Student Friendly Introduction to QFT · Problem 7.12

QED: Quantum Field Interaction Theory Applied to Electromagnetism

Physics — QFT · wicks-theorem-two-fermions-t2-less-t1

Problem

Use the full anti-commutation relations to re-order

\psi(x_1)\overline{\psi}(x_2)

two ways, first with earlier times on the right assuming

t_2 < t_1

, then with destruction operators on the right. Each of these relations is equal to the original relation

\psi(x_1)\overline{\psi}(x_2)

, and therefore they are equal to each other. Show that Wick's theorem (7-95) and (7-82) holds for

t_2 < t_1

, where the two operator fields are ordered using the

T

and

N

operators (which assume all fields commute/anti-commute.) Hint: Near the end, use (4-145) of Chap. 4 and (7-75).