FIELDS - Stony Brook University

PART ONE: SYMMETRY5OUTLINEIn this Outline we give a brief description of each item listed in the Contents.While the Contents and Index are quick ways to search, or learn the general layoutof the book, the Outline gives more detail for the uninitiated. (The PDF version alsoallows use of the “Find” command in PDF readers.)Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23general remarks on style, organization, focus, content, use, differences from othertexts, etc.Some field theory texts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36recommended alternatives or supplements (but see Preface).PART ONE: SYMMETRY.Relativistic quantum mechanics and classical field theory. Poincaré group specialrelativity. Enlarged spacetime symmetries: conformal and supersymmetry. Equationsof motion and actions for particles and fields/wave functions. Internal symmetries:global (classifying particles), local (field interactions).I. GlobalSpacetime and internal symmetries.A. Coordinatesspacetime symmetries1. Nonrelativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Poisson bracket, Einstein summation convention, Galilean symmetry (introductory example)2. Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46statistics, anticommutator; anticommuting variables, differentiation, integration3. Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51general structure of symmetries (including internal); Lie bracket, group,structure constants; brief summary of group theory4. Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Minkowski space, antiparticles, Lorentz and Poincaré symmetries, propertime, Mandelstam variables, lightcone bases5. Discrete: C, P, T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65charge conjugation, parity, time reversal, in classical mechanics and fieldtheory; Levi-Civita tensor

66. Conformal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68broken, but useful, enlargement of Poincaré; projective lightconeB. Indiceseasy way to group theory1. Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Hilbert-space notation2. Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76adjoint, Cartan metric, Dynkin index, Casimir, (pseudo)reality, directsum and product3. Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81with Levi-Civita tensors, Gaussian integrals; Pfaffian4. Classical groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84and generalizations, via tensor methods5. Tensor notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86index notation, simplest bases for simplest representationsC. Representationsuseful special cases1. More coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Dirac gamma matrices as coordinates for orthogonal groups2. Coordinate tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94formulations of coordinate transformations; differential forms3. Young tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99pictures for representations, their symmetries, sizes, direct products4. Color and flavor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101symmetries of particles of Standard Model and observed light hadrons5. Covering groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107relating spinors and vectorsII. SpinExtension of spacetime symmetry to include spin. Field equations for field strengthsof all spins. Most efficient methods for Lorentz indices in QuantumChromoDynamicsor pure Yang-Mills. Supersymmetry relates bosons and fermions, also useful for QCD.A. Two components2 2 matrices describe the spacetime groups more easily (2 4)1. 3-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110algebraic properties of 2 2 matrices, vectors as quaternions2. Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114in three (space) dimensions

PART ONE: SYMMETRY73. Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115basis for spinor notation4. Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117review of spin in simpler notation: many indices instead of bigger; tensornotation avoids Clebsch-Gordan-Wigner coefficients5. Lorentz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120still 2 2 matrices, but four dimensions; dotted and undotted indices;antisymmetric tensors; matrix identities6. Dirac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126example in free field theory; 4-component identities7. Chirality/duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128chiral symmetry, simpler with two-component spinor indices; more examples; dualityB. Poincarérelativistic solutions1. Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131conformal group as unified way to all massless free equations2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134reproduction of familiar cases (Dirac and Maxwell equations)3. Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137proof; lightcone methods; transformations4. Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141dimensional reduction; Stückelberg formalism for vector in terms of massless vector scalar5. Foldy-Wouthuysen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144an application, for arbitrary spin, from massless analog; transformationto nonrelativistic corrections; minimal electromagnetic coupling to spin1/2; preparation for nonminimal coupling in chapter VIII for Lamb shift6. Twistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148convenient and covariant method to solve massless equations; related toconformal invariance and self-duality; useful for QCD computations inchapter VI7. Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151via twistors; Penrose transformC. Supersymmetrysymmetry relating fermions to bosons, generalizing translations; general properties, representations

81. Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156definition of supersymmetry; positive energy automatic2. Supercoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157superspace includes anticommuting coordinates; covariant derivativesgeneralize spacetime derivatives3. Supergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160generalizing classical groups; supertrace, superdeterminant4. Superconformal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163also broken but useful, enlargement of supersymmetry, as classical group5. Supertwistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164massless representations of supersymmetryIII. LocalSymmetries that act independently at each point in spacetime. Basis of fundamentalforces.A. Actionsfor previous examples (spins 0, 1/2, 1)1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169action principle, variation, functional derivative, Lagrangians2. Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174quantizing anticommuting quantities; spin3. Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176actions in nonrelativistic field theory, Hamiltonian and Lagrangian densities4. Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180relativistic particles and fields, charge conjugation, good ultraviolet behavior, general forces5. Constrained systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186role of gauge invariance; first-order formalism; gauge fixingB. Particlesrelativistic classical mechanics; useful later in understanding Feynman diagrams; simple example of local symmetry1. Free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191worldline metric, gauge invariance of actions2. Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195gauge fixing, lightcone gauge3. Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197external fields

PART ONE: SYMMETRY94. Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198for classical particles; true vs. canonical energy5. Pair creation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .201and annihilation, for classical particle and antiparticleC. Yang-Millsself-coupling for spin 1; describes forces of Standard Model1. Nonabelian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204self-interactions; covariant derivatives, field strengths, Jacobi identities,action2. Lightcone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208a unitary gauge; axial gauges; spin 1/23. Plane waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .212simple exact solutions to interacting theory4. Self-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213and massive analog5. Twistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217useful for self-duality; lightcone gauge for solving self-duality6. Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220nonperturbative self-dual solutions, via twistors; ’t Hooft ansatz; ChernSimons form7. ADHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224general instanton solution of Atiyah, Drinfel’d, Hitchin, and Manin8. Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226more nonperturbative self-dual solutions, but staticIV. MixedGlobal symmetries of interacting theories. Gauge symmetry coupled to lower spins.A. Hidden symmetryexplicit and soft breaking, confinement1. Spontaneous breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232method; Goldstone theorem of massless scalars2. Sigma models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234linear and nonlinear; low-energy theories of scalars3. Coset space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237general construction, using gauge invariance, for sigma models4. Chiral symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240low-energy symmetry, quarks, pseudogoldstone boson, Partially Conserved Axial Current

105. Stückelberg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243scalars generate mass for vectors; free case6. Higgs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245same for interactions; Gervais-Neveu model; unitary gauge7. Dilaton cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247cosmology with gravity replaced by Goldstone boson of scale invarianceB. Standard modelapplication to real world1. Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259strong interactions, using Yang-Mills; C and P2. Electroweak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264unification of electromagnetic and weak interactions, using also Higgs3. Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267including all known fundamental leptons; Cabibbo-Kobayashi-Maskawatransformation; flavor-changing neutral currents4. Grand Unified Theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .269unification of all leptons and vector mesonsC. Supersymmetrysuperfield theory, using superspace; useful for solving problems of perturbation resummation (chapter VIII)1. Chiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275simplest (“matter”) multiplet2. Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277to introduce interactions; component expansion, superfield equations3. Covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280approach to gauge multiplet; vielbein, torsion; solution to Jacobi identities4. Prepotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282fundamental superfield for constructing covariant derivatives; solution toconstraints, chiral representation5. Gauge actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284for gauge and matter multiplets; Fayet-Iliopoulos term6. Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287of supersymmetry; spurions7. Extended . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289introduction to multiple supersymmetries; central charges

PART TWO: QUANTA.PART TWO: QUANTA11.Quantum aspects of field theory. Perturbation theory: expansions in loops, helicity,and internal symmetry. Although some have conjectured that nonperturbative approaches might solve renormalization difficulties found in perturbation, all evidenceindicates these problems worsen instead in complete theory.V. QuantizationQuantization of classical theories by path integrals. Backgrounds fields instead ofsources exclusively: All uses of Feynman diagrams involve either S-matrix or effectiveaction, both of which require removal of external propagators, equivalent to replacingsources with fields.A. Generalvarious properties of quantum physics in general context, so these items neednot be repeated in more specialized and complicated cases of field theory1. Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298Feynman’s alternative to Heisenberg and Schrödinger methods; relationto canonical quantization; unitarity, causality2. Semiclassical expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303JWKB in path integral; free particle3. Propagators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .307Green functions; solution to Schrödinger equation via path integrals4. S-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310scattering, most common use of field theory; unitarity, causality5. Wick rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .315imaginary time, to get Euclidean space, has important role in quantummechanicsB. Propagatorsrelativistic quantum mechanics, free quantum field theory1. Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319Stückelberg-Feynman propagator for spin 0; covariant gauge, lightconegauge2. Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322features, relations to classical Green functions, inner product3. Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326other spins, nature of quantum corrections4. Wick rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .329its relativistic use, in mechanics and field theory

12C. S-matrixpath integration of field theory produces Feynman diagrams/graphs; simpleexamples from scalar theories1. Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334definition of initial/final states; generating functional of background fields;perturbative evaluation2

FIELDS WARREN SIEGEL C. N. Yang Institute for Theoretical Physics State University of New York at Stony Brook Stony Brook, New Y