MATHEMATICAL FOUNDATIONS OF QUANTUM FIELD THEORY

The book is very different from other books devoted to quantum field theory, both in the style of exposition and in the choice of topics. Written for both mathematicians and physicists, the author explains the theoretical formulation with a mixture of rigorous proofs and heuristic arguments; references are given for those who are looking for more details. The author is also careful to avoid ambiguous definitions and statements that can be found in some physics textbooks.

In terms of topics, almost all other books are devoted to relativistic quantum field theory, conversely this book is concentrated on the material that does not depend on the assumptions of Lorentz-invariance and/or locality. It contains also a chapter discussing application of methods of quantum field theory to statistical physics, in particular to the derivation of the diagram techniques that appear in thermo-field dynamics and Keldysh formalism. It is not assumed that the reader is familiar with quantum mechanics; the book contains a short introduction to quantum mechanics for mathematicians and an appendix devoted to some mathematical facts used in the book.

Contents:
PrefaceIntroductionPrinciples of Quantum TheoryQuantum Mechanics of Single-Particle and Non-Identical Particle SystemsQuantum Mechanics of a System of Identical ParticlesOperators of Time Evolution S(t, t0) and Sα(t, t0)The Theory of Potential ScatteringOperators on the Fock SpaceWightman and Green FunctionsTranslation-Invariant HamiltoniansThe Scattering Matrix for Translation-Invariant HamiltoniansAxiomatic Scattering TheoryTranslation-Invariant Hamiltonians (Further Investigations)Axiomatic Lorentz-Invariant Quantum Field TheoryMethods of Quantum Field Theory in Statistical PhysicsAppendixBibliographyIndex
Readership: Physics and mathematics students and researchers interested in quantum field theory.

Albert Schwarz, a mathematician and theoretical physicist, was born in Soviet Union in 1934 and immigrated to the US in 1989. He started as a topologist, but later shifted to physics. His work was instrumental in the development of fruitful interaction between mathematics and physics in recent decades. His contributions include the analysis of topological solutions of equations of motions and of their role in quantum field theory, the discovery of gauge instantons, the construction of first examples of topological quantum field theories. He has worked also in many other directions (multiloop string amplitudes, geometric approach to quantization, applications of noncommutative geometry to string theory, etc.). More information can be found on his UC Davis personal page.

'... the book is balanced in the sense to be suitable to both people in mathematics and people in physics who are seeking the boundary knowledge for quantum field theory.' - ZBMath Open