normalorder

Normalorder, commonly called normal ordering, is a prescription in quantum mechanics and quantum field theory for rearranging products of creation and annihilation operators so that all creation operators appear to the left of all annihilation operators. It is typically denoted by colons, as in :a† a:.

The key property is that the vacuum expectation value of a normally ordered product vanishes: ⟨0| :O: |0⟩ = 0, whenever O is a finite product of field operators. This follows because annihilation operators kill the vacuum on the right, and any creation operators on the left cannot act on the bra vacuum. Because the state depends on the choice of vacuum, normal ordering is defined with respect to a particular reference state.

For bosonic operators, normal ordering simply rearranges factors with no extra signs; for fermionic operators, additional minus signs arise from permuting anticommuting operators.

Wick's theorem provides a practical framework: any time-ordered product can be written as a sum of normally ordered products with all possible contractions, where a contraction corresponds to a two-point function or propagator. Normal ordering thus helps separate "vacuum" contributions from interaction effects.

In curved spacetime and other settings with no unique vacuum, normal ordering becomes ambiguous; regulators or alternative schemes (e.g., point-splitting, zeta-function regularization) are used. Normal ordering remains a foundational tool in perturbative calculations, renormalization, and the construction of Hamiltonians in quantum many-body and quantum field theory.