Why do we have $[phi_1^+,:phi_2phi_3:]=:[phi_1^+,phi_2^-]phi_3:+:phi_2[phi_1^+,phi_3^-]:$?

Question

How $$[phi_1^+,:phi_2phi_3:]=:[phi_1^+,phi_2^-]phi_3:+:phi_2[phi_1^+,phi_3^-]:$$ 
with $phi_i=phi(x_i)$ field operators ($phi_i^+$ is the annihilation part while $phi_i^-$ is the creation part), can be demonstrated?

This formula seems to be used in the demonstration of the Wick theorem in Peskin&Schroeder's book (in the second line of the middle page equation (page 90)).

CAF · Answer

If you haven't done such a computation before, try splitting the fields $$phi_i = phi_i^+ + phi_i^-,$$ a decomposition into its positive and frequency parts containing, respectively, the annihilation and creation operators. Using the definition of normal ordering should give you your result after a little plug and chug.

Answered by CAF on July 16, 2021

Why do we have $[phi_1^+,:phi_2phi_3:]=:[phi_1^+,phi_2^-]phi_3:+:phi_2[phi_1^+,phi_3^-]:$?

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