Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part IV, Chapter 1 | Page 3 of 3

Wick's Theorem

Reducing time-ordered products to propagators and normal-ordered terms

1.10 Normal Ordering

Before stating Wick's theorem, we need two ingredients: normal ordering and contractions. Normal ordering, denoted by colons $:\!\cdots\!:$, places all creation operators to the left of all annihilation operators:

$$:\!a_{\mathbf{p}}^\dagger a_{\mathbf{q}}\!: \;= a_{\mathbf{p}}^\dagger a_{\mathbf{q}}, \qquad :\!a_{\mathbf{q}} a_{\mathbf{p}}^\dagger\!: \;= a_{\mathbf{p}}^\dagger a_{\mathbf{q}}$$

For fermionic operators, each swap needed to achieve normal ordering introduces a minus sign. The critical property of normal ordering is:

$$\boxed{\langle 0|:\!\text{any operator}\!:|0\rangle = 0}$$

This vanishes because the annihilation operator on the right annihilates $|0\rangle$, and the creation operator on the left annihilates $\langle 0|$. Normal ordering effectively "subtracts the vacuum" from operator products.

For a free scalar field $\phi(x) = \phi^+(x) + \phi^-(x)$, where $\phi^+$ contains annihilation operators and $\phi^-$ contains creation operators:

$$:\!\phi(x)\phi(y)\!: \;= \phi^-(x)\phi^-(y) + \phi^-(x)\phi^+(y) + \phi^-(y)\phi^+(x) + \phi^+(x)\phi^+(y)$$

Notice that all creation parts ($\phi^-$) are moved to the left of all annihilation parts ($\phi^+$).

1.11 Contractions and the Feynman Propagator

The contraction of two fields is defined as the difference between the time-ordered product and the normal-ordered product:

$$\boxed{\overline{\phi(x)\phi(y)} \equiv T[\phi(x)\phi(y)] - :\!\phi(x)\phi(y)\!:}$$

Since the contraction is a c-number (not an operator), it equals its vacuum expectation value. Let us compute it explicitly. For $x^0 > y^0$:

$$T[\phi(x)\phi(y)] = \phi(x)\phi(y) = [\phi^+(x),\phi^-(y)] + :\!\phi(x)\phi(y)\!:$$

since the only difference between the ordinary product and the normal-ordered product comes from the commutator $[\phi^+(x),\phi^-(y)]$ needed to push the creation operator to the left. Computing this commutator gives the positive-frequency Wightman function. Including both time orderings, the contraction is precisely the Feynman propagator:

$$\boxed{\overline{\phi(x)\phi(y)} = \langle 0|T[\phi(x)\phi(y)]|0\rangle = D_F(x-y) = \int\frac{d^4p}{(2\pi)^4}\frac{i}{p^2 - m^2 + i\epsilon}}$$

The contraction is a c-number, and this result is the cornerstone that connects operator algebra to Feynman diagrams: every contraction becomes a propagator line in a diagram.

💡Contractions as Virtual Particle Exchange

Each contraction $\overline{\phi(x)\phi(y)} = D_F(x-y)$ represents a virtual particle propagating from y to x (or x to y). The Feynman propagator sums over all momenta of this virtual particle. When we systematically contract all fields in a time-ordered product, we are enumerating all possible ways virtual particles can be exchanged — this is why Feynman diagrams emerge naturally from Wick's theorem.

1.12 Statement and Proof of Wick's Theorem

Wick's Theorem: The time-ordered product of free fields equals the normal-ordered product plus all possible contractions:

$$\boxed{T[\phi_1\phi_2\cdots\phi_n] = :\!\phi_1\phi_2\cdots\phi_n\!: + \sum_{\text{single contractions}}:\!\overline{\phi_i\phi_j}\;\phi_k\cdots\!: + \sum_{\text{double}}:\!\overline{\phi_i\phi_j}\;\overline{\phi_k\phi_l}\cdots\!: + \cdots}$$

where we sum over all distinct ways of pairing fields. Contracted pairs are replaced by $D_F$, and the remaining uncontracted fields are normal-ordered.

Proof by induction: The base case n = 2 is the definition of contraction:

$$T[\phi_1\phi_2] = :\!\phi_1\phi_2\!: + \overline{\phi_1\phi_2}$$

For the inductive step, assume Wick's theorem holds for n fields and prove it for n+1. Consider $T[\phi_1\cdots\phi_n\phi_{n+1}]$ where $t_{n+1}$ is the earliest time (we can always relabel). Then:

$$T[\phi_1\cdots\phi_{n+1}] = T[\phi_1\cdots\phi_n]\,\phi_{n+1}$$

since $\phi_{n+1}$ is at the earliest time and so already sits on the right. By the inductive hypothesis, $T[\phi_1\cdots\phi_n]$ is a sum of normal-ordered terms with various contractions. We must commute $\phi_{n+1} = \phi_{n+1}^+ + \phi_{n+1}^-$ inside each normal-ordered expression. The creation part $\phi_{n+1}^-$ already belongs on the left of any normal-ordered product. The annihilation part $\phi_{n+1}^+$ must be pushed through all creation operators in $:\!\cdots\!:$, generating commutators $[\phi_{n+1}^+, \phi_j^-]$ for each uncontracted field $\phi_j$with $t_j > t_{n+1}$. Each such commutator is exactly $\overline{\phi_{n+1}\phi_j}$. This produces all new single contractions involving $\phi_{n+1}$, completing the induction.

1.13 Example: Computing $\langle 0|T[\phi^4(x)]|0\rangle$

Let us apply Wick's theorem to a concrete calculation. Consider the vacuum expectation value of four fields at the same spacetime point, $\langle 0|T[\phi(x)\phi(x)\phi(x)\phi(x)]|0\rangle$. Write $\phi_i \equiv \phi(x)$ for brevity.

Wick's theorem gives:

$$T[\phi_1\phi_2\phi_3\phi_4] = :\!\phi_1\phi_2\phi_3\phi_4\!: + \overline{\phi_1\phi_2}\,:\!\phi_3\phi_4\!: + \overline{\phi_1\phi_3}\,:\!\phi_2\phi_4\!: + \overline{\phi_1\phi_4}\,:\!\phi_2\phi_3\!:$$
$$+ \;\overline{\phi_2\phi_3}\,:\!\phi_1\phi_4\!: + \overline{\phi_2\phi_4}\,:\!\phi_1\phi_3\!: + \overline{\phi_3\phi_4}\,:\!\phi_1\phi_2\!:$$
$$+ \;\overline{\phi_1\phi_2}\,\overline{\phi_3\phi_4} + \overline{\phi_1\phi_3}\,\overline{\phi_2\phi_4} + \overline{\phi_1\phi_4}\,\overline{\phi_2\phi_3}$$

Taking the vacuum expectation value, all normal-ordered terms vanish ($\langle 0|:\!\cdots\!:|0\rangle = 0$). Only the fully contracted terms survive:

$$\langle 0|T[\phi^4(x)]|0\rangle = 3\,[D_F(0)]^2$$

The factor of 3 counts the three distinct ways to pair four objects into two pairs: (12)(34), (13)(24), and (14)(23). Each contraction $\overline{\phi(x)\phi(x)} = D_F(0)$ is a propagator evaluated at zero separation.

💡From Wick's Theorem to Feynman Diagrams

The three fully contracted terms correspond to three distinct vacuum bubble diagrams: each is a figure-eight (two loops meeting at the vertex x). The combinatorial factor of 3 is a symmetry factor. In general:

  • Each contraction = a propagator line in a Feynman diagram
  • Each uncontracted field = an external line
  • The sum over all contractions = the sum over all Feynman diagrams

Wick's theorem is the rigorous bridge from the operator formalism to the diagrammatic language of Feynman. Every Feynman rule can be derived from it.

1.14 Application to Scattering Amplitudes

For a scattering process, we compute $\langle f|S|i\rangle$ where $|i\rangle$ and $|f\rangle$ are multi-particle Fock states. At first order in $\lambda\phi^4$ theory for 2 $\to$ 2 scattering:

$$\langle p_3 p_4|S^{(1)}|p_1 p_2\rangle = \frac{-i\lambda}{4!}\int d^4x\,\langle p_3 p_4|T[\phi_I^4(x)]|p_1 p_2\rangle$$

Using Wick's theorem, the four fields must each be contracted with one of the external particles (contracting two internal fields would leave unmatched external particles). Each external creation/annihilation operator contracts with a field to produce a plane wave factor $e^{\pm ip \cdot x}$. There are 4! = 24 ways to match the four fields to the four external particles, and the x-integral produces a momentum-conserving delta function:

$$\langle p_3 p_4|S^{(1)}|p_1 p_2\rangle = -i\lambda\,(2\pi)^4\delta^{(4)}(p_1 + p_2 - p_3 - p_4)$$

The 4! from the permutations exactly cancels the 1/4! in the coupling, giving the clean result $\mathcal{M} = -\lambda$at tree level. This is the simplest Feynman diagram: a single four-point vertex with no internal lines.

Key Concepts

  • Normal ordering $:\!\cdots\!:$ places creation operators left of annihilation operators; its vacuum expectation value vanishes
  • The contraction $\overline{\phi(x)\phi(y)} = D_F(x-y)$ equals the Feynman propagator
  • Wick's theorem: T-products = normal-ordered terms + all possible contractions
  • Vacuum expectation values of T-products receive contributions only from fully contracted terms
  • $\langle 0|T[\phi^4(x)]|0\rangle = 3[D_F(0)]^2$ counts the three distinct pairings
  • Each contraction becomes a propagator line in a Feynman diagram; Wick's theorem generates all diagrams systematically
  • The combinatorial factors from Wick's theorem produce the correct symmetry factors for Feynman diagrams
←Previous: The Dyson SeriesNext Chapter: Cross Sections→