本书是关于经典场和量子场路径积分的初级读本，写作风格简明扼要，重点介绍量子场的路径积分。本书介绍了场论最新发展的基本理论，可以作为一学期强化课程的教材使用。场论的路径积分方法有助于严格研究一些基本数学问题及其在强子物理、粒子物理和核物理中的应用，本书可供数学专业和理论物理专业的学生使用。本书是对作者《场、对称性和夸克》（Springer,1999）一书的补充，需要读者具有相对论量子力学方面的知识。
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目录
- Part I Non-Relativistic Quantum Theory
1 The Path Integral in Quantum Theory
1.1 Propagator of the Schrödinger Equation
1.2 Propagator as Path Integral
1.3 Quadratic Hamiltonians
1.3.1 Cartesian Metric
1.3.2 Non-Cartesian Metric
1.4 Classical Interpretation
2 Perturbation Theory
2.1 Free Propagator
2.2 Perturbative Expansion
2.3 Application to Scattering
3 Generating Functionals
3.1 Groundstate-to-Groundstate Transitions
3.1.1 Generating Functional
3.2 Functional Derivatives of Gs-Gs Transition Amplitudes
Part II Relativistic Quantum Field Theory
4 Relativistic Fields
4.1 Equations of Motion
4.1.1 Examples
4.2 Symmetries and Conservation Laws
4.2.1 Geometrical Space-Time Symmetries
4.2.2 Internal Symmetries
5 Path Integrals for Scalar Fields
5.1 Generating Functional for Fields
5.1.1 Euclidean Representation
6 Evaluation of Path Integrals
6.1 Free Scalar Fields
6.1.1 Generating Functional
6.1.2 Feynman Propagator
6.1.3 Gaussian Integration
6.2 Interacting Scalar Fields
6.2.1 Stationary Phase Approximation
6.2.2 Numerical Evaluation of Path Integrals
6.2.3 Real Time Formalism
7 Transition Rates and Green's Functions
7.1 Scattering Matrix
7.2 Reduction Theorem
7.2.1 Canonical Field Quantization
7.2.2 Derivation of the Reduction Theorem
8 Green's Functions
8.1 n-point Green's Functions
8.1.1 Momentum Representation
8.1.2 Operator Representations
8.2 Free Scalar Fields
8.2.1 Wick's Theorem
8.2.2 Feynman Rules
8.3 Interacting Scalar Fields
8.3.1 Perturbative Expansion
9 Perturbative φb4 Theory
9.1 Perturbative Expansion of the Generating Function
9.1.1 Generating Functional up to O(g)
9.2 Two-Point Function
9.2.1 Terms up to O(g0)
9.2.2 Terms up to O(g)
9.2.3 Terms up to O(g2)
9.3 Four-Point Function
9.3.1 Terms up to O(g)
9.3.2 Terms up to O(g2)
9.4 Divergences in n-Point Functions
9.4.1 Power Counting
9.4.2 Dimensional Regularization of φ4 Theory
9.4.3 Renormalization
10 Green's Functions for Fermions
10.1 Grassmann Algebra
10.1.1 Derivatives
10.1.2 Integration
10.2 Green's Functions for Fermions
10.2.1 Generating Functional for Fermions
10.2.2 Reduction Theorem for Fermions
10.2.3 Green's Functions
11 Interacting Fields
11.1 Feynman Rules
11.1.1 Fermion Loops
11.2 Wick's Theorem
11.3 Bosonization of Yukawa Theory
11.3.1 Perturbative Expansion
Part III Gauge Field Theory
12 Path Integrals for QED
12.1 Gauge Invariance in Abelian Free Field Theories
12.2 Generating Functional
12.3 Gauge Invariance in QED
12.4 Feynman Rules of QED
13 Path Integrals for Gauge Fields
13.1 Non-Abelian Gauge Fields
13.2 Generating Functional
13.3 Gauge Fixing of ￡
13.4 Faddeev-Popov Determinant
13.4.1 Explicit Forms of the FP Determinant
13.4.2 Ghost Fields
13.5 Feynman Rules
14 Examples for Gauge Field Theories
14.1 Quantum Chromodynamics
14.2 Electroweak Interactions
Units and Metric
A.1 Units
A.2 Metric and Notation
Functionals
B.1 Definition
B.2 Functional Integration
B.2.1 Gaussian Integrals
B.3 Functional Derivatives
Renormalization Integrals
Gaussian Grassmann Integration
References
Index
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