nLab Three Roles of Quantum Field Theory

In May 2011 Graeme Segal gave a series of lectures on the topic of quantum field theory at the Hausdorff Center for Mathematics in Bonn.

• Graeme Segal, Three Roles of Quantum Field Theory , Felix Klein Lectures 2011, Bonn (website)

  Abstract. Quantum field theory has many roles, and the lectures will be about three of them. The primary role is to provide a description of all of fundamental physics when gravity is firmly excluded. A second, at first surprising, role has emerged from string theory, which is a theory of gravitation: it turns out that a two-dimensional field theory can be regarded as a generalized manifold, and in particular can be a model for space-time. Thirdly, quite apart from physics, the concept of a field theory has taken on a new life as an organizing principle in other areas of mathematics - not only in geometry and representation theory, but even in connection with quantum computing.

The three roles can be seen together as aspects of noncommutative geometry, and that will be a central theme of the lectures. The talks will aim to show how powerful the field theory idea is by jumping between a variety of contexts, beginning from the origins of the structure in particle physics, with a little about so-called “Wick rotation”, and then moving towards more pure-mathematical applications to analysis, algebra, and the structure of manifolds.

• part I (mp4)

  Basics of classical mechanics from an AQFT-point of view: symplectic manifold, Hamiltonian, deformation quantization, C-star algebra, state, Lagrangian, action functional, covariant phase space, correlator, locality of the would-be path integral measure

• part II (mp4)

• part III (mp4)

• part IV (mp4)

• part V (mp4)

• part VI (mp4)

category: reference