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Path integral quantization of fields computes quantum amplitudes by summing over all possible field configurations, weighted by eiS[phi], where S is the classical action. It provides an alternative to canonical quantization that is manifestly Lorentz covariant, naturally handles gauge theories, and is the foundation for non-perturbative methods like lattice QFT.
Canonical quantization (promoting fields and their conjugate momenta to operators with commutation relations) works well for free fields and for QED, but becomes increasingly cumbersome for gauge theories. Path integral quantization provides an alternative approach that is manifestly Lorentz covariant and handles gauge invariance more naturally. The central object is the generating functional Z[J] = integral D[phi] ei(S[phi] + integral J phi d4x), where D[phi] denotes integration over all field configurations, S[phi] is the classical action, and J(x) is an external source.
The physical content is simple: the amplitude for any quantum process is obtained by summing over all possible ways it could happen, with each possibility weighted by eiS. The classical path (which extremizes S) dominates in the classical limit; quantum corrections come from nearby paths whose actions differ from the classical action by order hbar. Expanding the action to second order around the classical solution gives a Gaussian integral, which is the one-loop approximation. Higher terms give higher-loop corrections. The path integral thus provides a natural and systematic organization of perturbation theory.
For gauge theories, the path integral requires care. The naive integral over all gauge field configurations overcounts because gauge-equivalent configurations represent the same physics. The Faddeev-Popov procedure fixes this: it restricts the integral to one representative from each gauge orbit by inserting a gauge-fixing condition and a compensating functional determinant (which can be written as an integral over ghost fields). The resulting gauge-fixed path integral is well-defined and generates the correct Feynman rules, including ghost propagators and vertices.
The path integral also provides access to non-perturbative physics that is invisible to canonical perturbation theory. In the Euclidean (imaginary time) formulation, the path integral becomes Z = integral D[phi] e-S_E[phi], which resembles a statistical mechanics partition function. This connection enables lattice field theory (evaluating the path integral numerically on a discrete spacetime grid), the study of instantons (finite-action solutions of the Euclidean equations of motion that describe tunneling between topologically distinct vacua), and the identification of non-perturbative vacuum structure. The Euclidean path integral is the basis for essentially all non-perturbative calculations in QCD.