studying an expansion in power s of l/N for the equations of motion using the closed time path formal ism of quantum dynamics.