Wick Theorem for General Initial States

TL;DR

This paper generalizes Wick's theorem to arbitrary initial states, simplifying calculations of Green's functions in quantum many-body systems and connecting real-time and imaginary-time formalisms.

Contribution

It provides a compact proof of a generalized Wick theorem applicable to any initial state, enabling easier computation of Green's functions in quantum systems.

Findings

Derived a simplified proof of the generalized Wick theorem.

Established a connection between real-time and imaginary-time formalisms.

Introduced new self-energy diagrams for arbitrary initial states.

Abstract

We present a compact and simplified proof of a generalized Wick theorem to calculate the Green's function of bosonic and fermionic systems in an arbitrary initial state. It is shown that the decomposition of the non-interacting $n$-particle Green's function is equivalent to solving a boundary problem for the Martin-Schwinger hierarchy; for non-correlated initial states a one-line proof of the standard Wick theorem is given. Our result leads to new self-energy diagrams and an elegant relation with those of the imaginary-time formalism is derived. The theorem is easy to use and can be combined with any ground-state numerical technique to calculate time-dependent properties.