Wick's Theorem Preview

Wick’s theorem is the systematic rule that rewrites products of creation and annihilation operators as normal-ordered products plus all possible contractions. In this volume the theorem is only a preview: the full many-body and QFT versions require time ordering, Green functions, Gaussian states, and sign conventions that belong in more specialized treatments.

Green Functions in Many-Body QM owns the retarded, time-ordered, spectral, and Matsubara two-point functions that become contractions in many-body calculations.

The useful idea is already visible in ordinary Fock space. Whenever an annihilation operator must be moved past a creation operator, the commutator or anticommutator produces a c-number. Wick’s theorem organizes all such c-number terms so that complicated products can be reduced to normal-ordered pieces and two-point contractions.

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.

Why Wick’s Theorem Appears

Products of many operators appear almost everywhere in many-particle quantum mechanics:

• expectation values of density and correlation operators;
• perturbation expansions around a free or mean-field Hamiltonian;
• response functions and Green functions;
• mean-field decouplings of interaction terms;
• QFT transition amplitudes and path-integral generating functionals.

Directly expanding these products is possible but quickly becomes unreadable. The point of Wick’s theorem is not mystical: for a vacuum or Gaussian reference state, all higher-point correlations are determined by two-point contractions. The theorem gives the bookkeeping.

Perturbation Theory in Many-Body Systems develops the reference-state, excitation-rank, linked-cluster, and thermodynamic-limit structure that this contraction bookkeeping supports.

Diagrammatic Methods Preview shows how time-ordered contractions become propagator lines, vertices, self-energy insertions, and polarization bubbles once a complete convention is declared.

For bosonic operators $O_1,\ldots,O_n$ built from 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.

Contractions as Bookkeeping

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

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

The example is short, but it contains the main warning: Wick’s theorem for fermions is not obtained by copying the bosonic formula and forgetting signs.

Gaussian States and Limits of the Preview

Wick factorization is exact for vacuum states of quadratic Hamiltonians and, more generally, for Gaussian states. In such states, higher-point functions are determined by two-point functions. This is why Wick’s theorem is central in free-field QFT, noninteracting many-body theory, and perturbation theory around a quadratic reference problem.

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.

Many-Body and QFT Cross-Links

This page is a bridge, not the canonical home of Wick’s theorem. The full theorem becomes important when one studies:

• many-body perturbation theory and Green functions;
• quasiparticles and contractions relative to a Fermi sea or Bogoliubov vacuum;
• time-ordered correlation functions in QFT;
• Feynman propagators and diagrammatic expansions;
• path-integral Gaussian integrals and generating functionals.

The Fock-space version here should be read as the algebraic skeleton. It explains why contractions appear and how they relate to normal ordering. It does not replace the full proof, the diagrammatic rules, or the distributional care needed for relativistic fields.

Common Mistakes

• Using Wick factorization for a non-Gaussian interacting state without saying it is an approximation.
• Calling any two-point function a contraction without specifying the ordering and reference state.
• Forgetting fermionic signs when contracted operators cross other fermionic operators.
• Confusing normal ordering with time ordering.
• Treating an equal-time delta-function contraction as if it were automatically a relativistic propagator.
• Believing Wick’s theorem removes the need to track domains, cutoffs, or operator-valued distributions in field theory.

Cross-Links

• Creation and Annihilation Operators
• Bosonic Commutation Relations
• Fermionic Anticommutation Relations
• Green Functions in Many-Body QM
• Number Operators
• Field Operators
• Many-Particle Hamiltonians
• Normal Ordering
• Second Quantization: Bridge to QFT
• Formula Sheet
• Reference Bridge: Second Quantization
• Harmonic Oscillator to Fields

References

• G. C. Wick, “The Evaluation of the Collision Matrix”, Physical Review 80, 268-272, 1950.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Addison-Wesley, 1988.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Kluwer Academic/Plenum, 2000.
• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.
• 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.

Exercises

1. One contraction. Use Wick bookkeeping to expand $a a^\dagger a$ for one bosonic mode.

Solution

There is one nonzero contraction:

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

The normal-ordered product is

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

Therefore

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

2. Double-contraction check. Verify

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

by repeated use of $[a,a^\dagger]=1$.

Solution

Start from

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

The second term is

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

For the first term, use

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

Thus

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

Combining terms gives

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

3. Field contraction. For bosonic fields expanded as $\psi(\mathbf x)=\sum_i\varphi_i(\mathbf x)a_i$, show that

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

when the modes form a complete orthonormal basis.

Solution

Using the mode expansion,

$$\begin{aligned} C\!\left(\psi(\mathbf x),\psi^\dagger(\mathbf y)\right) &= \sum_{ij} \varphi_i(\mathbf x)\varphi_j^*(\mathbf y) \langle0\vert a_i a_j^\dagger\vert0\rangle \\ &= \sum_i \varphi_i(\mathbf x)\varphi_i^*(\mathbf y). \end{aligned}$$

Completeness gives

$$\sum_i \varphi_i(\mathbf x)\varphi_i^*(\mathbf y) = \delta^{(3)}(\mathbf x-\mathbf y).$$

4. Fermionic sign. Expand $c_i c_j^\dagger c_k$ into a normal-ordered term plus a contraction.

Solution

Move $c_j^\dagger$ left across $c_i$, producing one minus sign:

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

The contraction is

$$C(c_i,c_j^\dagger)=\delta_{ij}.$$

Therefore

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

5. Conceptual boundary. Why is this page not the canonical home for Wick’s theorem?

Solution

This page only explains the Fock-space algebra behind normal ordering and contractions. The full theorem requires a chosen ordering, a reference state, sign conventions, and often time-ordered products, propagators, diagrammatics, or Gaussian path integrals. Those belong in many-body theory and QFT treatments, where the theorem is used as a central technical tool rather than a short preview.