Mathematical Ideas and Notions of Quantum Field Theory

The development of quantum field theory and string theory in the last half-century has taken interactions between mathematics and physics to an unprecedented level, incorporating into this area of physics (often called ""high-energy physics"") such traditionally ""pure"" areas of mathematics as algebraic topology, category theory, differential and algebraic geometry, representation theory, combinatorics, and even number theory. This interaction has been highly fruitful in both directions, leading to the necessity for high-energy physicists to know the basics of modern mathematics and for mathematicians to know the basics of modern physics. However, in their attempts to learn these basics, many mathematicians are deterred by lack of rigor in physical texts and by a different manner of presentation. In particular, even the basic setting of quantum of field theory-necessary for understanding its more advanced (and more mathematically exciting) parts-is already largely unfamiliar to mathematicians. Nevertheless, many of the basic ideas of quantum field theory can in fact be presented rigorously, and even in cases when a rigorous exposition is impractical, difficult, or impossible, one can still attempt to explain the material in a mathematically natural way. Doing so is the main goal of the present book. It attempts to give a mathematical introduction to the basic ideas of quantum field theory, providing precise definitions and statements, motivation, examples, and rigorous proofs when possible. In other words, it provides a description of the basic settings of quantum field theory which is palatable to a mathematically minded audience.