Normal Ordering

Normal ordering is a convention for writing products of creation and annihilation operators with all creation operators placed to the left of all annihilation operators. It is denoted by colons:

$$:O:.$$

The rule is simple to state but easy to misuse:

• for bosons, move creation operators left in the specified monomial, with no sign from swaps;
• for fermions, do the same with a minus sign for each interchange of fermionic factors;
• the operation is defined relative to a chosen vacuum and a chosen creation-annihilation split.

Inside the colons, omit the commutator or anticommutator contraction terms; outside the colons, retain them to obtain an operator equality. For a long product those terms can multiply leftover operators, so the difference need not be a constant. Explicit constants remain constants: $:I:=I$. The prescription acts on a specified creation–annihilation expression; it is not an algebra homomorphism on arbitrarily rewritten operator expressions.

Helpful background. Bosonic Commutation Relations and Fermionic Anticommutation Relations supply the reordering identities.

Creation Operators to the Left

For one bosonic mode,

$$:a^\dagger a: = a^\dagger a, \qquad :a a^\dagger: = a^\dagger a.$$

The first product is already normal ordered. The second is not, because the annihilation operator appears to the left of the creation operator.

The operator identity is

$$a a^\dagger = a^\dagger a+1.$$

Thus

$$a a^\dagger = :a a^\dagger: +1.$$

The difference between the original product and the normal-ordered product is the c-number term produced by the commutator.

For several bosonic modes,

$$:a_i a_j^\dagger: = a_j^\dagger a_i,$$

and

$$a_i a_j^\dagger = :a_i a_j^\dagger: +\delta_{ij}.$$

Bosonic Normal Ordering

For bosons, creation operators commute with each other, and annihilation operators commute with each other:

$$[a_i^\dagger,a_j^\dagger]=0, \qquad [a_i,a_j]=0.$$

Normal ordering therefore only has to move annihilation operators past creation operators. For example,

$$:a_i a_j^\dagger a_k: = a_j^\dagger a_i a_k.$$

The original product is related to the normal-ordered product by

$$a_i a_j^\dagger a_k = a_j^\dagger a_i a_k +\delta_{ij}a_k.$$

The term $\delta_{ij}a_k$ is not part of the normal-ordered product; it is produced by the commutator needed to reorder the operators.

For bosonic field operators,

$$:\psi(\mathbf x)\psi^\dagger(\mathbf y): = \psi^\dagger(\mathbf y)\psi(\mathbf x),$$

and

$$\psi(\mathbf x)\psi^\dagger(\mathbf y) = :\psi(\mathbf x)\psi^\dagger(\mathbf y): +\delta^{(3)}(\mathbf x-\mathbf y).$$

The delta function is the continuum version of the Kronecker delta.

Fermionic Signs

For fermions, each swap of two fermionic operators introduces a minus sign. For one pair,

$$:c_i c_j^\dagger: = -c_j^\dagger c_i.$$

Using

$$\{c_i,c_j^\dagger\} = \delta_{ij},$$

one obtains

$$c_i c_j^\dagger = \delta_{ij} -c_j^\dagger c_i = :c_i c_j^\dagger: +\delta_{ij}.$$

The contraction term looks similar to the bosonic case, but the normal-ordered operator carries a sign.

For a longer fermionic product, normal ordering means applying the parity of the permutation required to move all creation operators left. For example,

$$:c_i c_j^\dagger c_k: = -c_j^\dagger c_i c_k.$$

If two identical fermionic creation operators or two identical fermionic annihilation operators appear, the product may vanish by anticommutation:

$$(c_i^\dagger)^2=0, \qquad c_i^2=0.$$

The signs are not optional bookkeeping. They are the operator form of antisymmetry.

Vacuum Expectation Values

Let $\lvert0\rangle$ be the vacuum annihilated by every annihilation operator:

$$d_i\lvert0\rangle=0.$$

Any nonconstant normal-ordered monomial has zero vacuum expectation value:

$$\langle0\vert :O:\vert0\rangle=0,$$

provided $:O:$ contains at least one creation or annihilation operator. The reason is that after normal ordering, every annihilation operator is on the right and kills the vacuum ket. If the product contains only creation operators, the vacuum bra kills it after taking the adjoint interpretation of the Fock inner product.

For example,

$$\langle0\vert a a^\dagger\vert0\rangle = 1,$$

but

$$\langle0\vert :a a^\dagger:\vert0\rangle = \langle0\vert a^\dagger a\vert0\rangle = 0.$$

This is why normal ordering is often described as subtracting vacuum contractions.

Normal ordering and a shift of vacuum energy

The number operator $N=a^\dagger a$ is already normal ordered. The symmetric oscillator expression gives

$$H_{\rm sym}=\frac{\hbar\omega}{2}(aa^\dagger+a^\dagger a), \qquad :H_{\rm sym}:=\hbar\omega a^\dagger a.$$

As an operator, $H_{\rm sym}=\hbar\omega(N+1/2)$. Its normal-ordered symmetric expression differs by the vacuum energy $\hbar\omega/2$. But if the starting expression is already $\hbar\omega(a^\dagger a+1/2)$, the literal reordering prescription leaves its explicit constant untouched. To define a vacuum-subtracted Hamiltonian without ambiguity, write $H_{\rm sub}=H-\langle0|H|0\rangle I$.

Subtracting the vacuum expectation is not generally the same as normal ordering. For $X=a+a^\dagger$,

$$X^4=:X^4:+6:X^2:+3I.$$

Thus $X^4-\langle0|X^4|0\rangle I=:X^4:+6:X^2:$, not just $:X^4:$. Lower-order operator terms survive. Wick’s theorem counts these contractions systematically.

Hamiltonians and Interaction Terms

Many-particle Hamiltonians are often written already in normal-ordered form:

$$H = \sum_{ij} h_{ij}d_i^\dagger d_j + \frac12 \sum_{ijkl} V_{ij;kl} d_i^\dagger d_j^\dagger d_l d_k.$$

The one-body term $d_i^\dagger d_j$ is normal ordered. The two-body term $d_i^\dagger d_j^\dagger d_l d_k$ is also normal ordered.

In field notation, the density

$$n(\mathbf x) = \psi^\dagger(\mathbf x)\psi(\mathbf x)$$

is normal ordered relative to the empty-particle vacuum. But products such as $\psi(\mathbf x)\psi^\dagger(\mathbf y)$ are not; reordering them produces delta functions.

Normal ordering is therefore a way to separate operator products into a normal-ordered part plus contraction terms. The full systematic expansion of products into normal-ordered products and contractions is the content of Wick-type formulas.

Choice of reference state

An ordinary unitary rotation among annihilation modes leaves empty-vacuum normal ordering unchanged. A transformation mixing creation with annihilation changes the split and generally changes the reference vacuum. A filled Fermi sea is naturally described using particle and hole operators, not the original empty vacuum.

Normal Ordering in Many-Body QM develops filled, coherent, Bogoliubov, and correlated references. Rewriting a Hamiltonian relative to such a reference generates zero-, one-, and residual many-body terms that must all be retained for an exact identity. Normal ordering by itself is neither a physical approximation nor a full renormalization procedure.

Exercises

1. Bosonic normal ordering. Normal order $a_i a_j^\dagger a_k$ and identify the extra term in the original product.

Solution

Move $a_j^\dagger$ left:

$$:a_i a_j^\dagger a_k: = a_j^\dagger a_i a_k.$$

Using $a_i a_j^\dagger=a_j^\dagger a_i+\delta_{ij}$,

$$a_i a_j^\dagger a_k = a_j^\dagger a_i a_k +\delta_{ij}a_k.$$

Thus the extra term is $\delta_{ij}a_k$.

2. A quartic number expression. Reorder $N^2=a^\dagger aa^\dagger a$ and compare it with its normal-ordered monomial. What does each give on $|n\rangle$?

Solution

Using $aa^\dagger=a^\dagger a+I$ gives $N^2=(a^\dagger)^2a^2+N$. The normally ordered monomial is $(a^\dagger)^2a^2$, with eigenvalue $n(n-1)$, whereas $N^2$ has eigenvalue $n^2$. Their difference is the operator N, not a constant.

3. Why the written expression matters. Apply the monomial normal-ordering prescription to $aa^\dagger-a^\dagger a$. Compare this with first replacing the commutator by I.

Solution

The first procedure gives $a^\dagger a-a^\dagger a=0$; the second gives $:I:=I$. There is no contradiction: the colon prescription reorders the specified free-operator expression and discards its contractions. It is not invariant under using the CCR before the prescription. An actual operator equality keeps the contraction: $aa^\dagger=:aa^\dagger:+I$.

References

• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Kluwer Academic/Plenum, 2000.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Addison-Wesley, 1988.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• M. Srednicki, Quantum Field Theory, Cambridge University Press, 2007.