Quantum field
Version September 2008
INTRODUCTION TO
QUANTUM FIELD THEORY by B. de Wit
Institute for Theoretical Physics
Utrecht University
Contents
1 Introduction
4
2 Path integrals and quantum mechanics
6
3 The classical limit
12
4 Continuous systems
22
5 Field theory
5.1 Second quantization
27
31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Correlation functions
6.1 Harmonic oscillator correlation functions;
6.2 Harmonic oscillator correlation functions;
6.2.1 Evaluating G0 . . . . . . . . . . .
6.2.2 The integral over qn . . . . . . . .
6.2.3 The integrals over q1 and q2 . . .
6.3 Conclusion . . . . . . . . . . . . . . . . .
operators . . path integrals
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
41
42
44
46
47
48
49
7 Euclidean Theory
52
8 Tunneling and instantons
8.1 The double-well potential . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 The periodic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
63
71
9 Perturbation theory
79
10 More on Feynman diagrams
89
11 Fermionic harmonic oscillator states
102
12 Anticommuting c-numbers
106
13 Phase space with commuting and anticommuting coordinates and quantization
113
14 Path integrals for fermions
127
15 Feynman diagrams for fermions
137
2
16 Regularization and renormalization
147
17 Further reading
157
3
1
Introduction
Physical systems that involve an infinite number of degrees of freedom can conveniently be described by some sort of field theory. Almost all systems in nature involve an extremely large number of degrees of freedom. For instance, a droplet of water
INTRODUCTION TO
QUANTUM FIELD THEORY by B. de Wit
Institute for Theoretical Physics
Utrecht University
Contents
1 Introduction
4
2 Path integrals and quantum mechanics
6
3 The classical limit
12
4 Continuous systems
22
5 Field theory
5.1 Second quantization
27
31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Correlation functions
6.1 Harmonic oscillator correlation functions;
6.2 Harmonic oscillator correlation functions;
6.2.1 Evaluating G0 . . . . . . . . . . .
6.2.2 The integral over qn . . . . . . . .
6.2.3 The integrals over q1 and q2 . . .
6.3 Conclusion . . . . . . . . . . . . . . . . .
operators . . path integrals
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
41
42
44
46
47
48
49
7 Euclidean Theory
52
8 Tunneling and instantons
8.1 The double-well potential . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 The periodic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
63
71
9 Perturbation theory
79
10 More on Feynman diagrams
89
11 Fermionic harmonic oscillator states
102
12 Anticommuting c-numbers
106
13 Phase space with commuting and anticommuting coordinates and quantization
113
14 Path integrals for fermions
127
15 Feynman diagrams for fermions
137
2
16 Regularization and renormalization
147
17 Further reading
157
3
1
Introduction
Physical systems that involve an infinite number of degrees of freedom can conveniently be described by some sort of field theory. Almost all systems in nature involve an extremely large number of degrees of freedom. For instance, a droplet of water