Normal Ordering in Many-Body QM

Normal ordering is more than a visual rule that places creation operators to the left. In many-body quantum mechanics it is a way to rewrite an operator around a chosen reference state. The reference may be the empty Fock vacuum, a filled Slater determinant, a coherent condensate, or a Bogoliubov quasiparticle vacuum. Changing that reference changes the contractions and therefore changes which constant, one-body, pairing, and residual interaction terms appear.

The central logical distinction is

$$\text{all induced terms retained} \quad\Longrightarrow\quad \text{exact operator identity},$$

whereas

$$\text{residual terms discarded} \quad\Longrightarrow\quad \text{reference-dependent approximation}.$$

This page develops that distinction for nonrelativistic many-body systems. The creation-left definition and elementary vacuum examples live in Normal Ordering. The systematic sum over contractions is introduced separately in Wick’s Theorem Preview.

Conventions

Unless stated otherwise:

• $a_p,a_p^\dagger$ are fermionic mode operators;
• $p,q,r,s$ label arbitrary one-particle modes;
• $i,j,k$ label modes occupied in a Slater-determinant reference;
• $a,b,c$ label modes unoccupied in that reference;
• $\lvert\Phi\rangle$ denotes the chosen many-body reference state;
• $:O:_0$ denotes normal ordering relative to the empty Fock vacuum;
• $\{O\}_\Phi$ denotes normal ordering relative to $\lvert\Phi\rangle$;
• the braces $\{O\}_\Phi$ are not an anticommutator;
• two-body matrix elements $\overline v_{pq;rs}$ are antisymmetrized;
• every one-particle basis is orthonormal unless noted otherwise.

For bosons, the corresponding commutators replace fermionic anticommutators. Sections devoted to bosonic displacement and quasiparticle references state their conventions explicitly.

Canonical Scope

This page owns the many-body use of normal ordering:

• normal ordering relative to a filled or quasiparticle reference;
• particle-hole creation and annihilation operators;
• exact zero-, one-, and two-body decompositions of Hamiltonians;
• lower-body terms induced by contractions;
• fermionic signs in reference-state expansions;
• Slater, coherent, Bogoliubov, and correlated references;
• the approximation introduced by dropping residual normal-ordered terms;
• the relation to Wick factorization without reproducing its full theorem.

Other pages retain their canonical roles:

• compact canonical-mode, number-shift, and Gaussian identities: Operator Identities;
• creation-left ordering and empty-vacuum examples: Normal Ordering;
• contraction bookkeeping and Gaussian factorization: Wick’s Theorem Preview;
• operator prefactors and two-body matrix elements: Two-Body Operators;
• particle-number sectors and anomalous averages: Number Operators and Conserved Quantities;
• field products and local densities: Field Operators in Many-Body Models;
• excitation-rank denominators, linked energies, and infrared counting: Perturbation Theory in Many-Body Systems.

What Normal Ordering Does

Normal ordering chooses a set of operators that annihilate a reference and then places their adjoints to the left of them. For the empty vacuum $\lvert0\rangle$,

$$a_p\lvert0\rangle = 0,$$

so $a_p^\dagger$ is a creator and $a_p$ is an annihilator. The elementary identity is

$$a_p a_q^\dagger = \delta_{pq} - a_q^\dagger a_p$$

for fermions. The first term is the contraction generated while reordering the second.

For a filled reference, some $a_p^\dagger$ operators annihilate the state by Pauli exclusion, while some $a_p$ operators create holes. The words creation and annihilation must then refer to excitations relative to the chosen state, not merely to microscopic particle number.

Empty-Vacuum Ordering

With the empty vacuum,

$$:a_p^\dagger a_q:_0 = a_p^\dagger a_q,$$

and

$$:a_p^\dagger a_q^\dagger a_s a_r:_0 = a_p^\dagger a_q^\dagger a_s a_r.$$

Every nonconstant vacuum-normal-ordered monomial has zero empty-vacuum expectation value:

$$\langle0\rvert :O:_0 \lvert0\rangle = 0.$$

This statement does not imply

$$\langle\Phi\rvert :O:_0 \lvert\Phi\rangle = 0$$

for a filled or correlated state. Empty-vacuum normal ordering is an algebraic convention tied to $\lvert0\rangle$.

Why the Reference Changes the Expansion

Suppose $\lvert\Phi\rangle$ contains occupied fermion orbitals. Then

$$a_i^\dagger\lvert\Phi\rangle = 0$$

for every occupied $i$, while

$$a_a\lvert\Phi\rangle = 0$$

for every unoccupied $a$. Relative to $\lvert\Phi\rangle$:

• $a_a^\dagger$ creates a particle excitation;
• $a_a$ annihilates a particle excitation;
• $a_i$ creates a hole excitation;
• $a_i^\dagger$ annihilates a hole excitation.

Thus an operator already normal ordered relative to $\lvert0\rangle$ need not be normal ordered relative to $\lvert\Phi\rangle$.

Normal ordering is a reference-dependent reorganization. Empty, Slater, coherent, and Bogoliubov references define different contractions. The decomposition remains an identity only while every induced operator rank is retained.

Particle-Hole Vacuum

For a Slater determinant with occupation numbers $n_p\in\{0,1\}$, define quasiparticle annihilators

$$\beta_p = \begin{cases} a_p,&n_p=0,\\ a_p^\dagger,&n_p=1. \end{cases}$$

They satisfy

$$\beta_p\lvert\Phi\rangle = 0.$$

Their adjoints create excitations:

$$\beta_p^\dagger = \begin{cases} a_p^\dagger,&n_p=0,\\ a_p,&n_p=1. \end{cases}$$

The particle-hole transformation preserves the canonical anticommutation relations:

$$\{\beta_p,\beta_q^\dagger\} = \delta_{pq}.$$

Reference normal ordering places every $\beta^\dagger$ to the left of every $\beta$. This definition immediately guarantees

$$\langle\Phi\rvert \{O\}_\Phi \lvert\Phi\rangle = 0$$

for every nonconstant normal-ordered quasiparticle monomial.

Reference Contractions

In the occupation basis of $\lvert\Phi\rangle$,

$$\langle\Phi\rvert a_p^\dagger a_q \lvert\Phi\rangle = n_p\delta_{pq},$$

and

$$\langle\Phi\rvert a_p a_q^\dagger \lvert\Phi\rangle = (1-n_p)\delta_{pq}.$$

Both contractions are needed. A filled mode has a nonzero $a_p^\dagger a_p$ contraction, whereas an empty mode has a nonzero $a_p a_p^\dagger$ contraction.

In an arbitrary basis, define

$$\rho_{pq} \equiv \langle\Phi\rvert a_p^\dagger a_q \lvert\Phi\rangle.$$

For a Slater determinant, $\rho$ is Hermitian and idempotent:

$$\rho^\dagger = \rho, \qquad \rho^2 = \rho.$$

The complementary contraction follows from the canonical anticommutator:

$$\langle\Phi\rvert a_p a_q^\dagger \lvert\Phi\rangle = \delta_{pq} - \rho_{qp}.$$

One-Body Strings Around a Reference

The simplest reference-normal-ordering identity is

$$a_p^\dagger a_q = \{a_p^\dagger a_q\}_\Phi + \rho_{pq}.$$

In the diagonal occupation basis,

$$a_p^\dagger a_q = \{a_p^\dagger a_q\}_\Phi + n_p\delta_{pq}.$$

For the total number operator,

$$\widehat N = N_\Phi + \sum_p \{a_p^\dagger a_p\}_\Phi,$$

where

$$N_\Phi = \sum_p n_p.$$

The normal-ordered correction counts particles above the reference and holes below it with opposite signs. A one-particle-one-hole excitation has the same total particle number as $\lvert\Phi\rangle$.

Fermionic Four-Operator Identity

For a Slater determinant or another number-conserving Gaussian reference,

$$\begin{aligned} a_p^\dagger a_q^\dagger a_s a_r ={}& \{a_p^\dagger a_q^\dagger a_s a_r\}_\Phi \\ &+ \rho_{pr} \{a_q^\dagger a_s\}_\Phi \\ &- \rho_{ps} \{a_q^\dagger a_r\}_\Phi \\ &- \rho_{qr} \{a_p^\dagger a_s\}_\Phi \\ &+ \rho_{qs} \{a_p^\dagger a_r\}_\Phi \\ &+ \rho_{pr}\rho_{qs} - \rho_{ps}\rho_{qr}. \end{aligned}$$

The last line is the double-contraction contribution. Taking the reference expectation gives

$$\begin{aligned} \langle a_p^\dagger a_q^\dagger a_s a_r \rangle_\Phi ={}& \rho_{pr}\rho_{qs} - \rho_{ps}\rho_{qr}. \end{aligned}$$

The minus sign is exchange. It is not a convention that may be dropped.

Deriving the Reference-Normal-Ordered Hamiltonian

Consider a fermionic Hamiltonian with one- and two-body terms:

$$\begin{aligned} H ={}& \sum_{p,q} t_{pq}a_p^\dagger a_q \\ &+ \frac14 \sum_{p,q,r,s} \overline v_{pq;rs} a_p^\dagger a_q^\dagger a_s a_r. \end{aligned}$$

The factor $1/4$ accompanies fully antisymmetrized matrix elements. Insert the one- and two-body normal-ordering identities and collect equal quasiparticle ranks. In the orbital basis where $\lvert\Phi\rangle$ occupies $i,j,\ldots$,

$$\begin{aligned} H ={}& E_\Phi + \sum_{p,q} f_{pq} \{a_p^\dagger a_q\}_\Phi \\ &+ \frac14 \sum_{p,q,r,s} \overline v_{pq;rs} \{a_p^\dagger a_q^\dagger a_s a_r\}_\Phi. \end{aligned}$$

The induced coefficients are

$$E_\Phi = \sum_i t_{ii} + \frac12 \sum_{i,j} \overline v_{ij;ij},$$

and

$$f_{pq} = t_{pq} + \sum_i \overline v_{pi;qi}.$$

No approximation has been made. The same operator has been reorganized into zero-body, one-body, and residual two-body pieces relative to $\lvert\Phi\rangle$.

Zero-Body Term

The constant is the reference expectation value:

$$E_\Phi = \langle\Phi\rvert H \lvert\Phi\rangle.$$

It is called zero-body because it contains no quasiparticle operators. It is not automatically disposable. It matters for:

• comparing variational references;
• phase competition and binding energies;
• thermodynamic potentials and partition functions;
• forces obtained by differentiating an energy;
• any calculation where absolute energy differences between models matter.

A constant may cancel from selected normalized expectation values or closed-system equations of motion, but that is a separate argument.

Induced One-Body Term

The coefficient

$$f_{pq} = t_{pq} + \sum_i \overline v_{pi;qi}$$

contains the original one-body matrix and the contraction of one incoming-outgoing pair in the interaction. It is often called a Fock matrix or in-medium one-body field.

This term does not mean that the microscopic interaction has become one-body. The residual two-body operator remains present. Normal ordering has only exposed how the interaction acts when one line is saturated by the occupied reference.

Residual Two-Body Term

The residual interaction is

$$H_{\mathrm{res}}^{[2]} = \frac14 \sum_{p,q,r,s} \overline v_{pq;rs} \{a_p^\dagger a_q^\dagger a_s a_r\}_\Phi.$$

It creates, annihilates, or scatters quasiparticle excitations relative to $\lvert\Phi\rangle$. Calling it residual does not imply that it is numerically small. Its size depends on the interaction, reference, observable, and regime.

Strongly correlated systems can have a poor single-determinant reference even when the algebraic normal-ordering decomposition is exact.

Hartree–Fock Stationarity

Let

$$\lvert\Phi_i^a\rangle = a_a^\dagger a_i \lvert\Phi\rangle$$

be a one-particle-one-hole excitation. The Hamiltonian matrix element is

$$\langle\Phi_i^a\rvert H \lvert\Phi\rangle = f_{ai}.$$

For a self-consistent Hartree–Fock reference, stationarity under occupied-unoccupied orbital rotations gives the Brillouin condition

$$f_{ai} = 0.$$

This removes the direct coupling from the reference to single excitations. It does not remove double excitations or make the residual two-body term vanish.

Exact Reorganization Versus Mean Field

The identity

$$H = E_\Phi + H_\Phi^{[1]} + H_\Phi^{[2]}$$

is exact for a Hamiltonian containing at most two-body interactions. A mean-field approximation begins only after one replaces the full problem by selected terms, for example

$$H \longrightarrow E_\Phi + H_\Phi^{[1]}.$$

Dropping $H_\Phi^{[2]}$ discards residual correlations. Conversely, perturbation theory, coupled-cluster theory, configuration interaction, and Green-function methods retain its effects in different controlled or approximate ways.

Normal ordering and mean-field decoupling are related, but they are not synonyms.

Three-Body Operators and Induced Lower Ranks

Let a fully antisymmetrized three-body operator be

$$\begin{aligned} W ={}& \frac1{36} \sum_{p,q,r,s,t,u} w_{pqr;stu} \\ &\times a_p^\dagger a_q^\dagger a_r^\dagger a_u a_t a_s. \end{aligned}$$

Normal ordering relative to a Slater determinant produces ranks from zero through three:

$$W = W_\Phi^{[0]} + W_\Phi^{[1]} + W_\Phi^{[2]} + W_\Phi^{[3]}.$$

With the displayed antisymmetrization convention,

$$W_\Phi^{[0]} = \frac16 \sum_{i,j,k} w_{ijk;ijk},$$ $$W_\Phi^{[1]} = \sum_{p,q} \left( \frac12 \sum_{i,j} w_{pij;qij} \right) \{a_p^\dagger a_q\}_\Phi,$$

and

$$\begin{aligned} W_\Phi^{[2]} ={}& \frac14 \sum_{p,q,r,s} \left( \sum_i w_{pqi;rsi} \right) \\ &\times \{a_p^\dagger a_q^\dagger a_s a_r\}_\Phi. \end{aligned}$$

The residual term is

$$\begin{aligned} W_\Phi^{[3]} ={}& \frac1{36} \sum_{p,q,r,s,t,u} w_{pqr;stu} \\ &\times \{a_p^\dagger a_q^\dagger a_r^\dagger a_u a_t a_s\}_\Phi. \end{aligned}$$

The normal-ordered two-body approximation retains $W_\Phi^{[0]}$, $W_\Phi^{[1]}$, and $W_\Phi^{[2]}$ but discards $W_\Phi^{[3]}$. That truncation can be accurate for suitable references and observables, but it is not an operator identity.

Why Reference Choice Can Help

A well-chosen reference incorporates large average occupancies into contractions. Contributions from a high-body interaction can then feed into lower-rank operators that are cheaper to store and manipulate.

This does not guarantee small residual terms. A reference is useful when it captures the dominant structure relevant to the target state or ensemble. Diagnostics include:

• the size of off-diagonal one-body couplings;
• occupation numbers far from $0$ or $1$;
• the norm or energy contribution of discarded residual ranks;
• sensitivity to changing the reference;
• comparison against larger-rank or exact calculations.

Fermionic Sign Discipline

Fermionic signs come from the parity of the permutation used to reorder operators. A safe calculation follows three rules:

1. retain the original operator order until a contraction is selected;
2. count every adjacent interchange of odd fermionic operators;
3. put each remaining string into one fixed canonical order.

For the four-operator string, the direct and exchange double contractions are

$$\rho_{pr}\rho_{qs}$$

and

$$-\rho_{ps}\rho_{qr}.$$

The exchange pairing crosses one additional fermionic line and therefore carries a minus sign.

Do not infer signs from a diagram’s visual symmetry alone. Index relabeling is safe only after the antisymmetry of matrix elements and the operator permutation have both been accounted for.

Wick Theorem Preview

For a Gaussian reference, Wick’s theorem states schematically that an operator product equals

$$\begin{aligned} \text{reference-normal product} &+ \text{all single contractions} \\ &+ \text{all double contractions} +\cdots . \end{aligned}$$

The four-operator identity above is one concrete instance. For fermions, every contraction pattern carries the permutation sign required to bring paired operators together and normal order the uncontracted remainder.

A contraction is defined only after specifying:

• the reference state;
• the operator ordering under consideration;
• whether the product is equal-time, time ordered, contour ordered, or thermal;
• the statistics and sign convention.

The full combinatorial theorem remains at Wick’s Theorem Preview. Here its role is to justify reference-dependent Hamiltonian decompositions.

Diagrammatic Methods Preview continues from this algebra to ordered propagators, interaction vertices, diagram signs, and Dyson organization. The decomposition on this page remains the canonical operator identity.

Gaussian References

Vacua of quadratic Hamiltonians are Gaussian states. Their higher correlation functions are determined by two-point contractions. Examples include:

• the empty particle vacuum;
• a Slater determinant or filled Fermi sea;
• a bosonic coherent or squeezed Gaussian state;
• an unprojected BCS or Hartree–Fock–Bogoliubov quasiparticle vacuum;
• a thermal state of a quadratic Hamiltonian, with a thermal ordering convention.

For such references, pair contractions are sufficient for Wick factorization. The state can still contain large fluctuations or entanglement in the original particle basis.

Correlated References and Density Cumulants

An interacting reference need not be Gaussian. Define its two-body reduced density matrix by

$$\Gamma_{pq;rs} \equiv \langle\Phi\rvert a_p^\dagger a_q^\dagger a_s a_r \lvert\Phi\rangle.$$

The connected two-body cumulant is

$$\lambda_{pq;rs} = \Gamma_{pq;rs} - \left( \rho_{pr}\rho_{qs} - \rho_{ps}\rho_{qr} \right).$$

For a Slater determinant,

$$\lambda_{pq;rs} = 0.$$

For a correlated reference it is generally nonzero. Pairwise Wick factorization then fails, and higher density cumulants enter generalized normal-ordering and Wick expansions.

Replacing a correlated reference by a Gaussian one with the same one-body density discards $\lambda$ and higher connected information. That replacement is an approximation, not a change of notation.

Generalized Normal Ordering

Generalized normal ordering can be defined relative to multiconfigurational or correlated references so that reference expectations of non-scalar generalized-normal products vanish. The corresponding generalized Wick theorem contains irreducible density matrices, not only pair contractions.

This machinery is valuable in multireference electronic-structure and nuclear many-body methods. It also carries more bookkeeping:

• one-body contractions need not be projectors;
• two- and higher-body cumulants survive;
• products of generalized-normal strings generate cumulant terms;
• truncations must specify which cumulant ranks are retained.

Ordinary Slater-determinant formulas must not be imported unchanged into a correlated-reference calculation.

Bosonic Displaced Reference

For a bosonic coherent reference $\lvert\boldsymbol\alpha\rangle$, define fluctuation operators

$$b_i = a_i-\alpha_i,$$

so that

$$b_i \lvert\boldsymbol\alpha\rangle = 0.$$

Then

$$a_i = \alpha_i+b_i.$$

Even a one-body number operator separates into several fluctuation ranks:

$$\begin{aligned} a_i^\dagger a_i ={}& |\alpha_i|^2 + \alpha_i^*b_i \\ &+ \alpha_i b_i^\dagger + b_i^\dagger b_i. \end{aligned}$$

An interacting Hamiltonian becomes

$$H = H^{[0]} + H^{[1]} + H^{[2]} + H^{[3]} + H^{[4]} +\cdots$$

in powers of $b,b^\dagger$. Stationarity of the appropriate mean-field functional, including chemical-potential or other constraints when required, cancels the linear term. Retaining only the quadratic part is a Bogoliubov approximation; retaining every power is the original Hamiltonian expressed in shifted variables.

The c-number substitution $a_i\to\alpha_i$ is therefore not an exact operator identity.

Number-Conserving Bosonic References

A condensate number state is not the same as a coherent state. For a sharp-$N$ state,

$$\langle a_i\rangle = 0$$

by the number selection rule, even when one mode has macroscopic occupation. Number-conserving Bogoliubov methods organize fluctuations without assuming a coherent superposition of total particle numbers.

Normal ordering around a coherent state is often efficient, but its broken-$U(1)$ language and its thermodynamic-limit interpretation should be stated. The dilute-gas setting is introduced in Weakly Interacting Bose Gas Preview, while Bogoliubov Theory owns the canonical quadratic diagonalization.

Bogoliubov Quasiparticle Vacuum

A fermionic Bogoliubov transformation mixes particle creation and annihilation operators:

$$\beta_\mu = \sum_p \left( U_{p\mu}^*c_p + V_{p\mu}^*c_p^\dagger \right).$$

The matrices $U,V$ must satisfy the conditions that preserve the canonical anticommutation relations.

For a quasiparticle vacuum $\lvert\Phi_B\rangle$,

$$\beta_\mu \lvert\Phi_B\rangle = 0.$$

Normal ordering relative to $\lvert\Phi_B\rangle$ places $\beta^\dagger$ operators left of $\beta$ operators. In the original particle basis, contractions include both a normal density

$$\rho_{pq} = \langle\Phi_B\rvert c_p^\dagger c_q \lvert\Phi_B\rangle$$

and an anomalous density

$$\kappa_{pq} = \langle\Phi_B\rvert c_p c_q \lvert\Phi_B\rangle.$$

Fermionic antisymmetry requires

$$\kappa_{pq} = -\kappa_{qp}.$$

A parity-even Hamiltonian can then contain quasiparticle terms of several even degrees. If the reference is chosen self-consistently, the quadratic part defines the quasiparticle spectrum and selected off-diagonal terms vanish.

Number Symmetry and Anomalous Contractions

The anomalous density carries particle-number charge $-2$ because

$$[\widehat N,c_p c_q] = -2c_p c_q.$$

It vanishes in an exact particle-number eigenstate. A nonzero $\kappa$ therefore signals an unprojected number-breaking reference, a phase reference, or an enlarged relational description.

Pairing correlations do not require a nonzero anomalous average. Neutral quantities such as

$$\langle c_p^\dagger c_q^\dagger c_s c_r \rangle$$

can reveal pairing in a fixed-number state. Number projection converts an unprojected quasiparticle vacuum into a non-Gaussian state, so the simple quasiparticle Wick theorem must then be reconsidered.

Reference Expectation Values

The defining practical property is

$$\langle\Phi\rvert \{O\}_\Phi \lvert\Phi\rangle = 0$$

for non-scalar strings under the chosen normal-ordering prescription. Consequently, the zero-body coefficient of an exactly normal-ordered operator is its reference expectation value.

Three statements should not be conflated:

1. $\langle0\rvert:O:_0\lvert0\rangle=0$ for empty-vacuum ordering;
2. $\langle\Phi\rvert\{O\}_\Phi\lvert\Phi\rangle=0$ for ordering relative to $\Phi$;
3. $\langle\Psi\rvert\{O\}_\Phi\lvert\Psi\rangle=0$ for an arbitrary state $\Psi$.

The third statement is generally false.

Basis Transformations

A unitary rotation among annihilation operators,

$$b_\alpha = \sum_p U_{p\alpha}^*a_p,$$

preserves the empty vacuum and does not mix creation with annihilation. Vacuum-normal-ordered polynomials remain vacuum normal ordered after transforming all coefficients consistently.

A particle-hole or Bogoliubov transformation does mix the original creation-annihilation split. It changes the natural reference vacuum and the contractions. The operator is unchanged, but its normal-ordered decomposition is not.

Truncating the one-particle space adds another issue: projection can change contractions and induce effective many-body terms. Basis rotation, reference change, and model-space truncation are distinct operations.

Continuum Fields and Coincident Points

For equal-time fields,

$$\psi(\mathbf x) \psi^\dagger(\mathbf y) = \delta(\mathbf x-\mathbf y) \pm \psi^\dagger(\mathbf y) \psi(\mathbf x),$$

where the upper sign is bosonic and the lower sign is fermionic. Reordering density products produces contact terms. For either statistics, with the operators ordered consistently,

$$\begin{aligned} n(\mathbf x)n(\mathbf y) ={}& \psi^\dagger(\mathbf x) \psi^\dagger(\mathbf y) \psi(\mathbf y) \psi(\mathbf x) \\ &+ \delta(\mathbf x-\mathbf y) \psi^\dagger(\mathbf x) \psi(\mathbf y). \end{aligned}$$

At coincident points the symbolic term $\delta(\mathbf0)$ warns that operator-valued distributions are being multiplied. A lattice spacing, momentum cutoff, point splitting, smearing, or another regulator may be required.

Normal ordering subtracts contractions associated with a chosen reference. It does not by itself define every singular continuum composite operator or replace renormalization.

Normal Ordering Versus Time Ordering

Normal ordering and time ordering answer different questions.

Ordering	Organizing rule	Main use
normal	creators left of reference annihilators	reference vacuum and Hamiltonian decompositions
time	later times left	real-time propagators and perturbation theory
imaginary time	larger imaginary time left	equilibrium Matsubara theory
symmetric	average over operator permutations	phase-space and measurement conventions

A time-ordered product is not generally normal ordered. Wick’s theorem relates the two for suitable Gaussian references by adding contractions. In relativistic QFT, those time-ordered contractions become propagators, not merely equal-time Kronecker or Dirac deltas.

Normal Ordering and Measured Correlations

Some measurement models naturally select normal-ordered correlations. For example, idealized particle-counting or photodetection expressions involve

$$G^{(2)}(1,2) = \langle \psi^\dagger(1) \psi^\dagger(2) \psi(2) \psi(1) \rangle.$$

The ordering is part of the observable model. It should not be changed merely because another ordering is algebraically convenient. Ordinary, normal-ordered, symmetrized, retarded, and time-ordered correlators generally encode different physical questions.

The broader taxonomy is developed in Correlation Functions Overview.

Computational Workflow

For a reference-normal-ordering calculation:

1. specify the microscopic operator and all prefactor conventions;
2. choose the reference state and declare whether it is Gaussian;
3. compute normal and, if needed, anomalous contractions;
4. define one canonical index and fermionic sign order;
5. generate every induced zero-, one-, two-, and higher-body term;
6. verify that the full expansion reproduces test matrix elements;
7. state any discarded residual rank explicitly;
8. estimate truncation error by changing the reference or retained rank;
9. keep zero-body terms when comparing energies or thermodynamics;
10. distinguish reference change from basis or model-space truncation.

For symbolic implementations, test low-dimensional Fock-space matrices. Random numerical coefficients are especially effective at exposing missing exchange signs or combinatorial factors.

Validation Checks

Useful exact checks include:

$$\langle\Phi\rvert H\lvert\Phi\rangle = E_\Phi,$$ $$\langle\Phi_i^a\rvert H\lvert\Phi\rangle = f_{ai},$$

and reconstruction of the original operator matrix from every retained normal-ordered rank. Also verify:

• Hermiticity of each reconstructed contribution;
• antisymmetry of fermionic coefficient tensors;
• conservation of exact charges when all terms are retained;
• covariance under unitary rotations within occupied and virtual subspaces;
• vanishing reference expectation of non-scalar normal products;
• convergence as discarded ranks are restored.

Common Mistakes

• Treating creation-left vacuum ordering as reference independent.
• Forgetting that an occupied-mode annihilator creates a hole.
• Using $\langle a_p^\dagger a_q\rangle$ but omitting the complementary $\langle a_p a_q^\dagger\rangle$ contraction.
• Dropping the exchange minus sign in a fermionic double contraction.
• Calling the induced one-body field an exact mean-field approximation while silently discarding the residual interaction.
• Removing the zero-body term before comparing reference energies.
• Applying Slater-determinant Wick factorization to a correlated reference with nonzero cumulants.
• Treating a coherent-state substitution as an operator identity.
• Assuming a number-projected BCS state remains a quasiparticle Gaussian vacuum.
• Confusing normal ordering with time ordering.
• Assuming normal ordering removes every ultraviolet divergence.
• Calling a normal-ordered three-body truncation exact after discarding its residual three-body term.

Quick Reference

Question	Diagnostic
What defines the ordering?	the operators that annihilate the stated reference
Is the decomposition exact?	yes, only if every induced rank is retained
What is the zero-body term?	the reference expectation value
Why does a two-body force induce one-body terms?	one particle line is contracted with the reference density
Where do fermionic signs enter?	permutation parity for contractions and residual strings
When is pairwise Wick factorization exact?	for a Gaussian reference
What changes for a correlated reference?	irreducible density cumulants enter
What does Hartree–Fock stationarity imply?	$f_{ai}=0$ for occupied-unoccupied rotations
Does a Bogoliubov vacuum conserve particle number?	generally no; anomalous contractions can be nonzero
Does normal ordering regularize local field products?	not in general; a regulator and renormalization may still be required

Summary

• Normal ordering is defined relative to a vacuum or reference state.
• A filled Slater determinant is a vacuum for particle and hole quasiparticle annihilators.
• Reference contractions turn microscopic interactions into zero-, one-, and residual higher-body terms.
• Retaining every induced term gives an exact operator identity.
• Mean-field and normal-ordered rank truncations begin when residual terms are discarded.
• Fermionic exchange signs follow from permutation parity and must be tracked explicitly.
• Pairwise Wick factorization is exact for Gaussian references, not arbitrary correlated states.
• Correlated references require density cumulants and generalized normal ordering.
• Coherent and Bogoliubov references can generate linear, anomalous, and pairing terms.
• Normal ordering is distinct from time ordering, regularization, and renormalization.

Exercises

Exercise 1: Particle and hole quasiparticles

For a Slater determinant $\lvert\Phi\rangle$, show that

$$\beta_a=a_a, \qquad \beta_i=a_i^\dagger$$

annihilate the reference for unoccupied $a$ and occupied $i$. Identify $\beta_a^\dagger$ and $\beta_i^\dagger$ physically.

Solution

An unoccupied mode contains no particle, so

$$a_a\lvert\Phi\rangle = 0.$$

An occupied fermion mode cannot accept another fermion, so

$$a_i^\dagger\lvert\Phi\rangle = 0.$$

Therefore both $\beta_a=a_a$ and $\beta_i=a_i^\dagger$ annihilate the reference. Their adjoints are

$$\beta_a^\dagger = a_a^\dagger, \qquad \beta_i^\dagger = a_i.$$

The first creates a particle above the reference. The second removes an occupied particle and therefore creates a hole.

Exercise 2: Number operator relative to a Fermi sea

Derive

$$\widehat N = N_\Phi + \sum_p \{a_p^\dagger a_p\}_\Phi.$$

Explain why $a_a^\dagger a_i\lvert\Phi\rangle$ has the same total particle number as $\lvert\Phi\rangle$.

Solution

For each mode,

$$a_p^\dagger a_p = \{a_p^\dagger a_p\}_\Phi + n_p.$$

Summing gives

$$\begin{aligned} \widehat N &= \sum_p a_p^\dagger a_p \\ &= \sum_p n_p + \sum_p \{a_p^\dagger a_p\}_\Phi \\ &= N_\Phi + \sum_p \{a_p^\dagger a_p\}_\Phi. \end{aligned}$$

The operator $a_i$ removes one particle from occupied mode $i$, while $a_a^\dagger$ adds one particle to unoccupied mode $a$. The net change is zero.

Exercise 3: Exchange from double contractions

For a Slater determinant, use pair contractions to evaluate

$$\langle a_p^\dagger a_q^\dagger a_s a_r \rangle_\Phi.$$

Solution

There are two complete pairings. The direct pairing contracts $p$ with $r$ and $q$ with $s$:

$$\rho_{pr}\rho_{qs}.$$

The exchange pairing contracts $p$ with $s$ and $q$ with $r$. It differs by one odd fermionic permutation, so it contributes

$$-\rho_{ps}\rho_{qr}.$$

Therefore

$$\langle a_p^\dagger a_q^\dagger a_s a_r \rangle_\Phi = \rho_{pr}\rho_{qs} - \rho_{ps}\rho_{qr}.$$

Exercise 4: Zero- and one-body terms from a pair force

Starting from

$$V = \frac14 \sum_{p,q,r,s} \overline v_{pq;rs} a_p^\dagger a_q^\dagger a_s a_r,$$

show that a Slater reference induces

$$V_\Phi^{[0]} = \frac12 \sum_{i,j} \overline v_{ij;ij}$$

and

$$V_\Phi^{[1]} = \sum_{p,q} \left( \sum_i \overline v_{pi;qi} \right) \{a_p^\dagger a_q\}_\Phi.$$

Solution

Insert the four-operator reference-normal-ordering identity. The double contractions give

$$\frac14 \sum_{p,q,r,s} \overline v_{pq;rs} \left( \rho_{pr}\rho_{qs} - \rho_{ps}\rho_{qr} \right).$$

In the occupation basis, $\rho_{pr}=n_p\delta_{pr}$. Using antisymmetry of $\overline v$ makes the two contraction patterns equal after their exchange sign is included. Hence

$$V_\Phi^{[0]} = \frac12 \sum_{i,j} \overline v_{ij;ij}.$$

The four single contractions similarly combine into

$$V_\Phi^{[1]} = \sum_{p,q,i} \overline v_{pi;qi} \{a_p^\dagger a_q\}_\Phi.$$

The remaining uncontracted term is the residual normal-ordered two-body operator.

Exercise 5: Brillouin condition

Show that

$$\langle\Phi_i^a\rvert H \lvert\Phi\rangle = f_{ai}$$

for $\lvert\Phi_i^a\rangle=a_a^\dagger a_i\lvert\Phi\rangle$. Why does a Hartree–Fock stationary point set this matrix element to zero?

Solution

In the reference-normal-ordered Hamiltonian, the zero-body term cannot connect states with different quasiparticle content. The residual two-body normal product cannot connect the reference directly to a single particle-hole excitation. The one-body term gives

$$\begin{aligned} \langle\Phi_i^a\rvert H \lvert\Phi\rangle ={}& \sum_{p,q} f_{pq} \langle\Phi_i^a\rvert \{a_p^\dagger a_q\}_\Phi \lvert\Phi\rangle \\ ={}& f_{ai}. \end{aligned}$$

An infinitesimal occupied-unoccupied orbital rotation changes the determinant in the direction of $\lvert\Phi_i^a\rangle$. Stationarity of the Hartree–Fock energy under every such rotation therefore requires $f_{ai}=0$.

Exercise 6: Three-body combinatorics

For the antisymmetrized three-body convention used above, explain the factors $1/6$, $1/2$, and $1$ multiplying the induced zero-, one-, and two-body coefficients.

Solution

The original operator carries $1/(3!)^2=1/36$. A zero-body term contracts all three creation operators with all three annihilation operators. There are $3!=6$ signed complete pairings that become equal after antisymmetry, giving

$$\frac{6}{36} = \frac16.$$

For a one-body remainder, choose one uncontracted creator and one uncontracted annihilator in $3\times3=9$ ways. The two contracted pairs have $2!=2$ equivalent pairings. Thus

$$\frac{9\times2}{36} = \frac12.$$

For a two-body remainder, choose one contracted creation-annihilation pair in $3\times3=9$ ways. Relative to the standard two-body prefactor $1/4$, the coefficient satisfies

$$\frac9{36} = \frac14,$$

so the induced antisymmetrized two-body matrix element carries no extra factor beyond the sum over the occupied index.

Exercise 7: Coherent displacement

Let $a=\alpha+b$ with $b\lvert\alpha\rangle=0$. Expand $a^\dagger a$ and compute its coherent-state expectation value.

Solution

Substitution gives

$$\begin{aligned} a^\dagger a ={}& (\alpha^*+b^\dagger) (\alpha+b) \\ ={}& |\alpha|^2 + \alpha^*b + \alpha b^\dagger + b^\dagger b. \end{aligned}$$

Because $b\lvert\alpha\rangle=0$ and $\langle\alpha\rvert b^\dagger=0$,

$$\langle\alpha\rvert a^\dagger a \lvert\alpha\rangle = |\alpha|^2.$$

The constant is the zero-body term relative to the displaced vacuum.

Exercise 8: Anomalous contraction and number symmetry

Let

$$\kappa_{pq} = \langle c_p c_q\rangle.$$

Show that $\kappa_{pq}$ is antisymmetric and explain why it vanishes in a particle-number eigenstate but can be nonzero in a Bogoliubov quasiparticle vacuum.

Solution

The fermionic anticommutator gives

$$c_p c_q = -c_q c_p,$$

so

$$\kappa_{pq} = -\kappa_{qp}.$$

The pair annihilator has number charge $-2$:

$$[\widehat N,c_p c_q] = -2c_p c_q.$$

A charged operator has zero expectation in a number eigenstate, hence $\kappa_{pq}=0$ there. An unprojected Bogoliubov vacuum mixes sectors whose particle numbers differ by even integers, so it can support a nonzero anomalous contraction while preserving fermion parity.

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