Constructive quantum field theory — Lecture 5 (slides)

Lecture 4. Interacting field at resolution \(t\) = solution of the FBSDE

In the forecast coordinate \(\psi_t =\mathbb{E}_t \Phi_T\) the forward equation has no drift and the trees are integrated exactly; the backward half is the force equation.

Lectures 5–6. Construct \(\Phi^4_3\) on \(\mathbb{T}^3\) with this FBSDE:

1. deterministic renormalisation: an explicit approximate force \(\bar{\mathcal{F}}_q\) and its defect (today);

2. a priori estimates for the exact FBSDE in the coordinates of \(\bar{\mathcal{F}}_q\), uniform in the lattice (Lecture 6);

3. removing the cutoffs and identifying the limit (Lecture 6).

Notes: Lectures 5–6 and the Supplement; Bailleul–Chevyrev–G. (in prep.)

For every coupling \(\lambda > 0\), every reference mass \(q_0 > 0\) and every finite part \(r\) of the counterterm, on a fixed torus, with no smallness, and an explicit counterterm with arbitrary finite part:

• the whole lattice sequence converges to a measure \(\nu\), an explicit density over an explicit SDE;

• the forecasts converge to the solution of an FBSDE from \(Y_{q_0} = 0\), driven by the limit of the lattice innovations, and \(\psi_q = m_q (\Phi_q)\);

• uniqueness in law for its scalar form, a variational formula with no infinite constant, quartic tails, integration by parts.

Not proved: the vector FBSDE alone, strong solutions, OS axioms in full, infinite volume.

Guiding principle. Do not estimate \(\Phi\) and \(F\). Estimate the error of an explicit approximate forecast.

The results of Lectures 5–6 are new: joint work with S.-J. Meyer + AI.

The problem and the exact FBSDE

Lecture 5

\(\Lambda_N = (N^{- 1} \mathbb{Z}/\mathbb{Z})^3\), \(\int f = N^{- 3} \sum_x f (x)\), \(\Delta = \Delta_N\) nearest neighbour. Resolution parameter \(q = q_0 + t^2 \in [q_0, \infty)\):

In Lecture 3 notation (\(T = \infty\), and \(\dot{Q} = \partial_q Q_q\)):

The bubble \(a_2 (q)\) is the coincident-point variance of the shell at resolution \(q\).

So \(D^2 U (y) = \lambda y^2\): the coupling is the curvature, the same \(\lambda\) as in Lecture 4.

The Wick ordering is now part of \(\mu_N\).

Reference mass \(q_0 > 0\) (constants \(\in \ker \Delta\) on the torus; \(r\) may be negative). Only \(q_0 + \mu_N\) matters: fix \(q_0\), vary \(r\).

These come from bounds on the error \(Z\) of an approximate forecast, uniform in \(N\): UV stability.

By Lecture 4, in the profile \(S_q = C_q\), \(\mathcal{R}_q = q - q_0\); at fixed \(N\) everything is a theorem.The Polchinski process solves

and \(\Phi_q \to \Phi_{\infty} \sim \nu_N\).

• a noisy observation of its endpoint, \(\Phi_q = (q - q_0) C_q \Phi_{\infty} + C_q B_{q - q_0}\), with \(B\) independent of \(\Phi_{\infty}\);

• forecast

  \(\displaystyle \psi_q =\mathbb{E}_q \Phi_{\infty} = \Phi_q + C_q F_q = m_q (\Phi_q),\)

  covariance \(\text{Cov}_q > 0\);

• \(W\) is the innovation: \(\mathrm{d} \psi_q = \text{Cov}_q \hspace{0.17em} \mathrm{d} W_q\);

• noise: \(\eta_q = K_q \Phi_q - (q - q_0) \psi_q =\mathbb{E}_q B_{q - q_0}\), sub-Gaussian whatever the interaction (L4).

One site, \(\nu \propto \mathrm{e}^{- \lambda \varphi^4 / 12} \mu^{1 / q_0}\), \(\lambda = 6\), \(q_0 = 1\). Left: one path of the observation \(\Phi_q\) and of the forecast \(\psi_q\), both converging to the hidden \(\Phi_{\infty}\). Right, over 200 paths: the forecast error \(\sim q^{- 1 / 2}\), the force term \(\psi_q - \Phi_q = C_q F_q \sim q^{- 1}\), and the error \(Z_q\) of the approximate forecast from \(\bar{\mathcal{F}} = \frac{\lambda}{3} y^3\), below the bound \(2 C_{\ast} (\lambda / 3)^{1 / 2} q^{- 3 / 2}\) of Theorem 12.8.

Since \(\Phi_q \mapsto \psi_q\) is invertible, define \(\mathcal{F}_q\) by

and \(H^{\text{ex}}_q (y) = (K_q + D\mathcal{F}_q (y))^{- 1}\). It is the \(\mathcal{F}\) of Lecture 4 (\(x = \psi + S\mathcal{F} (\psi)\)), and \(F_q = -\mathcal{F}_q (\psi_q)\).

Itô on \(\Phi = \psi + C\mathcal{F} (\psi)\). Its integrated form is Wetterich's equation, which we do not need. An identity about the unknown \(\nu_N\), not a construction.

Strategy: an approximate force \(\bar{\mathcal{F}}\) gives approximate forecast coordinates; the error should be driven by the failure of \(\bar{\mathcal{F}}\) to solve this equation.

\(\bar{\mathcal{F}}_q\) odd, \(D \bar{\mathcal{F}}_q\) symmetric, \(K_q + D \bar{\mathcal{F}}_q \geqslant \bar{c}_q\).

Proof: Itô on the coordinate equation, with the Polchinski process and \(\mathrm{d} C_q = - C_q^2 \mathrm{d} q\): the martingale part gives \(H\), the drift gives \(HZ + b\).

• \(Y\): forward and explicit once \(Z\) is known; coefficients \(b, H\) built from \(\bar{\mathcal{F}}\).

• \(Z\): backward. The terminal condition \(qZ_q \to 0\) is proved at fixed \(N\) (Lecture 6), because \(\bar{\mathcal{F}}_q \to\) bare force.

• The only source of \(Z\) is the drift \(b\), the force defect, transported by \(\exp \int H\).

Needed: (i) a defect integrable against the backward propagator (renormalisation); (ii) global coercivity of \(\bar{\mathcal{F}}\) (large fields); (iii) a priori estimates (Lecture 6).

Renormalisation: why two of them

Lecture 5

Free field at resolution \(q\): \(\text{Var} X_q (x) \lesssim q^{1 / 2}\), typical size \(q^{1 / 4} = g_q^{1 / 2} r_q \ll r_q\).

Rooted norm of a vertex \(T (x_1, \ldots, x_k)\) and field ball:

Natural size of the \(k\)th vertex of the loop corrections:

Seed (one loop): replace \(D\mathcal{F}\) by \(\lambda y^2\) and integrate from \(q = \infty\), with the linear part subtracted,

Leading term = bubble, \(s (D^3 \ell_q (0)) = 6 \lambda^2 a_2 (q)\).

Corrections. \(\Pi\) removes the linear jet, \(M_q = \mu_q + \Theta_q\) a linear path, \(U\prime = \frac{\lambda}{3} y^3\),

Force equation for \(\mathcal{F}_q = U\prime + M_q y + \ell_q\), \(\Pi \ell_q = \ell_q\):

\(\Pi\) and \(1 - \Pi\) split it:

• nonlinear part, integrable at \(q = \infty\) (\(k \geqslant 4\) legs), with \(\ell_{\infty} = 0\): a fixed point, \(\ell_q = - \int_q^{\infty} \Pi \mathcal{T}_s [M, \ell_s] \hspace{0.17em} \mathrm{d} s\); approximated by Picard steps, the seed being the first;

• linear part, divergent (tadpole, sunset): solved exactly by the mass equation; gives the counterterm.

Each step inserts the previous correction into the loop: one more loop, one more \(g_q\). Two steps give a defect of order \(g_q^3\).

The linear jet is not iterated: it is solved exactly by \(M\),

with \(\mu_{q_{\ast}} = b\), \(|b| \leqslant B \lambda q_{\ast}^{1 / 2}\), at an auxiliary scale \(q_{\ast}\), and \(\Theta_q \to 0\) as \(q \to \infty\). Bare mass \(\mu_N = \text{lim}_{q \to \infty} \mu_q\): an output of the flow.

For \(q_{\ast}\) large, uniformly in \(N\). Unique solution in a ball, \(| \mu_q | \leqslant C \lambda q^{1 / 2}\), \(v_1 (\Theta_q) \leqslant C \lambda^2 q^{- 1 / 2}\); increments \(\mathcal{N}_k (\ell^{[j]} - \ell^{[j - 1]}) \leqslant C_k g_q^{j + 1} qr_q^{2 - k}\); and the defect of \(\mathcal{F}^{\text{o}}\): \(S_4 (\mathcal{D}^{\text{o}}_q) \leqslant C \lambda^5 q^{- 3}\).

Explicit counterterm. \(\mu_N = r - \lambda \text{Tr} C + \frac{2}{3} \lambda^2 \int C^3\) is admissible: the measure does not depend on \(q_{\ast}\).

The two divergent self-energies of Appendix A.2 of Lecture 4:

• Tadpole: \(- \lambda \int_{q_{\ast}}^{\infty} \text{Tr} R_{0, q}^2 \hspace{0.17em} \mathrm{d} q \approx - \lambda \text{Tr} C \sim - N\). Wick ordering: \(\frac{\lambda}{12} \varphi^4 - \frac{1}{2} \lambda c \varphi^2 = \frac{\lambda}{12} H_4 (\varphi ; c) +\)const.

• Sunset: contracting the bubble in \(D^3 \ell (0)\) with \(R_0^2\) and integrating gives the two-loop self-energy,

  \(\displaystyle \frac{2}{3} \lambda^2 \int C^3 \approx \frac{\lambda^2}{24 \pi^2} \log N \quad (= 2 \gamma_T \text{ of Lecture 4}) .\)

Power counting: nothing else. In the flow: the integrals from \(q = \infty\) of all vertices with \(k \geqslant 4\) legs converge; only the linear jet needs a boundary condition.

\(\mu_q\) at fixed \(q\) has a limit as \(N \to \infty\); \(\mu_N\) does not. Never interchange the two limits.

The cross is \(R_{0, q}^2\). The bubble in \(D^3 \ell^{[2]} (0)\) produces the sunset.

Backward propagator \(\approx s / q\), so \(Z_q \approx \int_q^{\infty} \frac{s}{q} b_s \mathrm{d} s\), and on the ball \(|b_s | \lesssim s^{- 1} \textsf{S}_4 (\mathcal{D}_s) r_s^3\). After \(n\) steps, \(\textsf{S}_4 (\mathcal{D}_s) \lesssim g_s^{n + 1} s^{1 / 2} r_s^{- 4}\):

Two steps are the minimum, and suffice. The defect of one step is borderline in the backward equation, as the sunset is borderline in the mass. (Heuristic threshold for \(\Phi^4_3\): \(n > 1\); the general theory's sufficient criterion is \((n + 2) \varkappa > (3 + \varkappa) / 2\), \(\varkappa = \frac{1}{2}\), met by \(n = 2\).)

The flow at large fields

Lecture 5

All bounds hold on \(\mathcal{B}_q\) only. The free field exceeds \(r_q\) with small, nonzero probability, and nothing a priori controls the interacting forecast. We need \(\bar{\mathcal{F}}_q\) defined and uniformly coercive for all fields.

Perturbative control for \(|y| \lesssim r_q\); outside, a non-perturbative replacement that keeps the structure the estimates use: coercivity, carried by the bare cube.

One site: \(qy + \frac{\lambda}{3} y^3 = q \Phi_q\), noise coefficient \(H_q (y) = (q + \lambda y^2)^{- 1} \leqslant q^{- 1} \wedge (\lambda y^2)^{- 1}\). Large fields are more elliptic and less noisy.

\(\tau\) smooth odd, \(\tau (u) = u\) on \([- 1, 1]\), constant outside \([- 2, 2]\), \(0 \leqslant \tau \prime \leqslant 1\);

The cubic and linear parts are not clipped, and \(D \kappa_q\) keeps the force a gradient. Globally: \(s (D^{j - 1} \tilde{\ell}_q) \leqslant Cg_q qr_q^{2 - j}\) (\(j = 2, 3, 4\)), \(s (D \partial_q \tilde{\ell}_q) \leqslant Cg_q\).

Neumann series for \(H = (I + G (M + D \tilde{\ell}))^{- 1} G\), \(G = (K_q + \lambda y^2)^{- 1} \leqslant C_q\).

With \(DU = \frac{\lambda}{3} y^3\):

Up to a small Lipschitz perturbation, the forecast solves the cubic elliptic equation

driven by the field at resolution \(q\).

Summary of today, at every lattice:

• \(\mu_N\), \(M_q\) and a globally coercive \(\bar{\mathcal{F}}_q\), all deterministic;

• the exact FBSDE for \((Y, Z)\);

• a source whose bounded-field part is integrable (two steps), and whose large-field part is controlled by sixth powers of \(Y\) on large fields.

Next: estimates on \((Y, Z)\) uniform in \(N\).

The expansion needs \(g_q\) small, so \(\bar{\mathcal{F}}_q\) is built for \(q \geqslant q_{\ast}\). Below \(q_{\ast}\) nothing needs constructing: freeze it,

Then \(K_q + D \bar{\mathcal{F}}_q = K_{q_{\ast}} + D \bar{\mathcal{F}}_{q_{\ast}}\): coercivity is automatic, \(H_q = H_{q_{\ast}}\), and \(y_q (x) = y_{q_{\ast}} (C_{q_{\ast}} K_q x)\).

The defect is explicit: \(\mathcal{D}_s (y) = - y + \frac{1}{2} \text{Ctr}_{H_{q_{\ast}}^2} D^2 \bar{\mathcal{F}}_{q_{\ast}} (y)\), with \(\|\mathcal{D}_s (y)\|_p \leqslant C \|y\|_p\).

So the system holds on \([q_0, \infty)\) with \(Y_{q_0} = 0\): the initial value is deterministic, with no large mass. \(q_{\ast}\) stays only in the proofs, where the expansion and the closure of the loop need \(g_q\) small.

Programme: keep \(- \frac{\lambda}{3} \varphi^3\) unclipped, clip the higher orders, pay in the source of \(R = F - f\). Test it on one site: \(K_q = q\), \(S = 1 / q\).

So \(3 R / (\lambda x^3) \to 1\): the remainder cancels the cube. Field equation: \(qx = (q - q_0) m_q + \eta_q\), only linear confinement.

\(f^c = f_1 +\mathcal{C}f_2\), with \(f_1 = - \frac{\lambda}{3} (x^3 + 3 Sx)\), \(f_2\) the first tree, \(\mathcal{C}f_2 = \kappa_q \prime \cdot f_2 \circ \kappa_q\). For \(|x| \geqslant 2 r_q\), the Burgers defect is exactly

of size \((3 q / \lambda)^{1 / 2}\) at the edge of the clip: not integrable in \(q\).

The source and the dissipation cancel for the true \(R \sim \frac{\lambda}{3} x^3\): separate estimates give nothing.

Tails independent of \(R\): on one site from the posterior moment, but not uniform on the lattice (negative bare mass).

More orders do not help: the tree series has radius \(\frac{2}{3} r_q\).

Solve the transport part of the flow exactly:

• \(y_q (\Phi_q) = Y_q\): this is the forecast coordinate. \(F_q - f_q = K_q Z_q\), and \(Z\) is the posterior mean minus the mode.

• The defect is \(qb_q (y)\): bounded for all fields. The trees are in \(x \mapsto y\); the cube does not flow in \(y\), so only loops are clipped.

• The cube is coercive in \(y\): \(qy + \frac{\lambda}{3} y^3 = q \Phi_q\).

Not a no-go theorem: it rules out only estimating \(R\) as a small perturbation of the frozen cube. Integrating the cube in the original field is possible, but on the lattice its loop defect must still be renormalised; that route is not proved here.