Constructive quantum field theory — Lecture 4 (notes)

Introduction

In Lecture 3 we decomposed the Gaussian free field along a resolution parameter and took as primary objects the stochastic observables: martingales in the filtration of finite-resolution measurements [BCG26]. In this lecture we introduce interactions, and we tell one story from three points of view: renormalisation, optimal control and filtering. See also [Pol84, BK87, Sal99, BBD24] for the renormalisation group side and [BG20, BG23] for the variational side.

Structure.

What is formal.

Conventions.

Acknowledgements.

Main lecture

7Three ways to generate the interacting measure

Fix a finite volume \(\mathcal{M}=\mathbb{T}^d_L\) and a terminal resolution \(T < \infty\), which plays the role of an ultraviolet cutoff: the reference field \(X_T \sim \mu^{Q_T}\) is smooth. Let \(V_T\) be an interaction, a functional of smooth fields bounded below. Our main example is

with a mass counterterm \(\gamma_T\) to be fixed in Appendix A.2 (\(\gamma_T = 0\) for \(d = 2\)). The coupling is normalised by \(\lambda / 12\) so that the curvature of the quartic term is \(\lambda \varphi^2\), as in Lectures 5–6. The interacting measure at cutoff \(T\) is

How does the Gaussian scale decomposition of Lecture 3 generate \(\nu_T\)? There are three answers.

• Renormalisation (Section 7.1). Split the covariance into shells and integrate them out one at a time. At every resolution the field is again a Gibbs measure, and it is driven by an effective force.

• Optimal control (Section 7.2). Steer the Gaussian field by a drift, at the least cost. The effective potential is the value function, and the Polchinski equation is the dynamic programming equation.

• Filtering (Section 7.3). Observe the terminal field through noise, with increasing precision. The field at resolution \(t\) is the observation, the force is what it reveals about the terminal field, and backward in resolution the process has a linear dynamics with Gaussian transitions, started from the interacting measure.

Section 7.4 shows that the three constructions produce the same process. Throughout,

is the covariance still unresolved at resolution \(t\).

7.1Renormalisation: decomposing the covariance

Decompose the covariance as in Lecture 3: \(Q_T = Q_t + S_t\), and correspondingly

with independent terms. Integrating out the second one defines the effective potential.

\(V_t\) is the interaction felt by the field at resolution \(t\) once the fluctuations between \(t\) and \(T\) have been integrated out: it is the Wilson effective action. At \(t = 0\), \(V_0 (0) = - \log Z_T\) is the free energy. Here is the first key fact.

Proof. Under \(\mathbb{P}_0\), \(X_T = X_t + (X_T - X_t)\) with an independent increment of law \(\mu^{Q_{T, t}}\). Hence \(\mathbb{E}_{\mathbb{P}} [G (X_t)] = Z_T^{- 1} \mathbb{E}_0 [G (X_t) \mathrm{e}^{- V_T (X_T)}] = Z_T^{- 1} \mathbb{E}_0 [G (X_t) \mathrm{e}^{- V_t (X_t)}]\). The conditional law follows in the same way.\(\Box\)

So at every resolution the field is again a Gibbs measure, with reference covariance \(Q_t\) and potential \(V_t\). The measure-valued martingale \((\mu_t^{X_t})_t\) goes from \(\nu_T\) at \(t = 0\) to \(\delta_{X_T}\) at \(t = T\): it is a stochastic localisation of \(\nu_T\) in the sense of Eldan [Eld23], adapted to the covariance profile [BBD24, BCG26].

The field as a process.

Proof. The density process is \(M_t := \mathbb{E}_0 [\hspace{0.17em} \mathrm{d} \mathbb{P}/ \hspace{0.17em} \mathrm{d} \mathbb{P}_0 |\mathcal{F}_t] = Z_T^{- 1} \mathrm{e}^{- V_t (X_t)}\), by Proposition 7.2. It is a martingale, so the drift in Itô's formula vanishes and \(\hspace{0.17em} \mathrm{d} M_t = - M_t \langle \mathrm{D} V_t (X_t), \dot{Q}_t^{1 / 2} \hspace{0.17em} \mathrm{d} W_t \rangle\). By Girsanov's theorem, \(W^{\mathbb{P}}_t = W_t + \int_0^t \dot{Q}_s^{1 / 2} \mathrm{D} V_s (X_s) \hspace{0.17em} \mathrm{d} s\) is a \(\mathbb{P}\)-Brownian motion, which is (7.6). Differentiating (7.3) gives \(\mathrm{D} V_t (\phi) = \mu_t^{\phi} (\mathrm{D} V_T)\), which is the second identity.\(\Box\)

Read as an equation for an unknown pair \((\Phi, F)\), (7.6) is a forward–backward system [MY99]. The field is driven forward in resolution by a force that is determined backward, as the conditional expectation of a functional of the terminal field:

Here and below we drop the superscript and write \(W\) for the Brownian motion driving the interacting field. No equation for the potential is needed to state (7.7), and \(\nu_T = \text{Law} (\Phi_T)\) is recovered from the solution. This is the starting point of FBSDE constructions of Euclidean fields (Figure 7.1): one solves (7.7) with renormalised \(V_T\), with estimates uniform in \(T\) and in the volume, and takes the limit. The field coordinate \(\Phi\) is not an observable, only a semimartingale. Its martingale part is the Gaussian noise, and the observables are built from it and from \(F\).

Figure 7.1. The forward–backward structure of (7.7) on \(\mathbb{T}^2\), heat profile, at resolutions \(t_1 < t_2 < t_3 < T\), all built from the same noise. Top: the field gains the fluctuation of each shell, \(\dot{Q}_t^{1 / 2} \mathrm{d} W_t\), and reveals finer modes. Bottom: the effective force \(F_t =\mathbb{E}_t [- \mathrm{D} V_T (\Phi_T)]\) for the \(\Phi^4\) interaction, determined backward from the terminal interaction. It is shown to first order in \(\lambda\), where it is the Wick cube \(- \frac{\lambda}{3} H_3 (\Phi_t ; c_t)\), and each panel is in its own units.

7.2Optimal control: the Polchinski equation

The measure \(\nu_T\) also solves a minimisation problem. Perturb the Gaussian field by an adapted drift, and pay for the drift and for the interaction at the endpoint. The effective potential is the value function of this problem. Its equation in \(t\), which we first derive directly from (7.3), is the dynamic programming equation.

Proof. \(u_t := \mathrm{e}^{- V_t} = \mathrm{e}^{\frac{1}{2} \Delta_{Q_{T, t}}} u_T\) is a Gaussian convolution, where \(\Delta_Q := \text{Tr} (Q \mathrm{D}^2)\). Since \(\partial_t Q_{T, t} = - \dot{Q}_t\), it solves the backward heat equation \(\partial_t u_t = - \frac{1}{2} \text{Tr} (\dot{Q}_t \mathrm{D}^2 u_t)\). Insert \(\mathrm{D}^2 u = (\mathrm{D} V \otimes \mathrm{D} V - \mathrm{D}^2 V) u\) and divide by \(- u\).\(\Box\)

The nonlinear term generates new, generally nonlocal, interactions (trees), and the trace term contracts pairs of fields with the covariance of the shell \([t, t + \mathrm{d} t]\) (loops). The substitution \(u = \mathrm{e}^{- V}\), which linearises the equation, is a Cole–Hopf transform: we come back to this in Section 8.1, and in Appendix A.

The control problem.

the field driven by the control \(u\) under the Gaussian reference \(\mathbb{P}_0\).

The following result is standard in the stochastic optimal control setting, see e.g. Fleming and Soner's book [FS06]. A useful non-Markovian generalisation is given by Boué–Dupuis [BD98].

Proof. (Verification.) For any \(u\), by Itô's formula and (7.8),

with \(\mathrm{D} V_t = \mathrm{D} V_t (X^u_t)\) and a martingale \(M\). Integrating and taking expectations,

So the cost is \(\geqslant V_0 (0)\), with equality iff \(u = u^{\ast}\). With this control \(X^{u^{\ast}}\) solves the same SDE as \(\Phi\) in (7.6).\(\Box\)

Euler–Lagrange equation \(=\) FBSDE.

This is (7.7). The FBSDE is the Euler–Lagrange equation of the variational problem, and the Polchinski equation is its Hamilton–Jacobi equation.

Entropy.

Generating functionals.

Bounds on \(\mathcal{W}_T (f)\) uniform in \(T\), for \(f\) in a rich class, give tightness, moment bounds and the identification of limits. For \(\Phi^4_3\) the naive problem does not have uniform bounds: the leading-order optimal drift \(u_s \approx - \frac{\lambda}{3} \dot{Q}_s^{1 / 2} : X_s^3\): has a divergent cost. One renormalises by the change of variables \(u = - \frac{\lambda}{3} \dot{Q}^{1 / 2} : X^3 : + r\). After it, and after further paracontrolled changes of variables, the singular terms are explicit polynomials in the Wick powers of \(X\), cancelled by the counterterms, and the residual functional is coercive. Its control in Besov and paracontrolled norms, uniformly in \(T\), is the construction of \(\Phi^4_3\) in [BG20]; see also [BG21].

Pontryagin's maximum principle.

Pontryagin's principle [PBGM62] says that along an optimal pair \((x, u)\) there is a costate \(p\) such that

The state runs forward from its initial value, the costate backward from its terminal value: a two-point boundary value problem. In the smooth case the principle is the first-order condition. If \(\xi = \partial_{\varepsilon} x\) is the variation of the state under \(u \to u + \varepsilon v\), then \(\dot{\xi} = \partial_x b \xi + \partial_u bv\), \(\xi_t = 0\), and the costate equation is exactly what makes \(\frac{\mathrm{d}}{\mathrm{d} s} \langle p, \xi \rangle = \langle \partial_x \ell, \xi \rangle + \langle p, \partial_u bv \rangle\). Integrating over \([t, T]\) and using the terminal condition,

So the costate is the Lagrange multiplier of the constraint \(\dot{x} = b\), and an optimal control is a critical point of the Hamiltonian.

The costate is a gradient.

Its characteristics are Hamilton's equations for \(H_{\ast}\), and along them \(p_s = - \mathrm{D} V (s, x_s)\): the costate is the sensitivity of the optimal cost to a displacement of the state, i.e. the force that the future exerts on the present.

The Hamiltonian of the field.

is the Hamiltonian of a free particle with mass matrix \(\dot{Q}_s^{- 1}\): it does not depend on the state. Hence \(\dot{p} = 0\) and \(\dot{x} = \dot{Q}_s p\): the costate is conserved, and from \(t\) to \(T\) the state moves along the linear shear

Its trajectories are the characteristics of the transport part of (8.1). All the interaction sits in the terminal condition \(p = - \mathrm{D} V_T (x + S_t p)\), an implicit equation for the momentum, solved by a sum of tree graphs (Exercise 8.1). The corresponding Hamilton–Jacobi equation \(\partial_t V = \frac{1}{2} \langle \mathrm{D} V, \dot{Q}_t \mathrm{D} V \rangle\) is the Polchinski equation (7.8) without its trace term.

With noise.

The martingale term \(\Upsilon \mathrm{d} W\) is the price of adaptedness: the costate may not anticipate the noise. For the field \(\partial_x H = 0\), so the costate is a martingale, \(p_t =\mathbb{E}_t [- \mathrm{D} V_T (X_T)]\), and since the value function is the effective potential (Theorem 7.5), \(p_t = - \mathrm{D} V_t (X_t)\). The noise adds the Itô term to the Hamilton–Jacobi equation, which becomes the Polchinski equation (7.8).

In the notation of Theorem 7.5 the controlled field is \(\mathrm{d} X = \dot{Q}_t^{1 / 2} (u \hspace{0.17em} \mathrm{d} t + \mathrm{d} W)\) and the optimal control is \(u_t = \dot{Q}_t^{1 / 2} p_t\) (Exercise 7.3). Since \(p_t\) is the value \(F_t\) of the force, the FBSDE (7.7) is the stochastic Pontryagin system of this problem:

7.3Filtering: measuring the field

Stochastic localisation was introduced by Eldan [Eld13] to study measures in high dimension (see [Eld23] for a survey). The idea is to write a measure as a mixture of simpler ones, through a measure-valued martingale driven by noise. Proposition 7.2 says that the Polchinski process does exactly this for \(\nu_T\). What Eldan's construction adds to the picture of Section 7.1 is a statistical reading:

The terminal field as a hidden signal.

Read (7.14) as a statistical model. The signal \(\Phi_T\) is hidden, with prior \(\nu_T\), and the path \((\Phi_s)_{s \leqslant t}\) is an observation of it: a copy attenuated by \(J_t\) and corrupted by a Gaussian noise independent of the signal. Infinitesimally, the bridge is the Doob transform of \(X\) by the Gaussian density of the remaining increment \(\Phi_T - \Phi_t \sim \mu^{S_t}\), so that

with \(B\) a Brownian motion independent of \(\Phi_T\), in the filtration enlarged by \(\sigma (\Phi_T)\): at each resolution the field is pulled towards the hidden terminal field, and the pull is seen through noise. The filtering problem is to estimate \(\Phi_T\) from \(\mathcal{F}_t = \sigma (\Phi_s : s \leqslant t)\), the filtration of the Polchinski process itself. Let \(\psi_t =\mathbb{E}_t [\Phi_T]\), \(\text{Cov}_t = \text{Cov} (\Phi_T |\mathcal{F}_t)\), and let

be the regulator.

Proof. (a) By Proposition 7.2 the conditional law is \(\mu_t^{\Phi_t}\), the tilt of \(\mu^{S_t} (\cdot - \Phi_t)\) by \(\mathrm{e}^{- V_T}\). The ratio of the Gaussian densities of \(\mu^{S_t} (\cdot - x)\) and \(\mu^{Q_T}\) at \(y\) is \(\mathrm{e}^{\langle S_t^{- 1} x, y \rangle - \frac{1}{2} \langle y, \mathcal{R}_t y \rangle}\) up to a function of \(x\); equivalently, apply Bayes' formula to the likelihood of (7.14). (b) Integrate (7.6) from \(t\) to \(T\) and condition on \(\mathcal{F}_t\): since \(F\) is a martingale, \(\mathbb{E}_t \int_t^T \dot{Q}_s F_s \hspace{0.17em} \mathrm{d} s = S_t F_t\), so \(\psi_t = \Phi_t + S_t F_t\), and the posterior mean of the drift of (7.15) is \(\dot{Q}_t S_t^{- 1} (\psi_t - \Phi_t) = \dot{Q}_t F_t\). Comparing the two expressions (7.6) and (7.15) of \(\mathrm{d} \Phi_t\) gives the formula for \(W\). (c) \(O_t = G_t (\Phi_t)\) with \(G_t (x) = \mu_t^x (G)\) is a martingale, so the drift in Itô's formula vanishes, and differentiating (7.16) in \(x\) gives \(\mathrm{D} G_t (x) = S_t^{- 1} \text{Cov}_t (\Phi_T, G (\Phi_T))\).\(\Box\)

So the Polchinski process is a filter. The interacting measure is the prior on a hidden field; at resolution \(t\) one has observed the coarse field \(\Phi_t\), and \(\mu_t^{\Phi_t}\) is what is known about the fine one. The forecast \(\psi_t\) is the posterior mean of the hidden field, and \(\dot{Q}_t F_t\) is the posterior mean of its pull on the observed one. For the Gaussian prior \(\mu^{Q_T}\) the increments are independent, so the posterior mean is \(\Phi_t\) itself and \(F = 0\): \(\Phi_t\) is the estimate that ignores the interaction and \(\psi_t\) the one that includes it. The effective potential is a likelihood ratio: by the proof of Theorem 7.3, \(Z_T^{- 1} \mathrm{e}^{- V_t (\Phi_t)}\) is the likelihood ratio of the observation up to resolution \(t\) under the interacting and under the Gaussian prior, and the Polchinski equation is the equation of its logarithm.

Proof. Under \(\mathbb{P}_0\), \(X\) is a centred Gaussian process with independent increments, so its reversal is a Gaussian Markov process: for \(t < u\), the law of \(X_t\) given \((X_r)_{r \geqslant u}\) is \(N (Q_t Q_u^{- 1} X_u, Q_t - Q_t Q_u^{- 1} Q_t)\), and letting \(u \downarrow t\) gives the drift and the diffusion of (7.17). Since \(\mathrm{d} \mathbb{P}/ \mathrm{d} \mathbb{P}_0\) is a function of \(X_T\), the initial value of the reversed process, these transition kernels are the same under \(\mathbb{P}\), and only the initial law changes, from \(\mu^{Q_T}\) to \(\nu_T\).\(\Box\)

Generation by denoising.

the force is the score of \(\nu_t\) relative to that of the Gaussian reference \(\mu^{Q_t}\), as \(\nu_t \propto \mathrm{e}^{- V_t} \mu^{Q_t}\) requires. The forecast is the optimal denoiser, and Tweedie's formula

is again \(\psi_t = \Phi_t + S_t F_t\). This is the structure of score-based generative models (diffusion models), with \(\nu_T\) as the data distribution, the blurring as the noising process and the Polchinski process as the generating dynamics.

Remark 7.8. (The measurement coordinate and the literature) In the literature on stochastic localisation [Eld13, Eld23, BBD24] the observation is written in another coordinate. Let \(j_t := S_t^{- 1} \Phi_t\). Since \(S_t^{- 1} J_t =\mathcal{R}_t\), (7.14) becomes a linear measurement of the terminal field with signal-to-noise ratio \(\mathcal{R}_t\):

In this coordinate (7.16) is an exponential family with natural parameter \(j_t\) and log-partition function

and

up to a function of \(t\). Then \(\psi_t = \mathrm{D} \mathcal{K}_t (j_t)\), \(\text{Cov}_t = \mathrm{D}^2 \mathcal{K}_t (j_t)\), \(\hspace{0.17em} \mathrm{d} j_t = \dot{\mathcal{R}}_t \psi_t \hspace{0.17em} \mathrm{d} t + \dot{\mathcal{R}}_t^{1 / 2} \hspace{0.17em} \mathrm{d} W_t\) with the same innovation \(W\), and

the Polchinski equation in the variable \(j\) [BBD24] (\(\partial_t \mathcal{K}= - \frac{1}{2} \mathbb{E} \langle y, \dot{\mathcal{R}} y \rangle\) and \(\text{Tr} (\dot{\mathcal{R}} \mathrm{D}^2 \mathcal{K}) =\mathbb{E} \langle y, \dot{\mathcal{R}} y \rangle - \langle \psi, \dot{\mathcal{R}} \psi \rangle\), expectations under the posterior). The basic form of Eldan's scheme is the profile \(S_t = (Q_T^{- 1} + t)^{- 1}\), \(t \in [0, \infty)\) (the terminal resolution is then \(t = \infty\), and \(Q_T\) is the cutoff covariance), where \(\mathcal{R}_t = t\) and \(j_t = t \Phi_T + N_t\) with \(N\) a Brownian motion; Eldan's original construction [Eld13] uses an adapted control matrix. This is the profile of Lectures 5–6.

Literature: Polchinski, Eldan and diffusion models.

7.4Where the three views meet

The three constructions give one process. Renormalisation produces it as the field at resolution \(t\) with its effective drift (Theorem 7.3), control as the optimal state (7.11), and filtering as the observation written in its own filtration (Proposition 7.6). Each of them has a name for each object:

In the last column, \(\Phi_t\) is the estimate of the terminal field that ignores the interaction, and \(\psi_t\) is the estimate that includes it. Their difference is \(S_t F_t\): the force is what the interaction adds to the estimate, in units of the free propagator \(S_t\). The conditional covariance \(\text{Cov}_t\) replaces \(S_t\) as the propagator. Forces, propagators and covariances are the common language of the three views, and the next section writes the system in it. Only the last row of the table, and the formula for \(\text{Cov}_t\) in the control column, involve a potential.

Figure 7.2. A zero-dimensional Polchinski process (Exercise 7.2): \(Q_t = 1 - \mathrm{e}^{- t}\), \(T = 5\), \(V_T (x) = \frac{1}{2} x^4 - \frac{3}{2} x^2\). Left: the effective force \(F_t = - V_t'\) is almost linear at low resolution and develops the double-well structure as \(t \to T\). Centre: paths of the FBSDE (7.6), which split between the two wells (stochastic localisation). Right: histogram of \(\Phi_T\) (\(2 \cdot 10^4\) paths) against the density of \(\nu_T\).

Exercises

8Force coordinates and potentials

The three views agree on the forward–backward system (7.7), which from now on is the primary object, with a martingale force \(F_t = F_t (\Phi_t)\), \(\hspace{0.17em} \mathrm{d} F_t = \Upsilon_t \hspace{0.17em} \mathrm{d} W_t\). We look for the coordinates in which it is simplest. They are suggested by the table above: the posterior mean \(\psi\), which we call the forecast, and the force. They remove the predictable part of the forward dynamics, and they integrate exactly the bilinear (tree) part of the force flow.

Sections 8.1–8.4 tell this story in terms of forces, propagators and covariances. The potential enters only through the terminal force and, for a gradient force, through the identification of the response with the conditional covariance; the system in force coordinates makes sense for a force that is not a gradient. Sections 8.5–8.7 tell it again in terms of potentials: the value function generates the canonical transformation to force coordinates, the Legendre transform connects it to the filter, and the equation for the force is the Wetterich equation for the effective average action [Wet93, BTW02].

8.1The force and a Burgers equation

Differentiating (7.8), the effective force \(F_t = - \mathrm{D} V_t\), now as a function of the field, solves

an infinite-dimensional, backward, viscous Burgers equation. The Cole–Hopf transform \(F = \mathrm{D} \log u\) linearises it, which is the statement that \(u = \mathrm{e}^{- V}\) solves a heat equation. Two remarks on (8.1):

Figure 8.1. The Polchinski equation (7.8) and the Burgers equation (8.1) as graphs. A blob with \(n\) legs is the \(n\)-th derivative of \(V_t\), or of the force \(F_t\), whose root is the arrow; a line with a dot is the kernel \(\dot{Q}_t\). The first term joins two blobs and makes a tree; the second closes a loop on one blob. Integrating from \(V_T\) and iterating generates the connected Feynman graphs, with lines \(Q_{T, t}\).

The function \(F_t (\cdot)\) with \(F_t = F_t (\Phi_t)\) is the decoupling field of the FBSDE, and the Burgers equation (8.1) is its quasilinear PDE; \(\Upsilon_t = \mathrm{D} F_t (\Phi_t) \dot{Q}_t^{1 / 2}\) is the martingale representation integrand of the costate. The two terms of (8.1) have distinct origins. The transport term \(- \mathrm{D} F [\dot{Q} F]\) comes from the drift of the state, and generates the tree diagrams. The diffusion term \(- \frac{1}{2} \text{Tr} (\dot{Q} \mathrm{D}^2 F)\) is the Itô correction, and generates the loops.

8.2Removing the predictable drift

The drift \(\dot{Q}_t F_t\) of the state is predictable, and the costate is a martingale. Hence the drift that remains to be accumulated on \([t, T]\) is predicted, at resolution \(t\), by transporting the current state with the current costate along the free flow (7.13).

Proof. The identity is in the proof of Proposition 7.6(b), and uses no commutativity. For the differential, the drift \(\dot{Q}_t F_t\) of \(\Phi\) cancels against \(\dot{S}_t F_t\), and \(S_t \hspace{0.17em} \mathrm{d} F_t = S_t \Upsilon_t \hspace{0.17em} \mathrm{d} W_t\).\(\Box\)

So \(\psi\) is a closed observable, the field observable of [BCG26]. The field splits as

a martingale, minus the drift still expected on the unresolved scales, which vanishes at \(t = T\). The noise of \(\psi\) in (8.3) has two sources: the fresh fluctuation of the shell, \(\dot{Q}_t^{1 / 2} \mathrm{d} W_t\), and the revision of the forecast of the remaining drift, \(S_t \hspace{0.17em} \mathrm{d} F_t\). The coupling between the forward and the backward equations, which was in the drift of \(\Phi\), now sits in the volatility of \(\psi\). The coordinate \(\psi\) is the interaction picture with respect to the noiseless (classical) part of the dynamics.

8.3The FBSDE in the forecast coordinate

Assume that the map

is invertible, and write its inverse as

so that \(F_t = -\mathcal{F}_t (\psi_t)\): the costate is now a function of the forecast. Introduce the response, and recall the regulator of the filter:

Differentiating the two maps \(x \leftrightarrow \psi\) gives

Proof. The forward equation is (8.3) with \(\mathbb{1} + S_t \mathrm{D} F_t = H_t S_t^{- 1}\), and its covariance is

The backward equation is \(F_t =\mathbb{E}_t [F_T]\) with \(\psi_T = \Phi_T\). Since \(\mathcal{F}_t (\psi_t)\) is a martingale and \(\psi\) has no drift, the drift in Itô's formula for \(\mathcal{F}_t (\psi_t)\) is the left-hand side of (8.8).\(\Box\)

Compare (8.8) with the Burgers equation (8.1): the transport term has been absorbed exactly by the change of coordinates, and only the Itô term is left. If it is dropped, \(\mathcal{F}_t = \mathrm{D} V_T\) does not move: all the tree diagrams are contained in the coordinate change (8.5). The loops are generated only by the quadratic variation \(\mathcal{C}_t\) of the forecast, i.e. by the randomness of the noise and of the force revisions, and the nonlinearity has moved into the field dependence of \(\mathcal{C}_t\). None of this uses a gradient structure or a Gaussian convolution: Theorem 8.2 holds for any FBSDE of the form (7.7) with a Markov martingale force, gradient or not, for which the coordinate change is invertible. For a general terminal force \(F_T =\mathfrak{f} (\Phi_T)\), replace \(\mathrm{D} V_T\) by \(-\mathfrak{f}\) in (8.7) and (8.8).

A one-dimensional model: Burgers and Lax–Oleinik.

Along the characteristics \(x = y + sg' (y)\) the velocity is constant, \(u (s, y + sg' (y)) = g' (y)\): in the Lagrangian label \(y\) the transport term disappears, and nothing moves. As long as the characteristics do not cross, i.e. \(1 + sg'' > 0\), the map \(y \mapsto x\) is invertible, and \(u = \partial_x G\) with the potential given by the Lax–Oleinik formula (or Hopf–Lax formula)

whose minimiser \(y (s, x)\) is the Lagrangian label of the point \(x\). After the characteristics cross, the formula still selects one label, the entropy solution, and \(u\) has shocks. The viscous equation \(\partial_s u + u \partial_x u = \varepsilon \partial_x^2 u\) is linearised by the Cole–Hopf transform,

and \(G^{\varepsilon} \to G\) as \(\varepsilon \to 0\) by Laplace's method.

The dictionary with this section is: the unresolved covariance \(S_t\) is the time \(s\), the field \(x\) is the Eulerian point, the forecast \(\psi\) is the Lagrangian label \(y\), \(- F_t = \mathrm{D} V_t\) is the velocity \(u\) (and \(V_t\) is \(G\)), and \(V_T\) is the datum \(g\). The shear (7.13), i.e. (8.5) at tree level, \(x = \psi + S_t \mathrm{D} V_T (\psi)\), is the characteristic map; the noiseless Polchinski equation is the Hamilton–Jacobi equation \(\partial_s G + \frac{1}{2} (\partial_x G)^2 = 0\), whose gradient solves the inviscid Burgers equation; the tree-level potential (8.17) is the Lax–Oleinik formula; and the condition for the characteristics not to cross, \(S_t^{- 1} + \mathrm{D}^2 V_T > 0\), is \(1 + sg'' > 0\). The Polchinski equation is the viscous Hamilton–Jacobi equation, with \(\varepsilon = \frac{1}{2}\) in the metric \(\dot{Q}_t\), and (7.3) is the Cole–Hopf formula.

Theorem 8.2 is the Lagrangian description of the viscous problem. With noise the label is no longer a function of the Eulerian point: it is its posterior mean, the mean \(\psi\) of \(y\) under the Cole–Hopf measure \(\propto \mathrm{e}^{- g (y) - (x - y)^2 / 2 s}\), which is \(\mu_t^x\). In this label the velocity is no longer constant but a martingale, and its evolution (8.8) contains only the loops. The Lax–Oleinik form survives exactly: the Moreau formula (8.12) of Section 8.5 is the Lax–Oleinik formula with the datum \(V_T\) replaced by \(\Gamma_t\), and at finite cutoff no shocks form: the noise keeps the label map invertible, locally by Proposition 8.3 and globally by the convexity argument of Section 8.5. Figure 8.3 shows the construction for a double well.

8.4Conditional moments

For the Polchinski process the response has a direct meaning.

Proof. By (7.5), the law of \(\Phi_T\) given \(\mathcal{F}_t\) depends only on \(\Phi_t\), so it is also the law given \(\Phi_t\). The conditional law \(\mu_t^{\phi}\) is the tilt of \(\mu^{S_t} (\cdot - \phi)\) by \(\mathrm{e}^{- V_T}\); differentiating twice in \(\phi\) gives \(\text{Cov} (\mu_t^{\phi}) = S_t - S_t \mathrm{D}^2 V_t (\phi) S_t\). On the other hand \(H_t S_t^{- 1} = \mathbb{1} + S_t \mathrm{D} F_t = \mathbb{1} - S_t \mathrm{D}^2 V_t\). The last claim is the Brascamp–Lieb inequality.\(\Box\)

So the forecast and the response are the first two conditional moments of the stochastic localisation \(\mu_t^{\Phi_t}\) of Proposition 7.2: \((\psi_t, \text{Cov}_t) = (\mathbb{E}[\Phi_T \hspace{0.17em} | \hspace{0.17em} \Phi_t], \text{Cov} (\Phi_T \hspace{0.17em} | \hspace{0.17em} \Phi_t))\) (Figure 8.2). They are tied by Itô's formula: since \(\mathbb{E}_t [\Phi_T \otimes \Phi_T] = \psi_t \otimes \psi_t + \text{Cov}_t\) is a martingale,

Conditional variance is resolved at the rate \(\text{Cov}_t \dot{\mathcal{R}}_t \text{Cov}_t\), which for the Gaussian reference is \(S_t \dot{\mathcal{R}}_t S_t = \dot{Q}_t\).

Brascamp–Lieb and Cramér–Rao bounds.

The lower bound follows from \(\mathbb{E}_{\mu} [(y - m) \otimes \mathrm{D} \mathcal{U}(y)] = 1\) (integration by parts) and the Cauchy–Schwarz inequality, with the Fisher information \(\mathbb{E}_{\mu} [\mathrm{D} \mathcal{U} \otimes \mathrm{D} \mathcal{U}] =\mathbb{E}_{\mu} \mathrm{D}^2 \mathcal{U}\) (Exercise 8.9). The two bounds coincide when \(V_T\) is quadratic, and their gap is a Jensen gap for the inverse. Through Proposition 8.3 they are two-sided bounds on the Hessian of the effective potential, i.e. on the response \(H_t\), which is the propagator of the Wetterich equation below. Some consequences:

• If \(V_T\) is convex, then \(\operatorname{Cov}_t \leqslant S_t\): the interaction can only reduce the conditional variance. Equivalently \(\mathrm{D}^2 V_t \geqslant 0\): convexity is preserved along the Polchinski flow (as also follows from the Prékopa–Leindler inequality).

• If \(\mathrm{D}^2 V_T \geqslant \kappa\) with \(\kappa > 0\), then \(\operatorname{Cov}_t \leqslant (S_t^{- 1} + \kappa)^{- 1}\) uniformly in the conditioning: a uniform mass in the interaction is a uniform mass in every conditional law.

• For \(\Phi^4_3\) convexity is already lost through the Wick ordering: \(H_4 (\varphi ; c_T) = \varphi^4 - 6 c_T \varphi^2 + 3 c_T^2\), so (7.1) has curvature \(- \lambda c_T + 2 \gamma_T\) at \(\varphi = 0\), which diverges as \(T \to \infty\) (this happens already for \(\Phi^4_2\)). The upper bound is lost, and at low resolution the conditional law may spread over several wells (Figure 8.2). This is the large-field problem. In Lecture 5 it is handled by a clipped approximate force, whose global coercivity (Lemma 12.1 there) is a bound of Brascamp–Lieb type on the response.

Figure 8.2. Stochastic localisation for a two-component field: \(Q_t = (1 - \mathrm{e}^{- t}) \mathbb{1}\), \(T = 4\), \(V_T\) a tilted double well, and the contours show the density of \(\nu_T\). One path of (7.6) up to resolution \(t\) (navy), with its terminal value \(\Phi_T\) (open circle). The forecast \(\psi_t = \Phi_t + S_t F_t\) (gold) is the mean of the conditional law \(\mu_t^{\Phi_t}\): teal samples, and the ellipse is two standard deviations of \(\text{Cov}_t\). At low resolution the conditional law covers both wells; as \(t\) grows it localises on one of them.

8.5Potentials: the value function as a generator

We now use that the force is a gradient, \(F_t = - \mathrm{D} V_t\), whose graph is a Lagrangian submanifold, and that \(V_t\) is the value function of Theorem 7.5. In Hamilton–Jacobi theory a value function is more than a cost: it generates a canonical transformation. A complete family of solutions of the Hamilton–Jacobi equation generates the transformation that trivialises the dynamics. Force coordinates are this transformation.

Phase space and Lagrangian graphs.

The free action generates the shear.

The relation \(\psi = x + S_t p\) is the shear (7.13). Since \(\mathrm{D}_x \mathcal{A}_t = - p\) and \(\mathrm{D}_{\psi} \mathcal{A}_t = p\), at fixed \(t\)

so \(\mathcal{A}_t\) is the generating function of the canonical transformation \(\Sigma_t\). Moreover

for every \(\psi\) it solves the noiseless Polchinski equation, so it is a complete integral of that equation, with parameter \(\psi\). Jacobi's theorem then says that in the new coordinates the Hamiltonian \(H_{\ast}\) vanishes: \(\psi\) and \(p\) are constants of motion.

This is what force coordinates are:

The noise does not destroy this: the constants of motion become martingales, \(\psi_t\) by Proposition 8.1 and \(F_t\) by Theorem 7.3. The only dynamics left is their quadratic variation.

The value function generates the graph.

These two generating functions are related by the Moreau (infimal convolution) transform

whose optimality conditions \(x = \psi + S_t \mathrm{D} \Gamma_t (\psi)\), \(\mathrm{D} V_t (x) = \mathrm{D} \Gamma_t (\psi)\) are (8.5). It is well defined by the same convexity: as a function of \(j = S_t^{- 1} x\), the function \(\frac{1}{2} \langle x, S_t^{- 1} x \rangle - V_t (x)\) is the Legendre transform of \(\Gamma_t + \frac{1}{2} \langle \cdot, S_t^{- 1} \cdot \rangle\). The value function \(V_t\) itself need not be convex. We use (8.12) to fix the additive constant in \(\Gamma_t\). Figure 8.3 shows the construction in zero dimensions.

At the minimiser, \(V_t (x) = \Gamma_t (\psi) +\mathcal{A}_t (x, \psi)\): the value function is the cost \(\Gamma_t\) attached to the forecast, plus the cost of reaching the forecast by free motion. This is the rule for transporting a generating function along a canonical transformation:

since \(\mathrm{D} \Gamma_t (\psi) = - F_t = - p\).

No convexity is needed.

For \(\Gamma_t\) one only gets semiconvexity, \(S_t^{- 1} + \mathrm{D}^2 \Gamma_t =\operatorname{Cov}_t^{- 1} > 0\), i.e. \(\Gamma_t + \frac{1}{2} \langle \cdot, S_t^{- 1} \cdot \rangle\) is strictly convex, which is what (8.12) requires. At \(t = 0\), where \(S_0 = Q_T\), this semiconvexity is the convexity of the effective average action of Section 8.6, \(\Gamma^{\text{EAA}}_0\). \(\Gamma_t\) itself is convex if and only if \(\operatorname{Cov}_t \leqslant S_t\), which holds for convex \(V_T\) but may fail for a double well at low resolution. Convexity matters only without noise: the tree-level potential (8.17) is merely semiconcave, and unless \(S_t^{- 1} + \mathrm{D}^2 V_T > 0\) the characteristics cross, producing caustics and shocks of the force. The noise removes them.

Figure 8.3. The zero-dimensional model of Figure 7.2 at \(t = 1.2\). Left: the shear \(\Sigma_t\) (teal arrows) maps the graph of the force \(F_t = - V_t'\) (navy) to a graph over the forecast \(\psi\) (gold), which is the graph of \(- \Gamma_t'\). Right: the bare potential \(V_T\), the effective potential \(V_t\) and \(\Gamma_t\), each shifted to have minimum \(0\).

8.6The Legendre transform and Wetterich's equation

Proof. Integrate (8.8) in field space. With \(\mathcal{F}= \mathrm{D} \Gamma\), the derivative of \(H = (S^{- 1} + \mathrm{D}^2 \Gamma)^{- 1}\) in a direction \(v\) is \(- H \hspace{0.17em} \mathrm{D}^3 \Gamma [v] \hspace{0.17em} H\), so by the total symmetry of \(\mathrm{D}^3 \Gamma\),

Hence (8.8) reads \(\mathrm{D} \left( \partial_t \Gamma_t - \frac{1}{2} \text{Tr} (\dot{\mathcal{R}}_t H_t) \right) = 0\), and (8.13) holds up to a function of \(t\) alone. That this function is \(- \frac{1}{2} \text{Tr} (\dot{\mathcal{R}}_t S_t)\) with the normalisation (8.12) follows from the Polchinski equation by the envelope theorem (Exercise 8.4). For (8.14), note that \(\mathrm{D}^2 \Gamma^{\text{EAA}}_t +\mathcal{R}_t = \mathrm{D}^2 \Gamma_t + S_t^{- 1}\). For the endpoints: \(S_T = 0\); and at \(t = 0\), \(S_0 = Q_T\), so (8.12) identifies \(\frac{1}{2} \langle x, Q_T^{- 1} x \rangle - V_0 (x)\), up to a constant, with the cumulant generating function of \(\nu_T\) at \(Q_T^{- 1} x\), whose Legendre transform is \(\Gamma^{\text{EAA}}_0\) (Exercise 8.6).\(\Box\)

Figure 8.4. The Wetterich equation (8.14) as graphs. The double line is the full response \((\mathrm{D}^2 \Gamma^{\text{EAA}}_t +\mathcal{R}_t)^{- 1} = (S_t^{- 1} + \mathrm{D}^2 \Gamma_t)^{- 1}\) and the crossed circle is one insertion of \(\dot{\mathcal{R}}_t\). There is exactly one loop: the tree term of Figure 8.1 has been absorbed by the Legendre transform. Each field derivative brings down a higher vertex, so the hierarchy does not close at a finite order. The bottom row expands the response around \(S_t\).

Thus the Wetterich equation is the gradient-integrated form of the force equation (8.8), i.e. of the backward half of the FBSDE in forecast coordinates [Mor94, BCG26]. Every object in it is an observable of the Polchinski process:

The flow of \(\Gamma\) is driven by the difference between the interacting and the free conditional covariances, paired with the shell \(\dot{\mathcal{R}}_t\). It is one-loop exact, with the full propagator inside the trace. In the Gaussian case \(\mathrm{D}^2 \Gamma_t\) is constant and only the constant term of \(\Gamma\) flows. For the heat profile with \(T = \infty\), \(S_t = C (1 - \mathrm{e}^{- A / t^2})\) and

which behaves like a mass term \(t^2\) on the modes \(|k| \ll t\) (for \(m \ll t\)) and is exponentially small on the modes \(|k| \gg t\): the exponential regulator of the functional renormalisation group (Figure 8.5).

Proof. By (8.12) and the formula for \(\mathcal{K}\). At fixed \(\psi\), with \(j = \mathrm{D} \mathcal{K}^{\ast} (\psi)\), the envelope theorem and (7.19) give

and \(\partial_t \frac{1}{2} \langle \psi, S^{- 1} \psi \rangle = \frac{1}{2} \langle \psi, \dot{\mathcal{R}} \psi \rangle\). Hence \(\partial_t \Gamma_t = \frac{1}{2} \text{Tr} (\dot{\mathcal{R}} \text{Cov})\) up to a function of \(t\), which is \(- \frac{1}{2} \text{Tr} (\dot{\mathcal{R}} S)\) with the normalisation (8.12).\(\Box\)

So one Lagrangian graph has three generators: \(V_t\) in the field \(x\), \(\mathcal{K}_t\) in the coordinate \(j = S_t^{- 1} x\) of Remark 7.8, and \(\Gamma_t\) in the forecast \(\psi\). The Polchinski flow is the Hamilton–Jacobi equation of the log-partition function of the localisation; the Wetterich flow is its Legendre dual [Mor94], whose argument is the posterior mean, and whose regulator is the information gained. At \(t = 0\), where \(\mathcal{R}_0 = 0\), \(\mathcal{K}_0\) is the generating function \(\Lambda\) of the cumulants of \(\nu_T\) below.

Probabilistic meaning of \(\Gamma\).

the generating functional of the vertex functions introduced by Jona-Lasinio [JL64]. It is a probabilistic object, as Jona-Lasinio emphasised in his probabilistic reading of the renormalisation group [JL75, JL01]; for the link with large deviations see Section VIII of [JL01]. By Cramér's theorem the empirical mean of \(N\) independent samples of \(\nu_T\) takes the value \(\psi\) with probability

so \(\Gamma^{\text{EAA}}_0 - V_0 (0)\) is the rate function of the large deviations of averages of the field. In particular \(\Gamma^{\text{EAA}}_0\) is convex even when \(V_T\) is not, it is minimal at the mean of \(\nu_T\), and its Hessian there is the inverse covariance. Formally the same holds for the field averaged over a large volume, with the volume as the large parameter and the effective potential of quantum field theory (\(\Gamma\) per unit volume at constant \(\psi\), not \(V_t\)) as rate function; in that limit flat parts may appear (the Maxwell construction).

Figure 8.5. The regulator \(\mathcal{R}_t\) of the heat profile (\(T = \infty\), \(m = 1\)): close to the mass term \(t^2\) (dotted) on the modes \(|k| \ll t\), exponentially small on the modes \(|k| \gg t\).

8.7Variational meaning and the tree level

Constrained free energies.

for \(\nu\) of mean \(\psi\),

So \(\Gamma_t\), the effective average action up to the quadratic term \(\frac{1}{2} \langle \psi, Q_T^{- 1} \psi \rangle\), is the free energy of the unresolved shell constrained to the forecast \(\psi\), and (8.12) is the minimisation over the forecast. In the language of Theorem 7.5: \(V_t\) is the value function of the control problem, and \(\Gamma_t\) is its Moreau transform, i.e. the value function expressed in the forecast coordinate, the interaction picture of the classical part of the dynamics.

The tree level.

the solution of the noiseless Polchinski equation \(\partial_t V = \frac{1}{2} \langle \mathrm{D} V, \dot{Q} \mathrm{D} V \rangle\). It is the saddle point of the Gaussian convolution (7.3). Its minimiser solves the classical field equation \(S_t^{- 1} (y - x) + \mathrm{D} V_T (y) = 0\) in the shell \([t, T]\) with background \(x\), and expanding it in \(V_T\) produces exactly the tree diagrams.

Exercises

9Observables and field equations

In Lecture 3 the primary objects were the stochastic observables, martingales in the filtration of finite-resolution measurements [BCG26]. The filtering point of view makes this natural: an observable is a posterior mean \(\mathbb{E}_t [G (\Phi_T)]\). We have already met the force \(F_t\) and the forecast \(\psi_t\), and the field \(\Phi_t\) is not one of them. This section develops the language for the interacting theory, and then draws the main consequence of the filter: a field equation at every resolution, with a noise whose moments are controlled a priori.

9.1Observables of the interacting theory

For a functional \(G\) of the terminal field, the observable \(O_t =\mathbb{E}^{\mathbb{P}}_t [G (\Phi_T)]\) is \(O_t = G_t (\Phi_t)\), with

Formally, \(G_t\) solves the backward Kolmogorov equation of the interacting dynamics,

which is the Doob transform of the Gaussian heat equation by \(\mathrm{e}^{- V_t}\) (Exercise 9.1). Conversely, any solution of (9.2) yields an observable \(G_t (\Phi_t)\); at finite \(T\) it is closed, but as \(T \to \infty\) it need not be. The force is the case \(G = - \mathrm{D} V_T\), and is special because it also drives the dynamics.

Tangent observables.

Thus the tangent observables, such as the running \(: \varphi^4\): for \(\theta = \lambda\) (together with the derivative of the counterterm \(\partial_{\lambda} \gamma_T \int : \varphi^2\):), describe the response of the theory to a change of coupling. They are the observables conjugate to the coordinates of theory space.

9.2The field equation at every resolution

By (7.14) the field at resolution \(t\) is a blurred copy of the terminal field, \(\Phi_t = J_t \Phi_T + \zeta_t\). The noise \(\zeta_t\) is not observed, but its posterior mean is, \(\mathbb{E}_t \zeta_t = \Phi_t - J_t \psi_t\). Since \(\psi_t = \Phi_t + S_t F_t\) and \(1 - J_t = KS_t\), its rescaling \(S_t^{- 1} \mathbb{E}_t \zeta_t = K \Phi_t - J_t F_t\) is the noise observable \(\eta_t\) below. This gives a field equation at every resolution.

Proof. From (7.6) and \(\dot{J}_t = K \dot{Q}_t\), \(\mathrm{d} \eta_t = K (\dot{Q}_t F_t \mathrm{d} t + \dot{Q}_t^{1 / 2} \mathrm{d} W_t) - K \dot{Q}_t F_t \mathrm{d} t - J_t \mathrm{d} F_t\). The drift cancels, \(F\) is a martingale and \(\eta_0 = 0\). For (9.5) use Proposition 7.2: \(\Phi_t \sim \nu_t \propto \mathrm{e}^{- V_t} \mu^{Q_t}\). Gaussian integration by parts for \(\mu^{Q_t}\) in the direction \(Q_t k\) gives

Take \(k = Kh\), so that \(Q_t k = J_t h\), and recall \(F_t = - \mathrm{D} V_t (\Phi_t)\).\(\Box\)

Formally letting \(T \to \infty\), \(K \to A\), and (9.4) becomes an exact identity between observables at every resolution:

with \(AQ_t = \mathrm{e}^{- A / t^2}\) for the heat profile. This is the Euclidean field equation \(A \varphi = - V' (\varphi) +\)noise of Lectures 1–2, seen at resolution \(t\). It needs no products of distributions: the nonlinearity enters only through the force observable \(F_t\), smeared at scale \(1 / t\). For the GFF (\(F = 0\)), (9.5) is Gaussian integration by parts for \(\mu^{Q_t}\). For \(\Phi^4_3\), where \(F_t\) plays the role of the renormalised cube, the limit \(t \to \infty\) of (9.7) corresponds to the renormalised DS equation of [GH21]. Lecture 6 proves the field equation at every resolution for the limit (Theorem 18.9).

The field equation for the forecast.

As \(T \to \infty\) the two forms of the field equation at resolution \(t\) are

In the first, the force is smeared by \(\mathrm{e}^{- A / t^2}\) so the nonlinearity is seen only through a smoothing operator. In the second it acts on the same variable as the linear part, with no smoothing: this is the Euler–Lagrange equation of the action \(\frac{1}{2} \langle \psi, A \psi \rangle + \Gamma_t (\psi)\), forced by the noise \(\eta_t\). For \(\Phi^4\), at finite \(T\) and to first order in \(\lambda\), the Wetterich equation (8.13) gives

and since \(\partial_c H_4 = - 6 H_2\) and \(c_T - S_t (x, x) = c_t\), the one loop moves the Wick ordering from the cutoff to the resolution:

The cube acts directly on the forecast, and pairing with \(\psi_t\) gives

where \(4 a^4 - 12 c_t a^2 \geqslant - 9 c_t^2\) and \(c_t\) is finite at fixed \(t\): heuristically, a coercive bound up to a volume constant, driven by the sub-Gaussian noise of Section 9.3. With the bare Wick ordering \(c_T\) the quadratic term would diverge; in \(d = 3\) only the mass counterterm \(\gamma_T\), of order \(\lambda^2\), still does. In the form for \(\Phi_t\) the same positivity is hidden behind \(\mathrm{e}^{- A / t^2}\).

The integration by parts formula (9.5) also transfers: a functional of \(\psi_t\) is a functional of \(\Phi_t\), and \(\mathrm{D}_x \psi_t = \text{Cov}_t S_t^{- 1}\) (Proposition 8.3), so

The direction of differentiation is now random, the conditional covariance applied to \(\mathcal{R}_t h\).

Remark 9.2. (Comparison with stochastic quantisation) In parabolic or elliptic stochastic quantisation (Lecture 3, §4) the field equation is imposed in an auxiliary time, and the field at fixed time has the right law. Here the field equation (9.7) holds at each resolution, and the auxiliary parameter is the resolution itself. Taking conditional expectations of an SPDE solution with respect to a scale filtration gives identities of the same shape, but with a different noise covariance: the martingale \(J_t \hspace{0.17em} \mathrm{d} F_t\) in \(\mathrm{d} \eta_t\) is essential [BCG26].

Proof. \(G\) is \(\mathcal{F}_t\)-measurable, so \(\mathbb{E} [G \langle \eta_t, h \rangle] =\mathbb{E} [G \langle S_t^{- 1} \zeta_t, h \rangle]\). By (7.14), \(\Phi_s = J_s \Phi_T + \zeta_s\), the bridge covariance is \(\mathbb{E} [\zeta_s \otimes \zeta_t] = Q_s KS_t\) for \(s \leqslant t\), so the covariance of \(\langle S_t^{- 1} \zeta_t, h \rangle\) and \(\langle \zeta_s, k \rangle\) is \(\langle J_s h, k \rangle\). Apply Gaussian integration by parts in \(\zeta\), given \(\Phi_T\). The last claim is the martingale property of \(\eta\).\(\Box\)

Remark 9.4. (Closed versus local) At finite \(T\) the force is a closed observable, \(F_t =\mathbb{E}_t [F_T]\). For \(\Phi^4\), \(F_T (x) = - \frac{\lambda}{3} H_3 (\Phi_T (x) ; c_T) - 2 \gamma_T H_1 (\Phi_T (x) ; c_T)\) is local. As \(T \to \infty\) one expects the force to converge as an observable. In \(d = 2\) the limit is closed, \(F_{\infty} = - \frac{\lambda}{3} : \Phi_{\infty}^3\):. In \(d = 3\) it is expected not to be. If it were, \(F_t =\mathbb{E}_t [F_{\infty}]\) with \(F_{\infty} \in L^1\), the DS equation of Theorem 9.1 would pass to the limit (\(T \to \infty\), then \(t \to \infty\)), with \(F_{\infty}\) paired only with functionals of \(\Phi_{\infty}\). Conditioning on \(\sigma (\Phi_{\infty})\) would then give a logarithmic derivative of the \(\Phi^4_3\) measure that is an integrable function of the field, \(\langle Ah, \Phi_{\infty} \rangle -\mathbb{E} [F_{\infty} (h) | \Phi_{\infty}]\). This is not expected: the regularised logarithmic derivatives do not converge in \(L^2\) [ALZ06]. The limiting force would then be an observable in the proper sense, a compatible family with no terminal representative. Locality of the limiting force is the crux of the construction [BCG26].

9.3Gaussian bounds for the noise

The noise observable \(\eta_t\) is not Gaussian: it depends on the unknown law of the interacting field. But the filter says what it is, the conditional expectation of the rescaled bridge noise \(S_t^{- 1} \zeta_t\), so its moments are bounded by those of an explicit Gaussian, uniformly in the interaction.

Work at finite cutoff, where \(\Phi_t\) has a Lebesgue density \(p_t\), and let \(0 < t < T\). By Proposition 7.2,

and since \(J_t Q_t^{- 1} = K\),

The noise is the score of the law of the field at resolution \(t\), seen through \(J_t\).

Only deterministic covariance operators, all functions of \(A\), are commuted below. The variance that appears is the regulator (8.6) of the Wetterich equation: since \(S_t = Q_T (1 - J_t)\),

Proof. Write \(\Sigma_t = Q_t (1 - J_t)\), so that \(J_t \Sigma_t^{- 1} = S_t^{- 1}\), and \(g\) for the density of \(N (0, \Sigma_t)\). Then

and \(\mathrm{D} g (z) = - \Sigma_t^{- 1} z \hspace{0.17em} g (z)\), so

apply \(J_t\) and use (9.10). By conditional Jensen,

and \(S_t^{- 1} \zeta_t\) is centred Gaussian with covariance \(S_t^{- 1} \Sigma_t S_t^{- 1} =\mathcal{R}_t\).\(\Box\)

For the heat profile (compare (8.15)), \(K = A \mathrm{e}^{A / T^2}\) and \(J_t = \mathrm{e}^{- A (t^{- 2} - T^{- 2})}\) give

(Exercise 9.5). At a finite spatial cutoff all of this holds with terminal resolution \(T = \infty\), before any continuum measure is constructed.

A new observation.

The roughness of the noise.

Since \(x / (\mathrm{e}^x - 1) \leqslant \mathrm{e}^{- x / 2}\), \(\mathcal{R}_t \leqslant t^2 \mathrm{e}^{- A / 2 t^2}\): at resolution \(t\) the noise is no larger than \(t\) times a white noise smoothed at scale \(1 / t\). Pointwise, \(\operatorname{Var} \hspace{0.17em} \eta_t (x) \leqslant \mathcal{R}_t (x, x) \lesssim t^{d + 2}\), i.e. \(\eta_t (x) = O (t^{1 + d / 2})\). This is the size at scale \(1 / t\) of a distribution of regularity \(- 1 - d / 2\) (\(- 5 / 2\) in \(d = 3\)): seen at its own resolution, the noise is one derivative rougher than white noise (\(- d / 2\)) and two rougher than the GFF (\(1 - d / 2\)), the same as the GFF noise \(A \varphi = A^{1 / 2} \xi\) of the Dyson–Schwinger equation, whatever the interaction. It is not a bound on a limit distribution: \(\eta_t\) is a martingale in \(t\), not the smoothing of a fixed distribution, and on a fixed test function (9.12) only gives \(\langle h, \eta_t \rangle = O (t)\), while the free noise stays bounded. For \(\Phi^4_3\) no \(L^2\) limit of the logarithmic derivative is expected [ALZ06]. Compared with the free noise, the bound is sharp at and above the resolution, where \(\mathcal{R}_t / (A \mathrm{e}^{- A / t^2}) = (1 - \mathrm{e}^{- A / t^2})^{- 1} \to 1\), and larger by the factor \(t^2 / A\) on the resolved modes. Uniformly in the interaction it cannot be improved: for a pinning potential \(V_T = \frac{\kappa}{2} \| \varphi \|^2\), the covariance of \(\eta_t\) tends to \(\mathcal{R}_t\) as \(\kappa \to \infty\) (Exercise 9.5).

Remark 9.6. (Pathwise bounds) (sketch) Work at finite spatial cutoff, with \(T = \infty\) or \(t \leqslant T / 2\), so that \(\mathcal{R}_t\) obeys the bounds below uniformly in the cutoff. The norm

is convex, so Proposition 9.5 dominates \(\| \eta_t \|_{\varepsilon}\) by \(\| \Xi \|_{\varepsilon}\), with \(\Xi \sim N (0, \mathcal{R}_t)\). For this Gaussian field, \(\mathcal{R}_t (x, x) \lesssim t^{d + 2}\) and

so Dudley's bound and Gaussian concentration give Gaussian tails for \(\| \Xi \|_{\varepsilon}\) at the level \(t^{1 + d / 2} \sqrt{\log (2 + t)}\), uniformly in \(L\). Since \(\eta\) is a martingale, \(\| \eta_s \|_{\varepsilon}\) is a submartingale, and Doob's inequality extends the tails to \(\sup_{s \leqslant t}\). Summing over dyadic \(t\), almost surely

with \(C\) having Gaussian tails uniformly in the interaction and in the spatial cutoff. This is a pathwise bound at each resolution, in the sense above.

Control of the force.

so

Itô's isometry, \((a + b)^2 \leqslant 2 a^2 + 2 b^2\) and (9.12) give

For the heat profile with \(T = \infty\), \(KQ_t K = A \mathrm{e}^{- A / t^2} \leqslant t^2 / \mathrm{e}\), so the right-hand side is at most \(2 (1 + \mathrm{e}^{- 1}) t^2 \|h\|_{L^2}^2\). This controls a smoothed derivative of the force, \(\Upsilon_s = \mathrm{D} F_s (\Phi_s) \dot{Q}_s^{1 / 2}\), without any estimate on the terminal force.

Consequence for \(\Phi^4_3\).

9.4Infinite volume

On \(\mathbb{T}^d_L\) the free energy \(V_0 (0)\) and the effective potential \(V_t\) are extensive: they grow like \(L^d\), and (7.9) has no limit as \(L \to \infty\). Three objects are intensive and survive formally:

Differences of free energies.

Proof. By (7.12), the left-hand side is \(\inf_u J_f (u) - J_0 (u^{\ast})\), with

Writing \(u = u^{\ast} + w\) and expanding \(\frac{1}{2} \|u^{\ast} + w\|^2\), the terms \(V_T (\Phi_T)\) and \(\frac{1}{2} \int \|u^{\ast} \|^2\) cancel against \(J_0 (u^{\ast})\), and the cross term is computed above. The first-order condition in \(w\) is obtained as in (7.11).\(\Box\)

For a local interaction \(V_T (\varphi) = \int v_T (\varphi (x)) \mathrm{d} x\),

and \(b_T (a, 0) = 0\). So every term of (9.15) is local: there is no free energy and no extensive quantity, only the field \(\Phi\) and the perturbation that \(f\) induces on it. For \(f\) supported in a bounded region one expects the optimal \(w\), and with it \(I (w)\), to decay away from the support, at the rate of the decay of correlations of the measure. Both sides of (9.15) then make sense in infinite volume, with \(\Phi\) a solution of the infinite-volume FBSDE (9.14). This suggests the following formal description of the infinite-volume measures (at a fixed ultraviolet cutoff \(T\)):

The Laplace functionals in (9.15) determine the measure, and the condition involves only intensive quantities. When the FBSDE has several solutions, as is expected in the presence of phase transitions, each solution gives a measure with its own variational principle; the perturbations \(w\) of finite energy only probe a given solution locally. For \(\Phi^4_3\) this remainder term contains the divergent counterterms, and, as for (7.12), the change of variables of [BG20] is needed before (9.15) has bounds uniform in \(T\); in two dimensions, compare the infinite-volume analysis of [BG23].

The coercive estimates must then be local: one uses weighted norms and needs bounds whose constants do not depend on the volume. This infrared problem is the analogue, for the scale dynamics, of the “coming down from infinity” estimates for the parabolic dynamics.

Exercises

Appendices: additional material

Appendix AAlgebraic structure of the Polchinski flow

A.1A semigroup of renormalisation maps

For a positive covariance \(S\) define the renormalisation map

Since \(\mu^S \ast \mu^{S'} = \mu^{S + S'}\),

So the effective potentials are an orbit of an abelian semigroup, indexed by the cone of positive covariances, acting nonlinearly on interactions and linearly on Boltzmann factors \(\mathrm{e}^{- V}\). Its infinitesimal generators are the vector fields

and (7.8) reads \(\partial_t V_t = -\mathcal{P}_{\dot{Q}_t} (V_t)\).

Proof. (a) Differentiate \(\mathcal{R}_{sB} \mathcal{R}_{rB'} =\mathcal{R}_{rB'} \mathcal{R}_{sB}\) at \(s = r = 0\), or compute directly (Exercise A.1). (b) is (A.2).\(\Box\)

Different profiles with the same endpoints are different coordinate systems on the same renormalisation group trajectory. They change the process \(\Phi\) and the observables at intermediate resolution, but not the terminal theory. The flow has no holonomy [BCG26].

A.2Perturbation theory and counterterms

Expanding \(V_t =\mathcal{R}_{Q_{T, t}} (V_T)\) in cumulants of the Gaussian increment \(\zeta \sim \mu^{Q_{T, t}}\) gives

For (7.1), Exercise 6.1 of Lecture 3 shows that Gaussian convolution simply lowers the Wick ordering: since \(Q_{T, t} (x, x) = c_T - c_t\),

Here \(\gamma_T = O (\lambda^2)\) is kept for later use. At first order the effective potential is the Wick-ordered potential at the current resolution. This is the dual statement of Proposition 6.1 of Lecture 3: \(\int H_4 (X_t ; c_t)\) is an observable.

Proof. By the binomial formula for Hermite polynomials,

Then compute the variance in (A.4) with Lemma 3.8 of Lecture 2.\(\Box\)

Figure A.1. The terms \(k = 1, 2, 3, 4\) of (A.6), with \(k\) lines \(S\) joining two Wick-ordered vertices; Wick ordering has removed the lines from a vertex to itself. As \(T \to \infty\) in \(d = 3\), only the sunset, \(k = 3\), is a non-integrable divergence for the \(: \varphi^2 :\) term; it is cancelled by \(\gamma_T\).

Now let \(T \to \infty\) at fixed \(t\), so that \(S \to C - Q_t\), which near the diagonal is as singular as \(G\). In \(d = 2\) every \(S^k\) is locally integrable, so with \(\gamma_T = 0\) every term has a limit: Wick ordering suffices. In \(d = 3\), \(G (z) \sim (4 \pi |z|)^{- 1}\):

• \(k = 1, 2\): \(S\) and \(S^2\) are locally integrable. These terms are finite and nonlocal.

• \(k = 3\) (the sunset): \(S^3 \sim |z|^{- 3}\) is not integrable. Writing \(\phi (x) \phi (y) = \frac{1}{2} (\phi (x)^2 + \phi (y)^2) - \frac{1}{2} (\phi (x) - \phi (y))^2\), up to a field-independent constant the divergent part is local:

  \(\displaystyle - \frac{\lambda^2}{3} \left( \int S^3 \right) \int : \phi^2 :_t + \text{ (finite)},\)

  with \(\int S^3 \sim \frac{1}{16 \pi^2} \log T\). It is cancelled by

  \(\displaystyle \gamma_T := \frac{\lambda^2}{3} \int_{\mathbb{T}^3_L} Q_T (z)^3 \hspace{0.17em} \mathrm{d} z = \frac{\lambda^2}{48 \pi^2} \log T + O (1),\)

  (A.7)

  since \(\gamma_T - \frac{\lambda^2}{3} \int S^3\) converges (Exercise A.2).

• \(k = 4\): a divergent constant \(- \frac{\lambda^2}{12} L^3 \int S^4 \sim T\), the vacuum energy. It cancels in \(\nu_T\) but not in the free energy.

Power counting (Exercise A.3) shows that no further field-dependent counterterms are needed in \(d = 3\): \(\Phi^4_3\) is superrenormalisable. Thus renormalisation appears as a property of the flow: one chooses the terminal data \(V_T\) so that the effective potentials \(V_t\) at fixed resolution have limits. The divergences at second order are also related to the singularity of \(\Phi^4_3\) with respect to the GFF [BG21]: the measures \(\nu_t\) at fixed \(t\) are expected to stay absolutely continuous with respect to \(\mu^{Q_t}\), while the terminal one is not.

Exercises

Appendix BStochastic localisation

Proposition 7.2 and Section 7.3 say that the Polchinski process is a stochastic localisation of \(\nu_T\). This appendix records why that decomposition is useful, and what it proves.

Why it is useful.

• It is a martingale. \(\nu_T =\mathbb{E} \mu_t^{\Phi_t}\), and the expectation of any test function under \(\mu_t^{\Phi_t}\) is a martingale. Itô calculus then splits variance and entropy exactly into the part already revealed and the part still hidden.

• The pieces are simpler. The Gaussian factor adds \(\mathcal{R}_t\) to the Hessian of the potential. If \(\nu_T\) is log-concave, \(\mu_t^{\phi}\) is \(\mathcal{R}_t\)-uniformly log-concave, so Bakry–Émery and Brascamp–Lieb apply to it even when their constants are useless for \(\nu_T\) itself.

• One quantity controls the loss. What is revealed per unit resolution is governed by \(\operatorname{Cov}_t\), whose evolution (8.9) is explicit. An inequality for \(\nu_T\) reduces to showing that the localisation does not happen too fast.

The process can also be run as an algorithm: with an approximate force it is a sampler, as in the algorithmic stochastic localisation of [EMS22], and denoising diffusions are of the same kind [Mon23]. By (9.10), the noise of the field equation is, up to \(J_t\), the score that such algorithms estimate.

B.1Log-Sobolev inequalities along the localisation

Bauerschmidt and Bodineau [BB21] proved log-Sobolev inequalities by running the Polchinski flow (see the survey [BBD24]). In their language the renormalised measure at a given scale is the law of \(\Phi_t\), and the fluctuation measure is the conditional law \(\mu_t^{\Phi_t}\) of Proposition 7.2, i.e. the stochastic localisation of \(\nu_T\) [Eld23]. (Their scale parameter runs opposite to our resolution: their initial scale is our \(T\), where the renormalised measure is \(\nu_T\) and the fluctuation measure is trivial, and their integrated covariance is our \(S_t\).) In this dictionary their argument is an entropy computation along our filtration.

Entropy as quadratic variation.

and, stopping at resolution \(t\), the chain rule

the entropy already revealed at resolution \(t\) plus the expected entropy still hidden. The increments are controlled by the conditional law: differentiating the density of \(\mu_t^{\phi}\) in \(\phi\), \(\mathrm{D} G_t (\phi) = S_t^{- 1} \operatorname{Cov}_{\mu_t^{\phi}} (\rho, y)\).

The hypothesis gives, at each resolution, the gradient bound

inserted in the entropy formula it gives the theorem [BBD24]. For \(V_T = 0\) and a linear profile \(Q_t = tQ_T / T\) one may take \(\kappa = 0\), so \(1 / \gamma = T\) and the theorem is the Gaussian log-Sobolev inequality \(\operatorname{Ent} (\rho) \leqslant 2\mathbb{E} \langle D \sqrt{\rho}, Q_T D \sqrt{\rho} \rangle\). The hypothesis is a convexity condition on the effective potentials at every scale, not on \(V_T\), corrected by the scale dependence \(\ddot{Q}_t\) of the decomposition; \(\kappa_t\) may be negative at some scales: only the weights \(\mathrm{e}^{- 2 \int_t^T \kappa_r \mathrm{d} r}\) enter \(\gamma\).

In terms of the conditional covariance.

If \(\dot{Q}_t\) is invertible, multiplying on the left by \(S_t \dot{Q}_t^{- 1} = \dot{\mathcal{R}}_t^{- 1} S_t^{- 1}\) and on the right by its transpose \(\dot{Q}_t^{- 1} S_t\) shows that it is equivalent to the explicit bound

The second derivative \(\ddot{Q}_t\) is not estimated: it is a property of the Gaussian decomposition, and it enters only through the combination \(\kappa_t \dot{Q}_t - \frac{1}{2} \ddot{Q}_t\). When \(Q_t\) and \(S_t\) are functions of one operator (Fourier multipliers, say), the right-hand side is \(S_t - S_t^2 \dot{Q}_t^{- 2} (\kappa_t \dot{Q}_t - \frac{1}{2} \ddot{Q}_t)\) mode by mode. If \(\kappa_t \dot{Q}_t > \frac{1}{2} \ddot{Q}_t\) (for the linear profile: \(\kappa_t > 0\)), every conditional law is strictly more concentrated than the unresolved Gaussian fluctuation; if not, \(\operatorname{Cov}_t\) may exceed \(S_t\), as in a double well at low resolution. A concave profile (\(\ddot{Q} \leqslant 0\)) makes the bound smaller but forces \(\kappa\) down by the same amount: for the Gaussian with \(Q_t = 1 - \mathrm{e}^{- t}\) one has \(\ddot{Q} = - \dot{Q}\), the hypothesis holds with \(\kappa = - \frac{1}{2}\), and \(\operatorname{Cov}_t = S_t\). The hypothesis is thus an upper bound on the conditional covariance, uniform in the conditioning, measured in the metric \(\dot{\mathcal{R}}_t\). Closely related is the entropic stability of Chen and Eldan [CE22] (see also [BBD24]): if, for all \(r \leqslant t\) and all \(\phi\), \(\dot{\mathcal{R}}_r \operatorname{Cov}_{\mu_r^{\phi}} \dot{\mathcal{R}}_r \leqslant \alpha_r \dot{\mathcal{R}}_r\), then

the total entropy is controlled by the entropy that is still hidden at resolution \(t\). The low-resolution part of the localisation is where the measure may be multimodal, and the condition says that it does not hide too much. With the Brascamp–Lieb bounds of Section 8.4 the conditional covariance is controlled by the Hessian of \(V_T\) when this is convex; for \(\Phi^4_3\) it is not, and the condition is a statement about the effective potentials, i.e. about the renormalisation group flow itself.

Proof. Let \(M_r =\mathbb{E}_r f (\Phi_T)\) and \(\chi_r =\operatorname{Cov}_{\mu_r^{\Phi_r}} (f, y)\). As in Section B.1, \(\mathrm{d} \langle M \rangle_r = \langle \chi_r, \dot{\mathcal{R}}_r \chi_r \rangle \mathrm{d} r\), so \(v (r) := \mathbb{E}\operatorname{Var}_{\mu_r^{\Phi_r}} (f)\) has \(v' = -\mathbb{E} \langle \chi_r, \dot{\mathcal{R}}_r \chi_r \rangle\) and \(v (0) =\operatorname{Var}_{\nu_T} (f)\). By Cauchy–Schwarz, with \(h = \dot{\mathcal{R}}_r \chi_r\), \(\langle \chi_r, h \rangle^2 \leqslant \operatorname{Var}_{\mu_r^{\Phi_r}} (f) \langle h, \operatorname{Cov}_r h \rangle \leqslant \alpha_r \operatorname{Var}_{\mu_r^{\Phi_r}} (f) \langle \chi_r, \dot{\mathcal{R}}_r \chi_r \rangle\). Hence \(v' \geqslant - \alpha_r v\), which gives the first bound. For the second, the Hessian of the potential of \(\mu_t^{\phi}\) is at least \(\mathcal{R}_t\): apply Brascamp–Lieb to \(\mu_t^{\phi}\) and take the expectation.\(\Box\)

For \(V_T = 0\) one has \(\operatorname{Cov}_r = S_r\). In one mode, or for the linear profile, where \(\dot{Q}_r S_r^{- 1}\) is a number, \(\alpha_r = \dot{Q}_r S_r^{- 1}\) and \(\exp \int_0^t \alpha_r \mathrm{d} r = Q_T S_t^{- 1}\): the first bound is an equality for linear \(f\), and the second tends to the Gaussian Poincaré inequality as \(t \to T\) (Exercise B.2).

An application: the KLS conjecture.

The connection with Bauerschmidt–Bodineau.

Proposition 8.3 is the bridge: convexity of the renormalised potentials is an upper bound on the conditional covariance, namely (B.1). The two methods then use this information differently.

• Bauerschmidt–Bodineau. The Kolmogorov equation (9.2) transports \(G_t\) by the drift \(\dot{Q}_t F_t = - \dot{Q}_t \mathrm{D} V_t\). Its Jacobian plays the role of the curvature in the Bakry–Émery computation, and the scale dependence of the metric \(\dot{Q}_t\) adds \(\frac{1}{2} \ddot{Q}_t\). This yields the gradient bound for \(G_t\) at each scale, which is integrated in the entropy formula. The output is a log-Sobolev inequality with an explicit constant, directly at the terminal scale, also far from log-concavity. The hypothesis is uniform in \(\phi\), which suits field theories where the renormalisation group controls \(V_t\) for all fields, as for sine-Gordon at \(\beta < 6 \pi\) [BB21].

• Localisation. One bounds what is lost on \([0, t]\) (the proposition above, or entropic stability [CE22]) and uses a known inequality for \(\mu_t^{\phi}\). It works directly with moments of the conditional law, and can use bounds that hold only with high probability along the path, as in the KLS problem.

Exercises

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