Normal Ordering

Normal ordering is a linear reordering prescription for products in an operator algebra with distinguished creation and annihilation generators. It places every creation operator to the left of every annihilation operator. A normal-ordered product is conventionally enclosed in colons, as in .

For bosonic generators satisfying , normal ordering gives

For fermionic generators satisfying , each interchange introduces a minus sign, so

The prescription extends term by term to polynomials and formal power series in the generators. Rewriting an ordinary product in normal order generally produces additional commutator or anticommutator terms.

Wick's theorem expresses a time-ordered product as a sum of normal-ordered products and contractions.