Some immediate applications of the theory of elliptic functions and modular forms to problems of physical interest are presented, including the construction of the Green functions and functional determinants for the two-dimensional quantum field theories of the bc fields, the scalar field, and the spinor fields on the torus. In particular, it will be shown how the singular terms in the operator product expansion of holomorphic fields for the bc system essentially determine arbitrary correlation functions on the torus in terms of elliptic functions. Along the way, a brief but reasonably systematic introduction will be presented of two-dimensional conformal field theory methods.

This chapter presents Feynman’s formulation of quantum mechanics, based on a path integral representation of the evolution operator. The chapter presents detailed examples which make it possible to understand clearly Feynman’s “sum over paths,” and it contains a complete discussion of how to calculate Gaussian path integrals. It also discusses the Euclidean version of the path integral, as well as Wick’s theorem and Feynman diagrams. Finally, it discusses instantons in quantum mechanics.