本书是一部英文版的数学专著，中文书名或可译为《非交换几何、规范理论和重整化：一般简介与非交换量子场论的重整化》。
现在，非交换几何在数学上是一个新兴发展的领域，同时也呈现为前景可观的现代物理学框架。非交换空间上的量子场论确实需要全面的探索，并且得到新的有趣的特征。本书提供了一个对非交换几何、畸变量子化与量子场论的重整化：Wilson和BPHZ的对标量理论的方法以及对规范理论的代数方法的基本概念的教育性的介绍。本书能够帮助读者理解几个一般性的非交换量子场论的问题。基于作者的博士论文，本书给出了欧氏Moyal空间上的量子场论的重整化问题的概览，并且特别着重于Grosse-Wulkenhaar模型，以及与其相关的规范理论和其数学解释。本书适用于想要理解这个前沿的数学和物理研究领域的研究生和科研人员。

Acknowledgments
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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