Rigorous-RG Scheme Comparison and Continuum-Construction Status

Two rigorous RG schemes describe the same continuum observable only when a typed comparison relates their regulated inputs, trajectories, renormalization conditions, observable maps, and limits. An analytic conjugacy on uniform Banach domains is sufficient but rare; equality of a named limiting correlation is weaker and often all that is needed. Agreement of truncated beta functions proves neither.

Required background. Rigorous RG as a Dynamical System supplies the maps being compared. Universality, Critical Manifolds, and Observable Control supplies observable-level equivalence.

Helpful background. Constructive Existence by Model, Dimension, and Observable supplies dimension-specific status. Equivalence, Uniqueness, and Comparison Notions supplies the hierarchy of comparisons.

Three levels of scheme comparison

Let scheme $S_i$ have Banach spaces $\mathcal X_j^{(i)}$, domains $\mathcal D_j^{(i)}$, maps $\mathcal R_j^{(i)}$, reconstruction maps $\mathcal Q_j^{(i)}$ to regulated generating functionals, and observable maps $\mathcal O_i$.

Exact finite-cutoff equivalence requires the reconstructed generating functionals to agree after a specified change of variables and normalization. This can hold even when the coordinates look different.

Uniform conjugacy requires analytic, uniformly controlled maps $H_j$ with analytic inverses such that

$$H_{j+1}\circ\mathcal R_j^{(1)} =\mathcal R_j^{(2)}\circ H_j$$

on explicit domains for all relevant scales. Observable insertions must also intertwine, possibly with a field-strength factor.

Observable equivalence requires less: after each scheme tunes its bare parameters, one proves

$$\lim_{a\downarrow0}\mathcal O_1(a) =\lim_{\Lambda\uparrow\infty}\mathcal O_2(\Lambda)$$

in the same topology and normalization. This does not provide a conjugacy of the full flows. In every case, “same scheme” has a source, target, domain, and convergence mode.

An approximate comparison can be useful if its defect is controlled. Suppose $y_j=H_jx_j$ and define

$$e_j=H_{j+1}\mathcal R_j^{(1)}x_j -\mathcal R_j^{(2)}H_jx_j.$$

If the second scheme is Lipschitz with constants $L_j$, the orbit discrepancy satisfies

$$\|\delta_{j+1}\|\leq L_j\|\delta_j\|+\|e_j\|.$$

A comparison theorem must show that the transported defects remain summable, including amplification along relevant directions, and that the reconstruction mismatch tends to zero in the observable topology. Matching a finite Taylor jet gives only a pointwise estimate $e_j=O(g_j^k)$ near the origin; it supplies neither uniform domains nor the required products of $L_j$. This simple variation-of-constants estimate is an independent way to test whether “scheme independence” is a proved limit statement or only perturbative coordinate agreement.

A three-dimensional φ⁴ comparison fixture

Consider a finite-range lattice scheme with spacing $a$ and a continuum Wilsonian scheme with ultraviolet scale $\Lambda$. Fix common renormalization conditions for a two-point observable, for example

$$\Gamma_R^{(2)}(0)=m_R^2, \qquad \left.\frac{\partial\Gamma_R^{(2)}(p)}{\partial p^2}\right|_{p^2=\mu_R^2}=1.$$

Tune the lattice mass and field-strength counterterms as functions of $a$, and the continuum parameters as functions of $\Lambda$. A valid observable comparison must provide uniform bounds and prove both renormalized two-point functions converge, as tempered distributions or another named topology, to the same $G_R^{(2)}$. A coordinate ansatz

$$g_2=g_1+c_2g_1^2+O(g_1^3)$$

is only a formal relation unless the remainder is bounded on a nonperturbative domain and the observable maps are matched.

This is the worked decision required by Regulator Removal and Renormalized Predictions: compare one renormalized two-point object, then state separately whether the limit or only the coordinate matching is proved. Balaban-type scalar $\phi^4_3$ analysis supplies ultraviolet stability machinery Dimock 2013, Part III, §§ 1 and 6, pp. 1–8 and 38–49 of the Open PDF. Abdesselam constructs a complete nonperturbative RG trajectory between Gaussian and non-Gaussian fixed points for a modified-propagator three-dimensional model Abdesselam 2007, Theorem 1 and §§ 1–2, pp. 729–739. Neither source alone proves the conjugacy of a standard finite-range lattice scheme with an arbitrary continuum parametrization.

What a continuum construction additionally needs

Uniform trajectory bounds control effective interactions, but a Euclidean field still requires convergence of measures or all named Schwinger functions. One must identify the limiting covariance and counterterms, prove tightness or distributional convergence, control volume removal, and verify positivity and regularity appropriate to the desired reconstruction. A finite-volume partition-function bound is not automatically correlation convergence.

In four dimensions, the status is especially instructive. Weak lattice $|\phi|^4$ has rigorous logarithmic critical asymptotics and Gaussian torus scaling limits. Aizenman and Duminil-Copin prove Gaussianity of scaling limits for the stated critical four-dimensional Ising-type and lattice $\lambda\phi^4$ classes Aizenman and Duminil-Copin 2021, pp. 163–177, with 2024 corrigendum. These are strong continuum-status statements, but they do not construct an interacting four-dimensional $\phi^4$ QFT.

Adversarial truncated redefinition

Take two recursions

$$g' = g-\beta g^2+ag^3, \qquad \tilde g' = \tilde g-\beta\tilde g^2+\tilde a\tilde g^3,$$

and choose $\tilde g=g+cg^2$ so the coefficients match through one order. This calculation supplies a formal jet at $g=0$. It gives no radius of analyticity, no control of accumulated remainders over $O(g^{-1})$ scales, no map between polymer coordinates, and no observable identification. The strongest surviving statement is finite-order perturbative scheme agreement.

An independent check is to reconstruct a finite-cutoff generating functional on both sides. If the proposed $H_j$ conjugates coupling recursions but the reconstructed two-point functions differ by an uncontrolled contact term or field normalization, the physical comparison has failed.

Exercise

Assume $H_{j+1}\mathcal R_j^{(1)}=\mathcal R_j^{(2)}H_j$ exactly and $x_{j+1}^{(1)}=\mathcal R_j^{(1)}x_j^{(1)}$. Prove that $x_j^{(2)}=H_jx_j^{(1)}$ is an orbit of scheme 2.

Solution

$x_{j+1}^{(2)}=H_{j+1}x_{j+1}^{(1)}=H_{j+1}\mathcal R_j^{(1)}x_j^{(1)}=\mathcal R_j^{(2)}H_jx_j^{(1)}=\mathcal R_j^{(2)}x_j^{(2)}$. This algebraic statement uses exact conjugacy; an $O(g^k)$ equality accumulates errors and needs a separate stability estimate.

References

• Abdesselam, Abdelmalek. “A Complete Renormalization Group Trajectory Between Two Fixed Points.” Communications in Mathematical Physics 276 (2007): 727–772. Open PDF.
• Aizenman, Michael, and Hugo Duminil-Copin. “Marginal Triviality of the Scaling Limits of Critical 4D Ising and $\phi_4^4$ Models.” Annals of Mathematics 194 (2021): 163–235; corrigendum 199 (2024): 479. DOI; Corrigendum.
• Dimock, Jonathan. “The Renormalization Group According to Balaban—III. Convergence.” 2013. Open PDF.