Wick’s Theorem

Table of Contents

1. Introduction
2. Motivation and Importance
3. Operator Ordering in Quantum Field Theory
4. Normal Ordering
5. Time Ordering
6. The Need for Wick’s Theorem
7. Statement of Wick’s Theorem
8. Contractions of Operators
9. Feynman Propagator as a Contraction
10. Examples of Wick’s Theorem in Action
11. Proof Sketch of Wick’s Theorem
12. Application in Free Scalar Field Theory
13. Wick’s Theorem in Fermionic Fields
14. Sign Factors in Fermionic Wick Expansion
15. Wick’s Theorem and Feynman Diagrams
16. Relation to Generating Functionals
17. Time-Ordered Products and Green’s Functions
18. Higher-Order Perturbation Theory
19. Anomalies and Limitations
20. Wick’s Theorem Beyond Minkowski Space
21. Connection to Statistical Field Theory
22. Summary of Rules for Applying Wick’s Theorem
23. Implications in QED and QCD
24. Wick’s Theorem in Functional Formalism
25. Conclusion

1. Introduction

Wick’s Theorem is a fundamental result in quantum field theory (QFT) that simplifies the computation of time-ordered products of field operators. It systematically expresses these products as sums over contractions, facilitating the use of Feynman diagrams and perturbation theory.

2. Motivation and Importance

In perturbative QFT, we often compute vacuum expectation values (VEVs) of time-ordered products. Direct evaluation of these products is complex. Wick’s Theorem allows us to express these products in terms of known quantities—contractions—making calculations tractable.

3. Operator Ordering in Quantum Field Theory

Operators in QFT do not generally commute, and their ordering matters:

• Time ordering (\( T \)) arranges operators in decreasing time.
• Normal ordering (\( :\mathcal{O}: \)) puts annihilation operators to the right of creation operators.

4. Normal Ordering

Normal ordering eliminates vacuum expectation values:

\[
:\phi(x)\phi(y): = \phi(x)\phi(y) – \langle 0 | \phi(x)\phi(y) | 0 \rangle
\]

This operation sets all VEVs of normal-ordered products to zero.

5. Time Ordering

The time-ordered product \( T[\phi(x)\phi(y)] \) is defined as:

\[
T[\phi(x)\phi(y)] =
\begin{cases}
\phi(x)\phi(y) & \text{if } x^0 > y^0 \
\phi(y)\phi(x) & \text{if } y^0 > x^0
\end{cases}
\]

This is crucial in constructing Green’s functions and propagators.

6. The Need for Wick’s Theorem

Time-ordered products with many operators are complex to compute. Wick’s Theorem expresses them as sums of normal-ordered products with contractions, each corresponding to a propagator.

7. Statement of Wick’s Theorem

Let \( \phi_1, \phi_2, \dots, \phi_n \) be field operators. Then:

\[
T[\phi_1 \phi_2 \cdots \phi_n] = :\phi_1 \phi_2 \cdots \phi_n: + \text{(sum of contractions)}
\]

Each contraction represents a two-point correlator between fields.

8. Contractions of Operators

A contraction between two fields is defined as:

\[
\contraction{}{\phi}{(x)}{\phi}
\phi(x)\phi(y) = \langle 0 | T[\phi(x)\phi(y)] | 0 \rangle – :\phi(x)\phi(y):
\]

In free theory, this is equal to the Feynman propagator \( \Delta_F(x – y) \).

9. Feynman Propagator as a Contraction

\[
\contraction{}{\phi}{(x)}{\phi}
\phi(x)\phi(y) = \langle 0 | T[\phi(x)\phi(y)] | 0 \rangle = \Delta_F(x – y)
\]

This allows each contraction to be replaced by a known function.

10. Examples of Wick’s Theorem in Action

For two fields:

\[
T[\phi(x)\phi(y)] = :\phi(x)\phi(y): + \contraction{}{\phi}{(x)}{\phi} \phi(x)\phi(y)
\]

For four fields:

\[
T[\phi_1\phi_2\phi_3\phi_4] = :\phi_1\phi_2\phi_3\phi_4: + \text{(6 contractions)} + \text{(3 double contractions)}
\]

11. Proof Sketch of Wick’s Theorem

Wick’s Theorem is proved by:

1. Expressing fields in terms of creation and annihilation operators
2. Using commutation relations to move annihilation operators right
3. Rewriting terms as sums over contractions and normal-ordered products

12. Application in Free Scalar Field Theory

In a free theory, all multi-point functions reduce to products of two-point functions due to Wick’s Theorem. This underpins Feynman diagram expansion and makes free theory solvable.

13. Wick’s Theorem in Fermionic Fields

Wick’s Theorem applies to fermions with anti-commuting fields. The contractions are:

\[
\contraction{}{\psi}{(x)}{\bar{\psi}} \psi(x) \bar{\psi}(y) = \langle 0 | T[\psi(x)\bar{\psi}(y)] | 0 \rangle
\]

Extra minus signs appear due to anti-commutation.

14. Sign Factors in Fermionic Wick Expansion

Each swap of fermionic fields introduces a minus sign. The total sign of a term is determined by the parity of the permutation needed to order the fields.

15. Wick’s Theorem and Feynman Diagrams

Each term in Wick’s expansion corresponds to a Feynman diagram. Contractions correspond to lines, and normal-ordered products correspond to interaction vertices.

16. Relation to Generating Functionals

In path integral formalism, Wick’s Theorem is encoded in the structure of Gaussian integrals and generating functionals \( Z[J] \), where derivatives with respect to sources yield contractions.

17. Time-Ordered Products and Green’s Functions

Wick’s Theorem allows systematic construction of \( n \)-point functions:

\[
G_n(x_1, \dots, x_n) = \langle 0 | T[\phi(x_1)\cdots\phi(x_n)] | 0 \rangle
\]

which are built from pairwise contractions.

18. Higher-Order Perturbation Theory

In interacting theories, Wick’s Theorem is used after expanding the interaction exponential:

\[
T\left[e^{i\int \mathcal{L}_{\text{int}}(x)\, d^4x}\right]
\]

Wick’s expansion provides a systematic way to evaluate time-ordered products in the Dyson series.

19. Anomalies and Limitations

Wick’s Theorem holds in free theories and perturbative expansions but breaks down in strongly coupled systems and theories with operator mixing or singularities.

20. Wick’s Theorem Beyond Minkowski Space

In Euclidean field theory, the theorem remains valid but uses Euclidean Green’s functions. It also extends to statistical field theories and condensed matter systems.

21. Connection to Statistical Field Theory

In statistical mechanics, expectations are evaluated using Boltzmann weights. Wick’s Theorem connects these to Gaussian integrals over fields, just like in quantum theory.

22. Summary of Rules for Applying Wick’s Theorem

1. Write all possible pairwise contractions.
2. Replace each contraction with the corresponding propagator.
3. For fermions, include signs for permutation of fields.
4. Add all terms, including higher-order contractions.

23. Implications in QED and QCD

Wick’s Theorem is foundational in QED and QCD:

• Justifies Feynman rules
• Structures perturbation theory
• Enables loop expansions and renormalization

24. Wick’s Theorem in Functional Formalism

In the path integral approach, Wick’s Theorem emerges from Gaussian integration:

\[
\int \mathcal{D}\phi\, \phi(x_1)\cdots\phi(x_n) e^{iS[\phi]} \propto \text{sum over contractions}
\]

This reproduces the operator formalism’s results in a more general framework.

25. Conclusion

Wick’s Theorem is a cornerstone of quantum field theory. It provides the algorithmic foundation for evaluating time-ordered products and constructing Feynman diagrams. Its broad applicability across quantum mechanics, field theory, and statistical physics makes it an essential tool in theoretical physics.