Algebraic Preliminaries

This chapter reviews algebraic identities about gaussian integrals, in particular Wick's theorem, a result also relevant for gaussian probability distributions. It discusses the steepest descent method, which reduces a certain type of integrals to gaussian expectation values. It then defines and discusses a few properties of dierentiation and integration in a Grassmann, that is, antisymmetric algebra, relevant for theories with fermion particles. In particular, gaussian integrals are calculated and general integrals are again reduced to gaussian expectation values. The chapter also recalls...

This chapter reviews algebraic identities about gaussian integrals, in particular Wick's theorem, a result also relevant for gaussian probability distributions. It discusses the steepest descent method, which reduces a certain type of integrals to gaussian expectation values. It then defines and discusses a few properties of dierentiation and integration in a Grassmann, that is, antisymmetric algebra, relevant for theories with fermion particles. In particular, gaussian integrals are calculated and general integrals are again reduced to gaussian expectation values. The chapter also recalls the concept of Legendre transformation, generating functional, functional dierentiation and the algebraic defiition of the determinant of an operator.