Stochastic quantization and gauge covariance

Stochastic quantization introduces a random evolution whose intended equilibrium distribution is a Euclidean field measure. It replaces one construction question by several related ones: does the evolution exist, does it last for all auxiliary time, does it have an invariant distribution, and does that distribution define the desired physical theory?

Prerequisites: semigroups and probability measures.

After this page: solve a Gaussian example, distinguish local from global dynamics, and explain why a relaxation rate is not automatically a particle mass.

Scope: auxiliary time is denoted $s$. Euclidean physical time, when present, is $\tau$. The two are different variables.

A complete one-variable example

For the dimensionless action $S(x)=kx^2/2$, $k>0$, consider the Langevin equation

$$dX_s=-kX_s\,ds+\sqrt2\,dB_s,$$

where $B_s$ is ordinary Brownian motion. Multiplying by $e^{ks}$ and integrating yields

$$X_s=e^{-ks}X_0+\sqrt2\int_0^s e^{-k(s-u)}\,dB_u.$$

For a fixed starting value $X_0=x$, the mean is $xe^{-ks}$. The noise integral is Gaussian with variance

$$2\int_0^s e^{-2k(s-u)}\,du=\frac{1-e^{-2ks}}{k}.$$

The long-time distribution is therefore Gaussian with variance $1/k$, exactly the normalized density proportional to $e^{-S(x)}$. This calculation exhibits global dynamics and a limiting invariant distribution in a solvable model. Those properties are conclusions of the calculation, not automatic consequences of writing a stochastic equation.

The speed of sampling is not a physical observable

Change the mobility by a positive constant $c$:

$$dX_s=-ckX_s\,ds+\sqrt{2c}\,dB_s.$$

The equilibrium density stays the same, while the relaxation rate becomes $ck$. This can be checked either by repeating the variance calculation or by applying the generator $L_c=c(\partial_x^2-kx\partial_x)$ to $x$.

Consequently the speed of this auxiliary process is adjustable without changing its equilibrium theory. A Poincaré or mixing estimate can be a powerful tool for that measure, but identifying a physical Hamiltonian requires an additional reconstruction argument in the field’s Euclidean time direction. This example diagnoses an unsupported identification; it does not deny useful relations between dynamics and correlations under explicit hypotheses.

What changes for gauge fields?

The field has infinitely many modes, nonlinear products can be undefined at their natural regularity, and gauge-related configurations represent the same physical configuration. A regularized stochastic equation can require counterterms before its cutoff is removed. Gauge covariance of the limiting law is a constraint on that renormalization.

Chevyrev and Shen’s arXiv:2503.03060v2, introduction and section 1.2 study uniqueness of gauge-covariant renormalization for stochastic three-dimensional Yang–Mills–Higgs. Their introduction distinguishes a local process, possible finite-time blow-up and the still-open invariant-measure construction in the specified three-dimensional setting. Read the dossier for its precise theorem and source assessment.

The earlier Chandra–Chevyrev–Hairer–Shen arXiv:2201.03487v2 is a source for the underlying construction. Its abstract records a process on gauge orbits up to possible blow-up. An abstract-level reading is orientation, not a check of the long technical proof.

Four separate questions to carry into a paper

Question	What must be controlled
Local dynamics	Initial-state space, noise, counterterms and a positive existence time
Global dynamics	Continuation without loss of control or finite-time blow-up
Invariant measure	Tightness and consistency of long-time distributions, with the intended gauge symmetry
Physical theory and gap	Euclidean axioms, reconstruction, target dimension and physical spectral estimates

Progress in one row can enable another. It does not silently supply it. The tightness bridge explains one tool for obtaining limits of distributions.

Exercise: change the noise

Replace $\sqrt2$ in the first equation by $\sqrt{2D}$, with $D>0$, while keeping the drift $-kX_s$. What equilibrium variance results? How should the drift change to preserve variance $1/k$?

Solution: the variance tends to $D/k$. To preserve $1/k$, multiply the drift by the same factor $D$, giving $-DkX_s$. Changing only the noise changes the equilibrium measure; changing both in the matched way changes the clock. This is why an algorithm’s coefficients and its claimed stationary measure must be checked together.

Continue to three-dimensional gauge theory for the setting, or reflection positivity for the missing physical interpretation.