Constructive quantum field theory — Lecture 3 (notes)

Introduction

In Lectures 1–2 a Euclidean field was a probability measure on \(\mathcal{S}' (\mathcal{M})\), i.e. a random distribution \(\varphi\), and local operators were functions of \(\varphi\). This viewpoint hides the scale-by-scale organisation of the Wilson–Kadanoff picture [WK74], and forces us to work with singular objects from the start. Already for the GFF, the Wick power \(: \varphi^n\): is not a random distribution when \(n (d - 2) \geq d\), since its regularisations are not even tight (Proposition 3.11 of Lecture 2).

In this lecture we index everything by a resolution \(t \in [0, \infty)\). The field observed at resolution \(t\) is a smooth function \(\varphi_t\), and the information available at resolution \(t\) is the \(\sigma\)-algebra \(\mathcal{F}_t\) generated by \((\varphi_s)_{s \leq t}\). A physical quantity is then not a single random variable but a consistent family of finite-resolution estimates of it, i.e. a martingale. This is the notion of stochastic observable of [BCG26], on which this lecture is based; see also [Pol84, BK87, Sal99, BBD24] for the renormalisation group side and [BG20, BG24] for the variational side.

Structure.

What is formal.

Conventions.

Part I. Main lecture

4Setting: covariance profiles and observables

4.1Covariance profiles

Definition 4.1. A covariance profile is a family \((Q_t)_{t \in [0, \infty)}\) of bounded, symmetric, translation invariant operators on \(L^2 (\mathcal{M})\) with \(Q_0 = 0\), \(t \mapsto Q_t\) smooth, \(\dot{Q}_t := \partial_t Q_t \geq 0\), \(Q_t \to C\) as \(t \to \infty\), and such that \(\dot{Q}_t\) and \(Q_t\) have smooth kernels for every \(t < \infty\).

Our standard example is the heat profile

which smooths the Green function with the heat kernel at time \(t^{- 2}\). On \(\mathbb{R}^d\) the kernels are explicit,

so \(\dot{Q}_t\) is concentrated on distances \(\lesssim 1 / t\), and in Fourier space \(Q_t\) retains the modes \(|k| \lesssim t\): \(t\) is a momentum scale, or resolution. Other profiles are useful, even when they violate some smoothness requirement of Definition 4.1. A sharp Fourier cutoff is \(\hat{Q}_t (k) = \mathbb{1}_{|k| \leq t} (|k|^2 + m^2)^{- 1}\), which is not smooth in \(t\). In a finite range decomposition, \(\dot{Q}_t (x, y) = 0\) for \(|x - y| \geq 1 / t\) (Remark 8.4). We write

For \(d \geq 2\), \(c_t \to \infty\) as \(t \to \infty\) (Exercise 4.2).

Let \((W_t)_{t \geq 0}\) be a cylindrical Brownian motion on \(L^2 (\mathcal{M})\), \(\mathbb{E} [W_t (x) W_s (y)] = (t \wedge s) \delta (x - y)\), and

Proposition 4.2. For every \(t < \infty\), \(X_t\) is a smooth Gaussian field with law \(\mu^{Q_t}\), and \((X_t (x))_{t \geq 0}\) is a continuous martingale with \(\mathrm{d} \langle X (x), X (y) \rangle_t = \dot{Q}_t (x, y) \hspace{0.17em} \mathrm{d} t\). As \(t \to \infty\), \(X_t \to X_{\infty}\) in \(H^{- s}_{\text{loc}}\) for \(s > \frac{d}{2} - 1\), and \(\text{Law} (X_{\infty}) = \mu_m\). The increments \(X_T - X_t\) are independent of \(\mathcal{F}_t := \sigma (X_s : s \leq t)\), with law \(\mu^{Q_{T, t}}\).

So the GFF is the terminal value of a Gaussian martingale of smooth fields: the field at resolution \(t\) (Figure 4.1). The proof is a variant of Theorem 3.2 of Lecture 2 and is left as Exercise 4.1.

Figure 4.1. The field at resolution \(t\), \(X_t\), for the heat profile on \(\mathbb{T}^2\) (\(m = 4\), mean subtracted for display), at \(t = 8, 32, 128\) and at the lattice cutoff, all built from the same white noise. The martingale \(X_t\) reveals finer and finer details, while the large scales are fixed at low resolution.

4.2Stochastic observables

Fix a filtration \((\mathcal{F}_t)_{t \geq 0}\), for the moment that of \(X\), and write \(\mathbb{E}_t := \mathbb{E}[\hspace{0.17em} \cdot \hspace{0.17em} |\mathcal{F}_t]\).

Definition 4.3. ([BCG26]) An observable is a real or complex \((\mathcal{F}_t)\)-martingale \(O = (O_t)_{t \geq 0}\): \(O_s =\mathbb{E}_s [O_t]\) for \(s \leq t\). Its expectation is \(\langle O \rangle := \mathbb{E} [O_t]\), which is independent of \(t\). An observable field is a family \((O (x))_{x \in \mathcal{M}}\) of observables.

Every integrable \(\mathcal{F}_{\infty}\)-measurable random variable \(Y\) gives a closed observable \(O_t =\mathbb{E}_t [Y]\), with \(O_t \to Y\). The point of the definition is that interesting observables need not be closed: a martingale may diverge as \(t \to \infty\) (for instance \(X_t (x)\) for \(d \geq 2\)), and still carry a perfectly meaningful family of finite-resolution estimates. The next lemma is elementary but basic.

Lemma 4.4. (Asymptotic uniqueness) (a) If \(O^1, O^2\) are observables with \(\|O^1_t - O^2_t \|_{L^1} \to 0\), then \(O^1 = O^2\). (b) If \((N_t)\) is any adapted process and \(O\) an observable with \(\|N_t - O_t \|_{L^1} \to 0\), then \(O_s = \lim_{t \to \infty} \mathbb{E}_s [N_t]\) in \(L^1\).

Proof. (a) \(O^1_s - O^2_s =\mathbb{E}_s [O^1_t - O^2_t]\) for every \(t \geq s\), and conditional expectation is an \(L^1\) contraction. (b) Likewise, \(\|O_s -\mathbb{E}_s N_t \|_{L^1} \leq \|O_t - N_t \|_{L^1}\).\(\Box\)

So an observable is determined by its asymptotic behaviour. A germ for \(O\) is a family of functionals \(\ring{O}_t\) on smooth fields such that \(\|O_t - \ring{O}_t (X_t)\|_{L^1} \to 0\). The observable field \(O (x)\) is local if it admits germs of the form

Germs are not unique, and this freedom reflects the freedom of choosing counterterms. By Lemma 4.4(b), a germ determines the observable.

Exercises

5Interactions: the Polchinski equation and FBSDEs

5.1Effective potentials

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 Section 9.2 (\(\gamma_T = 0\) for \(d = 2\)). The interacting measure at cutoff \(T\) is

Definition 5.1. The effective potential at resolution \(t \in [0, T]\) is

\(V_t\) 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.

Proposition 5.2. (Marginals) Let \(\mathbb{P}\) be defined by \(\mathrm{d} \mathbb{P}/ \mathrm{d} \mathbb{P}_0 = Z_T^{- 1} \mathrm{e}^{- V_T (X_T)}\), where \(\mathbb{P}_0\) is the law of the Gaussian process \(X\). Then \(\text{Law}_{\mathbb{P}} (X_T) = \nu_T\), and for every \(t \leq T\)

Moreover, the conditional law of \(X_T\) given \(\mathcal{F}_t\) is

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].

Theorem 5.3. (Polchinski equation [Pol84]) (formal) \(V_t\) solves the backward Hamilton–Jacobi equation

with terminal condition \(V_T\).

Proof. \(u_t := \mathrm{e}^{- V_t} = \mathrm{e}^{\frac{1}{2} \Delta_{Q_{T, t}}} u_T\) is a Gaussian convolution, where \(\Delta_Q := \text{Tr} (QD^2)\). Since \(\partial_t Q_{T, t} = - \dot{Q}_t\), it solves the backward heat equation \(\partial_t u_t = - \frac{1}{2} \text{Tr} (\dot{Q}_t D^2 u_t)\). Insert \(D^2 u = (DV \otimes DV - D^2 V) u\) 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 5.4, and in Section 9.

5.2The Polchinski process

Theorem 5.4. (FBSDE for the interacting field) (formal) Under \(\mathbb{P}\) the process \(\Phi_t := X_t\), the interacting field at resolution \(t\), solves

with \(\Phi_0 = 0\) and a \(\mathbb{P}\)-cylindrical Brownian motion \(W^{\mathbb{P}}\). In particular the effective force \((F_t)_{t \leq T}\) is a \(\mathbb{P}\)-martingale: an observable field of the interacting theory.

Proof. The density process is \(M_t := \mathbb{E}_0 [\mathrm{d} \mathbb{P}/ \mathrm{d} \mathbb{P}_0 |\mathcal{F}_t] = Z_T^{- 1} \mathrm{e}^{- V_t (X_t)}\), by Proposition 5.2. By Itô's formula and (5.6), its drift vanishes and \(\mathrm{d} M_t = - M_t \langle DV_t (X_t), \dot{Q}_t^{1 / 2} \mathrm{d} W_t \rangle\). By Girsanov's theorem, \(W^{\mathbb{P}}_t = W_t + \int_0^t \dot{Q}_s^{1 / 2} DV_s (X_s) \mathrm{d} s\) is a \(\mathbb{P}\)-Brownian motion, which is (5.7). Differentiating (5.3) gives \(DV_t (\phi) = \mu_t^{\phi} (DV_T)\), which is the second identity. Alternatively, Itô's formula for \(DV_t (\Phi_t)\), together with the derivative of (5.6), shows that its drift vanishes. Here one uses that \(D^2 V_t\) is symmetric.\(\Box\)

Read as an equation for an unknown pair \((\Phi, F)\), (5.7) 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. The Polchinski equation is not needed to state (5.8), and \(\nu_T = \text{Law} (\Phi_T)\) is recovered from the solution. This is the starting point of FBSDE constructions of Euclidean fields (Figure 5.1): one solves (5.8) 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 5.1. The forward–backward structure of (5.8) 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 \(F_t =\mathbb{E}_t [- DV_T (\Phi_T)]\) 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 \(- 4 \lambda H_3 (\Phi_t ; c_t)\), and each panel is in its own units.

Remark 5.5. (Closed versus local) At finite \(T\) the force is a closed observable, \(F_t =\mathbb{E}_t [F_T]\). For \(\Phi^4\), \(F_T (x) = - 4 \lambda H_3 (\Phi_T (x) ; c_T) - 2 \gamma_T H_1 (\Phi_T (x) ; c_T)\) is local. As \(T \to \infty\) one expects the force to converge as an observable. In \(d = 2\) the limit is closed, \(F_{\infty} = - 4 \lambda : \Phi_{\infty}^3\):. 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 Section 10 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].

Figure 5.2. A zero-dimensional Polchinski process (Exercise 5.2): \(Q_t = 1 - \mathrm{e}^{- t}\), \(T = 5\), \(V_T (x) = \frac{1}{2} x^4 - \frac{3}{2} x^2\). 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 (5.7), which split between the two wells (stochastic localisation). Right: histogram of \(\Phi_T\) (\(2 \cdot 10^4\) paths) against the density of \(\nu_T\).

5.3Observables 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 5.3). Conversely, any solution of (5.10) 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 \(G = - DV_T\), 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.

5.4The force and a Burgers equation

Differentiating (5.6), the effective force \(F_t = - DV_t\), now as a function of the field, solves

an infinite-dimensional, backward, viscous Burgers equation. The Cole–Hopf transform \(F = D \log u\) linearises it, which is the statement that \(u = \mathrm{e}^{- V}\) solves a heat equation. Two remarks on (5.12):

1. it uses only the force, which is a local quantity (intensive), while \(V_t\) is extensive, of order of the volume. This matters in infinite volume (Section 7.2);

2. the Burgers flows for different covariances commute, and the curvature vanishes, when \(DF_t\) is symmetric, i.e. when \(F_t\) is a gradient [BCG26]. For a non-gradient force, as for the general Wilson–Itô diffusions of [BCG26], the flows do not commute.

Exercises

6The FBSDE in force coordinates: the Wetterich equation

In this section the forward–backward system (5.8) is the primary object. We look for the coordinates in which it is simplest. They remove the predictable part of the forward dynamics, and they integrate exactly the bilinear (tree) part of the force flow (5.12). In the gradient case the resulting equation for the force is the Wetterich equation for the effective average action [Wet93, BTW02]. Throughout,

is the covariance still unresolved at resolution \(t\), and \((\Phi, F)\) solves (5.8) with a martingale force \(F_t = F_t (\Phi_t)\), \(\mathrm{d} F_t = \Upsilon_t \hspace{0.17em} \mathrm{d} W_t\).

6.1State and costate

The FBSDE is the stochastic Pontryagin system of the control problem of Theorem 7.1. The controlled field \(\mathrm{d} X = \dot{Q}_t^{1 / 2} (u \hspace{0.17em} \mathrm{d} t + \mathrm{d} W)\) has Hamiltonian \(\langle p, \dot{Q}_t^{1 / 2} u \rangle - \frac{1}{2} \|u\|^2\), which does not depend on the state. By the stochastic maximum principle (Exercise 6.1), the adjoint process \(p\) has no drift, \(\mathrm{d} p_t = \Upsilon_t \hspace{0.17em} \mathrm{d} W_t\), with terminal value \(p_T = - DV_T (X_T)\), and the optimal control is \(u_t = \dot{Q}_t^{1 / 2} p_t\). Thus

The function \(F_t (\cdot)\) with \(F_t = F_t (\Phi_t)\) is the decoupling field of the FBSDE, and the Burgers equation (5.12) is its quasilinear PDE; \(\Upsilon_t = DF_t (\Phi_t) \dot{Q}_t^{1 / 2}\) is the martingale representation integrand of the costate. The two terms of (5.12) have distinct origins. The transport term \(- DF [\dot{Q} F]\) comes from the drift of the state, and generates the tree diagrams. The diffusion term \(- \frac{1}{2} \text{Tr} (\dot{Q} D^2 F)\) is the Itô correction, and generates the loops.

Without noise the state–costate pair follows the Hamiltonian flow of the free Hamiltonian \(h_s (p) = \frac{1}{2} \langle p, \dot{Q}_s p \rangle\): \(\dot{p} = 0\) and \(\dot{x} = \dot{Q}_s p\). From \(t\) to \(T\) this flow is the linear shear

Its trajectories are the characteristics of the transport part of (5.12) (Exercise 6.2).

6.2Removing 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 (6.1).

Proposition 6.1. (The forecast) (formal) The conditional mean of the terminal field is

i.e. \((\psi_t, F_t) = \Sigma_t (\Phi_t, F_t)\). It is a martingale, with

Proof. Integrate the forward equation from \(t\) to \(T\) and condition on \(\mathcal{F}_t\): since \(F\) is a martingale, \(\mathbb{E}_t \int_t^T \dot{Q}_s F_s \mathrm{d} s = S_t F_t\). 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 (6.3) 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.

6.3The FBSDE in the forecast coordinate

Assume that the map \(x \mapsto \psi = x + S_t F_t (x)\) is invertible, and write its inverse as

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

Differentiating the two maps \(x \leftrightarrow \psi\) gives \(\mathbb{1} + S_t DF_t (x) = (\mathbb{1} + S_t D \mathscr{F}_t (\psi))^{- 1} = H_t (\psi) S_t^{- 1}\).

Theorem 6.2. (FBSDE in force coordinates) (formal) In the coordinates \((\psi, \mathscr{F})\), the system (5.8) becomes

with \(\psi_0 = S_0 F_0 =\mathbb{E} [\Phi_T]\) and \(\psi_T = \Phi_T\). The forward equation has no drift, and its covariance is \(\mathcal{C}_t := H_t \dot{\mathscr{R}}_t H_t^{\ast}\). The decoupling field \(\mathscr{F}\) solves the pure diffusion equation

Proof. The forward equation is (6.3) with \(\mathbb{1} + S_t DF_t = H_t S_t^{- 1}\), and its covariance is

The backward equation is \(F_t =\mathbb{E}_t [F_T]\) with \(\psi_T = \Phi_T\). Since \(\mathscr{F}_t (\psi_t)\) is a martingale and \(\psi\) has no drift, the drift in Itô's formula for \(\mathscr{F}_t (\psi_t)\) is the left-hand side of (6.8).\(\Box\)

Compare (6.8) with the Burgers equation (5.12): the transport term has been absorbed exactly by the change of coordinates, and only the Itô term is left. If it is dropped, \(\mathscr{F}_t = DV_T\) does not move: all the tree diagrams are contained in the coordinate change (6.5). 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 6.2 holds for any FBSDE of the form (5.8) 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 \(DV_T\) by \(-\mathfrak{f}\) in (6.7) and (6.8).

6.4Conditional moments

For the Polchinski process the response has a direct meaning.

Proposition 6.3. (formal) If \(F_t = - DV_t\) then

In particular \(H_t > 0\) is symmetric, \(\mathbb{1} + S_t DF_t = \text{Cov}_t S_t^{- 1}\) is always invertible, and \(H_t \leq S_t\) when \(V_T\) is convex.

Proof. By (5.5), 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 \(\text{Cov} (\mu_t^{\phi}) = S_t - S_t D^2 V_t (\phi) S_t\). On the other hand \(H_t S_t^{- 1} = \mathbb{1} + S_t DF_t = \mathbb{1} - S_t D^2 V_t\). The last claim is the Brascamp–Lieb inequality.\(\Box\)

So the forecast coordinates are the first two conditional moments of the stochastic localisation \(\mu_t^{\Phi_t}\) of Proposition 5.2: \((\psi_t, \text{Cov}_t) = (\mathbb{E}_t \Phi_T, \text{Cov}_t \Phi_T)\) (Figure 6.1). 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 \(\text{Cov}_t \dot{\mathscr{R}}_t \text{Cov}_t\), which for the Gaussian reference is \(S_t \dot{\mathscr{R}}_t S_t = \dot{Q}_t\).

Figure 6.1. 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 (5.7) 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.

6.5The gradient sector: Wetterich's equation

Now let \(F_t = - DV_t\). The graph \(\{(x, - DV_t (x))\}\) is a Lagrangian submanifold of the (field, force) phase space, and the shear (6.1) is symplectic because \(S_t\) is symmetric. Its image is again Lagrangian. It is a graph over \(\psi\): the map \(x \mapsto x + S_t F_t (x)\) is \(S_t\) times the gradient of \(\frac{1}{2} \langle x, S_t^{- 1} x \rangle - V_t (x)\), which is strictly convex since its Hessian is \(S_t^{- 1} \text{Cov}_t S_t^{- 1} > 0\) (Proposition 6.3), so the map is injective. Hence the image is the graph of a gradient:

The generating function of this canonical transformation is the Moreau (infimal convolution) transform

whose optimality conditions \(x = \psi + S_t D \Gamma_t (\psi)\), \(DV_t (x) = D \Gamma_t (\psi)\) are (6.5). It is well defined by the same convexity: up to the change of variable \(j = S_t^{- 1} x\), it is a Legendre transform. We use (6.11) to fix the additive constant in \(\Gamma_t\). Figure 6.2 shows the construction in zero dimensions.

Figure 6.2. The zero-dimensional model of Figure 5.2 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 \(0\).

Theorem 6.4. (Wetterich equation) (formal) \(\Gamma_t\) solves

Equivalently, the effective average action \(\Gamma^{\text{EAA}}_t (\psi) := \Gamma_t (\psi) + \frac{1}{2} \langle \psi, Q_T^{- 1} \psi \rangle\) solves, up to a field-independent term,

It interpolates between the bare action \(\Gamma^{\text{EAA}}_T = \frac{1}{2} \langle \psi, Q_T^{- 1} \psi \rangle + V_T (\psi)\) (\(\mathscr{R}_T = \infty\)) and the 1PI effective action \(\Gamma^{\text{EAA}}_0\) of \(\nu_T\) (\(\mathscr{R}_0 = 0\)).

Proof. Integrate (6.8) in field space. With \(\mathscr{F} = D \Gamma\), the derivative of \(H = (S^{- 1} + D^2 \Gamma)^{- 1}\) in a direction \(v\) is \(- H \hspace{0.17em} D^3 \Gamma [v] \hspace{0.17em} H\), so by the total symmetry of \(D^3 \Gamma\),

Hence (6.8) reads \(D \left( \partial_t \Gamma_t - \frac{1}{2} \text{Tr} (\dot{\mathscr{R}}_t H_t) \right) = 0\), and (6.12) holds up to a function of \(t\) alone. That this function is \(- \frac{1}{2} \text{Tr} (\dot{\mathscr{R}}_t S_t)\) with the normalisation (6.11) follows from the Polchinski equation by the envelope theorem (Exercise 6.5). For (6.13), note that \(D^2 \Gamma^{\text{EAA}}_t + \mathscr{R}_t = D^2 \Gamma_t + S_t^{- 1}\). For the endpoints: \(S_T = 0\); and at \(t = 0\), \(S_0 = Q_T\), so (6.11) 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 6.7).\(\Box\)

Thus the Wetterich equation is the gradient-integrated form of the force equation (6.8), 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 \(\dot{\mathscr{R}}_t\). It is one-loop exact, with the full propagator inside the trace. In the Gaussian case \(D^2 \Gamma_t\) 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 6.3).

Figure 6.3. The regulator \(\mathscr{R}_t\) 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 \(|k| \gg t\).

6.6Variational meaning and the tree level

Constrained free energies.

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 (6.11) is the minimisation over the forecast. In the language of Theorem 7.1: \(V_t\) 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 \(\partial_t V = \frac{1}{2} \langle DV, \dot{Q} DV \rangle\). It is the saddle point of the Gaussian convolution (5.3). Its minimiser solves the classical field equation \(S_t^{- 1} (y - x) + DV_T (y) = 0\) in the shell \([t, T]\) with background \(x\), and expanding it in \(V_T\) produces exactly the tree diagrams.

Exercises

7Variational structure

7.1Finite volume: the Boué–Dupuis formula

Let \(\mathbb{H}_a\) be the set of progressively measurable drifts \(u : [0, T] \to L^2 (\mathcal{M})\) with \(\mathbb{E} \int_0^T \|u_s \|^2 \mathrm{d} s < \infty\), and set

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

Theorem 7.1. (Boué–Dupuis [BD98]; Polchinski as HJB) (formal)

The infimum is attained at the feedback control \(u^{\ast}_s = - \dot{Q}_s^{1 / 2} DV_s (X^{u^{\ast}}_s)\), and \(\text{Law} (X^{u^{\ast}}_T) = \nu_T\). The effective potential is the value function, and (5.6) is the Hamilton–Jacobi–Bellman equation of this control problem.

Proof. (Verification.) For any \(u\), by Itô's formula and (5.6),

with \(DV_t = DV_t (X^u_t)\) and a martingale \(M\). Integrating and taking expectations,

So the cost is \(\geq V_0 (0)\), with equality iff \(u = u^{\ast}\). With this control \(X^{u^{\ast}}\) solves the same SDE as \(\Phi\) in (5.7).\(\Box\)

Euler–Lagrange equation \(=\) FBSDE.

This is (5.8). The FBSDE is the Euler–Lagrange equation of the variational problem, and the Polchinski equation is its Hamilton–Jacobi equation. These are the three faces of the same object: the PDE (Polchinski), the stochastic dynamics (FBSDE), and the variational principle (Boué–Dupuis).

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 \(u_s \approx - 4 \lambda \dot{Q}_s^{1 / 2} : X_s^3\): has a divergent cost. One renormalises by the change of variables \(u = - 4 \lambda \dot{Q}^{1 / 2} : X^3 : + r\). 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 [BG21b].

7.2Infinite 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 (7.1) has no limit as \(L \to \infty\). Three objects are intensive and survive formally:

1. the force. For a local interaction \(V_T (\varphi) = \int v_T (\varphi (x)) \mathrm{d} x\), \(DV_T (\varphi) (x) = v_T' (\varphi (x))\), and (5.8) reads, on \(\mathbb{R}^d\),

   \(\displaystyle \Phi_t = \int_0^t \dot{Q}_s F_s \hspace{0.17em} \mathrm{d} s + \int_0^t \dot{Q}_s^{1 / 2} \mathrm{d} W_s, \qquad F_t (x) = -\mathbb{E}_t [v_T' (\Phi_T (x))] .\)

   (7.5)

   Here every quantity is local, and the equation makes sense without a volume cutoff. Infinite-volume limits of \(\nu_T^{(L)}\) are expected to be laws of solutions of (7.5). Uniqueness of such solutions (e.g. for small coupling, or by contraction in weighted spaces) is a uniqueness statement for the infinite-volume theory. Non-uniqueness is expected in the presence of phase transitions, e.g. for \(\Phi^4_2\) with a deep double-well potential [GJS75];

2. the Burgers flow (5.12) of the force, for the same reason;

3. differences of free energies. For a local \(f\), \(\mathcal{W}_T^{(L)} (f)\) in (7.4) is a difference of two extensive infima, and it stays bounded as \(L \to \infty\). Its variational characterisation, through the infinite-volume Euler–Lagrange equation (7.5), is the basis of the infinite-volume analysis of \(\Phi^4_2\) and \(\exp (\Phi)_2\) in [BG24].

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

Part II. Additional material

The following sections complement the main lecture. They develop the observables of the Gaussian free field and their products, the algebraic and perturbative structure of the Polchinski flow, and the Dyson–Schwinger equations at every scale.

8Observables of the Gaussian free field and their products

8.1Wick powers and exponentials

Point evaluations.

Wick powers.

Proposition 8.1. (Wick observables) For every \(d \geq 1\), \(n \geq 1\) and \(x \in \mathcal{M}\),

is a continuous martingale, with \(\mathrm{d} \hspace{0.17em} : X^n_t : (x) = n \hspace{0.17em} : X^{n - 1}_t : (x) \hspace{0.17em} \mathrm{d} X_t (x)\) and

In particular, for \(x \neq y\), \(\lim_t \mathbb{E}[: X^n_t : (x) : X^n_t : (y)] = n! \hspace{0.17em} G (x - y)^n\).

Proof. By Itô's formula, using \(\mathrm{d} \langle X (x), X (x) \rangle_t = \dot{c}_t \hspace{0.17em} \mathrm{d} t\),

The covariance is Lemma 3.8 of Lecture 2, applied to the Gaussian vector \((X_t (x), X_t (y))\).\(\Box\)

Compare with Theorem 3.9 of Lecture 2: there \(: \varphi^n\): existed as a random distribution only for \(n (d - 2) < d\), and by Proposition 3.11 not at all otherwise. As observables, Wick powers exist in every dimension and need no smearing. When \(n (d - 2) \geq d\) and \(0 \leq f \mathrel{\not\equiv} 0\), the smeared martingale \(t \mapsto : X_t^n : (f)\) is unbounded in \(L^2\) and, by the Paley–Zygmund argument of Lecture 2, not even tight as \(t \to \infty\): it has no terminal value, and the observable is the whole family, not a limit. The singularity has not disappeared. It has moved from the observables themselves to their products at coincident points, as we now see.

Exponentials.

8.2Products of observables

The pointwise product of two martingales is not a martingale. The natural definition of the product keeps the asymptotics.

Definition 8.2. Two observables \(O^1, O^2\) admit a product if \(O^1_t O^2_t \in L^1\) for all \(t\) and there is an observable \(O^1 \star O^2\) with \(\|(O^1 \star O^2)_t - O^1_t O^2_t \|_{L^1} \to 0\). By Lemma 4.4 it is unique, and \((O^1 \star O^2)_s = \lim_t \mathbb{E}_s [O^1_t O^2_t]\).

Proposition 8.3. Let \(O^1, O^2\) be continuous observables, square integrable on bounded intervals, with

Then \(O^1 \star O^2\) exists and

In particular \(\langle O^1 \star O^2 \rangle = O^1_0 O^2_0 +\mathbb{E} \langle O^1, O^2 \rangle_{\infty}\).

Proof. \(P_t := O^1_t O^2_t - \langle O^1, O^2 \rangle_t\) is a martingale on bounded intervals, and \(N_t := \mathbb{E}_t \langle O^1, O^2 \rangle_{\infty}\) is a closed martingale by (8.2). So \(P + N\) is a martingale and equals the right-hand side of (8.3). Its distance from \(O^1_t O^2_t\) in \(L^1\) is at most \(\mathbb{E} \int_t^{\infty} | \mathrm{d} \langle O^1, O^2 \rangle | \to 0\).\(\Box\)

Products of point evaluations.

which is the correct two-point function. We never had to discuss the regularity of \(X_{\infty}\). For \(x = y\) and \(d \geq 2\) the product does not exist: its expectation would be \(\lim_t \mathbb{E}X_t (x)^2 = \lim_t c_t = \infty\).

Operator product expansion.

Since both sides are martingales, this reads \(\textbf{X} (x) \star \textbf{X} (y) = G (x - y) \mathbb{1} + : \textbf{X}^2 : (x) + o (1)\), the remainder tending to zero at each fixed resolution \(t\): the Wick square is the regular part of the product, and \(G (x - y)\) is its singular coefficient. More generally (Exercise 8.2), for \(x \neq y\),

where \(: \textbf{X}^p (x) \textbf{X}^q (y) :_t := H_p (X_t (x) ; c_t) H_q (X_t (y) ; c_t)\) minus its contractions at scales \(\leq t\). This is Wick's theorem written at the level of observables. The coefficients \(G (x - y)^j\) carry all the singularities, and the observables on the right have regular limits as \(y \to x\).

Locality and factorisation.

Figure 8.1. Covariation of point observables, heat profile in \(d = 2\). Left: the kernels of \(\dot{Q}_s^{1 / 2}\) centred at \(x\) (navy) and at \(y\) (teal) at three resolutions, with their overlap (gold). The integral of the product of the two kernels is the covariation density \(\dot{Q}_s (x - y)\). Right: \(s \mapsto \dot{Q}_s (x - y)\) for three distances (\(m = 1\)). Its integral is \(G (x - y)\), and it is negligible for \(s \gg 1 / |x - y|\). As \(y \to x\) the mass moves to \(s \to \infty\), which is the divergence of the product at coincident points.

Remark 8.4. A finite range decomposition makes these statements exact: if \(\dot{Q}_t (x, y) = 0\) for \(|x - y| \geq 1 / t\), then \(X_T - X_t\) restricted to regions at distance \(\geq 1 / t\) are independent. Such decompositions of \((m^2 - \Delta)^{- 1}\) exist and are a basic tool of the rigorous renormalisation group [BBS19].

Exercises

9Algebraic structure of the Polchinski flow

9.1A 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 (5.6) reads \(\partial_t V_t = -\mathcal{P}_{\dot{Q}_t} (V_t)\).

Proposition 9.1. (Flatness) (formal) (a) \([\mathcal{P}_B, \mathcal{P}_{B'}] = 0\) for all positive \(B, B'\). (b) \(V_t\) depends on the profile only through \(Q_{T, t}\). In particular the free energy \(V_0 (0)\) and the law \(\nu_T\) depend only on \(Q_T\), not on the path \(t \mapsto Q_t\).

Proof. (a) Differentiate \(\mathcal{R}_{sB} \mathcal{R}_{rB'} =\mathcal{R}_{rB'} \mathcal{R}_{sB}\) at \(s = r = 0\), or compute directly (Exercise 9.1). (b) is (9.2).\(\Box\)

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].

9.2Perturbation 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 (5.1), Exercise 8.1 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 8.1: \(\int H_4 (X_t ; c_t)\) is an observable.

Proposition 9.2. (Second order) (formal) Write \(S = Q_{T, t}\) and \(: \phi^p :_t = H_p (\phi ; c_t)\). Then

Proof. By the binomial formula for Hermite polynomials,

Then compute the variance in (9.4) with Lemma 3.8 of Lecture 2.\(\Box\)

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: \(- 48 \lambda^2 \left( \int S^3 \right) \int : \phi^2 :_t +\)(finite), with \(\int S^3 \sim \frac{1}{16 \pi^2} \log T\). It is cancelled by

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

  (9.7)

  since \(\gamma_T - 48 \lambda^2 \int S^3\) converges (Exercise 9.2).

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

Power counting (Exercise 9.3) 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 [BG21b]: 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

10Dyson–Schwinger equations at every scale

Assume now that all the operators involved are functions of \(A\), hence commute, and let

For the heat profile, \(K = A \mathrm{e}^{A / T^2}\) and \(J_t = \mathrm{e}^{- A (t^{- 2} - T^{- 2})}\) is a heat semigroup, which tends to \(\mathrm{e}^{- A / t^2}\) as \(T \to \infty\). \(F_t\) is the force observable of Theorem 5.4.

Theorem 10.1. (DS equations at resolution \(t\)) (formal) The noise observable

is a centred \(\mathbb{P}\)-martingale, with \(\mathrm{d} \eta_t = K \dot{Q}_t^{1 / 2} \mathrm{d} W^{\mathbb{P}}_t - J_t \hspace{0.17em} \mathrm{d} F_t\). For every \(t \leq T\), every \(h\) and every functional \(G\),

At \(t = T\) this is the Dyson–Schwinger equation of \(\nu_T\),

Proof. From (5.7) and \(\dot{J}_t = K \dot{Q}_t\), \(\mathrm{d} \eta_t = K (\dot{Q}_t F_t \mathrm{d} t + \dot{Q}_t^{1 / 2} \mathrm{d} W^{\mathbb{P}}_t) - K \dot{Q}_t F_t \mathrm{d} t - J_t \mathrm{d} F_t\). The drift cancels, \(F\) is a martingale and \(\eta_0 = 0\). For (10.2) use Proposition 5.2: \(\Phi_t \sim \nu_t \propto \mathrm{e}^{- V_t} \mu^{Q_t}\). Gaussian integration by parts for \(\mu^{Q_t}\) in the direction \(Q_t k\) gives \(\mathbb{E}^{\nu_t} [D_{Q_t k} G] =\mathbb{E}^{\nu_t} [G (\langle k, \varphi \rangle + \langle Q_t k, DV_t (\varphi) \rangle)]\). Take \(k = Kh\), so that \(Q_t k = J_t h\), and recall \(F_t = - DV_t (\Phi_t)\).\(\Box\)

Formally letting \(T \to \infty\), \(K \to A\), and (10.1) 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 (\(F = 0\)), (10.2) 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 (10.4) corresponds to the renormalised DS equation of [GH21].

Remark 10.2. (Comparison with stochastic quantisation) In parabolic or elliptic stochastic quantisation (Lecture 2, §3.5–3.6) the field equation is imposed in an auxiliary time, and the field at fixed time has the right law. Here the field equation (10.4) 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].

Exercises

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