Stochastic Quantization and the Fokker–Planck Formulation of Euclidean Yang–Mills Theory

Stochastic quantization provides an alternative formulation of Euclidean quantumfield theory in which quantum expectation values are obtained as equilibrium averagesof a stochastic process evolving in an additional fictitious time, denoted by τ . InYang–Mills theory, the method replaces the direct construction of a gauge-fixedpath integral by a Langevin process in the space of gauge potentials or lattice linkvariables.The corresponding probability distribution obeys a functional Fokker–Planckequation whose stationary solution is formally proportional to the Euclidean Yang–Mills measure. The principal attraction of the method is that the original Langevinprocess can be written without explicitly imposing a conventional gauge condition.This avoids introducing a Faddeev–Popov determinant at the initial stage.Nevertheless, stochastic quantization does not automatically prove that all Gribovambiguities disappear. The original Parisi–Wu process contains both physicalmotion across gauge orbits and redundant motion along gauge orbits. A completenon-perturbative formulation requires control of gauge-orbit diffusion, equilibriumconvergence, ergodicity, normalizability, boundary conditions, and renormalization.The Fokker–Planck operator can be transformed into a Schrödinger-type Hamiltonian. Under detailed-balance conditions, the resulting operator is formally self-adjoint and non-negative:HFP = Q†Q ≥ 0.This factorization explains the positivity of the stochastic spectrum and identifiesthe equilibrium distribution with the ground-state probability density.