Wick's Theorem and Gaussian Factorization

Wick’s theorem expands a product of operators linear in creation and annihilation operators into normal-ordered products and pair contractions. For a Gaussian reference state it also reduces higher correlation functions to first and second moments. These are related statements, but an operator reordering identity does not imply Gaussian factorization in an arbitrary state.

Here we develop the equal-time algebra, bosonic examples, and a four-fermion sign rule. Green Functions in Many-Body QM explains the ordered two-point functions used in dynamical calculations.

For the empty Fock vacuum, assume

$$a_i\lvert0\rangle=0, \qquad c_i\lvert0\rangle=0.$$

A contraction is the vacuum two-point part of a pair of operators. For bosonic modes,

$$C(a_i,a_j^\dagger) \equiv \langle0\vert a_i a_j^\dagger\vert0\rangle = \delta_{ij}, \qquad C(a_i^\dagger,a_j)=0.$$

For fermionic modes,

$$C(c_i,c_j^\dagger) \equiv \langle0\vert c_i c_j^\dagger\vert0\rangle = \delta_{ij}, \qquad C(c_i^\dagger,c_j)=0,$$

but signs from fermionic swaps must still be tracked.

Helpful background. Normal Ordering distinguishes reordering a monomial from retaining all terms in an operator equality.

Wick expansion into disjoint pair contractions

Density correlations, perturbation series, and mean-field calculations contain long operator products. Repeated reordering generates the same families of pair contractions; Wick’s theorem sums them once with their correct coefficients.

For bosonic operators $O_1,\ldots,O_n$ linear in creation and annihilation operators, the schematic form is

$$O_1\cdots O_n = :O_1\cdots O_n: + \sum_{i<j} C(O_i,O_j) :O_1\cdots\widehat O_i\cdots\widehat O_j\cdots O_n: + \cdots ,$$

where a hat means that the operator is omitted. The remaining terms contain two contractions, three contractions, and so on until no additional disjoint pairs can be chosen.

For fermions, the same idea holds, but every term carries the sign needed to bring the contracted operators together and normal order the remaining fermionic operators. This sign accounting is one reason it is safer to learn Wick’s theorem from a full many-body or QFT treatment before using it in long calculations.

Two-operator contractions

Normal ordering separates the operator part from the c-number part. A contraction records the c-number part produced when one annihilation operator crosses one creation operator.

For one bosonic mode,

$$a a^\dagger = :a a^\dagger: + C(a,a^\dagger),$$

with

$$:a a^\dagger: = a^\dagger a, \qquad C(a,a^\dagger)=1.$$

Thus

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

The contraction is not a new physical interaction. It is a compact way to remember the c-number generated by the operator algebra.

For equal-time nonrelativistic fields in the empty bosonic vacuum,

$$C\!\left(\psi(\mathbf x),\psi^\dagger(\mathbf y)\right) = \langle0\vert \psi(\mathbf x)\psi^\dagger(\mathbf y) \vert0\rangle = \delta^{(3)}(\mathbf x-\mathbf y).$$

This is the field version of $C(a_i,a_j^\dagger)=\delta_{ij}$. In relativistic QFT, the contraction that appears in a time-ordered Wick theorem is usually a propagator, not merely an equal-time delta function. That distinction is essential.

Simple Oscillator Example

Consider the product

$$a a^\dagger a.$$

There is one nonzero contraction, between the first $a$ and $a^\dagger$. Wick bookkeeping gives

$$a a^\dagger a = :a a^\dagger a: + C(a,a^\dagger)a.$$

The normal-ordered piece is

$$:a a^\dagger a: = a^\dagger a^2,$$

so

$$a a^\dagger a = a^\dagger a^2+a.$$

The same result follows by the commutator:

$$a a^\dagger a = (a^\dagger a+1)a = a^\dagger a^2+a.$$

A slightly richer example is

$$a a^\dagger a a^\dagger.$$

The normal-ordered product is

$$:a a^\dagger a a^\dagger: = (a^\dagger)^2a^2.$$

There are three single contractions that leave one factor $a^\dagger a$, and one double contraction. Hence

$$a a^\dagger a a^\dagger = (a^\dagger)^2a^2 + 3a^\dagger a + 1.$$

This example shows why the theorem is useful. Instead of repeatedly commuting operators by hand, one counts allowed pairings and their leftover normal-ordered products.

Fermionic Sign Preview

For fermions, contractions again record anticommutator c-number terms, but operator swaps carry signs. For example,

$$c_i c_j^\dagger c_k = :c_i c_j^\dagger c_k: + C(c_i,c_j^\dagger)c_k.$$

The normal-ordered term is

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

and the contraction is $C(c_i,c_j^\dagger)=\delta_{ij}$, so

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

This agrees with the canonical anticommutation relation

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

For four linear fermionic factors in an even Gaussian state, the complete expectation rule is

$$\langle O_1O_2O_3O_4\rangle =\langle O_1O_2\rangle\langle O_3O_4\rangle -\langle O_1O_3\rangle\langle O_2O_4\rangle +\langle O_1O_4\rangle\langle O_2O_3\rangle.$$

Keep each pair in its original relative order. The crossing pairing $(1,3)(2,4)$ has a minus sign. Anomalous pairings such as $\langle cc\rangle$ must also be included when the reference has pairing correlations; they need not vanish in a BCS quasiparticle vacuum. Surace and Tagliacozzo, §§3.1–3.4, give a covariance-matrix account of fermionic Gaussian states and their Wick factorization. Exercise 1 checks the anomalous term in a two-mode state directly.

Gaussian expectations and non-Gaussian states

For a centered Gaussian state, higher moments of linear field operators are sums of products of two-point moments. For a displaced bosonic Gaussian state, first subtract each mean and then apply the centered formula; first moments are needed too. Normalizable thermal states of stable quadratic Hamiltonians on the full Fock space and quadratic quasiparticle vacua are standard examples. An excited bosonic number state is not Gaussian, and projecting a quadratic Gibbs state onto fixed total particle number generally destroys Gaussianity.

For interacting ground states, thermal states of interacting Hamiltonians, constrained Hilbert spaces, and strongly correlated states, naive Wick factorization is generally not exact. One may still use Wick-like decompositions as approximations, diagrammatic expansions, or definitions relative to a chosen reference state, but then the approximation must be stated.

The reference state matters. Normal ordering relative to the empty vacuum is different from normal ordering relative to a filled Fermi sea, a BCS quasiparticle vacuum, or a thermal Gaussian state. Changing the reference state changes the contraction.

Reference-state Hamiltonian decompositions, particle-hole vacua, and correlated-reference cumulants are treated in Normal Ordering in Many-Body QM.

Exercises

1. A paired state needs anomalous contractions. For two fermionic modes, use the ordering $|11\rangle=c_1^\dagger c_2^\dagger|00\rangle$ and the normalized state

$$|\Psi\rangle=u|00\rangle+v|11\rangle, \qquad u\geq0,\quad u^2+|v|^2=1.$$

Verify that $\alpha=u c_1-v c_2^\dagger$ and $\beta=u c_2+v c_1^\dagger$ are canonical annihilation operators for this state. Compute $\langle n_1n_2\rangle$, with $n_i=c_i^\dagger c_i$, both directly and from the four-factor Wick rule. What would go wrong if the anomalous contractions were omitted?

Solution

The CAR give $\{\alpha,\alpha^\dagger\}=\{\beta,\beta^\dagger\}=1$, with the cross anticommutators and squares zero. Since $c_1|11\rangle=|01\rangle$ and $c_2|11\rangle=-|10\rangle$, both $\alpha|\Psi\rangle$ and $\beta|\Psi\rangle$ vanish. This is a quasiparticle vacuum, so Gaussian factorization applies.

Directly, only $|11\rangle$ contributes to $n_1n_2$, giving $\langle n_1n_2\rangle=|v|^2$. The needed contractions are

$$\langle n_1\rangle=\langle n_2\rangle=|v|^2,\qquad \langle c_1^\dagger c_2^\dagger\rangle=uv^*,\qquad \langle c_1c_2\rangle=-uv,$$

while $\langle c_1^\dagger c_2\rangle=\langle c_1c_2^\dagger\rangle=0$. In the ordered product $c_1^\dagger c_1c_2^\dagger c_2$, Wick’s rule therefore gives

$$\langle n_1n_2\rangle =|v|^4-(uv^*)(-uv)+0 =|v|^4+u^2|v|^2 =|v|^2.$$

The crossing-pair minus sign combines with the minus sign in $\langle c_1c_2\rangle$ to give a positive contribution. Dropping it would predict $|v|^4$, which is wrong whenever both amplitudes are nonzero. For $u=v=1/\sqrt2$, the exact joint probability is $1/2$, whereas the incomplete factorization gives $1/4$.

2. A non-Gaussian number state. In a bosonic number state $|n\rangle$, compare $\langle a^\dagger a^\dagger aa\rangle$ with the factorization prediction for a centered Gaussian state having $\langle a^\dagger a\rangle=n$ and $\langle aa\rangle=0$.

Solution

The exact number-state result is $n(n-1)$. The two nonzero Gaussian pairings each give $n^2$, so factorization predicts $2n^2$. They disagree for every $n>0$. The normal-ordering operator identity still holds; it is the assumption of a Gaussian state that fails.

3. Fixed-number free fermions. Take two degenerate fermionic modes with exactly one particle, equally likely in either mode. Test Wick factorization of $\langle n_1n_2\rangle$.

Solution

The state is $(|10\rangle\langle10|+|01\rangle\langle01|)/2$. It has $\langle n_1\rangle=\langle n_2\rangle=1/2$, no off-diagonal or anomalous two-point function, and $\langle n_1n_2\rangle=0$. Gaussian factorization would give $1/4$. The exact number constraint correlates the occupations even though the Hamiltonian has no interaction.

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.
• J. Surace and L. Tagliacozzo, “Fermionic Gaussian states: an introduction to numerical approaches”, SciPost Physics Lecture Notes 54 (2022), §§3.1–3.4; arXiv:2111.08343.
• G. C. Wick, “The Evaluation of the Collision Matrix”, Physical Review 80, 268–272 (1950).