Quantum Field Theory An Integrated Approach

Cover
Contents
Preface and Acknowledgments
1. Introduction to Field Theory
	1.1 Examples of fields in physics
	1.2 Why quantum field theory?
2. Classical Field Theory
	2.1 Relativistic invariance
	2.2 The Lagrangian, the action, and the least action principle
	2.3 Scalar field theory
	2.4 Classical field theory in the canonical formalism
	2.5 Field theory of the Dirac equation
	2.6 Classical electromagnetism as a field theory
	2.7 The Landau theory of phase transitions as a field theory
	2.8 Field theory and statistical mechanics
	Exercises
3. Classical Symmetries and Conservation Laws
	3.1 Continuous symmetries and Noether’s theorem
	3.2 Internal symmetries
	3.3 Global symmetries and group representations
	3.4 Global and local symmetries: Gauge invariance
	3.5 The Aharonov-Bohm effect
	3.6 Non-abelian gauge invariance
	3.7 Gauge invariance and minimal coupling
	3.8 Spacetime symmetries and the energy-momentum tensor
	3.9 The energy-momentum tensor for the electromagnetic field
	3.10 The energy-momentum tensor and changes in the geometry
	Exercises
4. Canonical Quantization
	4.1 Elementary quantum mechanics
	4.2 Canonical quantization in field theory
	4.3 Quantization of the free scalar field theory
	4.4 Symmetries of the quantum theory
	Exercises
5. Path Integrals in Quantum Mechanics and Quantum Field Theory
	5.1 Path integrals and quantum mechanics
	5.2 Evaluating path integrals in quantum mechanics
	5.3 Path integrals for a scalar field theory
	5.4 Path integrals and propagators
	5.5 Path integrals in Euclidean spacetime and statistical physics
	5.6 Path integrals for the free scalar field
	5.7 Exponential decays and mass gaps
	5.8 Scalar fields at finite temperature
	Exercises
6. Nonrelativistic Field Theory
	6.1 Second quantization and the many-body problem
	6.2 Nonrelativistic field theory and second quantization
	6.3 Nonrelativistic fermions at zero temperature
	Exercises
7. Quantization of the Free Dirac Field
	7.1 The Dirac equation and quantum field theory
	7.2 The propagator of the Dirac spinor field
	7.3 Discrete symmetries of the Dirac theory
	7.4 Chiral symmetry
	7.5 Massless fermions
	Exercises
8. Coherent-State Path-Integral Quantization of Quantum Field Theory
	8.1 Coherent states and path-integral quantization
	8.2 Coherent states
	8.3 Path integrals and coherent states
	8.4 Path integral for a nonrelativistic Bose gas
	8.5 Fermion coherent states
	8.6 Path integrals for fermions
	8.7 Path-integral quantization of the Dirac field
	8.8 Functional determinants
	8.9 The determinant of the Euclidean Klein-Gordon operator
	8.10 Path integral for spin
	Exercises
9. Quantization of Gauge Fields
	9.1 Canonical quantization of the free electromagnetic field
	9.2 Coulomb gauge
	9.3 The gauge A0 =0
	9.4 Path-integral quantization of gauge theories
	9.5 Path integrals and gauge fixing
	9.6 The propagator
	9.7 Physical meaning of Z[J] and theWilson loop operator
	9.8 Path-integral quantization of non-abelian gauge theories
	9.9 BRST invariance
	Exercises
10. Observables and Propagators
	10.1 The propagator in classical electrodynamics
	10.2 The propagator in nonrelativistic quantum mechanics
	10.3 Analytic properties of the propagators of free relativistic fields
	10.4 The propagator of the nonrelativistic electron gas
	10.5 The scattering matrix
	10.6 Physical information contained in the S-matrix
	10.7 Asymptotic states and the analytic properties of the propagator
	10.8 The S-matrix and the expectation value of time-ordered products
	10.9 Linear response theory
	10.10 The Kubo formula and the electrical conductivity of a metal
	10.11 Correlation functions and conservation laws
	10.12 The Dirac propagator in a background electromagnetic field
	Exercises
11. Perturbation Theory and Feynman Diagrams
	11.1 The generating functional in perturbation theory
	11.2 Perturbative expansion for the two-point function
	11.3 Cancellation of the vacuum diagrams
	11.4 Summary of Feynman rules for φ4 theory
	11.5 Feynman rules for theories with fermions and gauge fields
	11.6 The two-point function and the self-energy in φ4 theory
	11.7 The four-point function and the effective coupling constant
	11.8 One-loop integrals
	Exercises
12. Vertex Functions, the Effective Potential, and Symmetry Breaking
	12.1 Connected, disconnected, and irreducible propagators
	12.2 Vertex functions
	12.3 The effective potential and spontaneous symmetry breaking
	12.4 Ward identities
	12.5 The low-energy effective action and the nonlinear sigma model
	12.6 Ward identities, Schwinger-Dyson equations, and gauge invariance
	Exercises
13. Perturbation Theory, Regularization, and Renormalization
	13.1 The loop expansion
	13.2 Perturbative renormalization to two-loop order
	13.3 Subtractions, counterterms, and renormalized Lagrangians
	13.4 Dimensional analysis and perturbative renormalizability
	13.5 Criterion for perturbative renormalizability
	13.6 Regularization
	13.7 Computation of regularized Feynman diagrams
	13.8 Computation of Feynman diagrams with dimensional regularization
	Exercises
14. Quantum Field Theory and Statistical Mechanics
	14.1 The classical Ising model as a path integral
	14.2 The transfer matrix
	14.3 Reflection positivity
	14.4 The Ising model in the limit of extreme spatial anisotropy
	14.5 Symmetries and symmetry breaking
	14.6 Solution of the two-dimensional Ising model
	14.7 Continuum limit and the two-dimensional Ising universality class
	Exercises
15. The Renormalization Group
	15.1 Scale dependence in quantum field theory and in statistical physics
	15.2 RG flows, fixed points, and universality
	15.3 General properties of a fixed-point theory
	15.4 The operator product expansion
	15.5 Simple examples of fixed points
	15.6 Perturbing a fixed-point theory
	15.7 Example of operator product expansions: φ4 theory
	Exercises
16. The Perturbative Renormalization Group
	16.1 The perturbative renormalization group
	16.2 Perturbative renormalization group for the massless φ4 theory
	16.3 Dimensional regularization with minimal subtraction
	16.4 The nonlinear sigma model in two dimensions
	16.5 Generalizations of the nonlinear sigma model
	16.6 The O(N) nonlinear sigma model in perturbation theory
	16.7 Renormalizability of the two-dimensional nonlinear sigma model
	16.8 Renormalization of Yang-Mills gauge theories in four dimensions
	Exercises
17. The 1/N Expansions
	17.1 The φ4 scalar field theory with O(N) global symmetry
	17.2 The large-N limit of the O(N) nonlinear sigma model
	17.3 The CPN-1 model
	17.4 The Gross-Neveu model in the large-N limit
	17.5 Quantum electrodynamics in the limit of large numbers of flavors
	17.6 Matrix sigma models in the large-rank limit
	17.7 Yang-Mills gauge theory with a large number of colors
	Exercises
18. Phases of Gauge Theories
	18.1 Lattice regularization of quantum field theory
	18.2 Matter fields
	18.3 Minimal coupling
	18.4 Gauge fields
	18.5 Hamiltonian theory
	18.6 Elitzur’s theorem and the physical observables of a gauge theory
	18.7 Phases of gauge theories
	18.8 Hamiltonian duality
	18.9 Confinement in the Euclidean spacetime lattice picture
	18.10 Behavior of gauge theories coupled to matter fields
	18.11 The Higgs mechanism
	18.12 Phase diagrams of gauge-matter theories
	Exercises
19. Instantons and Solitons
	19.1 Instantons in quantum mechanics and tunneling
	19.2 Solitons in (1+1)-dimensional φ4 theory
	19.3 Vortices
	19.4 Instantons and solitons of nonlinear sigma models
	19.5 Coset nonlinear sigma models
	19.6 The CPN-1 instanton
	19.7 The ’t Hooft–Polyakov magnetic monopole
	19.8 The Yang-Mills instanton in D=4 dimensions
	19.9 Vortices and the Kosterlitz-Thouless transition
	19.10 Monopoles and confinement in compact electrodynamics
	Exercises
20. Anomalies in Quantum Field Theory
	20.1 The chiral anomaly
	20.2 The chiral anomaly in 1+1 dimensions
	20.3 The chiral anomaly and abelian bosonization
	20.4 Solitons and fractional charge
	20.5 The axial anomaly in 3+1 dimensions
	20.6 Fermion path integrals, the chiral anomaly, and the index theorem
	20.7 The parity anomaly and Chern-Simons gauge theory
	20.8 Anomaly inflow
	20.9 θ vacua
	Exercises
21. Conformal Field Theory
	21.1 Scale and conformal invariance in field theory
	21.2 The conformal group in D dimensions
	21.3 The energy-momentum tensor and conformal invariance
	21.4 General consequences of conformal invariance
	21.5 Conformal field theory in two dimensions
	21.6 Examples of two-dimensional CFTs
	Exercises
22. Topological Field Theory
	22.1 What is a topological field theory?
	22.2 Deconfined phases of discrete gauge theories
	22.3 Chern-Simons gauge theories
	22.4 Quantization of abelian Chern-Simons gauge theory
	22.5 Vacuum degeneracy on a torus
	22.6 Fractional statistics
	Exercises
References
Index