Quantum field theory was originally developed to describe quantum electrodynamics and other fundamental problems in high-energy physics, but today has become an invaluable conceptual and mathematical framework for addressing problems across physics, including in condensed-matter and statistical physics. With this expansion of applications has come a new and deeper understanding of quantum field theory―yet this perspective is still rarely reflected in teaching and textbooks on the subject. Developed from a year-long graduate course Eduardo Fradkin has taught for years to students of high-energy, condensed-matter, and statistical physics, this comprehensive textbook provides a fully "multicultural" approach to quantum field theory, covering the full breadth of its applications in one volume.
Brings together perspectives from high-energy, condensed-matter, and statistical physics in both the main text and exercises
Takes students from basic techniques to the frontiers of physics
Pays special attention to the relation between measurements and propagators and the computation of cross sections and response functions
Focuses on renormalization and the renormalization group, with an emphasis on fixed points, scale invariance, and their role in quantum field theory and phase transitions
Other topics include non-perturbative phenomena, anomalies, and conformal invariance
Features numerous examples and extensive problem sets
Also serves as an invaluable resource for researchers
Author(s): Eduardo Fradkin
Edition: 1
Publisher: Princeton University Press
Year: 2021
Language: English
Pages: 732
City: New Jersey
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
