Wilsonian renormalization is usually formulated in momentum space, but on curved backgrounds momentum shells have no invariant meaning. We replace them by an intrinsic spectral cutoff, ordering modes by the covariant Laplacian and setting the cutoff resolution by the system size in renormalization group (RG) units. For a scalar field on $S^3$, this yields a covariant, momentum-free RG flow whose trace is an exact sum over spherical harmonics. The standard flat-space flow is recovered when the sphere is large compared with the coarse-graining scale. As a nontrivial test, the compact spectral flow realizes Wilson-Fisher universality without momentum shells: the interacting fixed point survives at finite resolution, has one relevant direction, and approaches its flat-space counterpart smoothly, with critical exponents only weakly affected by the compact spectrum.
Coarse graining from within: Wilson-Fisher universality on S3
Wilsonian renormalization is usually formulated in momentum space, but on curved backgrounds momentum shells have no invariant meaning. We replace them by an intrinsic spectral cutoff, ordering modes by the covariant Laplacian and setting the cutoff resolution by the system size in renormalization group (RG) units. For a scalar field on S3, this yields a covariant, momentum-free RG flow whose trace is an exact sum over spherical harmonics. The standard flat-space flow is recovered when the sphere is large compared with the coarse-graining scale. As a nontrivial test, the compact spectral flow realizes Wilson-Fisher universality without momentum shells: the interacting fixed point survives at finite resolution, has one relevant direction, and approaches its flat-space counterpart smoothly, with critical exponents only weakly affected by the compact spectrum.
We consider a functional relation between a given Wilsonian renormalization group (RG) flow, which has to be related to a specific coarse-graining procedure, and an infinite family of (UV cutoff) scale-dependent field redefinitions. Within this framework, one can define a family of Wilsonian proper-time (PT) exact RG equations associated to an arbitrary regulator function. New applications of these RG flow schemes to the Ising Universality class in three dimensions in the derivative expansion are shortly illustrated.
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.
This walkthrough derives, from substrate primitives, the structural backbone of Wilsonian renormalization applied to the canonical Event Density (ED) PDE — and shows what the RG actually says about ED as a coarse‑grained theory. In 1971, Kenneth Wilson introduced the renormalization group: instead of solving a system at one scale, track how its couplings flow as you integrate out short‑distance structure. Fixed points govern universal behavior; relevant and irrelevant operators determine which features survive coarse‑graining. The framework reshaped statistical physics and quantum field theory. Applied to the canonical ED PDE, three structural results emerge: (1) Form‑closure of the operator basis. Under Wilsonian coarse‑graining (momentum‑shell integration + rescaling), the ED operator basis is *closed*: no new operators appear at tree level or one loop. The ED PDE’s structure isa theorem, not an ansatz. (2) ED is not an RG fixed point at physical couplings. The beta functions are non‑trivial. With dynamical exponent z = 2, M₀ is marginal, P₀ and ζ are relevant, τ is irrelevant. The couplings flow; ED is regime‑dependent, not scale‑invariant. (3) Three distinct flow regimes. The RG flow produces a UV regime of free two‑channel diffusion, an intermediate ED‑activeregime where the full operator structure matters, and an IR regime of linear gapped diffusion. ED‑specific predictions live only in the intermediate window. The derivation chain: canonical two‑channel ED PDE (δ coupled to participation mode v) with finite operator basis → momentum‑shell split and rescaling → form‑closure at tree and one loop → beta‑function computation at z = 2 → fixed‑point catalog (Gaussian fixed point, two fixed lines, IR sink at infinity) → flow invariants → two crossover scales (ℓ_v for participation‑slaving onset; ℓ_ξ for gap‑dominated decay) → ED window ℓ_v < ℓ < ℓ_ξ → three effective PDEs in UV, intermediate, and IR regimes. The structural payoff: ED’s cross‑scale invariance is *operator‑structure* invariance within a bounded window, not coupling‑invariance across all scales. The “one PDE, three domains” claim means the same operator basis governs behavior across the intermediate regime, even though the couplings run. The RG walkthrough establishes ED’s self‑consistency: the canonical ED PDE is a closed, well‑behaved object under Wilsonian coarse‑graining, with a defined applicability window and controlled limits. This is not a derivation of a new equation — it is a structural validation that ED’s cross‑scale invariance survives RG analysis exactly as advertised.
Our recently developed Fourier Monte Carlo algorithm permits a nonperturbative calculation of momentum-shell renormalization-group flows by simulation which despite its apparent simplicity is illustrative both numerically as well as conceptually interesting. We study the example of a varphi(4) model with long-range lattice interaction. For this model we show that the topology of the renormalization flow is globally accessible in a particularly convenient way. The nontrivial fixed point of Wilson-Fisher type is observed its accompanying critical exponents are numerically determined from fitting its surrounding flow pattern to a linearized renormalization-group transformation. The results are compared to those obtained from perturbation theory, -expansion and earlier Monte Carlo simulations. Application of our method is also expected to be rewarding in other models with long-range interactions.
We consider the large N interacting vector O(N ) model on a sphere in 4 − ϵ Euclidean dimensions. The Gaussian theory in the UV is taken to be either conformally or non-conformally coupled. The endpoint of the RG flow corresponds to a conformally coupled scalar field at the Wilson-Fisher fixed point. We take a spherical entangling surface in de Sitter space and compute the entanglement entropy everywhere along the RG trajectory. In 4 dimensions, a free non-conformal scalar has a universal area term scaling with the logarithm of the UV cutoff. In 4 − ϵ dimensions, such a term scales as 1/ϵ. For a non-conformal scalar, a 1/ϵ term is present both at the UV fixed point, and its vicinity. For flow between two conformal fixed points, 1/ϵ terms are absent everywhere. Finally, we make contact with replica trick calculations. The conical singularity gives rise to boundary terms residing on the entangling surface, which are usually discarded. Consistency with our results requires they be kept. We argue that, in fact, this conclusion also follows from the work of Metlitski, Fuertes, and Sachdev, which demonstrated that such boundary terms will be generated through quantum corrections.
Previous attempts to accurately compute critical exponents from Wilson's momentum shell renormalization prescription suffered from the difficulties posed by the presence of an infinite number of irrelevant couplings. Taking the example of the one-dimensional long-ranged Ising model, we calculate the momentum shell renormalization flow in the plane spanned by the coupling constants (u0,r0) for different values of the momentum shell thickness parameter b by simulation using our recently developed Fourier Monte Carlo algorithm. We report strong anomalies in the b dependence of the fixed-point couplings and the resulting exponents ${y}_{\ensuremath{\tau}}$ and $\ensuremath{\omega}$ in the vicinity of a shell parameter ${b}^{\ensuremath{\ast}}<1$ characterizing a thin but finite momentum shell. Evaluation of the exponents for this value of b yields a dramatic improvement of their numerical accuracy, indicating a strong damping of the influence of irrelevant couplings for $b={b}^{\ensuremath{\ast}}$.