[Paper Review] Notes on Wick's theorem in many-body theory

Luca Guido Molinari|arXiv (Cornell University)|Oct 25, 2017

Quantum Mechanics and Applications5 references3 citations

TL;DR

This paper provides a pedagogical derivation of Wick's theorem in many-body theory, focusing on both static and time-ordered normal ordering of field operators with respect to a reference state. It establishes a systematic procedure to express products of creation and annihilation operators in normal-ordered form using c-number contractions, with a novel inductive proof for the time-ordered case, enabling efficient computation of Green's functions in free and interacting theories.

ABSTRACT

In this pedagogical note I present the operator form of Wick's theorem, i.e. a procedure to bring a product of 1-particle creation and destruction operators to normal order, with respect to some reference many-body state. Both the static and the time-ordered cases are presented. For the latter, in particular, I provide a simple proof.

Motivation & Objective

• To provide a clear, accessible derivation of Wick's theorem in the context of many-body quantum field theory.
• To formalize the procedure for normal-ordering products of field operators with respect to a reference many-body state, such as the Fermi sea or BCS ground state.
• To extend the standard Wick's theorem to time-ordered products, crucial for perturbative calculations in quantum field theory.
• To clarify the role of c-number contractions and their transformation properties under time-ordering and statistics (Bose/Fermi).

Proposed method

• Introduces a decomposition of field operators $A_i = A_i^- + A_i^+$, where $A_i^-$ annihilates the reference state and $A_i^+$ creates from it.
• Defines normal ordering $\mathcal{N}[\cdots]$ as ordering all $A_i^+$ operators to the left of $A_j^-$ operators.
• Introduces static contractions $\langle A_i A_j \rangle = \langle gs| A_i A_j |gs \rangle$ and time-ordered contractions $\overbrace{A_i(t_i) A_j(t_j)} = \langle gs| \mathcal{T} A_i(t_i) A_j(t_j) |gs \rangle$.
• Applies the canonical anticommutation (CAR) or commutation (CCR) relations to ensure that mixed brackets $[A_i^-, A_j^+]_\mp$ are c-numbers, a key condition for Wick's theorem.
• Uses induction to prove the time-ordered version of Wick's theorem, showing that $\mathcal{T}[A_1\cdots A_n]$ decomposes into sums of normal-ordered terms with increasing numbers of time-ordered contractions.
• Demonstrates that the theorem reduces the evaluation of $n$-particle Green's functions to permanents (bosons) or determinants (fermions) of one-particle propagators.

Experimental results

Research questions

• RQ1 How can products of field operators be systematically rewritten in normal-ordered form with respect to a non-vacuum reference state?
• RQ2 What is the structure of time-ordered contractions in many-body systems, and how do they relate to Feynman propagators?
• RQ3 Why does Wick's theorem hold only when mixed operator brackets $[A_i^-, A_j^+]_\mp$ are c-numbers?
• RQ4 How does the inductive proof of the time-ordered Wick's theorem ensure consistency with the causal structure of quantum field theory?
• RQ5 In what way do the results simplify the computation of $n$-particle Green's functions in free theories?

Key findings

• Wick's theorem for time-ordered products states that $\mathcal{T}[A_1\cdots A_n] = \sum_{k=0}^n \mathcal{N}[\text{all } k\text{-contractions}]$, with a rigorous inductive proof provided.
• Time-ordered contractions satisfy $\overbrace{A_i(t_i)A_j(t_j)} = \pm \overbrace{A_j(t_j)A_i(t_i)}$, reflecting the time-ordered nature of the expectation value.
• For non-interacting fermions, the $n$-particle Green's function reduces to the determinant of the one-particle propagator $G^0(x_i, y_j)$, as in Eq. (42).
• In the BCS superconducting state, anomalous contractions $F^0$ and $F^{\dagger 0}$ are non-zero, reflecting Cooper pair correlations.
• The theorem allows the evaluation of $n$-particle Green's functions in free theories as a sum over permutations weighted by the sign of the permutation (fermions) or permanent (bosons).
• The proof relies on the key lemma that $A_\ell N_{n,k} = \mathcal{N}[A_\ell N_{n,k}] + \mathcal{N}[\contraction{}{A}{{}_{\ell}}{N} A_\ell N_{n,k}]$, where the contraction is time-ordered due to maximal time ordering.