stylistic:tone, information:classification, quality:correction, quality:corruption
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 | ϕ I ⟩ {\displaystyle |\phi _{I}\rangle } at time t = 0 to some final state | ϕ F ⟩ {\displaystyle |\phi _{F}\rangle } at t = T, the total time T is divided into N small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let H be the Hamiltonian (i.e. generator of time evolution), then

Taking the limit N → ∞, the above product of integrals becomes the Feynman path integral:

where L is the Lagrangian involving ϕ and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian H 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.