Rigorous RG as a Dynamical System

A rigorous renormalization-group step is an analytic map between explicitly normed interaction spaces, together with an exact identity for the partition function or generating functional. Iteration is justified only while every coordinate remains in the map’s domain and the nonperturbative remainder satisfies a uniform contraction estimate. A recursion for a few couplings, by itself, is not that dynamical system.

Required background. The Constructive Program and Cutoff Removal supplies the regulated measures and limiting claims to be controlled. Domains, Signatures, Supports, and Regularity supplies the object–domain discipline used in the theorem.

Helpful background. Cluster Expansions and Correlation Inequalities supplies connected expansions, and Equivalence, Uniqueness, and Comparison Notions distinguishes coordinate changes from equality of continuum theories.

The normed RG map

Let $\Lambda_N=(\mathbb Z/L^N\mathbb Z)^d$ and decompose a positive covariance as $C=\sum_{j=1}^N C_j$. At scale $j$, write an interaction in coordinates

$$x_j=(V_j,K_j)\in\mathcal X_j=\mathcal V_j\times\mathcal W_j.$$

Here $\mathcal V_j$ is finite dimensional and contains the relevant and marginal local polynomial couplings. The Banach space $\mathcal W_j$ contains polymer activities, with a norm combining field derivatives, a large-field regulator, and decay in polymer size. A typical four-dimensional scalar coordinate is

$$V_j(\phi)=\sum_{x\in\Lambda_N} \left(g_j\tau_x^2+\nu_j\tau_x+z_j\tau_{\Delta,x}+u_j\right),$$

where $\tau_x=\tfrac12|\phi_x|^2$ and $\tau_{\Delta,x}=\tfrac12\phi_x\cdot(-\Delta\phi)_x$. The representation

$$Z_j(\phi)=e^{-u_j|\Lambda_N|}(I_j(V_j)\circ K_j)(\Lambda_N,\phi)$$

is exact; $\circ$ is the polymer circle product, not ordinary multiplication.

Gaussian integration with covariance $C_{j+1}$ followed by reblocking and local extraction defines

$$\mathcal R_j:\mathcal D_j\subset\mathcal X_j\longrightarrow\mathcal X_{j+1}, \qquad (V_j,K_j)\longmapsto(V_{j+1},K_{j+1}).$$

The domain $\mathcal D_j$ is a small, scale-dependent ball: $g_j$ is positive and small, the relevant coordinates are bounded relative to $g_j$, and $\|K_j\|_j\leq c g_j^3$. The single-step theorem is not merely a Taylor calculation. It states analyticity on this domain, preserves symmetries and locality, and gives estimates of the form

$$\|K_{j+1}\|_{j+1} \leq \kappa\|K_j\|_j+Cg_j^3, \qquad 0<\kappa<1,$$

with differentiated versions needed for observable insertions. The construction and its domain estimates are Brydges and Slade 2015, Theorems 1.11 and 1.13, pp. 605–613; the complete hierarchy of spaces and maps is summarized in Bauerschmidt, Brydges, and Slade 2019, Chapters 5 and 8–10, pp. 65–82 and 115–168.

One weak φ⁴ step

For weak lattice $|\phi|^4$, integrate a fluctuation $\zeta\sim\mathcal N(0,C_{j+1})$:

$$\mathbb E_{C_{j+1}}Z_j(\phi+\zeta)=Z_{j+1}(\phi).$$

Localize the connected second cumulant into $\mathcal V_{j+1}$ and put the unlocalized part into $K_{j+1}$. In normalized coordinates the quartic coupling has the directional estimate

$$g_{j+1}=g_j-\beta_jg_j^2+O(g_j^3),$$

while the mass coordinate has an $L^2$ relevant multiplier and $K_{j+1}=O(g_j^3)$ in its weighted norm. The $O(g_j^3)$ symbols are bounded analytic remainders on $\mathcal D_j$; they are not discarded terms. This is the rigorous finite-step object underlying the physical flow described in The Polchinski Exact RG Equation.

The proof mechanism has three independent parts. Finite-range support factors well-separated polymers. The localization operator removes all relevant and marginal Taylor jets without double counting. Gaussian and regulator estimates make the remaining connected activity contract after reblocking. Checking the exact integral identity tests normalization; checking the norm inequality tests iteration.

Composition across scales

Because the blocks, test-field norm, and regulator change with $j$, the rigorous object is generally a nonautonomous sequence of maps, not one self-map iterated on a fixed Banach space. For $k>j$, define the composition only on the admissible set

$$\mathcal D_{j:k} =\{x\in\mathcal D_j:\mathcal R_{\ell-1}\cdots\mathcal R_jx \in\mathcal D_\ell\ \text{for every }j<\ell\leq k\}.$$

Then $\mathcal R_{k-1}\cdots\mathcal R_j$ is analytic on the interior of $\mathcal D_{j:k}$, and repeated use of the one-step integral identity gives the exact partial integration of $C_{j+1}+\cdots+C_k$. This domain statement is essential: analyticity of every individual map on $\mathcal D_j$ says nothing about a point whose first image has already left $\mathcal D_{j+1}$.

The remainder estimate also has to close inductively. If $g_\ell$ stays in its prescribed interval and

$$\|K_{\ell+1}\|_{\ell+1} \leq \kappa\|K_\ell\|_\ell+Cg_\ell^3,$$

then a scale-dependent invariant ball $\|K_\ell\|_\ell\leq a g_\ell^3$ follows only after comparing $g_{\ell+1}$ with $g_\ell$ and choosing $a$ so that the inhomogeneous term fits inside the next ball. For correlation functions, one enlarges the coordinate by source-dependent activities and proves the corresponding differentiated bounds. Differentiating an unproved formal recursion does not control an inserted observable: the source derivatives must remain in their own normed domains through the same composition.

Failure boundary and continuum status

Start instead with $\|K_j\|_j$ outside the declared ball or with a quartic coupling whose real part does not stabilize the integral. The complex neighborhood needed for Cauchy estimates can cross a singular or nonintegrable region; the regulator no longer absorbs large fields, and the factor $\kappa$ is overwhelmed by nonlinear terms. The formal coupling recursion may still be writable, but no phase portrait inferred from it has the theorem’s status.

Even an exact map for every finite $N$ is not a continuum construction. Cutoff removal additionally needs a tuned all-scale trajectory, estimates uniform in $N$, convergence of specified correlations or measures in a named topology, and—if a Lorentzian QFT is claimed—reconstruction hypotheses. None follows from plotting finitely many iterates.

Exercise

Assume $\|K_{j+1}\|_{j+1}\leq\kappa\|K_j\|_j+Cg^3$ with constant $g$, $0<\kappa<1$, and $K_0=0$. Bound $K_j$ uniformly.

Solution

Iteration gives $\|K_j\|_j\leq Cg^3\sum_{r=0}^{j-1}\kappa^r\leq Cg^3/(1-\kappa)$. Thus the ball $\|K\|\leq ag^3$ is forward invariant if $a\geq C/(1-\kappa)$ and all other domain conditions are preserved. This scalar estimate does not replace the field-regulator and derivative estimates used to prove the one-step bound.

References

• Bauerschmidt, Roland, David C. Brydges, and Gordon Slade. Introduction to a Renormalisation Group Method. Lecture Notes in Mathematics 2242. Singapore: Springer, 2019. DOI; Open PDF.
• Brydges, David C., and Gordon Slade. “A Renormalisation Group Method. V. A Single Renormalisation Group Step.” Journal of Statistical Physics 159 (2015): 589–667. DOI; Open PDF.