A BBGKY-like Hierarchy for Quantum Field Theories

TL;DR

The paper introduces a Hamiltonian BBGKY-like framework for quantum field theories by leveraging a filtration of Hermitian polynomials to generate an infinite hierarchy of coupled evolution equations for reduced density matrices. A canonical hierarchy based on ladder-operator polynomials yields a tractable truncation scheme (via $N$) that preserves Hamiltonian structure and allows non-perturbative approximations of full quantum dynamics, with observables such as particle density and energy density computable from the lowest-level variables. The work outlines concrete forms for reduced densities $\Gamma^{(m,n)}$ and demonstrates how to express observables in terms of these objects, while noting ongoing challenges in Hamiltonian closures and boundary conditions. It lays a path toward semi-classical modeling of high-energy environments where quantum effects are pronounced, with future directions focusing on Hamiltonian closures, Poisson maps, and finite-truncation Hamiltonian structures.

Abstract

We present a Hamiltonian method of constructing BBGKY-like hierarchies for quantum field theories. With suitable choices, our method creates a hierarchical system of evolution equations for the k-th order reduced density matrices. These equations can be closed at finite order using methods developed for the classical BBGKY hierarchy to give non-perturbative approximations for the full quantum equations of motion. Classical observables can then be numerically computed from these approximate equations, providing an analytically tractable method of modeling high-energy environments where quantum effects play a pronounced role.