Constructive quantum field theory — Lecture 2 (slides)

Last time. Positivity of the energy \(\Rightarrow\) imaginary time; vacuum expectations are moments of \(\mu (\mathrm{d} \varphi) \hspace{0.17em} \backprime\backprime = " \hspace{0.17em} Z^{- 1} \mathrm{e}^{- \int [\frac{1}{2} | \nabla \varphi |^2 + \frac{1}{2} m^2 \varphi^2 + V (\varphi)]} \mathcal{D} \varphi\); relativistic QFTs \(\leftrightarrow\) measures on \(\mathcal{S}' (\mathbb{R}^d)\) satisfying the OS axioms.

Today. The Gaussian part, rigorously.

1. Construction on \(\mathbb{T}^d\) and \(\mathbb{R}^d\); regularity.

2. Reflection positivity; the free field; Källén–Lehmann again.

3. Local operators: Wick powers.

4. Gaussian integration by parts: the Dyson–Schwinger (DS) equation.

5. DS characterises the GFF.

6. The GFF is the stationary marginal of parabolic and elliptic stochastic quantisation.

Definition and construction

Lecture 2

\(\mathcal{M}=\mathbb{T}^d_L = (\mathbb{R}/ L\mathbb{Z})^d\) or \(\mathbb{R}^d\);\(\mathcal{S} (\mathcal{M})\) smooth, resp. Schwartz, functions;\(\mathcal{S}' (\mathcal{M})\) with the cylinder (\(=\) Borel) \(\sigma\)-algebra.

\(A\), \(C\) are continuous bijections of \(\mathcal{S} (\mathcal{M})\) (Fourier multipliers \((|k|^2 + m^2)^{\pm 1}\)), and \(\|C\|_{L^2 \to L^2} \leq m^{- 2}\).

Definition. The Gaussian free field \(\mu_m\) is the centred Gaussian measure on \(\mathcal{S}' (\mathcal{M})\) with

Formally \(\mu_m (\mathrm{d} \varphi) \backprime\backprime = "Z^{- 1} \mathrm{e}^{- \frac{1}{2} \langle \varphi, A \varphi \rangle} \mathcal{D} \varphi\): the Gaussian part of the Euclidean action.

Theorem. Let \((e_n)\) be a real orthonormal basis of eigenfunctions of \(- \Delta\) on \(\mathbb{T}^d_L\), \(- \Delta e_n = \kappa_n e_n\), \(\lambda_n = m^2 + \kappa_n\), and \((g_n)\) i.i.d. \(N (0, 1)\). For \(s > \frac{d}{2} - 1\),

converges a.s. and in \(L^2 (\mathbb{P}; H^{- s})\), and \(\text{Law} (\varphi) = \mu_m\).

Proof.

\(L^2\)-bounded \(H^{- s}\)-valued martingale \(\Rightarrow\) convergence. And \(\varphi (f) = \sum_n \lambda_n^{- 1 / 2} g_n \langle e_n, f \rangle\) is Gaussian with variance \(\sum_n \lambda_n^{- 1} \langle e_n, f \rangle^2 = \langle f, Cf \rangle\).

For \(\Psi (f) = \mathrm{e}^{- \frac{1}{2} \langle f, Cf \rangle}\): continuity from \(\langle f, Cf \rangle \leq m^{- 2} \|f\|_{L^2}^2\); positive definiteness from \(\Psi (f_j - f_k) = a_j a_k \mathrm{e}^{\langle f_j, Cf_k \rangle}\) and the Schur product theorem.

Regularity (sharp, Exercise 3.3):

The Cameron–Martin space \(H^1\) (finite action) has measure zero.

Reflection positivity and the free field

Lecture 2

\(\mathcal{M}=\mathbb{R} \times N\), \(N =\mathbb{R}^{d - 1}\) or \(\mathbb{T}^{d - 1}_L\); \(\Theta (\tau, \textbf{x}) = (- \tau, \textbf{x})\); \(A = - \partial_{\tau}^2 + B\), \(B = m^2 - \Delta_N\), \(\omega = B^{1 / 2}\).

Proposition. (a) \(C (\tau, \tau') = \mathrm{e}^{- | \tau - \tau' | \omega} (2 \omega)^{- 1}\) (operator-valued kernel).
(b) If \(\text{supp} f \subset \{\tau > 0\}\):

(c) \(\mu_m\) on \(\mathbb{R}^d\) satisfies OS0–OS4.

Proof. (a) \(\int \mathrm{e}^{\mathrm{i} kt} (k^2 + w^2)^{- 1} \frac{\mathrm{d} k}{2 \pi} = \frac{\mathrm{e}^{- w |t|}}{2 w}\) and spectral calculus. (b) For \(\tau < 0 < \tau'\), \(| \tau - \tau' | = | \tau | + \tau'\): the kernel factorises. (c) OS3: \(\hat{\mu} (f_i - \Theta f_j) = a_i a_j \mathrm{e}^{\langle \Theta f_i, Cf_j \rangle}\); Schur.

• OS reconstruction of \(\mu_m\) gives the free field of mass \(m\): Fock space over \(L^2 (N)\), \(H = \mathrm{d} \Gamma (\omega)\) (Nelson's free Markov field).

• The time-zero field \(\varphi (0, \cdot)\) exists and is Gaussian with covariance \((2 \omega)^{- 1}\): the ground state measure \(\Omega_0^2 \mathcal{D} \varphi\) of the free Hamiltonian in the Schrödinger representation.

• The GFF is the continuum limit of the lattice spin system of Lecture 1 with \(V = 0\) (Exercise 3.14).

• Markov property (Nelson): rooted in the locality of \(A\).

• Markov for \(\Lambda\): inside and outside are conditionally independent given the field near \(\partial \Lambda\).

• The GFF is Markov for every open \(\Lambda\) (Nelson 1973): \(\mathbb{E} [\varphi (f) \hspace{0.17em} | \hspace{0.17em} \text{outside}] = \varphi (\Pi f)\), and \(\Pi f = Au\) is supported on \(\partial \Lambda\) because \(A = m^2 - \Delta\) is local. Given the outside: harmonic extension \(+\) independent Dirichlet GFF (next slide).

• Markov \(+\) sharp \(\Sigma_0\) \(\Rightarrow\) RP: with \(\Sigma_0\) the time-zero field, \(\mu\) \(\Theta\)-invariant, \(\Theta\) trivial on \(\Sigma_0\):

  \(\displaystyle \mathbb{E} [\overline{\Theta F} F] =\mathbb{E} \left[ |\mathbb{E}[F \hspace{0.17em} | \hspace{0.17em} \Sigma_0] |^2 \right] \geq 0, \qquad \mathbb{E}[U_t \hspace{0.17em} \cdot \hspace{0.17em} | \hspace{0.17em} \Sigma_0] = \mathrm{e}^{- tH} \text{on } L^2 (\Sigma_0) .\)

  Nelson's reconstruction (1973), before Osterwalder–Schrader.

• Relevance: preserved by local perturbations \(\mathrm{e}^{- \int_{\Lambda} V}\) \(\Rightarrow\) \(P (\varphi)_2\) as classical statistical mechanics (Guerra–Rosen–Simon). Not stable under limits; OS use RP instead: not comparable (generalised free fields are RP, not Markov; the \(P^{- 1}\) field is Markov, not RP), but closed under weak limits.

Lattice GFF, \(m^2 \approx 10^{- 3}\), \(\Lambda\) a disc. Inside \(\Lambda\),

\(h =\mathbb{E} [\varphi \hspace{0.17em} | \hspace{0.17em} \text{outside}]\) depends only on the field at the boundary; \(\psi\) is a Dirichlet GFF, independent of the outside.

• Symanzik (1966, 1969): QFT through Schwinger functions as moments of a classical Markov field — the Euclidean programme.

• Paths: \(C (x, y) = \int_0^{\infty} \mathrm{e}^{- m^2 t} p_t (x, y) \hspace{0.17em} \mathrm{d} t\) is the Green function of a killed Brownian motion. For \(\mathrm{e}^{- \int V} \mu_m\): a gas of paths and loops interacting through local times; for \(\varphi^4\), intersections are penalised (lattice: Brydges–Fröhlich–Spencer 1982; triviality for \(d > 4\)).

• Loop soups (Lawler–Werner 2004): Poisson cloud of Brownian loops of intensity \(\alpha\); on a graph \(\mathbb{E} \hspace{0.17em} \mathrm{e}^{- \langle \chi, L \rangle} = \det (1 + C \chi)^{- \alpha}\).

• Le Jan's isomorphism: for \(\alpha = \frac{1}{2}\) the occupation field is \(\frac{1}{2} \varphi^2\); in \(d = 2, 3\) the renormalised one is \(\frac{1}{2} : \varphi^2\):.

• Lupu 2016: loop clusters \(=\) sign clusters of the cable-graph GFF. Werner 2016: spatial Markov property of loop soups. Qian–Werner 2019. Survey: Werner–Powell.

Wick rotation of \(W_2^{(M)}\): \(\int \mathrm{d} \Omega_M (\textbf{p}) \hspace{0.17em} \mathrm{e}^{- \omega_M (\textbf{p}) | \tau | + \mathrm{i} \textbf{p} \cdot \textbf{x}} = G_M (\tau, \textbf{x})\). Hence for any theory satisfying the axioms

The GFF is \(\rho = \delta_{m^2}\).

• Short distances. \(G_M\) decreases in \(M\) and \(G_M / G_m \to 1\) at \(0\) (\(d \geq 2\)), so \(\liminf_{x \to 0} S_2^T (x) / G_m (x) \geq \rho ([0, \infty)) > 0\): never less singular than free. Scale invariance \(\Rightarrow\) unitarity bound \(\Delta \geq \frac{d - 2}{2}\) (Exercise 3.2).

• Long distances. \(\text{supp} \rho \subset [m_0^2, \infty)\) \(\Leftrightarrow\) decay at least like \(\mathrm{e}^{- m_0 |x|}\): mass gap \(=\) clustering.

\(\rho = c_1^2 \delta_{m_1^2} + c_2^2 \delta_{m_2^2}\), \(0 < m_1 < m_2\):\(C_{\rho} = c_1^2 (m_1^2 - \Delta)^{- 1} + c_2^2 (m_2^2 - \Delta)^{- 1}\), the law of \(c_1 \varphi_1 + c_2 \varphi_2\) (independent GFFs). RP for each term \(\Rightarrow\) OS axioms.

What is lost:

• \((\Box + m_1^2) (\Box + m_2^2) \varphi = 0\), but no Klein–Gordon equation; \((\varphi (0), \pi (0))\) are not complete Cauchy data.

• DS: \(\mathbb{E} [D_h F] =\mathbb{E} [F \varphi (C_{\rho}^{- 1} h)]\) with \(C_{\rho}^{- 1} = \dfrac{(m_1^2 - \Delta) (m_2^2 - \Delta)}{c_1^2 (m_2^2 - \Delta) + c_2^2 (m_1^2 - \Delta)}\) nonlocal; \(\varphi\) is not Markov.

The local fourth-order action \(\frac{1}{2} \langle \varphi, (m_1^2 - \Delta) (m_2^2 - \Delta) \varphi \rangle\) gives a Markov field with

so \(\rho = (\delta_{m_1^2} - \delta_{m_2^2}) / (m_2^2 - m_1^2)\) has a negative weight.

• By Euclidean Källén–Lehmann it is not reflection positive: the mass \(m_2\) is a ghost of negative norm (Pais–Uhlenbeck, Lee–Wick).

• Same for any \(1 / \hat{C} =\) polynomial of degree \(\geq 2\): \(s \hspace{0.17em} \hat{C} (s) \to 0\), while \(s \int \frac{\rho (\mathrm{d} M^2)}{s + M^2} \uparrow \rho ([0, \infty))\).

• In Minkowski space, yes: all Wightman axioms hold, including locality (W3); positive energy, unique vacuum. Two free species: trivial and redundant, not inconsistent.

• “Nonlocal” is Euclidean: no Markov property \(\Leftrightarrow\) \(\varphi\) alone is not a complete set of local degrees of freedom (no Cauchy problem, no Lagrangian in \(\varphi\) only). Cure: the pair \((\varphi_1, \varphi_2)\) is local and Markov.

• Continuous \(\rho\) (a continuum of free species) is unphysical: no dynamical law of finite order, and too many localised states. Buchholz–Wichmann nuclearity (cf. Haag–Swieca) needs \(\sum_i n_i \mathrm{e}^{- \beta m_i} < \infty\): no sensible thermodynamics.

• Interacting theories also have a continuum in \(\rho\), but produced by a local interaction: the Euclidean field equation of \(\Phi^4_2\), with \((m^2 - \Delta) \varphi + 4 \lambda : \varphi^3\): in the DS equation, is local.

Local operators: Wick products

Lecture 2

Interactions \(\int V (\varphi (x)) \mathrm{d} x\) need local functions of \(\varphi (x)\): regularise, subtract, remove the regularisation.

Lemma. \((X, Y)\) centred Gaussian, \(\mathbb{E}X^2 = a\), \(\mathbb{E}Y^2 = b\):

Proof. \(\mathbb{E} [\mathrm{e}^{sX - s^2 a / 2} \mathrm{e}^{tY - t^2 b / 2}] = \mathrm{e}^{st \hspace{0.17em} \mathbb{E} [XY]}\); compare coefficients of \(s^n t^k\).

Mollifier \(\eta_{\varepsilon}\), \(\varphi_{\varepsilon} = \eta_{\varepsilon} \ast \varphi\), \(G_{\varepsilon, \delta} = \eta_{\varepsilon} \ast \eta_{\delta} \ast G\), \(c_{\varepsilon} = G_{\varepsilon, \varepsilon} (0)\),

Theorem. If \(n (d - 2) < d\) (all \(n\) in \(d = 2\); \(n \leq 2\) in \(d = 3\)), then \(: \varphi^n : (f) = \lim_{\varepsilon \to 0} : \varphi_{\varepsilon}^n : (f)\) exists in \(L^2 (\mu_m)\), is independent of \(\eta\), and

If \(n (d - 2) \geq d\) and \(0 \leq f \mathrel{\not\equiv} 0\): \(\mathbb{E}[: \varphi_{\varepsilon}^n : (f)^2] \to \infty\).

1. Uniform bound. \(K (z) = 1 + \log_+ \frac{1}{|z|}\) (\(d = 2\)), \(1 + |z|^{2 - d}\) (\(d \geq 3\)). With \(r = \max (\varepsilon, \delta)\):

• \(|z| \geq 4 r\): \(G_{\varepsilon, \delta} (z) \leq \sup_{|w| \geq |z| / 2} G (w) \leq CK (z)\);

• \(|z| < 4 r\): \(G_{\varepsilon, \delta} (z) \leq \| \eta \|_{\infty} r^{- d} \int_{B (0, 6 r)} G \leq C \frac{1}{|B (0, 6 r) |} \int_{B (0, 6 r)} K \leq CK (r) \leq C' K (z)\).

2. Convergence. By the lemma,

by dominated convergence: \(K^n \in L^1_{\text{loc}} \Longleftrightarrow n (d - 2) < d\).

3. Divergence. Otherwise \(G^n \gtrsim |z|^{- n (d - 2)} \not\in L^1_{\text{loc}}\); Fatou.

• Renormalisation: \(\varphi_{\varepsilon} (x)^n\) has no limit; subtracting \(c_{\varepsilon} \to \infty\) (via \(H_n\)) is the simplest counterterm. A different mollifier changes \(: \varphi^n\): by lower Wick powers with finite coefficients (Exercise 3.4).

• Chaos: \(: \varphi^n : (f)\) lies in the \(n\)-th Wiener chaos; \(L^2 (\mu_m) \simeq\) Fock space over \(H^{- 1} (\mathcal{M})\). Hypercontractivity \(\|X\|_{L^p} \leq (p - 1)^{n / 2} \|X\|_{L^2}\) gives convergence in all \(L^p\) (and in \(\mathcal{C}^{- \kappa}\) for \(d = 2\)).

• Towards \(\Phi^4_2\): on \(\mathbb{T}^2\), \(\int : \varphi^4 : \in \bigcap_p L^p\) and (Nelson) \(\mathrm{e}^{- \lambda \int : \varphi^4 :} \in L^1 (\mu_m)\).

• Other local operators: \(\partial^{\alpha} \varphi\); in \(d = 2\) the Wick exponentials \(: \mathrm{e}^{\mathrm{i} \beta \varphi}\):, in \(L^2\) iff \(\beta^2 < 4 \pi\) (sine-Gordon; Exercise 3.6).

The Dyson–Schwinger equation

Lecture 2

• If \(n (d - 2) \geq d\): \(\mathbb{E}[: \varphi_{\varepsilon}^n : (f)^2] \to \infty\). Worse: there is no random distribution at all, in any topology or on any space. Albeverio–Liang 2007: \(: \varphi^2 : (f)\), \(f > 0\), in \(d = 4\) (and \(d \geq 5\)) is a Hida distribution, not a random variable.

• Proof. \(X_{\varepsilon} = : \varphi_{\varepsilon}^n : (f) \in \mathcal{H}_n\), \(\sigma_{\varepsilon} = \|X_{\varepsilon} \|_2 \to \infty\), \(\mathbb{E}X_{\varepsilon}^4 \leq 9^n \sigma_{\varepsilon}^4\). Paley–Zygmund, \(\mathbb{P} (Z > \theta \mathbb{E}Z) \geq (1 - \theta)^2 (\mathbb{E}Z)^2 /\mathbb{E}Z^2\), with \(Z = X_{\varepsilon}^2\):

  \(\displaystyle \mathbb{P} \left( |X_{\varepsilon} | > \tfrac{1}{2} \sigma_{\varepsilon} \right) \geq \tfrac{9}{16} \hspace{0.17em} 9^{- n} \quad \Rightarrow \quad \text{not tight.}\)

• Bounded chaos: the law is controlled by \(\sigma = \|X\|_2\). Upper tails \(\|X\|_p \leq (p - 1)^{n / 2} \sigma\) (hypercontractivity); \(\mathbb{P} (|X| > \frac{1}{2} \sigma) \geq c_n\) (Paley–Zygmund); \(\mathbb{P} (|X| \leq t \sigma) \leq Cnt^{1 / n}\) (Carbery–Wright). So: tight \(\Leftrightarrow\) bounded in \(L^2\); in probability \(\Leftrightarrow\) in \(L^2\).

• Renormalising by \(\sigma_{\varepsilon}^{- 1}\) typically gives Gaussian limits (fourth moment theorem; Albeverio–Zhou 1996): the \(n\)-th power is lost.

Cylinder functions \(\mathcal{F}C^{\infty}_b \ni F (\varphi) = g (\varphi (f_1), \ldots, \varphi (f_n))\), \(g \in C^{\infty}_b\):

Lemma. \((Y, X_1, \ldots, X_n)\) centred Gaussian:\(\mathbb{E} [Y \hspace{0.17em} g (X)] = \sum_i \mathbb{E} [YX_i] \hspace{0.17em} \mathbb{E} [\partial_i g (X)]\).

Proof. For \(g = \mathrm{e}^{\mathrm{i} \langle u, \cdot \rangle}\): \(\mathbb{E} [Y \mathrm{e}^{\mathrm{i} \langle u, X \rangle}] = - \mathrm{i} \partial_{\theta} \mathbb{E} [\mathrm{e}^{\mathrm{i} \theta Y + \mathrm{i} \langle u, X \rangle}] |_{\theta = 0} = \mathrm{i} \sum_i u_i \mathbb{E} [YX_i] \mathbb{E} [\mathrm{e}^{\mathrm{i} \langle u, X \rangle}]\); then Fourier inversion and approximation.

Proposition. For all \(F \in \mathcal{F}C^{\infty}_b\), \(h \in \mathcal{S} (\mathcal{M})\):

Proof. \(\mathbb{E} [\varphi (Ah) \varphi (f_i)] = \langle Ah, Cf_i \rangle = \langle h, f_i \rangle\); apply the lemma.

With \(F = \varphi (f_2) \cdots \varphi (f_n)\) one gets the hierarchy

which with \(S_0 = 1\), \(S_1 = 0\) gives Wick's theorem (Exercise 3.7).

• Minkowski: \((\Box + m^2) \varphi = 0\) and CCR.Euclidean: \((- \Delta + m^2) \varphi\) “\(=\)” \(\delta / \delta \varphi\) inside expectations; the contact terms come from the CCR (time ordering).

• For a formal Gibbs measure \(\mu_V \propto \mathrm{e}^{- \int V (\varphi)} \mu_m\):

  \(\displaystyle \mathbb{E}_{\mu_V} [D_h F] =\mathbb{E}_{\mu_V} \left[ F \hspace{0.17em} ((m^2 - \Delta) \varphi + V' (\varphi)) (h) \right],\)

  the Euclidean version of \((\Box + m^2) \varphi + V' (\varphi) = 0\).

• \(V' (\varphi)\) must be a well-defined local operator: \(\Phi^4_2\): \(V' = 4 \lambda : \varphi^3\):; field equation for \(P (\varphi)_2\): Dimock–Glimm 1974, Albeverio–Röckner 1991. For \(\Phi^4_3\) the renormalised field equation goes back to Feldman–Rączka 1977; renormalised cubic term as a random distribution: Gubinelli–Hofmanová 2021.

• \(\varphi^3 (x)\): product at one point. Point splitting: regular part of \(\varphi (y_1) \varphi (y_2) \varphi (y_3)\) as \(y_i \to x\).

• Under the GFF (Wick):

  \(\displaystyle \varphi (y_1) \varphi (y_2) \varphi (y_3) = : \varphi (y_1) \varphi (y_2) \varphi (y_3) : + \sum_{i < j} G (y_i - y_j) \hspace{0.17em} \varphi (y_k)\)

  regular part \(\to : \varphi^3 : (x)\); mollified: \(: \varphi^3 := \lim_{\varepsilon} (\varphi_{\varepsilon}^3 - 3 c_{\varepsilon} \varphi_{\varepsilon})\).

• OPE (Wilson 1969, Zimmermann 1973): products at nearby points \(=\) local operators \(\times\) singular coefficients; composite operator \(=\) regular part. Field equation \((\Box + m^2) \varphi = - 4 \lambda N [\varphi^3]\).

• The coefficients are those of the interacting theory. \(\Phi^4_2\): Gaussian subtraction suffices (Schrader 1974). \(\Phi^4_3\): an extra \(\delta_{\varepsilon} \varphi_{\varepsilon}\), \(\delta_{\varepsilon} \sim \lambda \log (1 / \varepsilon)\): \(4 \lambda \delta_{\varepsilon}\) is the sunset mass counterterm (Gubinelli–Hofmanová 2021).

• Lecture 3: OPE of stochastic observables, explicit at every scale.

Theorem. Let \(\nu\) be a probability measure on \(\mathcal{S}' (\mathcal{M})\) with \(\varphi (f) \in L^1 (\nu)\) for all \(f\) and

Then \(\nu = \mu_m\).

Proof. Fix \(f\), take \(h = Cf \in \mathcal{S} (\mathcal{M})\) and \(F_s = \mathrm{e}^{\mathrm{i} s \varphi (f)}\), so \(D_h F_s = \mathrm{i} s \langle f, Cf \rangle F_s\). With \(\chi (s) =\mathbb{E}_{\nu} F_s\):

So \(\hat{\nu} (f) = \mathrm{e}^{- \frac{1}{2} \langle f, Cf \rangle}\) for all \(f\); linearity of \(f \mapsto \varphi (f)\) and Cramér–Wold give all finite-dimensional laws.

• The integrability assumption is exactly what is needed for DS to make sense: \(\varphi (Ah)\) ranges over all \(\varphi (g)\).

• DS for characteristic functions is a first-order linear ODE.

• \(m = 0\) on the torus: no probability measure solves DS (take \(h = 1\)); work modulo constants (Exercise 3.9).

• Finite dimensions: DS says \(\nabla (\mathrm{e}^S \nu) = 0\), so uniqueness is easy for any \(S\) (Exercise 3.8).

• Interacting theories: DS is a singular nonlinear relation; uniqueness is hard and largely open. Results for the exponential model in \(d = 2\): De Vecchi–Gubinelli–Turra.

Parabolic stochastic quantisation

Lecture 2

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: \(\mathbb{E} [\xi (u) \xi (v)] = \langle u, v \rangle_{L^2 (\mathbb{R} \times \mathcal{M})}\).

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

Proposition. \(\mathbb{E}_{\mu_m} [G \hspace{0.17em} \mathcal{L}F] = -\mathbb{E}_{\mu_m} [\langle DG, DF \rangle_{L^2 (\mathcal{M})}]\) for \(F, G \in \mathcal{F}C^{\infty}_b\); in particular \(\mathbb{E}_{\mu_m} [\mathcal{L}F] = 0\).

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.

Theorem. For \(t \in \mathbb{R}\), \(f \in \mathcal{S} (\mathcal{M})\) let \(\Phi_t (f) = \sqrt{2} \hspace{0.17em} \xi (K_{t, f})\), \(K_{t, f} (r, x) = \mathbb{1}_{r < t} (\mathrm{e}^{- (t - r) A} f) (x)\).

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 3.12).

(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 2

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 \(4 \pi\).

Theorem. Let \(\textsf{L} = m^2 - \Delta_{\mathbb{R}^2 \times \mathcal{M}} = - \Delta_{\mathbb{R}^2} + A\) and \(\zeta\) white noise on \(\mathbb{R}^2 \times \mathcal{M}\).

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.

Next lecture

• Lecture 3: the GFF along a scale parameter; stochastic observables; Polchinski equation and its FBSDE; Wetterich equation; variational structure.

• Later lectures (not yet settled): probably a construction of \(\Phi^4_3\) on the torus via the FBSDE.