pathintegralformalisme

The path integral formalism, or the Feynman path integral, is a formulation of quantum mechanics and quantum field theory in which transition amplitudes are obtained by summing over all possible histories connecting initial and final configurations. In non-relativistic quantum mechanics the propagator K(b,t_b; a,t_a) is written as a functional integral over all particle paths x(t) with x(t_a)=a and x(t_b)=b: K(b,t_b; a,t_a) = ∫ D[x(t)] exp(i S[x]/ħ).

Here S[x] denotes the action, S[x] = ∫_{t_a}^{t_b} dt L(x, ẋ,t), with L the Lagrangian. The integral weights each path by the exponential of i times the action divided by Planck's constant, encoding quantum interference among histories.

The formal measure D[x(t)] is defined by time slicing and taking a limit; in practice the path integral is regularized or discretized. The path integral formulation reproduces the Schrödinger equation and provides a bridge to the classical limit via stationary-phase approximation (paths near the classical trajectory dominate when ħ is small). In many contexts a Wick rotation t → -iτ yields a Euclidean path integral ∫ Dφ exp(-S_E[φ]/ħ), which is closely related to statistical mechanics and is convenient for non-perturbative methods.

In quantum field theory, path integrals extend to functional integrals over fields and are central to modern techniques such as Feynman diagrams, lattice gauge theory, and instanton calculations. They offer a manifestly covariant framework for quantization, gauge theories, and the study of non-perturbative phenomena, while also serving as a unifying language across condensed matter, statistical mechanics, and gravity.