Constructive quantum field theory — Lecture 3 (slides)

So far. A Euclidean field is a measure on \(\mathcal{S}' (\mathcal{M})\); the GFF \(\mu_m\) is its Gaussian part; local operators are functions of \(\varphi\). Wick powers \(: \varphi^n\): are random distributions iff \(n (d - 2) < d\), and not even tight otherwise.

Today. The GFF as an equilibrium; then index everything by a resolution \(t \in [0, \infty)\).

1. Stochastic quantisation: the GFF as the stationary marginal of a Langevin dynamics, and as the one-point marginal of an elliptic equation.

2. The GFF as the terminal value of a martingale of smooth fields; observables \(=\) martingales.

3. Observables of the GFF: Wick powers in every dimension; products and OPE.

Parabolic stochastic quantisation

Lecture 3

Finite dimensions: \(\mathrm{d} X_t = - \nabla S (X_t) \mathrm{d} t + \sqrt{2} \hspace{0.17em} \mathrm{d} B_t\) has generator \(\mathcal{L}= \Delta - \nabla S \cdot \nabla\) and

the Gibbs measure \(\mathrm{e}^{- S} \mathrm{d} x / Z\) is invariant.

Apply this to the Euclidean action, with a fictitious stochastic time \(t\):

\(\xi\) space-time white noise: \(\mathbb{E} [\xi (u) \xi (v)] = \langle u, v \rangle_{L^2 (\mathbb{R} \times \mathcal{M})}\). (stationary solutions)

For \(V = 0\): the stochastic heat equation (infinite-dimensional OU process).

History: Parisi–Wu 1981 (gauge theories without gauge fixing); physics review Damgaard–Hüffel 1987–88; first mathematical construction for \(P (\varphi)_2\) on \(\mathbb{T}^2\): Jona-Lasinio–Mitter 1985. Review: Gubinelli, Encyclopedia of Mathematical Physics 2025.

Itô's formula: for \(F = g (\varphi (f_1), \ldots, \varphi (f_n))\) the generator is

Proof. Apply DS to \(k \hspace{0.17em} \partial_i g\) with \(h = f_i\) and sum over \(i\):

Static (DS) and dynamic (Langevin) characterisations are two faces of the same integration by parts.

Proof of (a). \(\langle K_{t, f}, K_{s, g} \rangle = \int_{- \infty}^{t \wedge s} \langle f, \mathrm{e}^{- (t + s - 2 r) A} g \rangle \mathrm{d} r = \langle f, \mathrm{e}^{- |t - s|A} (2 A)^{- 1} g \rangle\); an \(\mathcal{S}'\)-valued version exists by Itô–Nawata/Mitoma regularisation.

(a), on \(\mathbb{T}^d\). The modes \(\Phi_t (e_n)\) are independent stationary OU processes \(\mathrm{d} X = - \lambda_n X \mathrm{d} t + \sqrt{2} \mathrm{d} B\) with variance \(\lambda_n^{- 1}\).

(b) Mild form from the semigroup identity \(K_{t, h} = K_{s, \mathrm{e}^{- (t - s) A} h} + \mathbb{1}_{(s, t)} \mathrm{e}^{- (t - \cdot) A} h\); weak form by stochastic Fubini (Exercise 4.3).

(c) By independence,

For \(m > 0\), \(\mathrm{e}^{- tA} f \to 0\) in \(\mathcal{S} (\mathcal{M})\), so \(\Psi_0 (\mathrm{e}^{- tA} f) \to 0\) a.s. If \(\Psi_0 \sim \mu_m\) the product is exactly \(\mathrm{e}^{- \frac{1}{2} \langle f, Cf \rangle}\) (invariance); an invariant \(\nu\) must have the same characteristic functional.

• As a field on \(\mathbb{R} \times \mathcal{M}\), the stationary solution has covariance \(2 (\partial_t + A)^{- 1} (- \partial_t + A)^{- 1} = 2 (- \partial_t^2 + A^2)^{- 1}\); its equal-time slice is \(2 \int_{\mathbb{R}} (\theta^2 + A^2)^{- 1} \frac{\mathrm{d} \theta}{2 \pi} = A^{- 1}\).

• The stochastic time \(t\) has nothing to do with the Euclidean time \(\tau\).

• Interacting case: a singular SPDE.

  • \(\Phi^4_2\): Da Prato–Debussche 2003 (also Jona-Lasinio–Mitter, Albeverio–Röckner).

  • \(\Phi^4_3\): Hairer 2014; Gubinelli–Imkeller–Perkowski 2015; Catellier–Chouk 2018.

  • Global bounds: Mourrat–Weber 2017; Gubinelli–Hofmanová 2019; construction of Euclidean \(\Phi^4_3\) via the dynamics: Gubinelli–Hofmanová 2021.

Elliptic stochastic quantisation

Lecture 3

The elliptic SPDE in \(d + 2\) dimensions

is related to the \(d\)-dimensional Gibbs measure: formally

• Mechanism: a hidden supersymmetry (Parisi–Sourlas; Klein–Landau–Perez 1984).

• Rigorous nonlinear results: \(d = 0\) (Albeverio–De Vecchi–Gubinelli 2020); polynomial on \(\mathbb{T}^2\) and exponential on \(\mathbb{R}^2\) (ADG 2021); elliptic \(\Phi^4\) in 4 and 5 dimensions (Gubinelli–Hofmanová 2019).

• For \(V = 0\) it is elementary, and it explains the \(4 \pi\).

Uniqueness: \(\textsf{L}\) maps \(\mathcal{S}\) onto \(\mathcal{S}\), so \(D (\textsf{L} u) = 0\) \(\forall u\) forces \(D = 0\).

Fourier transform in \(z\) only: \(\widehat{K_{z, f}} (q) = \mathrm{e}^{- \mathrm{i} q \cdot z} (|q|^2 + A)^{- 1} f\). By Plancherel and the spectral theorem,

since

Multiply by \(4 \pi\). Stationarity and isotropy: the covariance \(4 \pi \int \mathrm{e}^{\mathrm{i} q \cdot (z - z')} \langle f, (|q|^2 + A)^{- 2} g \rangle \frac{\mathrm{d} q}{(2 \pi)^2}\) depends only on \(|z - z' |\).

Field on \(\mathbb{R}^k \times \mathcal{M}\) with covariance \((- \Delta_{\mathbb{R}^k} + A)^{- p}\); its slice \(\{0\} \times \mathcal{M}\) has covariance

The GFF arises iff \(p = 1 + \frac{k}{2}\). For \(\mathcal{M}=\mathbb{R}^d\), every codimension-two slice of the elliptic field is a GFF.

• Wilson–Kadanoff: physics is organised scale by scale. The measure-on-\(\mathcal{S}'\) picture hides this, and forces singular objects from the start.

• Observe the field at resolution \(t\): a smooth function \(\varphi_t\). Information at resolution \(t\): \(\mathcal{F}_t = \sigma (\varphi_s : s \leqslant t)\).

• A physical quantity is not a random variable but a consistent family of finite-resolution estimates, i.e. a martingale.

Setting: covariance profiles and observables

Lecture 3

Heat profile:

\(\dot{Q}_t\) lives on distances \(\lesssim 1 / t\); \(Q_t\) keeps the modes \(|k| \lesssim t\): \(t\) is a momentum scale.

Others: sharp Fourier cutoff \(\mathbb{1}_{|k| \leqslant t} (|k|^2 + m^2)^{- 1}\); finite range decompositions, \(\dot{Q}_t (x, y) = 0\) for \(|x - y| \geqslant 1 / t\).

Notation: \(c_t = Q_t (x, x)\), \(Q_{T, t} = Q_T - Q_t\). For \(d \geqslant 2\), \(c_t \to \infty\) (\(\frac{1}{2 \pi} \log t\) in \(d = 2\), \(\asymp t^{d - 2}\) in \(d \geqslant 3\)).

\(W\) cylindrical Brownian motion on \(L^2 (\mathcal{M})\),

The GFF is the terminal value of a Gaussian martingale of smooth fields.

Proof (torus). \(X_t = \sum_n \left( \int_0^t \dot{q}_n (s)^{1 / 2} \mathrm{d} B^n_s \right) e_n\): independent Gaussian martingales, variance \(q_n (t) \uparrow q_n (\infty)\); then as in Lecture 2.

Heat profile on \(\mathbb{T}^2\), \(m = 4\): \(t = 8, 32, 128\) and the lattice cutoff, same white noise.

The martingale reveals finer and finer details; the large scales are fixed at low resolution.

\(\mathbb{E}_t =\mathbb{E} [\hspace{0.17em} \cdot \hspace{0.17em} |\mathcal{F}_t]\).

• \(Y \in L^1 (\mathcal{F}_{\infty})\) gives a closed observable \(O_t =\mathbb{E}_t Y\).

• The point: observables need not be closed. \(X_t (x)\), \(d \geqslant 2\), diverges, yet is a perfectly good family of estimates.

Proof. \(\mathbb{E}_s\) is an \(L^1\) contraction.

For a standard Brownian motion \(B\) and the Hermite polynomials \(H_n (a ; c)\) of variance \(c\), the martingales

are observables without a terminal value, and not even bounded in \(L^1\):

They sum up to the exponential martingale (\(\lambda \in \mathbb{R}, \lambda \neq 0\)):

since \(\lambda B_t - \lambda^2 t / 2 \sim - \lambda^2 t / 2 \to - \infty\). The mass escapes to rarer and rarer paths.

Proof. \(O_t = N_0 + M_t +\mathbb{E}_t \int_0^{\infty} \beta_s \mathrm{d} s\) is a martingale, and \(\|O_t - N_t \|_{L^1} \leqslant \mathbb{E} \int_t^{\infty} | \beta_s | \mathrm{d} s \to 0\); uniqueness is asymptotic uniqueness.

Example. \(N_t = B_t + \mathrm{e}^{- t} \sin B_t\). Itô: \(\beta_t = - \frac{3}{2} \mathrm{e}^{- t} \sin B_t\), and \(\mathbb{E}_t \sin B_s = \mathrm{e}^{- (s - t) / 2} \sin B_t\), so

Fix \(T > 0\), \(y \in \mathbb{R}\), the heat kernel \(p_s (w) = (2 \pi s)^{- 1 / 2} \mathrm{e}^{- w^2 / 2 s}\), and

• Martingale: \(\mathbb{E}_s \Pi^y_t = \Pi^y_s\) for \(s \leqslant t\), by Chapman–Kolmogorov \(p_{t - s} \ast p_{T - t} = p_{T - s}\).

• Density of \(B_T\): \(\int g (y) \Pi^y_t \mathrm{d} y =\mathbb{E}_t g (B_T)\) for bounded \(g\), by the Markov property.

• Singular: at fixed \(y\), \(\Pi^y_t \to 0\) almost surely as \(t \uparrow T\), since \(B_T \neq y\) a.s.; but \(\mathbb{E} \Pi^y_t = p_T (y) > 0\).

• Mass one: \(y \mapsto \Pi^y_t\) is a probability density for every \(t < T\), so \((\Pi^y)_y\) is a measure-valued observable, although each \(\Pi^y_t\) tends to zero.

High resolution means \(t \uparrow T\); reparametrise by \((T - t)^{- 1}\) to get \([T^{- 1}, \infty)\). The same works for any Markov process with a transition density.

An observable is determined by its asymptotics.

• A germ of \(O\): functionals \(\mathring{O}_t\) of smooth fields with \(\|O_t - \mathring{O}_t (X_t)\|_{L^1} \to 0\). A germ determines \(O\).

• \(O (x)\) is local if it has germs

  \(\displaystyle \mathring{O}_t (X_t) (x) = \mathring{\mathcal{O}}_t (X_t (x), \nabla X_t (x), \ldots, \nabla^k X_t (x)) .\)

• Germs are not unique: this freedom is the freedom of choosing counterterms.

Example. \(: X^n_t : (x) = H_n (X_t (x) ; c_t)\) is a local observable in every \(d\) (more later), although \(: \varphi^n\): does not exist as a random distribution when \(n (d - 2) \geqslant d\).

Observables and their products

Lecture 3

Recall \(\partial_c H_n (a ; c) = - \frac{1}{2} \partial_a^2 H_n\), \(\partial_a H_n = nH_{n - 1}\).

Proof. Itô: the drift is \((\frac{1}{2} \partial_a^2 + \partial_c) H_n \hspace{0.17em} \dot{c}_t = 0\).

• Lecture 2: \(: \varphi^n\): exists iff \(n (d - 2) < d\). As observables, Wick powers exist in every \(d\), unsmeared.

• For \(n (d - 2) \geqslant d\), \(0 \leqslant f \mathrel{\not\equiv} 0\), \(: X^n_t : (f)\) is unbounded in \(L^2\) and (Paley–Zygmund) not tight: no terminal value; the observable is the whole family.

• The singularity has moved to products at coincident points.

Proof. \(O^1 O^2 - \langle O^1, O^2 \rangle\) and \(\mathbb{E}_t \langle O^1, O^2 \rangle_{\infty}\) are martingales.

Fine resolution \(u = T - \varepsilon\), coarse resolution \(s < u\), \(\kappa_{\varepsilon} = (4 \pi \varepsilon)^{- 1 / 2}\). Since \(p_{\varepsilon} (w)^2 = \kappa_{\varepsilon} p_{\varepsilon / 2} (w)\), Chapman–Kolmogorov gives the exact identity

Taylor expand in the time variable, with \(\partial_{\tau} p_{\tau} = \frac{1}{2} \partial_y^2 p_{\tau}\):

• Singular coefficients (\(\kappa_{\varepsilon} \to \infty\); \(\varepsilon \kappa_{\varepsilon}, \varepsilon^2 \kappa_{\varepsilon} \to 0\)) times observables \(\Pi^y, \partial_y^2 \Pi^y, \ldots\): an operator product expansion.

• The order of limits matters: fixed coarse resolution \(s\), then \(\varepsilon \downarrow 0\). Without conditioning, \(\|(\Pi^y_{T - \varepsilon})^2 - \kappa_{\varepsilon} \Pi^y_{T - \varepsilon} \|_{L^1} \asymp \varepsilon^{- 1 / 2}\).

\(\langle \textbf{X} (x), \textbf{X} (y) \rangle_t = Q_t (x - y)\) is deterministic. Heat profile, \(d \geqslant 2\): the condition holds iff \(x \neq y\), and then

The correct two-point function, without ever discussing the regularity of \(X_{\infty}\). For \(x = y\): no product, since \(c_t \to \infty\).

OPE. At every fixed resolution, as \(y \to x\),

i.e. \(\textbf{X} (x) \star \textbf{X} (y) = G (x - y) \mathbb{1} + : \textbf{X}^2 : (x) + o (1)\): the Wick square is the regular part of the product.

For \(x \neq y\),

• Wick's theorem at the level of observables.

• The coefficients \(G (x - y)^j\) carry all singularities; the observables on the right have regular limits as \(y \to x\).

• \(O^1, O^2\) with germs supported in \(U^1_{1 / t}\), \(U^2_{1 / t}\), \(\text{dist} (U^1, U^2) > 0\): the covariation \(\langle DO^1_t, \dot{Q}_t \hspace{0.17em} DO^2_t \rangle \mathrm{d} t\) sees \(\dot{Q}_t\) only between the supports.

• Finite range decomposition: it vanishes for large \(t\), so the product exists and is asymptotically the pointwise product. Heat profile: exponentially small.

• Observables on disjoint regions multiply freely; products at the same point need renormalisation.

Bauerschmidt–Brydges–Slade, LNM 2242 (finite range decompositions)

• Stochastic quantisation: the GFF is the stationary marginal of the parabolic Langevin dynamics and the one-point marginal of the elliptic SPDE.

• The GFF is the terminal value of a martingale \(X_t\) of smooth fields; observables are martingales, possibly without terminal value.

• Wick observables exist in every dimension; products and the OPE live at finite resolution.