Structural aspects of quantum field theory and noncommutative geometry. Vol. 2. (English) Zbl 1275.81003

Grensing, Gerhard

Hackensack, NJ: World Scientific (ISBN 978-981-4472-69-2/set; 978-981-4472-73-9/hbk). xxii, 853-1516, index 12 p. (2013).

Volume II gives an up-to-date account of the mathematical methods of modern quantum field theory. Topics are included that are not usually found in textbooks of similar kind. This will make this book an invaluable reference work for mathematical physicists, but it is also appropriate to graduate courses on quantum field theory. It starts with Part VII, which recalls the Hamiltonian formalism, canonical transformations, symmetries and related conservation laws for systems with a finite number of degrees of freedom. Here, the concept of symplectic geometry plays a fundamental role. In the following Part VIII the symplectic geometry is applied to constrained systems like the Yang-Mills theory. It also discusses the formalism of Becchi, Rouet and Stora. Part IX addresses the Weyl quantization while Part X studies anomalies and index theorems. Non-commutative geometry is the subject of Part XI while quantum groups and Hopf algebras are analyzed in Part XII. The final Part XIII is rather brief. It wants to exhibit that non-commutative geometry and the subject of quantum groups are intimately related. As in Volume I [Zbl 1275.81002], there are many glimpses of advanced topics, which serve to give readers an idea of the power of mathematics in quantum field theory and thus in our way to understand particle physics.

Reviewer: Gert Roepstorff (Aachen)

MSC:

81-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
81R60	Noncommutative geometry in quantum theory
81T13	Yang-Mills and other gauge theories in quantum field theory
81R50	Quantum groups and related algebraic methods applied to problems in quantum theory
81V05	Strong interaction, including quantum chromodynamics
81V10	Electromagnetic interaction; quantum electrodynamics
81V22	Unified quantum theories
83E50	Supergravity
16T05	Hopf algebras and their applications

Keywords:

quantum field theory; non-commutative geometry; symplectic geometry; Weyl quantization; anomalies; index theorems; quantum groups; Hopf algebra

Citations:

Zbl 1275.81002