One-particle irreducibility with initial correlations

TL;DR

This paper explores the structure of Green functions in quantum field theory with initial correlations, using Hopf algebra to extend the understanding of one-particle irreducibility beyond the vacuum state.

Contribution

It introduces a Hopf algebraic framework to analyze Green functions with initial correlations, advancing the theoretical understanding of non-Gaussian states in QFT.

Findings

Derived the structure of Green functions with initial correlations.

Extended the concept of one-particle irreducibility to correlated initial states.

Provided a mathematical framework for complex many-body systems.

Abstract

In quantum field theory (QFT), the vacuum expectation value of a normal product of creation and annihilation operators is always zero. This simple property paves the way to the classical treatment of perturbative QFT. This is no longer the case in the presence of initial correlations, that is if the vacuum is replaced by a general state. As a consequence, the combinatorics of correlated systems such as the ones occurring in many-body physics is more complex than that of quantum field theory and the general theory has made very slow progress. Similar observations hold in statistical physics or quantum probability for the perturbation series arising from the study of non Gaussian measures. In this work, an analysis of the Hopf algebraic aspects of quantum field theory is used to derive the structure of Green functions in terms of connected and one-particle irreducible Greeen functions for perturbative QFT in the presence of initial correlations.