Stochastic quantization of gauge theories
We give a well-posedness theory for the Langevin dynamics of Yang–Mills in temporal gauge, constructing the stochastic quantization measure on the space of distributional connections.
1 · Langevin dynamics
Stochastic quantization descends from equilibrium statistical fields to a gradient flow driven by space-time white noise. For gauge theories the dynamics must respect gauge symmetry; working in temporal gauge reduces the problem to a singular parabolic SPDE on the space of connections.
2 · Regularity structures
The forcing is more singular than the equation can absorb classically: solutions live in a Hölder space of negative index. We renormalize the dynamics using regularity structures, subtracting the finite family of counterterms dictated by the symmetry of the Yang–Mills action.
3 · The invariant measure
The renormalized dynamics admits an invariant measure which we identify with the formal Yang–Mills measure pushed forward through the gauge fixing. In dimension two this recovers the constructive theory; in dimension three the measure is constructed only locally in time.