ABSTRACT
Observables in Quantum Mechanics are usually represented by non-commuting linear operators. In this chapter it is shown how to reconstruct relations between quantum operators starting from the Green's functions of field operators, i.e. the vacuum expectation values of time-ordered fields. In particular, starting from the Feynman Path Integral formulation of the Green's functions, it is shown how to derive: (i) the equations of motion in operator form; (ii) the canonical commutation rules of Quantum Mechanics; (iii) the symmetry properties of the theory and the corresponding Noether's theorem characterising the conserved quantities, in particular the Ward's identities of Quantum Electrodynamics, or QED. An important role, in Quantum Field Theory, is played by the vacuum state. In this Chapter one may find a synthetic demonstration of two basic theorems: (i) the Federbush-Johnson theorem: any local operator which annihilates the vacuum is the null operator and; (ii) the Coleman theorem: the symmetries of the vacuum are the symmetries of the world.
