Spectral Form Factor as an OTOC Averaged over the Heisenberg Group

TL;DR

This work proves that the two-point spectral form factor ($SFF$) in bosonic quantum mechanics can be obtained as an average of the two-point out-of-time-ordered correlator ($OTOC$) over the Heisenberg group generated by position and momentum, and extends the construction to quantum field theories as a two-field path integral over mode-wise Heisenberg groups. The authors develop the formalism using coherent-state techniques in Jaynes-Cummings-type settings and analyze large-$N$ limits of matrix quantum mechanics to connect late-time SFF behavior with saddle-point structures. The approach provides a practical framework to compute spectral statistics from chaos diagnostics and offers insights into the relation between early-time chaos and late-time spectral universality, with potential implications for holography and black-hole information. The results suggest a phase-space averaging interpretation of spectral universality and outline promising directions for applications in large-$N$ CFTs, gauge theories, and ergodic-system self-averaging phenomena.

Abstract

We prove that in bosonic quantum mechanics the two-point spectral form factor can be obtained as an average of the two-point out-of-time ordered correlation function, with the average taken over the Heisenberg group. In quantum field theory, there is an analogous result with the average taken over the tensor product of many copies of the Heisenberg group, one copy for each field mode. The resulting formula is expressed as a path integral over two fields, providing a promising approach to the computation of the spectral form factor. We develop the formula that we have obtained using a coherent state description from the JC model and also in the context of the large-$N$ limit of CFT and Yang-Mills theory from the large-$N$ matrix quantum mechanics.