Marcillaud de Goursac, Axel; De Goursac, Axel Marcillaud; Goursac, Axel Marcillaud de: Noncommutative geometry, gauge theory and renormalization. 2009
Inhalt
- Noncommutative Geometry
- Topology and C*-algebras
- Definitions
- Spectral theory
- Duality in the commutative case
- GNS construction
- Vector bundles and projective modules
- Measure theory and von Neumann algebras
- Noncommutative differential geometry
- Epsilon-graded algebras
- General theory of the epsilon-graded algebras
- Commutation factors and multipliers
- Definition of epsilon-graded algebras and properties
- Relationship with superalgebras
- Noncommutative geometry based on epsilon-derivations
- Application to some examples of epsilon-graded algebras
- Renormalization of QFT
- Wilsonian renormalization of scalar QFT
- Scalar field theory
- Effective action and equation of the renormalization group
- Renormalization of the usual phi-4 theory in four dimensions
- Relation with BPHZ renormalization
- Renormalization of gauge theories
- QFT on Moyal space
- Presentation of the Moyal space
- Deformation quantization
- The Moyal product on Schwartz functions
- The matrix basis
- The Moyal algebra
- The symplectic Fourier transformation
- UV/IR mixing on the Moyal space
- Renormalizable QFT on Moyal space
- Gauge theory on Moyal space
- Definition of gauge theory
- Gauge theory associated to standard differential calculus
- U(N) versus U(1) gauge theory
- UV/IR mixing in gauge theory
- The effective action
- Properties of the effective action
- Symmetries of vacuum configurations
- Equation of motion
- Solutions of the equation of motion
- Minima of the action
- Extension in higher dimensions
- Interpretation of the effective action