  We present a replica path integral approach describing the quantum chaotic
dynamics of the SYK model at large time scales. The theory leads to the
identification of non-ergodic collective modes which relax and eventually give
way to an ergodic long time regime (describable by random matrix theory). These
modes, which play a role conceptually similar to the diffusion modes of dirty
metals, carry quantum numbers which we identify as the generators of the
Clifford algebra: each of the $2^N$ different products that can be formed from
$N$ Majorana operators defines one effective mode. The competition between a
decay rate quickly growing in the order of the product and a density of modes
exponentially growing in the same parameter explains the characteristics of the
system's approach to the ergodic long time regime. We probe this dynamics
through various spectral correlation functions and obtain favorable agreement
with existing numerical data.
