Table of Contents
- Introduction
- Motivation and Importance
- Operator Ordering in Quantum Field Theory
- Normal Ordering
- Time Ordering
- The Need for Wick’s Theorem
- Statement of Wick’s Theorem
- Contractions of Operators
- Feynman Propagator as a Contraction
- Examples of Wick’s Theorem in Action
- Proof Sketch of Wick’s Theorem
- Application in Free Scalar Field Theory
- Wick’s Theorem in Fermionic Fields
- Sign Factors in Fermionic Wick Expansion
- Wick’s Theorem and Feynman Diagrams
- Relation to Generating Functionals
- Time-Ordered Products and Green’s Functions
- Higher-Order Perturbation Theory
- Anomalies and Limitations
- Wick’s Theorem Beyond Minkowski Space
- Connection to Statistical Field Theory
- Summary of Rules for Applying Wick’s Theorem
- Implications in QED and QCD
- Wick’s Theorem in Functional Formalism
- 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:
- Expressing fields in terms of creation and annihilation operators
- Using commutation relations to move annihilation operators right
- 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
- Write all possible pairwise contractions.
- Replace each contraction with the corresponding propagator.
- For fermions, include signs for permutation of fields.
- 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.
