Renormalization group (RG) is a core concept in physics from statistical mechanics to quantum field theory, yet its schemes differ widely between field theory and many-body physics. We formulate the Wilsonian RG as a quantum channel, whose Kraus representation yields pure conditional trajectories for pure inputs and recovers mixed-state flows upon averaging. We then develop an algebraic extension applicable beyond the usual momentum-space, factorized setting: the low-energy algebra is constructed from the one-particle spectrum, and coarse graining is a conditional expectation onto it. We further construct a resource theory of RG whose free states are fixed points. Under stated assumptions, the resulting relative-entropy monotone is proportional to $c-c_{\rm IR}$ to quadratic order along a single stable RG direction near a two-dimensional IR fixed point. We also introduce a complementary measure of the UV information discarded by coarse graining, establish its monotonicity for successive coarse-graining maps, and discuss its holographic realization.
Renormalization group (RG) is a core concept in physics from statistical mechanics to quantum field theory, yet its schemes differ widely between field theory and many-body physics. We formulate the Wilsonian RG as a quantum channel, whose Kraus representation yields pure conditional trajectories for pure inputs and recovers mixed-state flows upon averaging. We then develop an algebraic extension applicable beyond the usual momentum-space, factorized setting: the low-energy algebra is constructed from the one-particle spectrum, and coarse graining is a conditional expectation onto it. We further construct a resource theory of RG whose free states are fixed points. Under stated assumptions, the resulting relative-entropy monotone is proportional to c−cIR to quadratic order along a single stable RG direction near a two-dimensional IR fixed point. We also introduce a complementary measure of the UV information discarded by coarse graining, establish its monotonicity for successive coarse-graining maps, and discuss its holographic realization.
We show that the real-space renormalization group (RG), as a map from the observable algebra to the subalgebra of long-distance observables, is an error correction code, best described by a conditional expectation. It is comprised of a coarse-graining step followed by an isometric embedding. The coarse-graining is the error map and the long-distance observables are the correctable operators. We show that if there is a state that is preserved under renormalization the coarse-graining step is the Petz dual of the isometric embedding (the Petz map). We demonstrate that a set of states are preserved under this map if and only if their pairwise relative entropies do not change when we restrict to the long-distance observables. We study the operator algebra quantum error correction in the GNS Hilbert space which applies to any quantum system including the local algebra of quantum field theory. We show that the recovery map is an isometric embedding of the correctable subalgebra. Similar to the RG, the composition of the error map followed by the recovery map forms a conditional expectation (a projection in the GNS Hilbert space). In gauge/gravity dualities, the bulk relative entropy of holographic states is the same as their boundary relative entropies which implies that the holographic map is an error correction code, and hence a conditional expectation. It follows that the boundary to the bulk map is a Petz map.
Renormalization provides a framework for relating microscopic models of physical systems to effective descriptions at larger length scales. This procedure is studied for the boundary states of non-chiral two-dimensional topologically ordered models. The initial data consist of renormalization fixed points built from representations of finite-dimensional C∗-Hopf algebras, which are then perturbed by uniform on-site noise quantum channels and repeatedly coarse-grained. The resulting flows admit an intrinsic algebraic description in terms of completely positive maps on the C∗-Hopf algebra or, equivalently, positive linear functionals on its enveloping C∗-Hopf algebra. Their iteration is governed by convolution powers, and convergent trajectories yield new matrix product density operator fixed points, described by finite ∗-quantum hypergroups. This provides a concrete physical interpretation of such structures. For finite group algebras and their duals, we provide explicit classifications via Goursat's lemma for groups. Finally, we formulate and prove a quantum generalization of Goursat's lemma for finite-dimensional C∗-Hopf algebras, a result of independent interest, which gives an explicit structural description of all convergent renormalization trajectories.
In the successful framework of the Renormalization Group (RG) approach to Quantum Field Theories (QFT), establishing general constraints on the RG flow has become a central problem. This is formally addressed by the monotonicity theorems, such as Zamolodchikov’s c-theorem in d=2, the a-theorem in d=4, and the F-theorem in d=3. Essentially, these theorems state that if a RG trajectory leads from a UV fixed point to a IR one, each described by a Conformal Field Theory (CFT), then a suitable positive quantity decreases. In both even and odd dimensions, this quantity can be extracted, at fixed points, from the free energy of the CFT on a sphere, leading to the conjecture of a generalized F-theorem valid in continuous dimension d. In this thesis we investigate this framework in a multicritical multiscalar theory with a generic sextic interaction for N massless scalar fields below its upper critical dimension d=3. This model captures, among many others, the critical features of the Tricritical Ising universality class. A first step is to establish whether the RG flow admits a gradient structure, i.e. that there exists a monotonic function A whose gradient yields the beta function up to a positive-definite metric in the space of couplings. This is one of the strongest realizations of monotonicity, as it guarantees the irreversibility of the RG flow and gives an intrinsic geometric structure to the theory space. Given the known beta function up to next-to-leading order, we solve the gradient flow equation perturbatively, finding a parametric family of solutions for the monotonic A-function together with the metric at and below the upper critical dimension, so that the RG flow admits a gradient structure at this order. As this procedure leaves some parameters undetermined, to turn it into a physical statement these functions must be matched to suitable observables of the CFTs describing the fixed points. We therefore compute the CFT candidates suggested by the generalized F-theorem, namely the sphere free energy of the CFT and the Zamolodchikov metric. Specializing to the O(N)-symmetric fixed point, we consistently match the RG and CFT results, fixing all the free parameters in a unique way. This matching is the central result of this work: it establishes that the RG flow admits a gradient structure perturbatively, with F as the monotonic function and Zamolodchikov's metric as the metric on the theory space, recovered from the geometric construction for a specific and unique choice of the free parameters. Importantly, this result provides a nontrivial, perturbative confirmation of the F-theorem in its strongest form, at and below the upper critical dimension, for renormalizable three-dimensional multiscalar theories.
Recently, a novel real-space renormalization group (RG) algorithm was introduced. By maximizing an information-theoretic quantity, the real-space mutual information, the algorithm identifies the relevant low-energy degrees of freedom. Motivated by this insight, we investigate the information-theoretic properties of coarse-graining procedures for both translationally invariant and disordered systems. We prove that a perfect real-space mutual information coarse graining does not increase the range of interactions in the renormalized Hamiltonian, and, for disordered systems, it suppresses the generation of correlations in the renormalized disorder distribution, being in this sense optimal. We empirically verify decay of those measures of complexity as a function of information retained by the RG, on the examples of arbitrary coarse grainings of the clean and random Ising chain. The results establish a direct and quantifiable connection between properties of RG viewed as a compression scheme and those of physical objects, i.e., Hamiltonians and disorder distributions. We also study the effect of constraints on the number and type of coarse-grained degrees of freedom on a generic RG procedure.
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.
Recently, a novel real-space renormalization group (RG) algorithm was introduced.By maximizing an information-theoretic quantity, the real-space mutual information, the algorithm identifies the relevant lowenergy degrees of freedom.Motivated by this insight, we investigate the information-theoretic properties of coarse-graining procedures for both translationally invariant and disordered systems.We prove that a perfect real-space mutual information coarse graining does not increase the range of interactions in the renormalized Hamiltonian, and, for disordered systems, it suppresses the generation of correlations in the renormalized disorder distribution, being in this sense optimal.We empirically verify decay of those measures of complexity as a function of information retained by the RG, on the examples of arbitrary coarse grainings of the clean and random Ising chain.The results establish a direct and quantifiable connection between properties of RG viewed as a compression scheme and those of physical objects, i.e., Hamiltonians and disorder distributions.We also study the effect of constraints on the number and type of coarse-grained degrees of freedom on a generic RG procedure.