ABSTRACT
This chapter provides a detailed discussion of the formulation of quantum field theory—seen as the limiting case of the theory of quantum mechanical systems of many-degrees of freedom—within the approach based on Feynman path integrals introduced in Chapter 2. The use of the lattice approximation to obtain the Action from the Lagrangian density—which is a function of the field and its space derivatives—is illustrated in the case of a scalar field https://www.w3.org/1998/Math/MathML" display="inline"> ϕ ( x , t ) . The expressions of the fundamental elements of the formalism, notably the generating functional and the N-point Green's functions, the derivation of which is thoroughly described, turn out to involve path integrals extended to the space of field configurations satisfying proper boundary conditions. The calculation scheme allowing to obtain the Green's functions from functional differentiation of the generating functional is illustrated through its applications to the harmonic oscillator and the free scalar field. The creation and annihilation operators and the spectrum of excited states of these theories are also analysed.
