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A Unified Theory of Characteristic Equations for the Renormalization Group: From Critical Phenomena to Quantum Gravity and Information Theory

shifa liu

Published Dec 31, 2025• 0 citations
Publisher

Abstract

This paper establishes a comprehensive, rigorous, and unified theory of characteristic equations for the renormalization group (RG). Extending the methodology developed in companion works on knot theory and fluid mechanics, we formulate the RG flow as a dynamical system on an infinite-dimensional theory space and demonstrate that the linearized RG operator L at a fixed point gives rise to a universal characteristic equation det(L − λI) = 0, whose eigenvalues λn determine all critical exponents, scaling laws, relaxation rates, and oscillation frequencies of the system. We prove that the RG flow is a semi-group, that its fixed points correspond to critical theories, and that the characteristic spectrum is invariant under S-duality. We derive rigorous characteristic equations for ϕ4 theory, Gross-Neveu models, quantum electrodynamics, holographic RG, and asymptotically safe gravity. We establish variational principles, topological extensions with θ-angles, non-equilibrium Keldysh-RG, information-theoretic entropy production, and connections to tensor networks, quantum computing, and machine learning. We prove a completeness theorem: any RG flow can be uniquely decomposed into a superposition of characteristic eigenmodes. The theory is then elevated to a Grothendieck topos, providing the ultimate mathematical foundation. Numerical predictions are systematically compared with experimental data, achieving agreement within 0.1% for the 3D Ising model, 3D XY model, superfluid helium, and QCD coupling constant. The theory is closed under the most general mathematical framework known: the internal logic of the RG Characteristic Topos. Furthermore, a completeness theorem (Theorem 22.13) establishes that the characteristic equation fully determines the RG equivalence class of any system, and the internal logic of the topos is self-referential closed (Theorem 22.16), ensuring that the entire theory is both externally and internally consistent. Three previously open constructive problems — the non-perturbative holographic inverse problem, the entropy–black hole entropy equivalence, and the CMB non-Gaussianity inverse problem — are resolved explicitly as Theorems 23.2–23.4, thereby completing the constructive aspects of the theory. The three constructive resolutions are mutually consistent within the topos-theoretic framework: the holographic inverse problem (Theorem 23.2) provides the spectral input for the entropy–black hole equivalence (Theorem 23.3), while the CMB non-Gaussianity formula (Theorem 23.4) reduces to the standard slow-roll result as a direct consequence of the characteristic spectrum. The consistency is formally established in Theorem 23.5. Thus the theory is now complete at all levels: mathematical foundations (topos, spectral theory), constructive algorithms (inverse problems, quantum simulation), and observational predictions.

DOI
10.5281/zenodo.22255173

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This paper establishes a comprehensive, rigorous, and unified theory of characteristic equations for the renormalization group (RG). Extending the methodology developed in companion works on knot theory and fluid mechanics, we formulate the RG flow as a dynamical system on an infinite-dimensional theory space and demonstrate that the linearized RG operator L at a fixed point gives rise to a universal characteristic equation det(L − λI) = 0, whose eigenvalues λn determine all critical exponents, scaling laws, relaxation rates, and oscillation frequencies of the system. We prove that the RG flow is a semi-group, that its fixed points correspond to critical theories, and that the characteristic spectrum is invariant under S-duality. We derive rigorous characteristic equations for ϕ4 theory, Gross-Neveu models, quantum electrodynamics, holographic RG, and asymptotically safe gravity. We establish variational principles, topological extensions with θ-angles, non-equilibrium Keldysh-RG, information-theoretic entropy production, and connections to tensor networks, quantum computing, and machine learning. We prove a completeness theorem: any RG flow can be uniquely decomposed into a superposition of characteristic eigenmodes. The theory is then elevated to a Grothendieck topos, providing the ultimate mathematical foundation. Numerical predictions are systematically compared with experimental data, achieving agreement within 0.1% for the 3D Ising model, 3D XY model, superfluid helium, and QCD coupling constant. The theory is closed under the most general mathematical framework known: the internal logic of the RG Characteristic Topos. Furthermore, a completeness theorem (Theorem 22.13) establishes that the characteristic equation fully determines the RG equivalence class of any system, and the internal logic of the topos is self-referential closed (Theorem 22.16), ensuring that the entire theory is both externally and internally consistent. Three previously open constructive problems — the non-perturbative holographic inverse problem, the entropy–black hole entropy equivalence, and the CMB non-Gaussianity inverse problem — are resolved explicitly as Theorems 23.2–23.4, thereby completing the constructive aspects of the theory. The three constructive resolutions are mutually consistent within the topos-theoretic framework: the holographic inverse problem (Theorem 23.2) provides the spectral input for the entropy–black hole equivalence (Theorem 23.3), while the CMB non-Gaussianity formula (Theorem 23.4) reduces to the standard slow-roll result as a direct consequence of the characteristic spectrum. The consistency is formally established in Theorem 23.5. Thus the theory is now complete at all levels: mathematical foundations (topos, spectral theory), constructive algorithms (inverse problems, quantum simulation), and observational predictions.
The concept of renormalization group (RG) flow is one of the most novel and broad-reaching aspects of quantum field theory (QFT). The RG flow is implemented by constructing effective descriptions of a QFT at decreasing energy scales. One reason that RG flow is useful is that often one is interested in low-energy properties of theories with complicated short-distance structures. RG flows are subject to C theorems in relativistic QFT. The C theorems order the space of Lorentz-invariant QFTs. RG flows generically begin at scale-invariant fixed points known as conformal field theories (CFTs) and end in trivial massive theories. With tuning, the RG flows may end at non-trivial CFTs. Each CFT has an associated dimensionless C value. The C theorem states that under RG flow from a UV to an IR fixed point, the C value decreases. In this Dissertation I present the F-theorem, which is a C theorem in three spacetime dimensions. I show that the correct quantity to consider is the Euclidean free energy of the CFT conformally mapped to the three-sphere, known as the F value. After motivating the F-theorem, I develop tools for calculating the F value in a variety of CFTs, with and without supersymmetry, including free field theories and gauge theories with large numbers of flavors. I also show that the F value is itself a useful quantity for probing the gauge/gravity duality and understanding other aspects of CFT, such as the scaling dimensions of monopole operators. The F theorem is closely related to quantum entanglement entropy. At conformal fixed points, the F value is equal to minus the renormalized entanglement entropy (REE) in flat Minkowski space across a circle. Away from the fixed points, the REE is a monotonically decreasing function along the RG flow. I compute the REE in a variety of holographic and non-holographic theories. I conclude the Dissertation by discussing a somewhat surprising result: the REE is not stationary at conformal fixed points.
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.
This article develops a joint state-law fixed-point method for self-consistent quantum models. A physical model is represented by a density operator and a finite-dimensional vector of effective law parameters; admissible dynamics consist of a state-update channel and a law-update map. A recursive solution is their joint fixed point. This formulation separates stationarity of a state-dependent Hamiltonian from genuine co-determination of states and laws and preserves the option of completely positive trace-preserving subsystem dynamics. Three general results are obtained: existence for continuous maps on compact convex state-law domains, uniqueness and iterative stability under a contraction condition, and persistence of non-degenerate fixed points under weak deformation. An exactly solvable two-state model displays a supercritical pitchfork bifurcation, critical exponent one-half, and a divergent zero-field susceptibility at the feedback threshold. In renormalization-group theory space, the same perturbative result yields first-order shifts of fixed-point coordinates and critical exponents. An action-level construction shows how recursive consistency operators can enter the standard functional flow without adding ad hoc terms to the Wetterich equation. Covariant gravity and laboratory phase-shift formulas are presented as conditional templates whose predictions require a specified kernel and normalization. The framework therefore contributes a broadly applicable method linking quantum channels, nonlinear dynamics, fixed-point theory, and renormalization-group deformations, while making no claim to a completed theory of quantum gravity.
If we study the history of science we see produced two phenomena which are, so to speak, each the inverse of the other. Sometimes it is simplicity which is hidden under what is apparently complex; sometimes, on the contrary, it is simplicity which is apparent, and which conceals extremely complex realities. Henri Poincaré Devoting only a short chapter to the renormalization group (RG) is a challenge owing to the wealth of both fundamental concepts and computational tools associated with this term. The RG is indeed encountered in many different domains of theoretical physics, ranging from quantum electrodynamics to second-order phase transitions to fractal growth and diffusion processes – one should rather speak of renormalization group S ! We refer to Brown (1993) and Fisher (1998) for an historical account, to Goldenfeld (1992) and Lesne (1998) and references therein for an overview of the domains of application and variants of RGs. We here emphasize that the RG is a way, maybe the most successful and with no doubt the most systematic and constructive, to derive effective low-dimensional descriptions that capture large-scale and/or long-time behavior . The RG can be extended far beyond the specific scope of critical phenomena, to an iterated multiscale approach allowing the construction of robust and minimal macroscopic models describing the universal large-scale features and asymptotics of a complex system. This generalized viewpoint brings out the close logical and even technical connections that bridge, within a unified framework, perturbative RG for singular series expansions, spin-block RG and momentum-shell RG for critical phenomena, RG for the asymptotic analysis of differential and partial differential equations, and probabilistic RG for the derivation of statistical laws and limit theorems.
This work presents an extended, mathematically complete formulation of the Information Geometry Unification Principle (IGUP), a foundational framework in which quantum mechanics, statistical thermodynamics, general relativity, and spacetime geometry emerge from a single information-theoretic ontology. We begin with rigorous axioms defining information as the primary physical substance, then systematically construct the Bures–Fisher–Rao metric on the manifold of quantum density matrices. We derive the universal information action functional and perform full variational derivations to obtain the unified field equations. We show explicitly how standard physical theories including the Schrödinger equation, Heisenberg picture, Liouville mechanics, classical thermodynamics, and Einstein field equations are recovered in appropriate limiting regimes. We further derive geometric expressions for entanglement, entropy, temperature, energy, and force. The theory resolves longstanding conceptual tensions between unitary quantum evolution and gravitational irreversibility, provides a natural resolution to the black hole information paradox, and yields eight concrete, experimentally testable predictions. This extended version includes full tensor calculus, covariant derivatives, curvature expansions, and limit theorems, amounting to a self-contained theory of everything based on Geometry=Information\text{Geometry} = \text{Information}Geometry=Information.
We develop a comprehensive renormalization group (RG) approach to criticality in open Floquet systems, where dissipation enables the system to reach a well-defined Floquet steady state of finite entropy, and all observables are synchronized with the drive. We provide a detailed description of how to combine Keldysh and Floquet formalisms to account for the critical fluctuations in the weakly and rapidly driven regime. A key insight is that a reduction to the time-averaged static sector is not possible close to the critical point. This guides the design of a perturbative dynamic RG approach, which treats the time-dependent, dynamic sector associated to higher harmonics of the drive, on an equal footing with the time-averaged sector. Within this framework, we develop a weak drive expansion scheme, which enables to systematically truncate the RG flow equations in powers of the inverse drive frequency ${\mathrm{\ensuremath{\Omega}}}^{\ensuremath{-}1}$. This allows us to show how a periodic drive inhibits scale invariance and critical fluctuations of second-order phase transitions in rapidly driven open systems. Although criticality emerges in the limit ${\mathrm{\ensuremath{\Omega}}}^{\ensuremath{-}1}=0$, any finite drive frequency produces a scale that remains finite all through the phase transition. This is a universal mechanism that relies on the competition of the critical fluctuations within the static and dynamic sectors of the problem.