Feynmanpathintegral

Feynman path integral refers to the path integral formulation of quantum mechanics introduced by Richard Feynman. In this approach, the quantum amplitude for a system to evolve from an initial configuration to a final one is obtained by summing over all possible histories or trajectories that connect the two endpoints. The central object is the propagator K(x_b, t_b; x_a, t_a) = ∫ Dx(t) exp(i/ħ ∫_{t_a}^{t_b} dt L[x, ẋ, t]), where L is the Lagrangian and the integral runs over all paths x(t) with x(t_a) = x_a and x(t_b) = x_b. For a non-relativistic particle, L = 1/2 m ẋ^2 − V(x, t).

The integral is often defined by a time-slicing procedure: divide the time interval into N segments, insert resolutions of the identity, perform simple integrals, and take the limit N → ∞. The action S[path] = ∫ L dt appears in the exponential as exp(i S[path]/ħ). The expression is a sum over histories rather than a probability measure, so interference between paths plays a fundamental role.

Connections and uses: the path integral formulation is equivalent to the Schrödinger equation and provides intuitive semiclassical approximations via stationary phase, which recover the classical action. A Wick rotation to imaginary time yields Euclidean path integrals, linking quantum mechanics to statistical mechanics and enabling certain numerical methods. In quantum field theory, path integrals generalize to functional integrals over field configurations and underpin perturbation theory and renormalization. Gauge theories require gauge fixing and determinants (Faddeev–Popov). While powerful as a formal and computational tool, the path integral is treated carefully in rigorous contexts.