Introduction
1 Introduction to Noncommutative Geometry
1.1 Topology and C*-algebras
1.1.1 Definitions
1.1.2 Spectral theory
1.1.3 Duality in tile commutative case
1.1.4 GNS construction
1.1.5 Vector bundles and projective modules
1.2 Measure theory and yon Neumann algebras
1.2.1 Definition of von Neumann algebras
1.2.2 Duality in the commutative case
1.3 Noncommutative differential geometry
1.3.1 Algebraic geometry
1.3.2 Differential calculi
1.3.3 Hochschild and cyclic homologies
1.3.4 Spectral triples
2 Epsilon-graded algebras noncommutative geometry
2.1 General theory of the ε-graded algebras
2.1.1 Commutation factors and multipliers
2.1.2 Definition of ε-graded algebras and properties
2.1.3 Relationship with superalgebras
2.2 Noncommutative ε-graded geometry
2.2.1 Differential calculus
2.2.2 ε-connections and gauge transformations
2.2.3 Involutions
2.3 Application to some examples of ε-graded algebras
2.3.1 ε-graded commutative algebras
2.3.2 ε-graded matrix algebras with elementary grading
2.3.3 ε-graded matrix algebras with fine grading
3 An Introduction to Renormalization of QFT
3.1 Renormalization of scalar theories in the wilsonian approach
3.1.1 Scalar field theory
3.1.2 Effective action and equation of the renormalization grour
3.1.3 Renormalization of the usual ψ4 theory in four dimensions
3.2 BPHZ renormalization
3.2.1 Power-counting
3.2.2 BPHZ subtraction scheme
3.2.3 Beta functions
3.3 Renormalization of gauge theories
3.3.1 Classical theory and BRS formalism
3.3.2 Algebraic renormalization
4 QFT on Moyal space
4.1 Presentation of the Moyal space
4.1.1 Deformation quantization
4.1.2 The Moyal product on Schwartz functions
4.1.3 The matrix basis
4.1.4 The Moyal algebra
4.1.5 The symplectic Fourier transformation
4.2 UV/IR m/x.ing on the Moyal space
4.3 Renormalizable QFT on Moyal space
4.3.1 Renormalization of the theory with harmonic term
4.3.2 Principal properties
4.3.3 Vacuum configurations
4.3.4 Possible spontaneous symmetry breaking?
4.3.5 Other renormalizable QFT on Moyal space
5 Gauge theory on the Moyal space
5.1 Definition of gauge theory
5.1.1 Gauge theory associated to standard differential calculus
5.1.2 U(N) versus U(1) gauge theory
5.1.3 UV/IR mixing in gauge theory
5.2 The effective action
5.2.1 Minimal coupling
5.2.2 Computation of the effective action
5.2.3 Discussion on the effective action
5.3 Properties of the effective action
5.3.1 Symmetries of vacuum configurations
5.3.2 Equation of motion
5.3.3 Solutions of the equation of motion
5.3.4 Minima of the action
5.3.5 Extension in higher dimensions
5.4 Interpretation of the effective action
5.4.1 A superalgebra constructed from Moyal space
5.4.2 Differential calculus and scalar theory
5.4.3 Graded connections and gauge theory
5.4.4 Discussion and interpretation
Conclusion
Bibliography
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