I am trying to understand wick's theorem and normal ordering mostly from Peskin and Schroeder. Now I have this problem with how normal ordering is defined. It seems to me that if you take the normal ordering of a commutator it should always be zero.(adsbygoogle = window.adsbygoogle || []).push({});

Here is what I understand normal ordering to be. If there is some operator [itex]\hat{A}[/itex] which is a product of bosonic operators [itex]a_p[/itex] and [itex] a_p^{\dagger}[/itex] , then normal ordering of of [itex] \hat{A}[/itex] is [itex] N(\hat{A})[/itex] where the creation operators are all moved to the left and the annihilation operators are all moved to the right. This essentially would mean that inside a normal ordering all operators can be commuted. For example,

[tex] N(a_pa_k^{\dagger}) = N(a_k^{\dagger}a_p) = a_k^{\dagger}a_p [/tex]

Now taking the normal ordering of a commutator,

[tex] N([a_p,a_k^{\dagger}]) = N(a_pa_k^{\dagger} - a_k^{\dagger}a_p) = N(a_pa_k^{\dagger}) -N(a_k^{\dagger}a_p) = 0 [/tex]

But if I had used the fact that the commutator of [itex] [a_p,a_k^{\dagger}] = \delta^{(3)}(p-k) [/itex] which is a c-number, then

[tex] N([a_p,a_k^{\dagger}]) = N(\delta^{(3)}(p-k)) = \delta^{(3)}(p-k) [/tex]

So there is a contradiction. Can someone explain to me what is the right way to think about normal ordering?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Normal ordering of bosonic commuators

**Physics Forums | Science Articles, Homework Help, Discussion**