●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 sible 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
編輯手記

本書是一部英文版的數學專著，中文書名或可譯為《非交換幾何、規範理論和重整化：一般簡介與非交換量子場論的重整化》。現在，非交換幾何在數學上是一個新興發展的領域，同時也呈現為前景可觀的現代物理學框架。非交換空間上的量子場論確實需要全面的探索，並且得到新的有趣的特征。本書提供了一個對非交換幾何、畸變量子化與量子場論的重整化：Wilson和BPHZ的對標量理論的方法以及對規範理論的代數方法的基本概念的教育性的介紹。本書能夠幫助讀者理解幾個一般性的非交換量子場論的問題。基於作者的博士論文，本書給出了歐氏Moyal空間上的量子場論的重整化問題的概覽，並且特別著重於Grosse-Wulkenhaar模型，以及與其相關的規範理論和其數學解釋。本書適用於想要理解這個前沿的數學和物理研究領域的研究生和科研人員。