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 of Lecture 5 (\(\gamma_T = 0\) 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 6–7. 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.\(\Box\)

Remark 7.4. (A Doob \(h\)-transform) Under \(\mathbb{P}_0\), \(X\) is the GFF martingale of Lecture 3, \(X_t =\mathbb{E}_0 [X_T |\mathcal{F}_t]\), a Gaussian process with independent increments and generator \(L_t = \frac{1}{2} \text{Tr} (\dot{Q}_t \mathrm{D}^2)\). The function

is space-time harmonic for \(X\), \((\partial_t + L_t) h_t = 0\), and this is the Polchinski equation (7.8) for \(V_t = - \log h_t - \log Z_T\). Since \(h_0 (0) = 1\), the density of \(\mathbb{P}\) with respect to \(\mathbb{P}_0\) on \(\mathcal{F}_t\) is \(h_t (X_t)\): the Polchinski process is the Doob \(h\)-transform of the GFF martingale by the terminal weight \(\mathrm{e}^{- V_T}\). Its generator is

and the force is the logarithmic gradient of \(h\), \(F_t = \mathrm{D} \log h_t (\Phi_t)\). Conditioning a Brownian motion on its endpoint is the simplest example: \(h_t\) is then the Gaussian density of the remaining increment, and the drift is that of the bridge (7.15). The interacting field is a GFF martingale conditioned, in the sense of Doob, to end with the law \(\nu_T\). The same transform acts on the backward Kolmogorov equation (9.2) of Lecture 5 (Exercise 9.1 there).

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. Read backward, the same process solves the linear equation (7.17): a Gaussian blurring of \(\Phi_T \sim \nu_T\), which does not see the interaction (dashed arrow). Vertical arrows: at each resolution the force is the function \(- \mathrm{D} V_t\) of the field. 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. \(h_t = Z_T^{- 1} \mathrm{e}^{- V_t} = \mathrm{e}^{\frac{1}{2} \Delta_{Q_{T, t}}} h_T\) (Remark 7.4) 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 \(\partial_t h_t = - \frac{1}{2} \text{Tr} (\dot{Q}_t \mathrm{D}^2 h_t)\). Insert \(\mathrm{D}^2 h = (\mathrm{D} V \otimes \mathrm{D} V - \mathrm{D}^2 V) h\) and divide by \(- h\).\(\Box\)

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 \(h \propto \mathrm{e}^{- V}\), which linearises the equation, is a Cole–Hopf transform: we come back to this in Section 8.1, and in Appendix A of Lecture 5.

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 \(\mathcal{H}_{\ast}\), 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 \(\partial_x \mathcal{H}= 0\), 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 7.6), \(p_t = - \mathrm{D} V_t (X_t)\). The noise adds the Itô term to the Hamilton–Jacobi equation, which becomes the Polchinski equation (7.8).

In the notation of Theorem 7.6 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 7.9. (Time reversal compensates the drift up to the score) Forward, in the direction of increasing resolution, \(\Phi\) has drift \(b_t = \dot{Q}_t F_t\), which contains the whole interaction. Backward, by (7.17), the reversed process runs in the direction of decreasing resolution and has at \(s = T - t\) the drift \(\beta_t (x) = - \dot{Q}_t Q_t^{- 1} x\), which does not see the interaction at all. Written as a velocity in the direction of increasing resolution, the backward drift is \(b_{\ast} = - \beta\). The two velocities \(b\) and \(b_{\ast}\) do not agree, and the time-reversal formula [And82, HP86] says what separates them:

\(\dot{Q}_t\) times the score of the density \(p_t\) of the field at that scale. Reversing the direction of resolution compensates the drift exactly up to the score. For the free field \(b_t = 0\) and \(\mathrm{D} \log p_t (x) = - Q_t^{- 1} x\), so the whole backward drift is the Gaussian score; with an interaction, the score carries everything that the backward dynamics ignores. In Nelson's stochastic mechanics [Nel67] \(b_{\ast}\) is the backward velocity, and the half difference \(\frac{1}{2} (b - b_{\ast}) = \frac{1}{2} \dot{Q}_t \mathrm{D} \log p_t\) is the osmotic velocity. The same score is the noise of the field equation of Lecture 5, the observable \(\omega_t = K \Phi_t - J_t F_t\) of Section 9.3 there: by (9.14) of Lecture 5,

so the noise at resolution \(t\) is, up to the smoothing \(J_t\), the part of the drift that time reversal does not compensate.

Remark 7.10. (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. After the change of variable \(q = q_0 + t^2\), and with \(Q_T\) replaced by the full covariance \(C\), this is the mass-regulator profile of Exercise 8.8, used in Lectures 6–7.

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 7.7). 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 \(F = \mathrm{D} \log h\) linearises it, which is the statement that \(h \propto \mathrm{e}^{- V}\) 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 7.7(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 6 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 7.6. 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 \(\mathcal{H}_{\ast}\) 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 the formula for \(\mathcal{K}_0\) in Remark 7.10 (the Cameron–Martin formula) identifies \(\frac{1}{2} \langle x, Q_T^{- 1} x \rangle - V_0 (x) + V_0 (0)\) with the cumulant generating function of \(\nu_T\) at \(Q_T^{- 1} x\). By (8.12), its Legendre transform is \(\Gamma^{\text{EAA}}_0 - V_0 (0)\) (Exercise 8.6).

Near \(t = T\) the two traces \(\text{Tr} (\dot{\mathcal{R}}_t (S_t^{- 1} + \mathrm{D}^2 \Gamma_t)^{- 1})\) and \(\text{Tr} (\dot{\mathcal{R}}_t S_t) = \text{Tr} (S_t^{- 1} \dot{Q}_t)\) in (8.13) diverge separately (like \(n / (T - t)\) for \(Q_t = t\) in dimension \(n\)); only their difference, which is \(- \text{Tr} (\dot{Q}_t \mathrm{D}^2 V_t)\) by Exercise 8.4, is integrable. For the same reason the field-independent term dropped in (8.14) is not integrable up to \(T\): \(\Gamma^{\text{EAA}}_t\) is defined up to an infinite constant.\(\Box\)

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 7.10, 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. Strictly, the Gibbs principle gives only (8.12) with \(\Gamma_t\) replaced by the right-hand side \(\tilde{\Gamma}_t\) of (8.16). Since the mean is \(\psi\), \(\tilde{\Gamma}_t (\psi) + \frac{1}{2} \langle \psi, S_t^{- 1} \psi \rangle = \inf \{ \int V_T \mathrm{d} \nu + \text{Ent} (\nu | \mu^{S_t}) : \hspace{0.17em} \text{mean } \psi\}\), an infimum of a convex functional under a linear constraint, hence convex in \(\psi\); so is \(\Gamma_t + \frac{1}{2} \langle \psi, S_t^{- 1} \psi \rangle\) (Proposition 8.5). Both are recovered from the same \(V_t\) by Legendre duality, and therefore \(\tilde{\Gamma}_t = \Gamma_t\). In the language of Theorem 7.6: \(V_t\) 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

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