LargeNquantum time evolution beyond leading order

For quantum theories with a classical limit (which includes the large N limits of typical field theories), we derive a hierarchy of evolution equations for equal time correlators which systematically incorporate corrections to the limiting classical evolution. Explicit expressions are given for next-to-leading order, and next-to-next-to-leading order time evolution. The large N limit of N-component vector models, and the usual semiclassical limit of point particle quantum mechanics are used as concrete examples. Our formulation directly exploits the appropriate group structure which underlies the construction of suitable coherent states and generates the classical phase space. We discuss the growth of truncation error with time, and argue that truncations of the $\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}-N$ evolution equations are generically expected to be useful only for times short compared to a “decoherence” time which scales like $N^{1/2}.$