2026-10-01 · chapter Yang–Mills and the mass gapPart 6 Constructive field theorystochastic quantisation, regularity structures

Stochastic quantisation and regularity structures — the road of making the measure "by moving it", and why it stops short of four dimensions

To draw points from a finite-dimensional density $e^{-S(x)}/Z$ ($Z$ is the normalising constant), one keeps shaking the point with noise while letting it go down the slope $S$.

$W_t$ is Brownian motion (1-09). Formula (1) means: advance in small steps $\Delta t$ by the rule

($N$ is a standard normal random number drawn afresh each time), and take the limit $\Delta t\to0$. The ball in Figure 1 also moves by this rule (the Euler–Maruyama method). Formula (1) is called the Langevin equation, and the time evolution it defines the Langevin dynamics.

If many points run at once, the density $\rho(t,x)$ of the swarm obeys the Fokker–Planck equation $\partial_t\rho=\nabla\cdot(\nabla\rho+\rho\nabla S)$. The right-hand side has the form of the divergence of a flow, so the total number of points is conserved (a conservation law obeyed by the density). For $\rho=e^{-S}/Z$ the inside of the bracket is 0 and nothing moves. This is the invariant measure. As a metaphor, it is the balance between the flow down the slope and the flow spread by the noise. Precisely, when $S$ grows fast enough at infinity, the law of $X_t$ approaches $e^{-S}/Z$ wherever it starts. Known

Making the measure of a field as the invariant measure of an equation of this form is stochastic quantisation, proposed in the context of gauge theory by Parisi–Wu (1981) (Chevyrev 2022 §1.2). The formula in 6-02 is the same thing with time rescaled, and the Monte Carlo method (5-07) rests on the same idea. The example is $S_h(x)=h(x^2-1)^2$, with wells at $x=\pm1$ and a hill of height $h$.

Computed$Z$, $\langle x^2\rangle$ and $T$ in this figure are values computed on this machine. The ball is moved on the spot by the Euler–Maruyama method. The simulation in Table 1 is 2000 chains started from $x=-1$ (step 0.002, times 20–120), and $T$ is the mean hitting time from $-1$ to $+1$, $\int_{-1}^{1}e^{S(y)}\int_{-\infty}^{y}e^{-S(z)}dz\,dy$.

$\langle x^2\rangle$ agrees within 0.3%. On the other hand $T$ grows exponentially when $h$ is large (80.9 and 437.5 for $h=6,8$), and for $h=4$ the chain average $\langle x\rangle$ is still −0.33 at time 10, biased towards the left well. Computed

How it grows for large $h$ is given by Kramers' formula: from the curvatures at the bottom of the wells and the top of the hill, $S_h''(\pm1)=8h$ and $S_h''(0)=-4h$,

Known Forming $T\cdot h/e^{h}$ from the computed $T$ gives 1.27, 1.20, 1.17 for $h=4,6,8$, slowly approaching the coefficient $\pi/(2\sqrt2)\approx1.11$. Computed That an invariant measure exists and that it is approached quickly are separate problems, and in field theory too, estimates uniform in time are the hard part (§08).

For a field on the torus $T^d$ with $S(\phi)=\int\big(\tfrac12|\nabla\phi|^2+\tfrac12m^2\phi^2+\tfrac\lambda4\phi^4\big)dx$, formula (1) becomes

($\Delta$ is the Laplacian, not the $\Delta$ of the mass gap). $\xi$ is space-time white noise, a random distribution formally satisfying $\mathbb E[\xi(t,x)\xi(s,y)]=\delta(t-s)\,\delta(x-y)$ (1-09). The linear stochastic heat equation with $\lambda=0$ splits into independent $d\hat\phi_k=-(|k|^2+m^2)\hat\phi_k\,dt+\sqrt2\,dW_k$ for each Fourier component, and the stationary distribution is the Gaussian free field with covariance $(-\Delta+m^2)^{-1}$ (the starting measure of 6-03). Known

In the parabolic scaling, which counts time as two lengths, white noise has regularity just below degree $-(d+2)/2$, and solving with the heat equation raises it by 2 (Hairer 2014 §8). The linear solution $\Psi$ lies in

For $0\lt\alpha\lt1$, $\mathcal C^{\alpha}$ is the smoothness $|f(x)-f(y)|\lesssim|x-y|^\alpha$; for $\alpha\lt0$ it is the space of distributions whose values smeared with a bump function of width $r$ stay of order $r^{\alpha}$. For $d=1$ it is $\tfrac12-\kappa$ (the roughness of Brownian motion), for $d=2$ $-\kappa$ (no longer a function), and for $d=3$ $-\tfrac12-\kappa$.

The trouble is $\phi^3$. Products of distributions have the following limit (Hairer 2014 §1, Chevyrev 2022 §1.3; $\alpha,\beta$ are not integers). Known

For $\Psi\in\mathcal C^{-\kappa}$ with $d=2$, $\Psi^2$ cannot be defined; the mean of the cut-off $\Psi_\varepsilon^2$ diverges like $\log(1/\varepsilon)$, and subtracting a constant (Wick products; 6-03) lets the limit be taken. The margin in Table 2 is "the degree of the non-linear term − the degree of the noise"; if positive, the non-linear term stays a correction smoother than the noise at fine scales.

ComputedTable 2 Accounting of degrees (excluding $\kappa$; output in rational numbers by trees.py). $d=4$ is critical, $d=5$ supercritical.

The margin is $4-d$ for $\phi^4$ and $(4-d)/2$ for $A\partial A$ ($A^3$ is the same as $\phi^3$), positive for $d\lt4$. Hairer takes this as the condition of local subcriticality, and checks that for $\phi^4$ it is equivalent to $d\lt4$ (Hairer 2014 Assumption 8.3, §8). Known

In two dimensions the solution is split into "the roughest part of the noise" and the rest, setting $\phi=\Psi+v$.

${:}\Psi^n{:}$ is the Wick product in $\mathcal C^{-\kappa}$, which can be made first by probabilistic computation. The right-hand side is in $\mathcal C^{-\kappa}$, so $v\in\mathcal C^{2-\kappa}$, and $v\cdot{:}\Psi^2{:}$ has a positive sum of degrees and falls under (4), so it can be solved by an ordinary fixed-point theorem (for short times, the right-hand side of the equation written in integral form is a contraction, and there is exactly one fixed point). Returning to $\phi$, it is the limit of the equation with $\phi^3$ replaced by $\phi^3-3C_\varepsilon\phi$ ($C_\varepsilon$ of order $\log(1/\varepsilon)$). Da Prato–Debussche (2003) obtained strong solutions in two dimensions in this way (solutions satisfying the equation for each given realisation of the noise). Known

The construction of the process goes back further. Jona-Lasinio–Mitter (1985), on a square with Dirichlet boundary, made a Markov process (a stochastic process whose future distribution is determined only by the present state) converging to the $\phi^4_2$ measure as time goes to infinity, as a weak solution (one where the noise may be replaced too, satisfying the equation in law) of the equation smoothed by a power of an operator (Summers 2016 §5). Hairer 2014 §1 lists Albeverio–Röckner (1991) and Da Prato–Debussche (2003) as earlier work in two dimensions. Known

With $a=1-d/2$, $v\in\mathcal C^{2+3a}$, so $v\cdot{:}\Psi^2{:}$ needs $5a+2\gt0$, that is $d\lt14/5$. In three dimensions a single split is not enough (Hairer 2014 §8). Computed

A smooth function can be written near each point by Taylor polynomials. Hairer (2014) replaced these polynomials by a set of distributions built from the noise for each problem (in the words of the abstract, "a model built for the problem at hand").

$\tau$ are symbols such as $\Psi$, $\mathcal I(\Psi^3)\Psi$ ($\mathcal I$ is the operation of solving with the heat equation) and polynomials $X$, and $\Pi_x\tau$ is the distribution obtained by recentring it at the point $x$. Products that cannot be defined move into the algebra of symbols, and what the analysis needs is only the finitely many $\Pi_x\tau$ that can be made by probabilistic computation. The operation subtracting their divergences appears as the action of a "renormalisation group" (same abstract; the algebra is Bruned–Hairer–Zambotti 2019). For three-dimensional $\phi^4$

with $C_1^\varepsilon\sim\varepsilon^{-1}$ and $C_2^\varepsilon\sim\log\varepsilon$ (ibid. (9.21), Remark 10.23). The symbols of negative degree are the 8 symbols $\Xi,\Psi,\Psi^2,\Psi^3,\Psi^2X_i,\mathcal I(\Psi^3)\Psi,\mathcal I(\Psi^2)\Psi^2,\mathcal I(\Psi^3)\Psi^2$ (ibid. §10.5, the proof of Theorem 10.22). There is also another road, paracontrolled distributions (Gubinelli–Imkeller–Perkowski 2015), and later results on invariant measures and the axioms are in Table 4 and 6-03. Known

Magnifying as $\phi_\delta(t,x)=\delta^{(d-2)/2}\phi(\delta^2t,\delta x)$ (the gauge field $A$ is magnified the same way), the law of the noise and $\Delta$ do not change; only the coefficients of the non-linear terms change. The right of formula (8) is the Yang–Mills coupling $g$. With the action $\frac1{g^2}\int|F_A|^2$ and $A$ replaced by $gA$, the $A\partial A$ term of formula (9) in §07 gets $g$ and the $A^3$ term gets $g^2$.

For $d\lt4$ the non-linear terms vanish at fine scales $\delta\to0$, and the equation becomes a perturbation of the stochastic heat equation. This is subcriticality, which corresponds to super-renormalisability (Chevyrev 2022 §1.3: subcritical is equivalent to $d\lt4$). The general theory of regularity structures assumes local subcriticality (Hairer 2014 Assumption 8.3). Known A figure setting the margin $4-d$ beside other quantities that switch at four is the dimension knob in 1-12 §10.

$d=4$ is critical, and the general theory covers neither model. For $\phi^4$, Hairer 2014 §1 writes that it is not expected to have a non-trivial meaning in four dimensions, and Aizenman–Duminil-Copin later showed that the limit from the lattice is trivial (6-08). Known Yang–Mills is asymptotically free, and the coupling is thought to become logarithmically small at fine scales. This is a result about the perturbative β function (4-06), not a non-perturbative theorem. Physics

Seen through the number of symbols

The approach to criticality can also be seen in the number of symbols of §05. Build symbols by the following three rules, and count those with negative degree. $\kappa$ is treated as a positive infinitesimal, and the directions of $X_i$ and the component indices of the field are not distinguished.

1. Ingredients. The noise $\Xi$ (degree $-(d+2)/2-\kappa$), and for each symbol $\tau$ already made, $\mathcal I(\tau)$ with $\mathcal I$ (the operation of §05 of solving with the heat equation) applied (degree +2) and its derivative $\partial\mathcal I(\tau)$ (degree +1). $\Psi=\mathcal I(\Xi)$.
2. Products. Ingredients are placed in seats matching the form of the non-linear term and multiplied (degrees add). For $\phi^4$, one to three $\mathcal I(\cdot)$ in the three seats of $\phi^3$. For Yang–Mills, up to one each of $\mathcal I(\cdot)$ and $\partial\mathcal I(\cdot)$ in the two seats of $A\partial A$, or one to three $\mathcal I(\cdot)$ in the three seats of $A^3$.
3. Empty seats. If even one seat is empty, a polynomial $X$ (degree +1, any power) may be placed there.

Example ($\phi^4$ for $d=3$): $\Xi$ is $-\tfrac52-\kappa$ and $\Psi$ is $-\tfrac12-\kappa$. $\Psi^3$, filling the three seats with $\Psi$, is $-\tfrac32-3\kappa$, and $\mathcal I(\Psi^3)$, with $\mathcal I$ applied to it, is $\tfrac12-3\kappa$. $\mathcal I(\Psi^3)\Psi$, putting $\mathcal I(\Psi^3)$ and $\Psi$ in two seats, is $-4\kappa$, just below 0, so it is counted. Putting $X$ in the empty third seat gives $1-4\kappa$, positive, so it is not counted. This is repeated until no new negative symbols appear.

ComputedThe numbers in this figure and Table 3 were computed on this machine (trees.py; the number of "shapes" by the rules above; — means not computed).

$\phi^4$ agrees at $d=3$ with the 8 of §05. $\mathcal I(\Psi^3)\Psi^2$, added at $d=14/5$, is the borderline where the split of §04 breaks down, and the two added at $d=22/7$ are the borderline that Hairer 2014 §9.2 flags as $\alpha=-18/7$. Yang–Mills, with half the margin, grows much faster, and at $d=4$ stacking $\mathcal I(\tau)\Psi^2$ does not change the degree, so there are infinitely many. Computed For $\phi^4$, Chandra–Moinat–Weber take the noise in $\mathcal C^{-3+\delta}$ to imitate non-integer dimensions, and give a priori estimates over the whole subcritical range. Known

The components of the gauge field $A=(A_1,\dots,A_d)$ are Lie algebra elements (matrices), and $[X,Y]=XY-YX$ is the Lie bracket (1-03). $d_A=d+[A,\cdot\,]$ is the covariant derivative, $F_A$ its curvature (the field strength), and $d_A^*$ the adjoint of $d_A$ (the operator corresponding to the transpose with respect to the inner product) (1-07). Applying formula (1) to the action $S(A)=\int|F_A|^2$ and recounting time by a factor 2 gives $\partial_tA=-d_A^{*}F_A+\xi$. Even without noise, this equation is not parabolic (the type that, like the heat equation $\partial_tu=\Delta u$, smooths out fine fluctuations in every direction over time). This is because the action does not change under gauge transformations, so no smoothing force acts in the direction of the orbits (Chevyrev 2022 §1.2). Adding the term tangent to the orbits $-d_Ad^*A$ ($d^*A=-\sum_i\partial_iA_i$) (Zwanziger 1981, Donaldson 1985; for Ricci flow, DeTurck 1983)

makes it parabolic (ibid. (1.3); in shorthand $\partial_tA=\Delta A+A\partial A+A^3+\xi$). Formally, $B=A^g$, the solution $A$ moved by a time-dependent gauge transformation $g$, also satisfies the same equation with the noise $\mathrm{Ad}_g\xi=g\xi g^{-1}$ (rotated by sandwiching with $g$), and $\mathrm{Ad}_g\xi$ has the same law as $\xi$. This gauge covariance in law is the ground for defining a process on the orbits. Known

In renormalisation, a mass term $(C^\varepsilon_{\mathrm{bphz}}+\mathring C)A$ is added to the equation with the noise smoothed at width $\varepsilon$. $C^\varepsilon_{\mathrm{bphz}}$ (the BPHZ constant) is the part depending on $\varepsilon$; in two dimensions it converges to a finite value by cancellation, and in three dimensions it diverges like $\varepsilon^{-1}$ (Chevyrev 2022 Remark 1.10). $\mathring C$ (the bare mass) is the finite part independent of $\varepsilon$, and can be chosen freely. Both this mass term and the smoothing of the noise break covariance. Chandra–Chevyrev–Hairer–Shen (CCHS) showed, in two dimensions (2022) and three dimensions (2024), that choosing $\mathring C$ to be a certain value $\check C$ makes the two breakings cancel in the limit. Known

Whether $\check C$ is fixed uniquely has been answered in two stages. CCHS showed that there is a unique choice in the sense of a certain coupling (a construction that places the solutions from two gauge-equivalent initial values on the same probability space with rotated noise and compares them) (Chevyrev 2022 Theorem 1.14). That there is only one mass renormalisation giving solutions gauge covariant in law was shown in two dimensions by Chevyrev–Shen (Comm. Pure Appl. Math. 2026, Theorem 2.15 of arXiv:2302.12160) and in three dimensions by Chevyrev–Shen (Arch. Ration. Mech. Anal. 2026, arXiv:2503.03060), strengthening the result of CCHS. Known

Example: for $G=U(1)$ the equation is $\partial_tA=\Delta A+CA+\xi$, and on $T^d$ only $C=0$ is covariant (with $C\ne0$, equivalent initial values with $A_1$ shifted by the constant $2\pi$ give different expectations of $e^{i\int A_1}$; ibid. §1.4). Known

A state space for the solutions to live in is also needed. In two dimensions line integrals along segments make sense, and the space of orbits is a Polish space (a complete metric space with a countable dense subset; convenient for handling probability measures) (CCHS 2022). In three dimensions line integrals of the Gaussian free field already diverge ($\mathbb E|A(\ell)|^2\asymp\int_0^1\!\int_0^1|\ell|/|t-s|\,ds\,dt=\infty$), so one uses a non-linear space in which gauge equivalence is defined for fields smoothed by the Yang–Mills flow (the deterministic flow obtained by removing the noise from formula (9)) (Chevyrev 2022 §3.1, 6-06). What was obtained is a Markov process on the space of orbits (§04), in a form that leaves open the possibility of blow-up in finite time. In two dimensions Chevyrev–Shen (Comm. Pure Appl. Math. 2026) showed that the Yang–Mills measure on $T^2$ is its unique invariant measure, by moment estimates using solvability, gauge fixing on the lattice, and Bourgain's argument (6-02). Gauge fixing, avoided in the dynamics, is used on the lattice in the proof of the invariant measure. Known

KnownTable 4 What stochastic quantisation has reached (finite volume is, as a rule, the torus).

The open problems of the survey are the existence of the invariant measure (in three dimensions, the first construction of a Yang–Mills measure even in finite volume), $T^d\to\mathbb R^d$ (not trivial even in two dimensions), non-trivial principal bundles, and $N\to\infty$ (Chevyrev 2022 §4). Chevyrev–Shen (Comm. Pure Appl. Math. 2026) attribute the restriction to two dimensions to the moment estimates required by Bourgain's argument, and leave extension to higher dimensions open. Reflection positivity is another task (6-01). Known

What can be carried over: the viewpoint of seeing the measure as the invariant measure of a dynamics compatible with gauge. The state space that compares after smoothing by the Yang–Mills flow (on the lattice side too, Lüscher 2010 writes that theory and numerics suggest the flowed field becomes a smooth renormalised field Physics). That the DeTurck term was originally a device of Ricci flow is a clue for thinking, in 9-03, about "what corresponds to Ricci flow".

What cannot be carried over: the premise of local subcriticality. In four dimensions the margin is 0, and the symbols do not stay finite in number (Table 3). Even in three dimensions, the existence of the invariant measure, infinite volume and reflection positivity remain. The mass gap is a property of the spectrum of the infinite-volume Hamiltonian (4-11), and does not come directly from an invariant measure in finite volume. How four dimensions stops is summarised in 6-10.

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Next chapter to read: 6-10 Where four dimensions stops. Stochastic quantisation as a candidate for a new concept is looked at in 9-03.

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