Structural aspects of quantum field theory and noncommutative geometry
By: Gerhard Grensing, University of Kiel
Publisher: World Scientific
Published: 2013
Place of publication: [Hackensack] New Jersey
Language: eng
ISBN-13: 9789814472708; 9781299715028
Subjects: Quantum field theory; SCIENCE; Waves & Wave Mechanics
Description: This book is devoted to the subject of quantum field theory. It is divided into two volumes. The first can serve as a textbook on the main techniques and results of quantum field theory, while the second treats more recent developments, in particular the subject of quantum groups and noncommutative geometry, and their interrelation. The first volume is directed at graduate students who want to learn the basic facts about quantum field theory. It begins with a gentle introduction to classical field theory, including the standard model of particle physics, general relativity, and also supergravity. The transition to quantized fields is performed with path integral techniques, by means of which the one-loop renormalization of a self-interacting scalar quantum field, of quantum electrodynamics, and the asymptotic freedom of quantum chromodynamics is treated. In the last part of the first volume, the application of path integral methods to systems of quantum statistical mechanics is covered. The book ends with a rather detailed investigation of the fractional quantum Hall effect, and gives a stringent derivation of Laughlin's trial ground state wave function as an exact ground state. The second volume covers more advanced themes. In particular Connes' noncommutative geometry is dealt with in some considerable detail; the presentation attempts to acquaint the physics community with the substantial achievements that have been reached by means of this approach towards the understanding of the elusive Higgs particle. The book also covers the subject of quantum groups and its application to the fractional quantum Hall effect, as it is for this paradigmatic physical system that noncommutative geometry and quantum groups can be brought together.
Notes: Includes bibliographical references and index.
Table of contents: title: 1. Relativistic free fields: bosons. 1.1. Maxwell's equations in relativistic notation. 1.2. Klein-Gordon equation. 1.3. Group velocity and special relativity. 1.4. Nonrelativistic limit and the Schrödinger equation. 1.5. Classical interpretation of fields. 1.6. Normal coordinates. 1.7. Quantized harmonic oscillator. 1.8. Quantization of the Klein-Gordon field; title: 2. Lagrange formalism for fields. 2.1. Functionals. 2.2. Euler-Lagrange equations for fields. 2.3. Variational principle. 2.4. Appendix: improper bases; title: 3. Relativistic invariance. 3.1. Minkowski space and its symmetry group. 3.2. Transformation law of fields. 3.3. Appendix: Lie groups and Lie algebras; title: 4. Special relativity. 4.1. Inertial frames and causality. 4.2. Lengths and time intervals. 4.3. Addition theorem for velocities. 4.4. Rotating frames. 4.5. Accelerated inertial frames. 4.6. Appendix: Product integral; title: 5. Relativistic free fields: fermions. 5.1. Dirac's equation. 5.2. Relativistic invariance of Dirac's equation. 5.3. Variational principle for the Dirac equation. 5.4. On the origin of gauge invariance. 5.5. Nonrelativistic limit. 5.6. 'Classical' interpretation of fermions. 5.7. Clifford algebras and spin groups; title: 6. Relativistic free fields and spin. 6.1. Scalar field. 6.2. Dirac field. 6.3. Maxwell field. 6.4. Spin. 6.5. Transformation law of fields and induced representations; title: 7. Neutral fermions. 7.1. Charge conjugation. 7.2. Majorana spinors. 7.3. Neutrinos; title: 8. Symmetries and conservation laws; title: 9. Differential and integral calculus for anticommuting variables. 9.1. Real Grassmann variables. 9.2. Fourier transformation. 9.3. Complex Grassmann variables. 9.4. Appendix: Pfaffians; title: 10. Dynamical principles: internal symmetries. 10.1. Internal gauge theories. 10.2. Yang-Mills theory. 10.3. Gauge theories and elementary particle physics; title: 11. Dynamical principles: external symmetries. 11.1 Introduction. 11.2 Dynamics of a relativistic point particle in a gravitational field. 11.3. Differential geometry: a first course. 11.4. Einstein's theory of gravity. 11.5. Differential geometry: a second course. 11.6. Differential geometry: a third course. 11.7. Accelerated observers and inertial systems. 11.8. Gravity as a gauge theory of the Poincaré group; title: 12. Supergravity. 12.1. Super Poincaré group. 12.2. Supersymmetry and differential geometry: Cartan connexions. 12.3. Rarita-Schwinger fermions. 12.4. Supergravity as a gauge theory of the super Poincaré group. 12.5. Summary. 12.6. Appendix: Majorana spinors in higher dimensions.; title: 13. Cosmology. 13.1. Gauss' normal coordinates. 13.2. Symmetric spaces. 13.3. Realization of maximally symmetric spaces. 13.4. Robertson-Walker metric. 13.5. General relativistic hydrodynamics. 13.6. Friedmann equations. 13.7. Models of the universe. 13.8. Present status of the universe. 13.9. Observational astronomy and cosmological parameters. 13.10. Cosmological constant problem. 13.11. Quintessence. 13.12. Appendix: Geometric optics in the presence of gravity. 13.13. Appendix: Local scale invariance and Weyl geometry; title: 14. Quantization of free fields. 14.1. Scalar field. 14.2. Dirac field. 14.3. Maxwell field; title: 15. Quantum mechanical perturbation theory. 15.1. Interaction picture. 15.2. Time independent perturbation theory. 15.3. Formal theory of scattering. 15.4. In and out picture. 15.5. Gell-Mann & Low formula. 15.6. Transcription to quantum field theory. 15.7. Reduction formulae; title: 16. Perturbative quantum electrodynamics. 16.1. QED Hamiltonian in the Coulomb gauge. 16.2. QED scattering operator and states. 16.3. Wick's theorem. 16.4. Scattering matrix elements. 16.5. Feynman rules for QED. 16.6. Cross sections. 16.7. Elementary processes. 16.8. Appendix: Gamma 'gymnastics'; title: 17. Path integral quantization. 17.1. Feynman path integral. 17.2. Gauge invariance and the midpoint rule. 17.3. Canonical transformations and the path integral; title: 18. Path integral quantization of the harmonic oscillator. 18.1. Harmonic oscillator. 18.2. Driven harmonic oscillator; title: 19. Expectation values of operators. 19.1. Expectation values for a finite time interval. 19.2. Expectation values for an infinite time interval. 19.3. Driven harmonic oscillator revisited; title: 20. Perturbative methods. 20.1. Perturbation theory. 20.2. Imaginary time and quantum statistical mechanics. 20.3. Ground state energy of the quartic anharmonic oscillator; title: 21. Nonperturbative methods. 21.1. Expansion in terms of Planck's constant. 21.2. Small deviations. 21.3. Stationary phase approximation: particle in an external potential. 21.4. Wentzel-Kramers-Brillouin and stationary phase approximation: compatibility. 21.5. Stationary phase approximation: charged particle in an external magnetic field. 21.6. Particle in an external gravitational field and heat kernel expansion. 21.7. Partition functions and functional determinants; title: 22. Holomorphic quantization. 22.1. Coherent states: bosons. 22.2. Coherent state path integral: bosons. 22.3. Coherent states: fermions. 22.4. Path integral for fermions. 22.5. Driven harmonic oscillator: bosonic and fermionic; title: 23. Ghost fermions. 23.1. Schrödinger representation. 23.2. Vector space realization. 23.3. Dirac states and their duals. 23.4. Feynman type path integral. 23.5. Poisson structures for fermions; title: 24. Quantum fields on a lattice. 24.1. Lattice bosons. 24.2. Lattice fermions. 24.3. Lattice gauge fields.; title: 25. Self interacting bosonic quantum field. 25.1. Partition function and perturbation theory. 25.2. Effective action. 25.3. Effective action and perturbation theory. 25.4. Dimensional regularization. 25.5. Renormalization. 25.6. 'Cosmological' constant. 25.7. Renormalization group equations. 25.8. Asymptotia. 25.9. Coleman-Weinberg effective potential; title: 26. Quantum electrodynamics. 26.1. Path integral for the free Dirac field. 26.2. Path integral for the free electromagnetic field. 26.3. Path integral representation of quantum electrodynamics. 26.4. Ward's identity. 26.5. Regularization. 26.6. Renormalization and the Callan-Symanzik function. 26.7. Application: anomalous magnetic moment. 26.8. Structure of the physical vacuum; title: 27. Quantum chromodynamics. 27.1. Faddeev-Popov device. 27.2. Becchi-Rouet-Stora transformation. 27.3. Zinn-Justin equations. 27.4. Feynman rules. 27.5. Regularization. 27.6. Asymptotic freedom. 27.7. Conclusion; title: 28. Nonrelativistic second quantization. 28.1. Field operators and the Fock space construction. 28.2. Multilinear algebra and the Fock space construction. 28.3. Second quantized form of the N-particle Hamiltonian; title: 29. Quantum statistical mechanics. 29.1. Thermodynamics and the partition function. 29.2. Canonical ensemble. 29.3. Constant mode expansion of the canonical partition function; title: 30. Grand canonical ensemble. 30.1. Path integral representation of second quantized fields. 30.2. Grand canonical partition function as a functional integral. 30.3. Perturbation theory in Direct space. 30.4. Perturbation theory in Fourier space. 30.5. Connection with thermodynamic quantities. 30.6. Noninteracting case; title: 31. Bose-Einstein condensation. 31.1. Spontaneous symmetry breaking and condensation. 31.2. Condensation and Feynman rules. 31.3. Schwinger-Dyson-Beliaev equations. 31.4. Hugenholtz-Pines relation. 31.5. Nonperturbative approach. 31.6. Superfluidity; title: 32. Superconductivity. 32.1. Introduction. 32.2. Effective action; title: 33. Relativistic quantum field theory at nonzero temperature. 33.1. Relativistic ideal gas. 33.2. Symmetry restoration; title: 34. Fractional quantum Hall effect. 34.1. Classical Hall effect. 34.2. Landau problem. 34.3. Second quantization and the integer effect. 34.4. Chern-Simons theory and Ginzburg-Landau effective theory. 34.5. Laughlin theory. 34.6. Excitations. 34.7. Braid statistics. 34.8. Chern-Simons theory and braid statistics. 34.9. Edge excitations. 34.10. Virasoro and Kac-Moody algebras. 34.11. Laughlin ground state and vertex operators. 34.12. Laughlin's trial wave function as an exact ground state; title: 35. Hamiltonian formalism and symplectic geometry. 35.1. Introduction. 35.2. Canonical transformations. 35.3. Generating functions. 35.4. Vector fields as generators of diffeomorphisms. 35.5. One parameter subgroups of canonical transformations.; title: 36. Conventional symmetries. 36.1. Symmetries and conservation laws: Lagrange formalism. 36.2. Symmetries and conservation laws: Hamilton formalism. 36.3. Gauge invariance; title: 37. Accidental symmetries. 37.1. Hydrogen qtom or quantum mechanical Kepler problem. 37.2. Three-dimensional harmonic oscillator; title: 38. Anomalous symmetries. 38.1. Generalized Noether charges and anomalies. 38.2. Cochains and boundaries. 38.3. BRS operator. 38.4. Landau problem: 1. Variation. 38.5. Cohomology of Lie groups and algebras; title: 39. Constrained systems and symplectic reduction. 39.1. Linear reduction. 39.2. Nonlinear reduction. 39.3. Constraints and reduction. 39.4. Symmetry and Marsden-Weinstein reduction. 39.5. Dirac brackets. 39.6. Dirac brackets and Poisson structures; title: 40. Quantum reduction of constrained systems. 40.1. Gauge theories as constrained systems. 40.2. Finite dimensional analogue of gauge theories. 40.3. Quantum mechanical time evolution of constrained systems. 40.4. Quantization of constrained systems. 40.5. Geometry of systems with first class constraints. 40.6. Geometry of Yang-Mills fields. 40.7. Yang-Mills theory and Poisson-Dirac brackets. 40.8. Faddeev's path integral formula for constrained systems; title: 41. BRS quantization of constrained systems. 41.1. BRS invariance. 41.2. Extended BRS formalism. 41.3. Fradkin-Vilkovisky theorem. 41.4. Zinn-Justin equations; title: 42. Weyl quantization of bosons. 42.1. Weyl order: real representation. 42.2. Weyl order: complex representation. 42.3. Groenewold-Moyal bracket. 42.4. Generalized Weyl formalism. 42.5. Berezin's path integral. 42.6. Other ordering schemes and symbols: real representation. 42.7. Other ordering schemes and symbols: complex representation. 42.8. Generating functions and their quantum counterparts. 42.9. Weyl ordering and the path integral. 42.10. Appendix: Pseudodifferential operators and Weyl; title: 43. Weyl quantization of bosons and canonical transformations. 43.1. Symplectic vector spaces and symplectic transformations. 43.2. Complex structures and complexifications. 43.3. Complex realization of the symplectic group. 43.4. Heisenberg-Weyl group and quantization. 43.5. Metaplectic operator. 43.6. Bargmann transform. 43.7. Symplectic transformations and quantum mechanics; title: 44. Geometric quantization and spin. 44.1. Generalized coherent states: SU(2). 44.2. Coherent states: noncompact picture. 44.3. Coherent state path integral: noncompact picture. 44.4. Coherent states: compact picture. 44.5. Coherent state path integral: compact picture. 44.6. Spin models.; title: 45. Weyl quantization of fermions. 45.1. Canonical symmetry: Weyl and spinorial operator. 45.2. Weyl ordered operators. 45.3. Fermionic Heisenberg-Weyl transformation of wave functions. 45.4. Antiholomorphic representation. 45.5. Complex realization of rotations. 45.6. Quantum mechanical representation of canonical transformations. 45.7. Fermionic Weyl formalism. 45.8. Groenewold-Moyal bracket for fermions. 45.9. Generalized Weyl formalism. 45.10. Berezin's path integral for fermions. 45.11. Partition function in the Weyl approach; title: 46. Anomalies and index theorems. 46.1. Axial anomaly. 46.2. Axial gauge anomaly. 46.3. Physical consequences of anomalies. 46.4. Anomalies and geometry. 46.5. Gravitational anomalies. 46.6. Supersymmetric relativistic point particle with spin. 46.7. Appendix: Spin and spin[symbol] structures. 46.8. Appendix: Geometric gauge fixing conditions; title: 47. Integrated anomalies. 47.1. Pure non-abelian Chern-Simons theory. 47.2. Nonabelian Schwinger model. 47.3. Chiral nonabelian Schwinger model; title: 48. Noncommutative geometry: algebraic tools. 48.1. Basic algebraic tools. 48.2. Noncommutative differential geometry. 48.3. Cyclic cohomology. 48.4. Graded cyclic cohomology. 48.5. Berezin integration and graded cyclic cohomology; title: 49. Noncommutative geometry: analytic tools. 49.1. Spectral triples. 49.2. Spectral triples and universal differential calculus. 49.3. Dixmier trace. 49.4. Wodzicki residue. 49.5. Real structures. 49.6. Order one and orientation. 49.7. Regularity and finiteness. 49.8. Axiomatic foundation. 49.9. Internal symmetries. 49.10. Appendix: Review of C*-algebra basics; title: 50. Noncommutative geometry: particle physics. 50.1. Fermionic action. 50.2. Bosonic action. 50.3. Outlook; title: 51. A glance at noncommutative quantum field theory. 51.1. Noncommutative spaces. 51.2. Landau problem: 2. Variation. 51.3. Noncommutative quantum field theory; title: 52. Hopf algebras. 52.1. Motivation. 52.2. Algebras. 52.3. Coalgebras. 52.4. Bialgebras. 52.5. Hopf algebras. 52.6. Hopf *-algebras; title: 53. Quasitriangular Hopf algebras. 53.1. Almost cocommutative Hopf algebras. 53.2. Quasitriangular Hopf algebras. 53.3. Ribbon Hopf algebras. 53.4. Matrix realizations of the universal R-operator and Artin's braid group. 53.5. Quasitriangular Hopf algebras and *-structures; title: 54. Quantum groups: basic example. 54.1. Motivation. 54.2. Uq(sl2) as an algebra. 54.3. Uq(sl2) as a Hopf algebra. 54.4. Uq(sl2) as a quasitriangular Hopf algebra. 54.5. Uq(sl2) as a quasitriangular Ribbon Hopf algebra. 54.6. Elements of q-analysis. 54.7. Real forms of Uq(sl2). 54.8. Representation theory of Uq(sl2); title: 55. Quantum groups and the noncommutative torus. 55.1. Landau problem: 3. Variation. 55.2. Weyl quantization and quantum groups; title: 56. Quantum Hall effect with realistic boundary conditions.