Editing Normal order (section)

== Uses in quantum field theory==

The [[vacuum expectation value]] of a normal ordered product of creation and annihilation operators is zero. This is because, denoting the [[vacuum state]] by <math>|0\rangle</math>, the creation and annihilation operators satisfy
:<math>\langle 0 | \hat{a}^\dagger = 0 \qquad \textrm{and} \qquad \hat{a} |0\rangle = 0</math>
(here <math>\hat{a}^\dagger</math> and <math>\hat{a}</math> are creation and annihilation operators (either bosonic or fermionic)).

Let <math>\hat{O}</math> denote a non-empty product of creation and annihilation operators. Although this may satisfy
:<math>\langle 0 | \hat{O} | 0 \rangle \neq 0,</math>
we have
:<math>\langle 0 | :\hat{O}: | 0 \rangle = 0.</math>

Normal ordered operators are particularly useful when defining a quantum mechanical [[Hamiltonian (quantum mechanics)|Hamiltonian]]. If the Hamiltonian of a theory is in normal order then the ground state energy will be zero:
<math>\langle 0 |\hat{H}|0\rangle = 0</math>.

===Free fields===
With two free fields φ and χ,

:<math>:\phi(x)\chi(y):\,\,=\phi(x)\chi(y)-\langle 0|\phi(x)\chi(y)| 0\rangle</math>

where <math>|0\rangle</math> is again the vacuum state. Each of the two terms on the right hand side typically blows up in the limit as y approaches x but the difference between them has a well-defined limit. This allows us to define :φ(x)χ(x):.

===Wick's theorem===
{{Main|Wick's theorem}}

'''Wick's theorem''' states the relationship between the time ordered product of <math>n</math> fields and a sum of 
normal ordered products. This may be expressed for <math>n</math> even as

:<math>\begin{align}
T\left[\phi(x_1)\cdots \phi(x_n)\right]=&:\phi(x_1)\cdots \phi(x_n):
+\sum_\textrm{perm}\langle 0 |T\left[\phi(x_1)\phi(x_2)\right]|0\rangle :\phi(x_3)\cdots \phi(x_n):\\
&+\sum_\textrm{perm}\langle 0 |T\left[\phi(x_1)\phi(x_2)\right]|0\rangle \langle 0 |T\left[\phi(x_3)\phi(x_4)\right]|0\rangle:\phi(x_5)\cdots \phi(x_n):\\ 
\vdots \\
&+\sum_\textrm{perm}\langle 0 |T\left[\phi(x_1)\phi(x_2)\right]|0\rangle\cdots \langle 0 |T\left[\phi(x_{n-1})\phi(x_n)\right]|0\rangle
\end{align}</math>

where the summation is over all the distinct ways in which one may pair up fields. The result for <math>n</math> odd looks the same
except for the last line which reads

:<math>
\sum_\text{perm}\langle 0 |T\left[\phi(x_1)\phi(x_2)\right]|0\rangle\cdots\langle 0 | T\left[\phi(x_{n-2})\phi(x_{n-1})\right]|0\rangle\phi(x_n).
</math>

This theorem provides a simple method for computing vacuum expectation values of time ordered products of operators and was the motivation behind the introduction of normal ordering.