_Normal ordering_ refers to an ordering of products [creation and annihilation operators](creation%20and%20annihilation%20operators.md) where all [creation operators](creation%20and%20annihilation%20operators.md) are grouped to the left of [annihilation operators.](creation%20and%20annihilation%20operators.md) A normal ordering of an expression is denoted by placing colons ($:$) on either side. Such that, for instance, in the simplest example for a pair of operators $\hat{a}^\dagger,$ $\hat{a}$ such that [$[\hat{a},\hat{a}^\dagger]=0$](Commutators%20in%20quantum%20mechanics.md#Commutation%20relations%20in%202nd%20quantization) for when these are [Bosonic creation and annihilation operators](creation%20and%20annihilation%20operators.md#Bosonic%20creation%20and%20annihilation%20operators), $:\hat{a}_{k_1}\hat{a}_{k_2}^\dagger:\;=\hat{a}_{k_2}^\dagger\hat{a}_{k_1}$ denotes the normal ordered version of this expression. The subscripts are arbitrary and are there to show that this also applies for groups of creation and annihilation operators corresponding to different physical properties $k_1$ and $k_2$ and so on where you can normal order any arbitrary product of creation and annihilation operators. If $\hat{a}^\dagger$ and $\hat{a}$ were [Fermionic](Normal%20ordering.md#Fermionic%20normal%20ordering) then the requirement would be that [$\{\hat{a},\hat{a}^\dagger\}=0$.](Anti-commutators%20in%20quantum%20mechanics.md) for $:\hat{a}_{k_1}\hat{a}_{k_2}^\dagger:\;=\hat{a}_{k_2}^\dagger\hat{a}_{k_1}$ to hold. %%Look carefully at the link to the stack exchange below as well as responses to it. normal ordering isnt just about operators but applies to polynomial functions in quantum mechanics it seems. It's a general thing.%% %%suggested reading pg 225 of Walter Greiner's book along with his explanation of Wick's theorem in there.%% # Properties of Normal ordering %%Consider a product of two or more [normal ordered](Normal%20ordering.md) [creation and annihilation operators,](creation%20and%20annihilation%20operators.md) $:p:.$ This defines a normal ordering if and only if,%% %%best definition is apparently here: Solitons: Differential equations, symmetries, and infinite-dime]nsional algebras. elaborated on here as well https://physics.stackexchange.com/questions/345898/how-exactly-is-normal-ordering-an-operator-defined%% %%The use of the term polynomial in the textbook source is interesting. are products of creation and annihiliaton operators still linear? seems like now.%% # Bosonic normal ordering [Boson](Boson.md) ## Bosonic pair distribution operator %%page 465 of the Sakurai%% # Fermionic normal ordering [Fermion](Fermion.md) #QuantumMechanics/MultiParticleQuantumSystems #QuantumMechanics/QuantumFieldTheory