Operator Dynamics in Quantum Chaos: Part I - Internal Degrees of Freedom

We study operator growth in quantum chaos by considering the SYK model, a toy model of holography containing only internal degrees of freedom which evolve via q-local interactions. First, we note that the product length of an operator is directly related to its sensitivity to small perturbations. This reveals that in the SYK model, the commutator-squared/out-of-time-ordered correlator - a new diagnostic of quantum chaos - is literally measuring the effective length of the operator. It is known that this quantity grows exponentially in time with a "Lyapunov exponent", and thus we conclude that lengths of operators grows exponentially in time (amongst the internal degrees of freedom). Motivated by this, we group the operators into families defined by their lengths, thereby explicitly solving for the coarse-grained dynamics of an operator in the large N, large q limit. We also note that one can understand the time evolution of operators by relating it to the quantum mechanics of a particle on a graph with a nontrivial topology. Lastly, we may make some comments on the bulk interpretation of operator growth in SYK.