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

How $$[\phi_1^+,:\phi_2\phi_3:]=:[\phi_1^+,\phi_2^-]\phi_3:+:\phi_2[\phi_1^+,\phi_3^-]:$$ 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)).

• The equation that you wrote is not correct and is not the same equation that is in Peskin&Scroeder (if you are talking about the set of equations that comes after (4.41). Notice that you are missing some $\pm$ on top of $\phi_1,\phi_2$. – Gonenc Mogol Feb 26 '18 at 23:34
• Of course, in P&S it is $ϕ^+_1$ but I still have a problem to demonstrate that. – ketherok Feb 28 '18 at 8:41
• At a first look, even with the corrections, I’m not sure either if the equation can be correct. On the rhs, all creation ops are to the left via the normal ordering. But on the lhs this need not be true. – CAF Mar 1 '18 at 19:11
• Therefore, what formula is used in P&S? – ketherok Mar 2 '18 at 15:05
• @ketherok I don't have access to P&S right now to check – CAF Mar 4 '18 at 15:10

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.
• The equation is equivalent to $[\phi_1,:\phi_2\phi_3:]=:[\phi_1,\phi_2\phi_3]:$ This is not because there is not much freedom for placing the normal order symbol, that this equation should be right. – ketherok Feb 24 '18 at 8:22
• @ketherok Yes the equivalence is there, I just meant terms like $:[A,B]:$ on the right hand side can’t be there because they vanish. Actually from what you wrote you can simply derive a formula for $[A,BC]$ and insert between the normal ordered signs. – CAF Feb 24 '18 at 8:38
• Actually, I did not demonstrate the last equation I wrote. I have just written the right-hand side of the original equation in an other way. Why the right-hand side $:[\phi_1,\phi_2\phi_3]:$ is non zero ? – ketherok Feb 24 '18 at 10:18