Editing Quantum field theory (section)

===Path integrals===
{{Main|Path integral formulation}}

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 <math>|\phi_I\rang</math> at time {{math|''t'' {{=}} 0}} to some final state <math>|\phi_F\rang</math> at {{math|''t'' {{=}} ''T''}}, the total time {{math|''T''}} is divided into {{math|''N''}} small intervals. The overall amplitude is the product of the amplitude of evolution within each interval, integrated over all intermediate states. Let {{math|''H''}} be the [[Hamiltonian (quantum mechanics)|Hamiltonian]] (''i.e.'' [[time evolution operator|generator of time evolution]]), then{{r|zee|page1=10}}
:<math>\lang \phi_F|e^{-iHT}|\phi_I\rang = \int d\phi_1\int d\phi_2\cdots\int d\phi_{N-1}\,\lang \phi_F|e^{-iHT/N}|\phi_{N-1}\rang\cdots\lang \phi_2|e^{-iHT/N}|\phi_1\rang\lang \phi_1|e^{-iHT/N}|\phi_I\rang.</math>
Taking the limit {{math|''N'' → ∞}}, the above product of integrals becomes the Feynman path integral:{{r|peskin|zee|page1=282|page2=12}}
:<math>\lang \phi_F|e^{-iHT}|\phi_I\rang = \int \mathcal{D}\phi(t)\,\exp\left\{i\int_0^T dt\,L\right\},</math>
where {{math|''L''}} is the Lagrangian involving {{math|''ϕ''}} and its derivatives with respect to spatial and time coordinates, obtained from the Hamiltonian {{math|''H''}} via [[Legendre transformation]]. The initial and final conditions of the path integral are respectively
:<math>\phi(0) = \phi_I,\quad \phi(T) = \phi_F.</math>
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.