A differential identity for Green functions

TL;DR

The paper derives a differential identity for Green functions involving exponential operators, enabling systematic calculations of Green functions in many-body quantum systems, including fermionic cases with external potentials.

Contribution

It introduces a new differential identity for Green functions under exponential operators, applicable to scalar and fermionic fields, advancing theoretical tools in many-body physics.

Findings

Derived a hierarchy of Green functions using the identity

Applied the identity to electron systems in external potentials

Extended the identity to fermionic fields

Abstract

If P is a differential operator with constant coefficients, an identity is derived to calculate the action of exp(P) on the product of two functions. In many-body theory, P describes the interaction Hamiltonian and the identity yields a hierarchy of Green functions. The identity is first derived for scalar fields and the standard hierarchy is recovered. Then the case of fermions is considered and the identity is used to calculate the generating function for the Green functions of an electron system in a time-dependent external potential.