Editing Normal order (section)

==Alternative definitions==

The most general definition of normal ordering involves splitting all quantum fields into two parts (for example see Evans and Steer 1996)
<math>\phi_i(x)=\phi^+_i(x)+\phi^-_i(x)</math>.  
In a product of fields, the fields are split into the two parts and the <math>\phi^+(x)</math> parts are moved so as to be always to the left of all the <math>\phi^-(x)</math> parts.  In the usual case considered in the rest of the article, the <math>\phi^+(x)</math> contains only creation operators, while the <math>\phi^-(x)</math> contains only annihilation operators.  As this is a mathematical identity, one can split fields in any way one likes.  However, for this to be a useful procedure one demands that the normal ordered product of ''any'' combination of fields has zero expectation value

:<math>\langle :\phi_1(x_1)\phi_2(x_2)\ldots\phi_n(x_n):\rangle=0</math>

It is also important for practical calculations that all the commutators (anti-commutator for fermionic fields) of all <math>\phi^+_i</math> and <math>\phi^-_j</math> are all c-numbers.  These two properties means that we can apply '''Wick's theorem''' in the usual way, turning expectation values of time-ordered products of fields into products of c-number pairs, the contractions.  In this generalised setting, the contraction is defined to be the difference between the time-ordered product and the normal ordered product of a pair of fields.

The simplest example is found in the context of [[thermal quantum field theory]] (Evans and Steer 1996).  In this case the expectation values of interest are statistical ensembles, traces over all states weighted by  <math>\exp (-\beta \hat{H})</math>.  For instance, for a single bosonic quantum harmonic oscillator we have that the thermal expectation value of the number operator is simply the [[Bose–Einstein statistics|Bose–Einstein distribution]]

:<math>\langle\hat{b}^\dagger \hat{b}\rangle
 = \frac{\mathrm{Tr} (e^{-\beta \omega \hat{b}^\dagger \hat{b}} \hat{b}^\dagger \hat{b} )}{\mathrm{Tr} (e^{-\beta \omega \hat{b}^\dagger \hat{b} })}
 = \frac{1}{e^{\beta \omega}-1}
</math>

So here the number operator <math>\hat{b}^\dagger \hat{b}</math> is normal ordered in the usual sense used in the rest of the article yet its thermal expectation values are non-zero.  Applying Wick's theorem and doing calculation with the usual normal ordering in this thermal context is possible but computationally impractical. The solution is to define a different ordering, such that the <math>\phi^+_i</math> and <math>\phi^-_j</math> are ''linear combinations'' of the original annihilation and creations operators.  The combinations are chosen to ensure that the thermal expectation values of normal ordered products are always zero so the split chosen will depend on the temperature.