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Towards the mathematics of quantum field theory / Frederic Paugam.

By: Material type: TextTextSeries: Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. folge | A series of modern surveys in mathematics ; 59Publication details: Switzerland : springer, 2014.Description: xvi, 487 p. : illustrations ; 25 cmISBN:
  • 9783319045634 (alk. paper)
Subject(s): DDC classification:
  • 530.143 23 P323
Contents:
1. Introduction -- Part I Mathematical Preliminaries: 2. A categorical toolbox-- 3. Parametrized and functional differential geometry-- 4. Functorial analysis-- 5. Linear groups-- 6. Hopf algebras-- 7. Connections and curvature-- 8. Lagrangian and Hamiltonian systems-- 9. Homotopical algebra-- 10. A glimpse at homotopical geometry-- 11. Algebraic analysis of linear partial differential equations-- 12. Algebraic analysis of non-linear partial differential equations-- 13. Gauge theories and their homotopical poisson reduction-- Part II Classical Trajectories and Fields -- 14. Variational problems of experimental classical physics-- 15. Variational problems of experiment quantum physics-- 16. Variational problems of theoretical physics-- Part III Quantum Trajectories and Fields -- 17. Quantum mechanics-- 18. Mathematical difficulties of perturbative functional integrals-- 19. The Connes-Kreimer-van suijlekon view of renormalization-- 20. Nonperturbative quantum field theory-- 21. Perturbative renormalization a la Wilson-- 22. Causal perturbative quantum field theory-- 23. Topological deformation quantizations-- 24. Factorization spaces and quantization-- References-- Index.
Summary: The aim of this book is to introduce mathematicians (and, in particular, graduate students) to the mathematical methods of theoretical and experimental quantum field theory, with an emphasis on coordinate-free presentations of the mathematical objects in play. This should in turn promote interaction between mathematicians and physicists by supplying a common and flexible language for the good of both communities, even if the mathematical one is the primary target. This reference work provides a coherent and complete mathematical toolbox for classical and quantum field theory, based on categorical and homotopical methods, representing an original contribution to the literature. The first part of the book introduces the mathematical methods needed to work with the physicists' spaces of fields, including parameterized and functional differential geometry, functorial analysis, and the homotopical geometric theory of non-linear partial differential equations, with applications to general gauge theories. The second part presents a large family of examples of classical field theories, both from experimental and theoretical physics, while the third part provides an introduction to quantum field theory, presents various renormalization methods, and discusses the quantization of factorization algebras. The book is primarily intended for pure mathematicians (and in particular graduate students) who would like to learn about the mathematics of quantum field theory.
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Holdings
Item type Current library Call number Status Date due Barcode Item holds
Books ISI Library, Kolkata 530.143 P323 (Browse shelf(Opens below)) Available 135857
Total holds: 0

Includes bibliographical references (pages 469-480) and index.

1. Introduction --
Part I Mathematical Preliminaries:
2. A categorical toolbox--
3. Parametrized and functional differential geometry--
4. Functorial analysis--
5. Linear groups--
6. Hopf algebras--
7. Connections and curvature--
8. Lagrangian and Hamiltonian systems--
9. Homotopical algebra--
10. A glimpse at homotopical geometry--
11. Algebraic analysis of linear partial differential equations--
12. Algebraic analysis of non-linear partial differential equations--
13. Gauge theories and their homotopical poisson reduction--

Part II Classical Trajectories and Fields --
14. Variational problems of experimental classical physics--
15. Variational problems of experiment quantum physics--
16. Variational problems of theoretical physics--

Part III Quantum Trajectories and Fields --
17. Quantum mechanics--
18. Mathematical difficulties of perturbative functional integrals--
19. The Connes-Kreimer-van suijlekon view of renormalization--
20. Nonperturbative quantum field theory--
21. Perturbative renormalization a la Wilson--
22. Causal perturbative quantum field theory--
23. Topological deformation quantizations--
24. Factorization spaces and quantization--
References--
Index.

The aim of this book is to introduce mathematicians (and, in particular, graduate students) to the mathematical methods of theoretical and experimental quantum field theory, with an emphasis on coordinate-free presentations of the mathematical objects in play. This should in turn promote interaction between mathematicians and physicists by supplying a common and flexible language for the good of both communities, even if the mathematical one is the primary target. This reference work provides a coherent and complete mathematical toolbox for classical and quantum field theory, based on categorical and homotopical methods, representing an original contribution to the literature. The first part of the book introduces the mathematical methods needed to work with the physicists' spaces of fields, including parameterized and functional differential geometry, functorial analysis, and the homotopical geometric theory of non-linear partial differential equations, with applications to general gauge theories. The second part presents a large family of examples of classical field theories, both from experimental and theoretical physics, while the third part provides an introduction to quantum field theory, presents various renormalization methods, and discusses the quantization of factorization algebras. The book is primarily intended for pure mathematicians (and in particular graduate students) who would like to learn about the mathematics of quantum field theory.

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