Constructive quantum field theory — Lecture 3 (notes)

Introduction

We start from stochastic quantisation (Section 4): the GFF is the stationary marginal of a Langevin dynamics in an auxiliary time, and the one-point marginal of an elliptic equation in two additional dimensions. This is a first way to describe a Euclidean field by an equation for the field itself.

We then change the point of view (Section 5): instead of a single random distribution, we look at the field observed at every finite resolution, and take as primary objects the stochastic observables of [BCG26]: martingales in the filtration of finite-resolution measurements.

Sections, equations and exercises keep the numbering of the whole course: references to Sections 1–3 are to the notes of Lectures 1–2.

Structure.

Conventions.

4Stochastic quantisation

Stochastic quantisation describes a Euclidean field as the marginal of a stochastic dynamics. For the GFF the dynamics are linear and everything can be computed explicitly; we give two versions, parabolic and elliptic.

4.1The GFF as stationary marginal of parabolic stochastic quantisation

Langevin dynamics.

with \(\xi\) space-time white noise. This is (parabolic) stochastic quantisation. For \(V = 0\) it is the stochastic heat equation, an infinite-dimensional Ornstein–Uhlenbeck process.

History.

Space-time white noise on \(\mathbb{R} \times \mathcal{M}\) is a centred Gaussian family \((\xi (u))_{u \in L^2 (\mathbb{R} \times \mathcal{M})}\) with \(\mathbb{E} [\xi (u) \xi (v)] = \langle u, v \rangle\), formally \(\mathbb{E} [\xi (t, x) \xi (s, y)] = \delta (t - s) \delta (x - y)\). Let \(W_t (h) = \xi (\mathbb{1}_{[0, t]} \otimes h)\), \(t \geqslant 0\), a cylindrical Brownian motion. For a (weak) solution \(\Phi\) of (4.1) with \(V = 0\) and \(F = g (\varphi (f_1), \ldots, \varphi (f_n)) \in \mathcal{F}C^{\infty}_b\), Itô's formula shows that \(F (\Phi_t) - \int_0^t \mathcal{L}F (\Phi_r) \mathrm{d} r\) is a martingale, with

Proof. We may assume \(F = g (\varphi (f_1), \ldots)\) and \(G = k (\varphi (f_1), \ldots)\) with the same \(f_i\). Apply (3.10) of Lecture 2 to the cylinder function \(k \hspace{0.17em} \partial_i g\) with \(h = f_i\) and sum over \(i\): \(\sum_i \mathbb{E} [k \hspace{0.17em} \partial_i g \hspace{0.17em} \varphi (Af_i)] = \sum_{i, j} \langle f_i, f_j \rangle \mathbb{E} [\partial_j k \hspace{0.17em} \partial_i g + k \hspace{0.17em} \partial_{ij} g]\). Insert this into \(\mathbb{E} [G\mathcal{L}F]\).\(\Box\)

So the static characterisation (DS) and the dynamic one (invariance under the Langevin flow) are two faces of the same integration by parts formula.

Proof. (a) Since \(A\) is self-adjoint and commutes with \(\mathrm{e}^{- uA}\),

In particular \(K_{t, f} \in L^2 (\mathbb{R} \times \mathcal{M})\), and multiplying by \(2\) gives the covariance. Stationarity follows. For fixed \(t\), \(f \mapsto \Phi_t (f)\) is linear and \(\mathbb{E} [\Phi_t (f)^2] = \langle f, Cf \rangle \leqslant m^{- 2} \|f\|_{L^2}^2\), so it is continuous from \(\mathcal{S} (\mathcal{M})\) to \(L^2 (\mathbb{P})\). By the regularisation theorem for random linear functionals on nuclear spaces (Itô–Nawata, Mitoma) it has an \(\mathcal{S}' (\mathcal{M})\)-valued version, which is centred Gaussian with covariance \(C\), i.e. has law \(\mu_m\). On the torus one can see this directly: the coordinates \(\Phi_t (e_n)\) are independent stationary OU processes \(\mathrm{d} X = - \lambda_n X \mathrm{d} t + \sqrt{2} \mathrm{d} B\) with variance \(\lambda_n^{- 1}\), and (3.3) of Lecture 2 applies.

(b) The mild formula is the identity \(K_{t, h} = K_{s, \mathrm{e}^{- (t - s) A} h} + \mathbb{1}_{(s, t)} (r) \mathrm{e}^{- (t - r) A} h\) together with the linearity of \(\xi\). The weak formulation follows from the mild one by a stochastic Fubini argument (Exercise 4.3).

(c) Since \(\Psi_0\) is independent of \(\xi\),

For \(m > 0\), \(\mathrm{e}^{- tA} f \to 0\) in \(\mathcal{S} (\mathcal{M})\) as \(t \to \infty\), so \(\Psi_0 (\mathrm{e}^{- tA} f) \to 0\) a.s. because \(\Psi_0 \in \mathcal{S}' (\mathcal{M})\). By dominated convergence the first factor tends to \(1\), while the second tends to \(\mathrm{e}^{- \frac{1}{2} \langle f, Cf \rangle}\). If \(\Psi_0 \sim \mu_m\), the first factor equals \(\mathrm{e}^{- \frac{1}{2} \langle f, \mathrm{e}^{- 2 tA} Cf \rangle}\) and the product is exactly \(\mathrm{e}^{- \frac{1}{2} \langle f, Cf \rangle}\), which proves invariance. If \(\nu\) is invariant, start from \(\Psi_0 \sim \nu\): then \(\text{Law} (\Psi_t) = \nu\) for all \(t\), so the characteristic functional of \(\nu\) equals that of \(\mu_m\), and \(\nu = \mu_m\).\(\Box\)

Figure 4.1. Parabolic stochastic quantisation of the GFF on a circle of length \(8\), \(m = 1\), started from \(\Phi_0 = 0\). Left: a sample of the solution. Right: each Fourier mode is an Ornstein–Uhlenbeck process, whose variance relaxes to \(\lambda_k^{- 1}\) at rate \(2 \lambda_k\) (Theorem 4.2(c)). Dots: Monte Carlo estimates with \(4000\) samples.

Remark 4.3. As a Gaussian field on \(\mathbb{R} \times \mathcal{M}\), the stationary solution has covariance \(2 (\partial_t + A)^{- 1} (- \partial_t + A)^{- 1} = 2 (- \partial_t^2 + A^2)^{- 1}\), and its equal-time slice has covariance \(2 \int_{\mathbb{R}} (\theta^2 + A^2)^{- 1} \frac{\mathrm{d} \theta}{2 \pi} = A^{- 1}\). The stochastic time \(t\) is unrelated to the Euclidean time \(\tau\). For interacting theories, (4.1) is a singular SPDE. It was solved by Da Prato and Debussche for \(\Phi^4_2\) [DPD03] (see also [JLM85, AR91]), and for \(\Phi^4_3\) using regularity structures and paracontrolled calculus [Hai14, GIP15, CC18]. Global-in-time bounds [MW17, GH19] then allow constructing the Euclidean \(\Phi^4_3\) theory as the invariant measure of the dynamics [GH21].

4.2The GFF as one-point marginal of elliptic stochastic quantisation

Parisi and Sourlas [PS79] discovered that the elliptic SPDE in \(d + 2\) dimensions

is related to the \(d\)-dimensional Gibbs measure: formally, the law of \(\Psi (z, \cdot)\), \(z \in \mathbb{R}^2\), is \(\propto \exp \left( - 4 \pi \left[ \tfrac{1}{2} \langle \varphi, A \varphi \rangle + \int V (\varphi) \right] \right) \mathcal{D} \varphi\) (dimensional reduction). The underlying mechanism is a hidden supersymmetry [PS79, KLP84]. In [KLP84] the dimensional reduction of the supersymmetric representation is proved rigorously, but not the supersymmetric representation of the solution of the SPDE; the full nonlinear statement was proved for \(d = 0\) in [ADG20] and for some two-dimensional models (polynomial interactions on \(\mathbb{T}^2\), exponential interactions on \(\mathbb{R}^2\)) in [ADG21]; elliptic \(\Phi^4\) equations in \(4\) and \(5\) dimensions are solved in [GH19]. For \(V = 0\) the statement is elementary, and it explains the factor \(4 \pi\). We normalise the noise so that the slice is exactly \(\mu_m\).

Proof. (a) \(\textsf{L}\) is a continuous bijection of \(\mathcal{S} (\mathbb{R}^2 \times \mathcal{M})\), so \(\Psi (u) := \sqrt{4 \pi} \zeta (\textsf{L}^{- 1} u)\) defines an \(\mathcal{S}'\)-valued variable with \(\Psi (\textsf{L} u) = \sqrt{4 \pi} \zeta (u)\). If \(D \in \mathcal{S}'\) satisfies \(D (\textsf{L} u) = 0\) for all \(u \in \mathcal{S}\), then \(D = 0\) since \(\textsf{L}\) is onto: the solution is unique.

(b), (c) Take the Fourier transform in \(z\) only, with measure \(\mathrm{d} q / (2 \pi)^2\). The transform of \(K_{z, f}\) is the \(L^2 (\mathcal{M})\)-valued function \(q \mapsto \mathrm{e}^{- \mathrm{i} q \cdot z} (|q|^2 + A)^{- 1} f\). By Plancherel, the spectral theorem for \(A\) and Tonelli,

using the scalar identity

In particular \(K_{z, f} \in L^2\), and its continuity in \(z\) follows by dominated convergence. Since the transform of \(\int v (z) K_{z, f} \mathrm{d} z\) is \(\hat{v} (q) (|q|^2 + A)^{- 1} f\), the Bochner integral \(\int v (z) K_{z, f} \mathrm{d} z\) equals \(\textsf{L}^{- 1} (v \otimes f)\). The Wiener integral \(\zeta\) is an isometry \(L^2 \to L^2 (\mathbb{P})\) and commutes with Bochner integrals, which gives (b). Multiplying by \(4 \pi\) gives the covariance in (c). The \(\mathcal{S}' (\mathcal{M})\)-valued version is obtained as in Theorem 4.2. Stationarity and isotropy hold because \(\mathbb{E} [\Psi (z, f) \Psi (z', g)] = 4 \pi \int \mathrm{e}^{\mathrm{i} q \cdot (z - z')} \langle f, (|q|^2 + A)^{- 2} g \rangle \frac{\mathrm{d} q}{(2 \pi)^2}\) depends only on \(|z - z' |\).\(\Box\)

Remark 4.5. (Slices of Gaussian fields) Both stochastic quantisations exhibit the GFF as a slice of a Gaussian field in higher dimension. For the field on \(\mathbb{R}^k \times \mathcal{M}\) with covariance \((- \Delta_{\mathbb{R}^k} + A)^{- p}\), the slice \(\{0\} \times \mathcal{M}\) has covariance \(c_{k, p} A^{k / 2 - p}\) with \(c_{k, p} = \Gamma (p - \frac{k}{2}) / ((4 \pi)^{k / 2} \Gamma (p))\) (Exercise 4.4). The GFF arises iff \(p = 1 + k / 2\): \((k, p) = (2, 2)\) is elliptic SQ, while \((k, p) = (1, 1)\) gives instead the time-zero field \((2 \omega)^{- 1}\) of Proposition 3.4 of Lecture 2. When \(\mathcal{M}=\mathbb{R}^d\), the elliptic field \(\Psi\) is Euclidean invariant in \(\mathbb{R}^{d + 2}\), and its restriction to every codimension-two affine subspace is a GFF.

Exercises

5Setting: covariance profiles and observables

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 \(n (d - 2) \geqslant d\), since its regularisations are not even tight (Proposition 3.11 of Lecture 2).

From now on 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 \((\varphi_s)_{s \leqslant t}\). 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, BG23] for the variational side.

In Section 4 the variable \(t\) was the auxiliary time of a stochastic dynamics; from now on \(t\) is a resolution.

5.1Covariance profiles

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 5.1. A sharp Fourier cutoff is \(\hat{Q}_t (k) = \mathbb{1}_{|k| \leqslant t} (|k|^2 + m^2)^{- 1}\), which is not smooth in \(t\). In a finite range decomposition, \(\dot{Q}_t (x, y) = 0\) for \(|x - y| \geqslant 1 / t\) (Remark 6.4). We write

For \(d \geqslant 2\), \(c_t \to \infty\) as \(t \to \infty\) (Exercise 5.2).

Let \((W_t)_{t \geqslant 0}\) 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

So the GFF is the terminal value of a Gaussian martingale of smooth fields: the field at resolution \(t\) (Figure 5.1). The proof is a variant of Theorem 3.2 of Lecture 2 and is left as Exercise 5.1.

Figure 5.1. 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.

5.2Stochastic observables

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

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 \(d \geqslant 2\)), and still carry a perfectly meaningful family of finite-resolution estimates. The next lemma is elementary but basic.

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

Observables as equivalence classes.

Random variables as observables.

Example 5.5. (Brownian observables) Let \(B\) be a standard Brownian motion and \((\mathcal{F}_t)\) its filtration. With \(H_n (a ; c)\) the Hermite polynomials of variance \(c\), the martingales

are observables without a terminal value, and they are not even bounded in \(L^1\): \(\mathbb{E} |B_t^2 - t| = t\mathbb{E} |Z^2 - 1| \to \infty\), \(Z \sim N (0, 1)\). They are the prototypes of the Wick powers of Section 6: at a fixed point, \(X_t (x)\) is a continuous Gaussian martingale with quadratic variation \(c_t\), i.e. a Brownian motion run with the clock \(c_t\). The exponential martingale

makes the loss of terminal information explicit: \(Z_t \to 0\) almost surely, while \(\mathbb{E}Z_t = 1\) for every \(t\). The almost sure limit is zero, the observable is not: the mass escapes to rarer and rarer paths, and the limit forgets all the finite-resolution expectations.

Proof. \(O_t = N_0 + M_t +\mathbb{E}_t \left[ \int_0^{\infty} \beta_s \mathrm{d} s \right]\) is a sum of martingales, and \(\|O_t - N_t \|_{L^1} \leqslant \mathbb{E} \int_t^{\infty} | \beta_s | \mathrm{d} s \to 0\). Uniqueness is Lemma 5.4(a).\(\Box\)

The correction is the expected drift still to come. For instance, for \(N_t = B_t + \mathrm{e}^{- t} \sin B_t\) Itô's formula gives \(\beta_t = - \frac{3}{2} \mathrm{e}^{- t} \sin B_t\), and since \(\mathbb{E}_t \sin B_s = \mathrm{e}^{- (s - t) / 2} \sin B_t\),

so the observable in the class of \(N\) is \(O_t = B_t\). This correction is the germ of the forward–backward equations of Lecture 4: there the forecast is obtained from the field in the same way, by adding the expected drift still to come.

Example 5.7. (A delta at a Brownian endpoint) Fix \(T > 0\), \(y \in \mathbb{R}\), let \(p_s (w) = (2 \pi s)^{- 1 / 2} \mathrm{e}^{- w^2 / 2 s}\) be the heat kernel, and

Here the resolution runs over \([0, T)\) (reparametrise by \((T - t)^{- 1}\) to get \([T^{- 1}, \infty)\)). The heat semigroup gives \(\mathbb{E}_s \Pi^y_t = \Pi^y_s\), and \(\int g (y) \Pi^y_t \mathrm{d} y =\mathbb{E}_t g (B_T)\) for bounded \(g\). At fixed \(y\), \(\Pi^y_t \to 0\) almost surely as \(t \uparrow T\), but \(\mathbb{E} \Pi^y_t = p_T (y) > 0\). So \(\delta (B_T - y)\) is not a random variable, but it is an observable: the family of its conditional densities. The same construction works for any Markov process with a transition density.

So an observable is determined by its asymptotic behaviour. A germ for \(O\) is a family of functionals \(\mathring{O}_t\) on smooth fields such that \(\|O_t - \mathring{O}_t (X_t)\|_{L^1} \to 0\). 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 5.4(b), a germ determines the observable.

Exercises

6Observables of the Gaussian free field and their products

6.1Wick powers and exponentials

Point evaluations.

Wick powers.

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 \(n (d - 2) \geqslant d\) and \(0 \leqslant f \mathrel{\not\equiv} 0\), 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.

6.2Products of observables

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

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 (6.2). So \(P + N\) is a martingale and equals the right-hand side of (6.3). Its distance from \(O^1_t O^2_t\) in \(L^1\) is at most \(\mathbb{E} \int_t^{\infty} | \mathrm{d} \langle O^1, O^2 \rangle | \to 0\).\(\Box\)

Brownian deltas.

Since \(B_t \sim N (0, t)\), \(\mathbb{E}p_{(T - t) / 2} (B_t - \bar{y}) = p_{(T + t) / 2} (\bar{y})\), and, the product being nonnegative, its expectation is its \(L^1\) norm: for \(y \neq z\)

so \(\Pi^y_t \Pi^z_t\) is asymptotically equivalent to the zero observable, and by Definition 6.2 the product exists and \(\Pi^y \star \Pi^z = 0\). At \(y = z\) the exponential suppression is lost, and \(\mathbb{E} (\Pi^y_t)^2 = \kappa_{T - t} p_{(T + t) / 2} (y) \to \infty\): the square does not exist, since a martingale \(O\) with \(\|O_t - (\Pi^y_t)^2 \|_{L^1} \to 0\) would force \(\mathbb{E} (\Pi^y_t)^2 \to \mathbb{E}O_0 < \infty\). The square can, however, be resolved at a fixed coarser resolution. Let \(u = T - \varepsilon\) be a fine resolution and \(s < u\) a coarser one. Since \(p_{\varepsilon} (w)^2 = \kappa_{\varepsilon} p_{\varepsilon / 2} (w)\),

by Taylor expansion in the time variable and \(\partial_{\tau} p_{\tau} = \frac{1}{2} \partial_y^2 p_{\tau}\). This is an operator product expansion: singular coefficients (\(\kappa_{\varepsilon} \to \infty\), while \(\varepsilon \kappa_{\varepsilon}, \varepsilon^2 \kappa_{\varepsilon} \to 0\)) times the observables \(\Pi^y\), \(\partial_y^2 \Pi^y\), \(\ldots\). The order of the limits matters: the remainder is small only after conditioning at a fixed resolution \(s\). Without conditioning, \((\Pi^y_{T - \varepsilon})^2 - \kappa_{\varepsilon} \Pi^y_{T - \varepsilon} = \kappa_{\varepsilon} [p_{\varepsilon / 2} - p_{\varepsilon}] (B_{T - \varepsilon} - y)\) has \(L^1\) norm of order \(\varepsilon^{- 1 / 2}\).

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 \(d \geqslant 2\) 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 6.2), 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 \(\leqslant t\). 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 \(y \to x\).

Higher orders.

Subtracting \(G\) leaves the bilocal Wick product \(X_t (x_-) X_t (x_+) - Q_t (x_-, x_+)\), which is smooth in \(\rho\) at fixed \(t\), and the symmetry \(\rho \mapsto - \rho\) kills the odd orders. Each coefficient is a finite sum of Wick bilinears, hence an observable. The constant in the remainder depends on \(t\); the limit \(t \to \infty\) is a separate question. A systematic study of the operator product expansion in this framework is due to P. Paulovics (in preparation).

Locality and factorisation.

Figure 6.1. 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 6.4. A finite range decomposition makes these statements exact: if \(\dot{Q}_t (x, y) = 0\) for \(|x - y| \geqslant 1 / t\), then \(X_T - X_t\) restricted to regions at distance \(\geqslant 1 / t\) are independent. Such decompositions of \((m^2 - \Delta)^{- 1}\) exist and are a basic tool of the rigorous renormalisation group [BBS19].

Outlook.

Exercises

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