to get or This equation is used in defining the contractions used in Wick's theorem. 3. An example with multiple operators is: 4. A simple example shows that normal ordering cannot be extended by linearity from the monomials to all operators in a self-consistent way: The implication is that normal ordering is not a linear function on operators. Multiple bosons If we now consider different bosons there are operators: : the boson's creation operator. : the boson's annihilation operator. Here . These satisfy the commutation relations: where and denotes the Kronecker delta. These may be rewritten as: Examples 1. For two different bosons () we have 2. For three different bosons () we have Notice that since (by the commutation relations) the order in which we write the annihilation operators does not matter. Bosonic operator functions Normal ordering of bosonic operator functions , with occupation number operator , can be accomplished using (falling) factorial powers and Newton series instead of Taylor series: It is easy to show that factorial powers are equal to normal-ordered (raw) powers and are therefore normal ordered by construction, such that the Newton series expansion of an operator function , with -th forward difference at , is always normal ordered. Here, the eigenvalue equation relates and . As a consequence, the normal-ordered Taylor series of an arbitrary function is equal to the Newton series of an associated function , fulfilling if the series coefficients of the Taylor series of , with continuous , match the coefficients of the Newton series of , with integer , with -th partial derivative at . The functions and are related through the so-called normal-order transform according to which can be expressed in terms of the Mellin transform , see for details. Fermions Fermions are particles which satisfy Fermi–Dirac statistics. We will now examine the normal ordering of fermionic creation and annihilation operator products. Single fermions For a single fermion there are two operators of interest: : the fermion's creation operator. : the fermion's annihilation operator. These satisfy the anticommutator relationships where denotes the anticommutator. These may be rewritten as To define the normal ordering of a product of fermionic creation and annihilation operators we must take into account the number of interchanges between neighbouring operators. We get a minus sign for each such interchange. Examples 1. We again start with the simplest cases: This expression is already in normal order