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The only graduate-level textbook on quantum field theory that fully integrates perspectives from high-energy, condensed-matter, and statistical physics
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
Price: $120.00
Pages: 760
Publisher: Princeton University Press
Imprint: Princeton University Press
Publication Date:
23 March 2021
ISBN: 9780691149080
Format: Hardcover
Eduardo Fradkin is the Donald Biggar Willett Professor of Physics at the University of Illinois, Urbana-Champaign. An internationally recognized leader in theoretical physics and a member of the National Academy of Sciences, he is the author of Field Theories of Condensed Matter Physics.
- 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 the Wilson 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