Up to now we have investigated systems possessing one (q) or several (q α α = 1,...,D) degrees of freedom and studied their quantization using path integrals. In this chapter we will pass over to (relativistic) field theories, starting with the case of a scalar neutral field o(x). When discussing canonical quantization we already noticed that the field function o(x, t) can be viewed as a generalized coordinate vector q i (t) depending on the “continuous index” x in the place of the discrete index i. Thus a field is a system with an infinite number of degrees of freedom. Its dynamics is described by a Lagrange function which in local field theories can be written as an integral over the Lagrange density: $$L = \int {{d^3}} xL(\phi (x),{\partial _\mu }\phi (x))$$ (12.1) .