# Spectral Form Factor as an OTOC Averaged over the Heisenberg Group

## TL;DR

This paper establishes a method to compute the spectral form factor in bosonic quantum mechanics and quantum field theory by averaging out-of-time ordered correlators over the Heisenberg group, offering a new path integral approach.

## Contribution

It introduces a novel formula linking spectral form factors to OTOCs averaged over the Heisenberg group, applicable in quantum mechanics and field theories.

## Key findings

- Derived a path integral expression for spectral form factors.
- Connected spectral form factors with OTOCs in large-N theories.
- Applied the formula to the JC model and large-N CFTs.

## 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.