The Global Approach to Quantum Field Theory

The book shows how classical field theory, quantum mechanics, and quantum field theory are related. The description is global from the outset. Quantization is explained using the Peierls bracket rather than the Poisson bracket. This allows one to deal immediately with observables, bypassing the canonical formalism of constrained Hamiltonian systems and bigger-than-physical Hilbert (or Fock) spaces. The Peierls bracket leads directly to the Schwinger variational principle and the Feynman functional integral, the latter of which is taken as defining the quantum theory. Also included are the theory of tree amplitudes and conservation laws, which are presented classically and later extended to the quantum level. The quantum theory is developed from the many-worlds viewpoint, and ordinary path integrals and the topological issues to which they give rise are studied in some detail. The theory of mode functions and Bogoliubov coefficients for linear fields is fully developed, and then the quantum theory of nonlinear fields is confronted. The effective action, correlation functions and counter terms all make their appearance at this point, and the S-matrix is constructed via the introduction of asymptotic fields and the LSZ theorem. Gauge theories and ghosts are studied in great detail. Many applications of the formalism are given: vacuum currents, anomalies, black holes, fourth-order systems, higher spin fields, the (lambda phi) to the fourth power model (and spontaneous symmetry breaking), quantum electrodynamics, the Yang-Mills field and its topology, the gravitational field, etc. Special chapters are devoted to Euclideanization and renormalization, space and time inversion, and the closed-time-path or "in-in" formalism. Emphasis is given throughout to the role of the functional-integral measure in the theory. Six helpful appendices, ranging from superanalysis to analytic continuation in dimension, are included at the end.I: CLASSICAL DYNAMICAL THEORY ; 1. Fundamentals ; 2. Dynamics and Invariance Transformations ; 3. Small Disturbances and Green's Functions ; 4. The Peierls Bracket ; 5. Finite Disturbances. Tree Theorems. Asmptotic Fields ; 6. Conservation Laws ; II: THE HEURISTIC ROAD TO QUANTIZATION. THE QUANTUM FORMALISM AND ITS INTERPRETATION ; 7. Classical Theory of Measurement ; 8. Quantum Theory of Measurement ; 9. Interpretation of the Quantum Formalism I ; 10. The Schwinger Variational Principle and the Feynman Functional Integral ; 11. The Quantum Mechanics of Standard Canonical Systems ; 12. Interpretation of the Quantum Formalism II ; III: EVALUATION AND APPROXIMATION OF FEYNMAN FUNCTIONAL INTEGRALS ; 13. The Functional Integral for Standard Canonical Systems ; 14. Approximation and Evaluation of the Path Integral ; 15. The Nonrelativistic Particle in a Curved Space ; 16. The Heat Kernel ; IV: LINEAR SYSTEMS ; 17. Linear Boson Fields in Stationary Backgrounds ; 18. Quantization of Linear Boson Fields ; 19. Linear Fermion Fields. Stationary Backgrounds ; 20. Quantization of Linear Fermion Fields ; 21. Linear Fields in Nonstationary Backgrounds ; 22. Linear (or Linearized) Fields Possessing Invariant Flows ; V: NONLINEAR FIELDS ; 23. The Effective Action, the S-matrix, and Slavnov-Taylor Identities ; 24. Gauge Theories I. General Formalism ; 25. Gauge Theories II. Background Field Methods. Scattering Theory ; 26. Case-I Gauge Theory without Ghosts. Description of Cases II and III ; VI: TOOLS FOR QUANTUM FIELD THEORY. APPLICATIONS ; 27. The Heat Kernel ; 28. Vacuum Currents. Anomalies ; 29. More Vacuum Phenomena ; 30. Black Hole Vacua. Hawking Radiation ; 31. The Closed-time-path or "In-in" Formalism ; VII: SPECIAL TOPICS ; 32. Euclideanization and Renormalization ; 33. Canonical Transformations. Space Inversion and Time Reversal ; 34. Quantum Electrodynamics ; 35. The Yang-Mills and Gravitational Fields ; VIII: EXAMPLES. SIMPLE EXERCISES IN THE USE OF THE GLOBAL FORMALISM ; X0 The Nonrelativistic Particle in Flat Space ; X1 A Simple Fermi System ; X2 A Fermi doublet ; X3 Fermi Multiplet ; X4 The Fermi Oscillator ; X5 The Bose Oscillator ; X6 A Fourth-order System ; X7 A Model for Ghosts ; X8 Free Scalar Field in Flat Spacetime ; X9 Massive Vector Field in Four-dimensional Flat Spacetime ; X10 Massive Antisymmetric Tensor Field ; X11 Massive Symmetric Tensor Field in Flat Spacetime ; X12 Massive Spinor Field in Flat Spacetime ; X13 Massive spin-3/2 Field in Flat Spacetime ; X14 Electromagnetic Field in Flat Spacetime ; X15 Massless Symmetric Tensor Field in Flat Spacetime ; X16 Massless Spinor Field in Four-dimensional Flat Spacetime ; X17 Massless Spin-3/2 Field in Four-dimensional Flat Spacetime ; X18 Renormalization Group and Spontaneous Symmetry Breaking in the (lambda phi) to the fourth power Model ; X19 The Relativistic Particle in Minkowski Spacetime ; X20 A Simple Soluble Nonlinear Model ; X21 Quantum Mechanics on a Circle ; X22 Quantum Mechanics on a Klein Bottle ; X23 Ghosts for Ghosts ; X24 Massless Antisymmetric Tensor Field