# The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations

## TL;DR

This paper reformulates the Dirac-Frenkel principle using reduced density matrices to derive effective evolution equations like the Bogoliubov-de-Gennes equations, establishing their optimality and well-posedness in quantum many-body systems.

## Contribution

It introduces a new formulation of the Dirac-Frenkel principle for reduced density matrices and derives key equations for fermionic and bosonic systems, showing their optimality and well-posedness.

## Key findings

- Derivation of Bogoliubov-de-Gennes and Hartree-Fock-Bogoliubov equations from the principle
- Proof of well-posedness of the Bogoliubov-de-Gennes equations in energy space
- Demonstration that the approximation is optimal within quasifree states

## Abstract

The derivation of effective evolution equations is central to the study of non-stationary quantum many-body sytems, and widely used in contexts such as superconductivity, nuclear physics, Bose-Einstein condensation and quantum chemistry. We reformulate the Dirac-Frenkel approximation principle in terms of reduced density matrices, and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov-de-Gennes and Hartree-Fock-Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the class of quasifree states. Furthermore, we prove well-posedness of the Bogoliubov-de-Gennes equations in energy space and discuss conserved quantities.