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Unifying a range of topics that are currently scattered throughout the literature, this book offers a unique and definitive review of mathematical aspects of quantization and quantum field theory. The authors present both basic and more advanced topics of quantum field theory in a mathematically consistent way, focusing on canonical commutation and anti-commutation relations. They begin with a discussion of the mathematical structures underlying free bosonic or fermionic fields, like tensors, algebras, Fock spaces, and CCR and CAR representations (including their symplectic and orthogonal invariance). Applications of these topics to physical problems are discussed in later chapters. Although most of the book is devoted to free quantum fields, it also contains an exposition of two important aspects of interacting fields: diagrammatics and the Euclidean approach to constructive quantum field theory. With its in-depth coverage, this text is essential reading for graduate students and researchers in departments of mathematics and physics.

Geometric quantization. --- Quantum theory --- Geometry, Quantum --- Quantization, Geometric --- Quantum geometry --- Geometry, Differential --- Mathematics.

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This is a new approach to the theory of non-holomorphic modular forms, based on ideas from quantization theory or pseudodifferential analysis. Extending the Rankin-Selberg method so as to apply it to the calculation of the Roelcke-Selberg decomposition of the product of two Eisenstein series, one lets Maass cusp-forms appear as residues of simple, Eisenstein-like, series. Other results, based on quantization theory, include a reinterpretation of the Lax-Phillips scattering theory for the automorphic wave equation, in terms of distributions on R2 automorphic with respect to the linear action of SL(2,Z).

Geometric quantization. --- Forms, Modular. --- Formes [Modulaires] --- Forms [Modular ] --- Geometric quantization --- Geometry (quantum) --- Quantization (geometric) --- Quantum geometry --- Vormen [Modulaire ] --- Number theory. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra

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Based on lectures given at the renowned Villa de Leyva summer school, this book provides a unique presentation of modern geometric methods in quantum field theory. Written by experts, it enables readers to enter some of the most fascinating research topics in this subject. Covering a series of topics on geometry, topology, algebra, number theory methods and their applications to quantum field theory, the book covers topics such as Dirac structures, holomorphic bundles and stability, Feynman integrals, geometric aspects of quantum field theory and the standard model, spectral and Riemannian geometry and index theory. This is a valuable guide for graduate students and researchers in physics and mathematics wanting to enter this interesting research field at the borderline between mathematics and physics.

Geometric quantization. --- Quantum field theory --- Relativistic quantum field theory --- Field theory (Physics) --- Quantum theory --- Relativity (Physics) --- Geometry, Quantum --- Quantization, Geometric --- Quantum geometry --- Geometry, Differential --- Mathematics.

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This text analayzes in considerable generality the quantization-dequantization integral transform scheme of Weyl and Wigner, and considers several phase operator theories.

Geometric quantization. --- Phase space (Statistical physics) --- Weyl groups. --- Quantum theory --- Weyl's groups --- Group theory --- Space, Phase (Statistical physics) --- Generalized spaces --- Geometry, Quantum --- Quantization, Geometric --- Quantum geometry --- Geometry, Differential --- Mathematics.

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The Białowieża workshops on Geometric Methods in Physics, taking place in the unique environment of the Białowieża natural forest in Poland, are among the important meetings in the field. Every year some 80 to 100 participants both from mathematics and physics join to discuss new developments and to interchange ideas. The current volume was produced on the occasion of the XXXI meeting in 2012. For the first time the workshop was followed by a School on Geometry and Physics, which consisted of advanced lectures for graduate students and young researchers. Selected speakers of the workshop were asked to contribute, and additional review articles were added. The selection shows that despite its now long tradition the workshop remains always at the cutting edge of ongoing research. The XXXI workshop had as a special topic the works of the late Boris Vasilievich Fedosov (1938–2011) who is best known for a simple and very natural construction of a deformation quantization for any symplectic manifold, and for his contributions to index theory.

Mathematical physics. --- Geometric quantization. --- Geometry, Quantum --- Quantization, Geometric --- Quantum geometry --- Geometry, Differential --- Quantum theory --- Physical mathematics --- Physics --- Mathematics --- Group theory. --- Global analysis. --- Group Theory and Generalizations. --- Global Analysis and Analysis on Manifolds. --- Quantum Computing. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Quantum computers. --- Computers --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic