Constructive quantum field theory — Lecture 6 (notes)

Introduction

Lecture 5 proved, at every lattice cutoff \(N\) and every upper scale \(T\),

for the backward energy \(E_T\) and the large-field action \(J_T\) of (14.6). Three things remain.

1. Show that the boundary term vanishes: \(TZ_T \to 0\) as \(T \to \infty\) at fixed \(N\) (Section 17).

2. Turn the uniform energy bounds into compactness of the processes and of the measures (Section 18).

3. Identify the limit: the deterministic coefficients, the noise, and the equations (Section 19).

The structure of (iii) is instructive: convergence of the fields alone does not suffice to pass to the limit in an FBSDE, because the backward martingale might converge to something with extra noise. Here the statistical meaning of the backward variable, a posterior mean with a positive covariance, saves us.

17The boundary at infinite resolution

17.1The boundary lemma

At a fixed lattice, the terminal resolution is \(q = \infty\), and the field at resolution \(q\), \(\Phi_q = C_q P_q\), converges to the terminal field \(\Phi_{\infty}\). The approximate action \(\bar{\Gamma}_q\) was built to converge to the bare action: \(\mu_q \to \mu_N\), \(\Theta_q \to 0\), and the nonlinear corrections were integrated from \(q = \infty\), so that they vanish there. This is the boundary condition of the backward equation.

Lemma 17.1. (Boundary lemma) At fixed \(N\), for every \(p < \infty\), as \(q \to \infty\),

The constants depend on \(N\); this is not a uniform estimate.

Proof. All norms on \(H_N\) are equivalent, and we do not track the dependence on \(N\).

The field at resolution \(q\). By (11.5), \(\Phi_q = C_q P_q = (q - q_0) C_q \Phi_{\infty} + C_q B_{q - q_0}\), and \((q - q_0) C_q - I = - C_q K_{q_0} = O (q^{- 1})\), \(\|C_q B_{q - q_0} \|_{L^p} = O (q^{- 1 / 2})\). So \(\| \Phi_q - \Phi_{\infty} \|_{L^p} \le C_{N, p} q^{- 1 / 2}\).

The approximate forecast. Global convexity gives \(\|Y_q \| \le C \|P_q \| / q\), bounded in every \(L^p\) uniformly in \(q\). The coordinate equation \(\Phi_q = Y_q + C_q D \bar{\Gamma}_q (Y_q)\) and the bounds \(| \mu_q | \le C_N\), \(\| \Theta_q \|_{\text{row}} = O_N (q^{- 1 / 2})\) and \(\|D \tilde{h}_q (y)\|_{\infty} \le C \textsf{S}_4 (h^{[2]}_q) \|y\|_{\infty}^3 \le C \gamma^2 q^{- 1 / 2} \|y\|_{\infty}^3\) (Taylor from the fourth order, since the low jets vanish) show that \(D \bar{\Gamma}_q (Y_q)\) is bounded in \(L^p\), so \(\|Y_q - \Phi_q \|_{L^p} = O (q^{- 1})\). Moreover \(| \mu_q - \mu_N | = O_N (q^{- 1})\), and therefore

Thus the explicit force at the approximate forecast converges to the bare force at the terminal field.

The error. Subtract the approximate coordinate identity \(P_q = K_q Y_q + D \bar{\Gamma}_q (Y_q)\) from the exact posterior identity (11.10), \(P_q = K_q \psi_q +\mathbb{E}_q [DV_N (\Phi_{\infty})]\):

Let \(G = DV_N (\Phi_{\infty})\) and \(G^0_q = DV_N (\Phi_q)\), which is \(\mathcal{F}_q\)-measurable. Then \(\|\mathbb{E}_q G - G\|_{L^p} \le \|\mathbb{E}_q (G - G^0_q)\|_{L^p} + \|G^0_q - G\|_{L^p} \le 2 \|G - G^0_q \|_{L^p} = O (q^{- 1 / 2})\), by the cubic Lipschitz bound for \(DV_N\) and the first step. With (17.2) and \(\|qC_q \| \le 1\) this proves the claim.\(\Box\)

The proof uses no estimate uniform in \(N\), and no concentration of the posterior: only that the observed estimator \(\Phi_q = C_q P_q\), which is \(\mathcal{F}_q\)-measurable, converges to the hidden field. In Lecture 3 language, the forecast and the field at resolution \(q\) have the same limit at infinite resolution, and the force \(F_q = K_q (\psi_q - \Phi_q)\) converges to the bare force \(- DV_N (\Phi_{\infty})\) (Exercise 17.1; recall \(F_q =\mathbb{E}_q [- DV_N (\Phi_{\infty})]\) by (11.9), so this is the conditional-expectation step of the proof).

17.2UV stability

Now let \(T \to \infty\) at fixed \(N\) in (1): the paths and the law \(\nu_N\) do not depend on \(T\), \(E_T\) and \(J_T\) increase to \(E_{\infty}\) and \(J_{\infty}\), and \(\delta_{N, T, p} \to 0\) by Lemma 17.1. Only then take the supremum over \(N\).

Theorem 17.2. (UV stability [FBSDE26]) For \(q_0\) large enough, depending only on \(\gamma\),

and \(\sup_N \mathbb{E} \exp \{c (E^N_{\infty} + J^N_{\infty})^{1 / 5} \} < \infty\) for small \(c > 0\).

Proof. Monotone convergence in (1), then the representation formula (16.2) on \([q, \infty)\), whose terminal term vanishes by Lemma 17.1. The exponential moment follows from the polynomial growth by Markov's inequality with \(p \asymp R^{1 / 5}\).\(\Box\)

The order of limits matters. The constant \(C_{N, p}\) in Lemma 17.1 diverges with \(N\), and we never take a supremum over \(N\) of the remainder \(C_{N, p} / T\). The infinite-resolution boundary is used only qualitatively, at each lattice separately.

Exercises

18Compactness

18.1One Hilbert space

To compare lattices, partition \(\mathbb{T}^3\) into cubic cells \(\square_x\) of side \(N^{- 1}\) centred at \(x \in \Lambda_N\), and let

\(\iota_N : H_N \to H = L^2 (\mathbb{T}^3)\) is an isometry, \(j_N = \iota_N^{\ast}\), and \(\textsf{P}_N \to I\) strongly. Fields are embedded by \(\iota_N\), operators by \(\iota_N \cdot j_N\). The embedded innovation has covariance \(\textsf{P}_N\); in the limit it becomes a cylindrical Brownian motion on \(H\), obtained as the limit of the lattice innovations.

18.2Spatial suprema by a maximum principle

The energy bounds control \(L^4\) and \(L^6\) norms of large fields. Spatial suprema at finite resolution come for free from the elliptic structure of (14.4).

Lemma 18.1. (Supremum after closure) Let \(\varpi = \frac{3}{8}\), \(\mathcal{Y}_T = \sup_{q_0 \le q \le T} q^{- \varpi} \|Y_q \|_{\infty}\) and \(\mathcal{X}_T = \sup_{q_0 \le q \le T} q^{- \varpi} \|X_q \|_{\infty}\). For \(q_0\) large,

Idea. Remove the backward forcing by \(\mathcal{H}_q = C_q F^Z_q\), with \(\|\mathcal{H}_q \|_{\infty} \le Cq^{- 1 / 4} \|F^Z_q \|_2 \le C \gamma^{1 / 2} E_T^{1 / 2}\) (by Cauchy–Schwarz in \(q\)). Then \(U = V -\mathcal{H}\) solves, at each frozen \(q\), \(K_q U + \frac{\gamma}{3} (U + d)^3 = f\) with \(d =\mathcal{H}+ X - \varrho (Y)\) and \(f = \int_{q_0}^q (Y_s - X_s) \mathrm{d} s\). Constant functions of height \(\max \{\|d\|_{\infty}, q^{- 1} \|f\|_{\infty} \}\) are super- and subsolutions, so the lattice maximum principle bounds \(\|U\|_{\infty}\). This gives \((1 - 2 \varepsilon_q) y_q \le 2 (\|\mathcal{H}_q \|_{\infty} + x_q) + q^{- 1} \int_{q_0}^q (y_s + x_s) \mathrm{d} s\) for \(y_q = \|Y_q \|_{\infty}\), \(x_q = \|X_q \|_{\infty}\), where \(\varepsilon_q = C (g_q + g_q^2)\) bounds \(\|D \varrho_q \|_{\infty \to \infty}\) (Lecture 5). The weight \(q^{- \varpi}\) absorbs the memory since \(q^{- 1 - \varpi} \int_{q_0}^q s^{\varpi} \mathrm{d} s \le (1 + \varpi)^{- 1} < 1\). The Gaussian supremum is a chaining argument: \(\text{Var} X_q (x) \le Cq^{1 / 2}\) and the increments in space and scale are Hölder.\(\Box\)

In particular finite-resolution spatial suprema of \(Y\) have all moments uniformly in \(N\). At the terminal resolution there is no such bound: \(\Phi_{\infty}\) is a distribution.

18.3Regularity in space and scale

On a finite scale interval \(I = [q_0, Q]\) write (14.4) as the linear elliptic equation

The energies control \(\mathbb{E} \int_I \|Y_q \|_6^6 \mathrm{d} q\) (since \(|Y|^6 \le r_q^6 + |t_q |^6\)) and \(\mathbb{E} \int_I \|Z_q \|_2^2 \mathrm{d} q\), hence \(\mathbb{E} \int_I \|f_q \|_2^2 \le C_I\), and testing with \(V\),

So the non-Gaussian part \(V\) of the approximate forecast is in \(H^1\), and the approximate forecast \(Y = V + X - \varrho (Y)\) inherits the regularity of the free field.

Proposition 18.2. (Forward compactness) For \(0 < \sigma, \theta < \frac{1}{2}\),

The embedded approximate forecasts are tight in every space–time \(L^p\), and polynomials of them converge along strongly convergent subsequences. The forward equation gives the Hölder bound \(\sup_N \mathbb{E} \| \iota_N Y^N \|^8_{C^{1 / 4} (I ; H)} < \infty\), and averaging in time against the \(H^{\sigma}\) bound gives uniform Fourier tails \(\lim_m \sup_N \mathbb{E} \sup_{q \in I} \|(I - \textsf{F}_m) \iota_N Y^N_q \|_2^2 = 0\) (\(\textsf{F}_m\) the projection on frequencies \(|k| \le m\)); hence they are also tight in \(C (I ; H)\).

The spatial bound combines the \(H^1\) bound on \(V\), the \(H^{\sigma}\) regularity of \(X\) (\(\sigma < \frac{1}{2}\)), and the contraction property of \(\varrho\); the scale regularity uses the forward equation, whose drift \(b (Y) + H (Y) Z\) has bounded integrated square by Proposition 13.2 and the energies, and whose martingale part has bounded bracket density. The supremum bound of Lemma 18.1 upgrades \(L^2\) compactness to every \(L^p\).

18.4The terminal field

The approximate forecast converges at infinite resolution in negative Sobolev spaces, uniformly in \(N\) (to the same limit as the forecast \(\psi\)). The martingale part of \(Y\) has bracket \(\int H^2\), and the positive inequality \(H_q^2 \le \frac{C}{q} C_q\) gives

where \(q + c |k|^2\) bounds the lattice symbol from below (aliases of the cell embedding only increase the frequency). The drifts are integrable at infinity in \(H\): the bounded-field drift is \(O (q^{- 5 / 2})\), the large-field drift is controlled by \(\int q^{7 / 2} \|b^{\text{l}} \|^2 \le C \gamma^3 J_{\infty}\) and Cauchy–Schwarz, and \(\int \|H_q Z_q \| \mathrm{d} q \le C \int q^{- 1} \|Z_q \| \le C \gamma^{1 / 2} E_{\infty}^{1 / 2}\). More precisely, the squared drift tails beyond \(Q\) are \(O (Q^{- 3})\) (bounded-field), \(O (Q^{- 5 / 2}) \gamma^3 J_{\infty}\) (large-field) and \(O (Q^{- 3 / 2}) \gamma E_{\infty}\) (the \(HZ\) term), all at most \(Q^{- (s - \frac{1}{2})}\) for \(s < \frac{3}{2}\). By Doob's inequality, for \(\frac{1}{2} < s < \frac{3}{2}\),

Letting \(t \to \infty\) at fixed \(N\) (Lemma 17.1) this holds with \(Y_t\) replaced by \(\Phi_{\infty}\). Together with the tightness at finite resolution, this gives:

Corollary 18.3. (Tightness of the lattice measures) For every \(s > \frac{1}{2}\) the embedded laws \(\nu_N \circ \iota_N^{- 1}\) are tight in \(H^{- s} (\mathbb{T}^3)\), and \(\sup_N \mathbb{E} \| \iota_N \Phi^N_{\infty} \|^p_{H^{- s}} < \infty\) for every \(p\).

For the moments: the drift integrals have all moments by (17.4), and the \(H^{- s}\) martingale has a deterministic bracket bound by (18.3) at \(Q = q_0\), so the Burkholder–Davis–Gundy inequality gives all moments.

The threshold \(s > \frac{1}{2}\) is that of the free field in \(d = 3\): the interacting field is no rougher than the Gaussian one, and at every finite resolution its non-Gaussian part \(V\) is smoother (\(H^1\)).

19Identifying the limit

19.1The deterministic coefficients

The coefficients \(M^N_q\), \(h^{[j], N}_q\), \(H^N_q (y)\), \(b^N_q (y)\) are constructed at each lattice. Their uniform bounds (Proposition 12.1) do not by themselves identify a limit: a weakly convergent vertex cannot in general be contracted against a varying covariance. The three tools are:

• Weak vertex measures. Derivative kernels with bounded rooted norms are signed measures with bounded mass; they converge weakly along subsequences, and the centred kernel \(\Theta_q\) converges in a weak topology controlled by its first moment. The limiting coefficients solve the same fixed-point problem, whose solution is unique in its ball; so the whole sequence converges.

• Compact sandwich. If \(G_N \to G\) in Hilbert–Schmidt norm, and \(\textsf{B}_N\) are symmetric, uniformly bounded and converge weakly to \(\textsf{B}\), with \(\sup_N \|G_N \textsf{B}_N \| \le \theta < 1\) and \(\|G \textsf{B} \| \le \theta\), then \((I + G_N \textsf{B}_N)^{- 1} G_N \to (I + G \textsf{B})^{- 1} G\) in Hilbert–Schmidt norm. The free resolvent is the compact factor that turns weak convergence of the Hessian into strong convergence of the response.

• Contraction against a moving covariance. The rows of the inverses are Hölder (any exponent \(a < \frac{1}{2}\)); with the Hilbert–Schmidt convergence given by the sandwich, the squared kernels \((H^N_q)^2\) converge uniformly to a continuous kernel. Weakly convergent vertices of bounded variation can therefore be contracted against them, and the trace steps of the recursion pass to the limit.

With these, \(H^N_q (Y^N_q) \to H_q (Y_q)\) and \(b^N_q (Y^N_q) \to b_q (Y_q)\) in \(L^2 (I ; \text{HS})\) and \(L^2 (I ; H)\) along strongly converging forecasts with the uniform supremum moments of Lemma 18.1.

19.2The noise

The delicate point is the backward martingale. Fields converging in law do not imply that the stochastic integrals converge to stochastic integrals of the limits.

Remark 19.1. (Hidden noise) Let \(B\) be a real Brownian motion and \(L^n_t = \int_0^t (1 + \sin (ns)) \mathrm{d} B_s\). Then \((L^n, B)\) converges in law to \((B + 2^{- 1 / 2} \tilde{B}, B)\), with \(\tilde{B}\) an independent Brownian motion: the integrands converge weakly to \(1\), but their squares converge weakly to \(\frac{3}{2}\). A limiting FBSDE obtained this way would have an extra martingale, orthogonal to the driving noise.

To exclude this one needs strong compactness of the integrands. For the forward equation the integrand \(H (Y)\) is an explicit function of \(Y\), and strong convergence follows from the coefficient limits. For the backward equation the integrand is \(\Xi = \text{Cov} - H (Y)\), and we need strong compactness of the posterior covariance \(\text{Cov}_q\). It comes from its exact dynamics.

Lemma 19.2. (Riccati equation of the posterior covariance) At each \(N\), \(\text{Cov}_q = D^2 \mathcal{K}_q (P_q) \ge 0\) and

Proof. Differentiate the equation of Proposition 11.3(c) twice in \(p\): \((\partial_q + \frac{1}{2} \Delta_p + D\mathcal{K} \cdot D_p) D^2 \mathcal{K}= - (D^2 \mathcal{K})^2\), and apply Itô's formula along \(\mathrm{d} P = D\mathcal{K} (P) \mathrm{d} q + \mathrm{d} W\).\(\Box\)

This is (6.9) for \(\partial_q \mathscr{R} = 1\): conditional variance is resolved at the rate \(\text{Cov}^2\). For a smooth test \(f\), the process \(\langle f, \text{Cov}_q f \rangle \ge 0\) is a nonnegative supermartingale, with drift \(- \| \text{Cov}_q f\|^2\), and initial value \(\text{Var}_{\nu_N} \langle \Phi_{\infty}, f \rangle\), bounded uniformly in \(N\) by Corollary 18.3. A nonnegative supermartingale has controlled oscillations: after stopping at the first time it exceeds a level \(K\) (an event of probability at most \(C_f / K\)), its decreasing part \(A\) and the bracket of its martingale part \(J\) satisfy \(\mathbb{E}A_Q \le C_f\) and \(\mathbb{E} [J]_Q \le K^2 + 2 KC_f\), which gives time regularity in \(W^{a, 1}\), \(a < \frac{1}{2}\), and hence strong compactness of each finite block \(\langle f_i, \text{Cov} f_j \rangle\). Spatial tails come from the isometry (recall \(\psi_{q_0} =\mathbb{E} \Phi_{\infty} = 0\), \(\nu_N\) being even), with symmetry exchanging input and output tails, \(\mathbb{E} \int_I \|(I - \textsf{F}_m) \text{Cov}_q \|_{\text{HS}}^2 \mathrm{d} q =\mathbb{E}\|(I - \textsf{F}_m) \psi_Q \|_2^2\), and from the terminal averaging \(\psi_Q =\mathbb{E}_Q \int_L^{L + 1} (Y_s + Z_s) \mathrm{d} s\), using the \(H^{\sigma}\) bound of Proposition 18.2 and \(\mathbb{E} \|Z_s \|_2^2 \le Cs^{- 5 / 2}\) (first \(m \to \infty\), then \(L \to \infty\)). Altogether \(\text{Cov}^N\) is tight in strong \(L^2 (I ; \text{HS})\). Since \(S \mapsto \int SS^{\ast}\) is continuous from \(L^2 (I ; \text{HS})\) to \(C (I ; \text{trace class})\), the brackets of \(\psi^N\) are tight in \(C (I)\); a continuous martingale with tight brackets and uniformly small maximal Fourier tails (from the terminal averaging above and Doob's inequality) is tight in \(C (I ; H)\). Hence \(\psi^N\), and \(Z^N = \psi^N - Y^N\), are tight in \(C (I ; H)\). Strong convergence of the integrands gives convergence of the brackets \(\int \text{CovCov}^{\ast}\) and of the covariations with \(W\), and then:

Proposition 19.3. (Identification of the martingales [FBSDE26]) Along a subsequence, on a common filtered probability space, the limit of the forecast satisfies \(\psi_q = \int_{q_0}^q \text{Cov}_s \hspace{0.17em} \mathrm{d} W_s\) with \(W\) a cylindrical Brownian motion for the joint filtration, and \(Y\), \(Z\), \(X\) satisfy

The argument is: the limiting \(\psi\) is a continuous square-integrable martingale for the joint filtration (fixed-time martingale identities pass to the limit with uniform integrability); its bracket and its covariation with \(W\) are those of \(\int \text{Cov} \hspace{0.17em} \mathrm{d} W\); so \(\psi - \int \text{Cov} \hspace{0.17em} \mathrm{d} W\) has zero bracket and vanishes. No extra martingale survives. The same argument for the forward equation identifies its noise with the same \(W\), and identifies the limiting Gaussian part \(X\) as \(\int C \hspace{0.17em} \mathrm{d} W\), not a new Gaussian field.

19.3The result

Theorem 19.4. (Subsequential construction of \(\Phi^4_3\) [FBSDE26]) Let \(\lambda > 0\) and \(q_0\) large enough. Along a subsequence of lattices:

So the posterior structure survives the continuum limit: \(\psi = Y + Z\) is a closed observable, the conditional mean of the terminal field; \(Y\) is computed from the explicit action; and the error \(Z\) is small at high resolution. This is the FBSDE of Lecture 3, in forecast coordinates, for a subsequential limit \(\nu\) of the renormalised lattice measures, with the full noise retained.

19.4The renormalised field equation

The terminal law satisfies an integration by parts formula in which the renormalised cube appears only tested in expectation.

Theorem 19.5. (Integration by parts) Let \(h, h_1, \ldots, h_k\) be smooth and \(G (\varphi) = g (\langle \varphi, h_1 \rangle, \ldots, \langle \varphi, h_k \rangle)\) with \(g \in C_b^{\infty} (\mathbb{R}^k)\), and let \(D_h G (\varphi) = \frac{\mathrm{d}}{\mathrm{d} a} G (\varphi + ah) |_{a = 0}\). Along the subsequence of Theorem 19.4 (recall \(C = (m^2 - \Delta_N)^{- 1}\) with \(m^2 = q_0\)),

Proof. At fixed \(N\), Gaussian integration by parts in the density \(\mathrm{e}^{- V_N - \frac{1}{2} \langle \varphi, K_{q_0} \varphi \rangle}\) gives, with \(x = \Phi^N_{\infty}\) and \(h_N = j_N h\),

Cell averaging commutes with lattice translations, so \(K_{q_0} j_N h = j_N (q_0 - \Delta^{\text{tr}}_N) h\) with \(\Delta^{\text{tr}}_N\) the finite-difference Laplacian acting on smooth functions, which converges to \(\Delta h\) in every Sobolev norm. The right side converges by Corollary 18.3 (weak convergence in \(H^{- s}\) and uniform moments), and the derivative term is a finite sum with converging coefficients \(\langle j_N h, j_N h_i \rangle \to \langle h, h_i \rangle\).\(\Box\)

This is the terminal case of the Dyson–Schwinger equation of Theorem 10.1, (10.3), for the limiting theory. It does not say that \(\mu_N \Phi + \lambda \Phi^3\) converges as a random distribution: only its expectation against cylinder functions is identified. This is consistent with the expectation of Lecture 3 (Figure 5.1 and the remark following it) that the limiting force is an observable without a terminal representative; it is not proved here that no such representative exists. At every finite resolution, on the other hand, the force is explicit: \(F_q = K_q Z_q - D \bar{\Gamma}_q (Y_q)\) (Exercises 17.1 and 19.1).

19.5What is not proved, and where to go next

• Uniqueness. The limit is subsequential. Uniqueness of the limiting FBSDE, or of \(\nu\), would give convergence of the whole sequence and independence of the regularisation.

• Small mass (strong coupling). The construction needs \(\gamma q_0^{- 1 / 2}\) small. By scaling, it is equivalent to the weak-coupling regime \(\lambda / m \ll 1\) (unit mass on a torus of side \(m\)). Removing it requires a priori estimates that do not rely on the smallness of the feedback coefficient \(\gamma^2 / q_0\), and a coefficient construction outside the perturbative ball (\(g_{q_0} = \gamma q_0^{- 1 / 2}\) small in Proposition 12.1).

• Axioms. Reflection positivity holds at every lattice, for reflections in lattice and mid-lattice planes (nearest-neighbour action), and passes to \(\nu\) for every reflection in a coordinate plane, by approximating the plane with lattice planes (standard; Exercise 19.4). Rotation invariance (absent on the torus) and clustering need infinite volume.

• Infinite volume. The force, its Burgers flow, and differences of free energies are intensive (Section 7.2). The estimates of Lecture 5 must then be made local, with weights in space and constants per unit volume. A new difficulty appears in the backward equation: its integrand \(\Xi\) is a nonlocal operator, and the backward error at a point can receive noise from arbitrarily far away. Uniform-in-volume coefficient bounds and local energy moments are themselves open; they could give an infinite-volume Gibbs subsequence. Controlling the “noise from infinity” (together with local compactness of the Hessian blocks) is the main additional step needed to retain the full FBSDE without an orthogonal martingale [FBSDE26].

Summary of the course.

Exercises

References

[FBSDE26]

M. Gubinelli. The massive \(\Phi^4_3\) field through forward–backward equations. Course notes, October 2026.