simplest theories that contain no interactions (so-called free theory). In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes. To perform calculations on any realistic interacting theory, perturbation theory would be necessary. The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a quartic interaction term could be introduced to the Lagrangian of the real scalar field: where is a spacetime index, , etc. The summation over the index has been omitted following the Einstein notation. If the parameter is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory. Path integrals The path integral formulation of QFT is concerned with the direct computation of the scattering amplitude of a certain interaction process, rather than the establishment of operators and state spaces. To calculate the probability amplitude for a system to evolve from some initial state at time to some final state at , the total time is divided into small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let be the Hamiltonian (i.e. generator of time evolution), then Taking the limit , the above product of integrals becomes the Feynman path integral: where is the Lagrangian involving and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian via Legendre transformation. The initial and final conditions of the path integral are respectively In other words, the overall amplitude is the sum over the amplitude of every possible path between the initial and final states, where the amplitude of a path is given by the exponential in the integrand. Two-point correlation function In calculations, one often encounters expression likein the free or interacting theory, respectively. Here, and are position four-vectors, is the time ordering operator that shuffles its operands so the time-components and increase from right to left, and is the ground state (vacuum state) of the interacting theory, different from the free ground state . This expression represents the probability amplitude for the field to propagate from to , and goes by multiple names, like the two-point propagator, two-point correlation function, two-point Green's function or two-point function for