Pfadintegralformulierung

Pfadintegral, or path integral, is a formulation of quantum mechanics and quantum field theory in which the probability amplitude for a transition is obtained by integrating over all possible histories between initial and final configurations. Introduced by Richard Feynman in 1948, it offers an alternative to the operator-based Hilbert space approach and provides a natural framework for incorporating classical action principles.

In quantum mechanics, the transition amplitude from position x_i at time t_i to position x_f at time t_f is written as ⟨x_f,t_f|x_i,t_i⟩ = ∫ Dx(t) e^{i S[x]/ħ}, where S[x] = ∫_{t_i}^{t_f} L(x, ẋ, t) dt is the action along a path x(t) subject to x(t_i)=x_i and x(t_f)=x_f. For quantum fields, the integral extends over field configurations ϕ(x,t), yielding ⟨…⟩ = ∫ Dϕ e^{i S[ϕ]/ħ}.

The formalism makes the classical limit transparent: stationary-phase paths satisfy the Euler–Lagrange equations, recovering classical mechanics. With imaginary time, t → -iτ, the path integral becomes a real exponential ∫ Dϕ e^{-S_E[ϕ]/ħ}, known as the Euclidean or imaginary-time path integral, which underpins statistical mechanics and lattice field theory.

Path integrals are widely used to derive propagators and Feynman diagrams in perturbation theory, to study nonperturbative effects via instantons, and in finite-temperature and condensed-matter contexts. Mathematically, defining measures on function spaces requires regularization and renormalization; gauge theories introduce extra factors to handle redundant degrees of freedom.

See also: Feynman diagrams, Wick rotation, functional integral, lattice QCD.