1 Introduction↩︎

The Renormalization Group (RG) framework, since its inception by Wilson and Wegner [1], [2], has served as a cornerstone of modern theoretical physics, providing a systematic paradigm for understanding how macroscopic phenomena emerge from microscopic degrees of freedom. Central to this framework is the Functional Renormalization Group (FRG), which offers a powerful non-perturbative approach via the Wetterich equation [3] to describe the scale-dependence of the effective average action \(\Gamma_\Lambda\). Despite its profound success in modern physics, a comprehensive geometric principle governing the trajectory of the RG flow in the space of couplings remains an open and fundamental question.

In parallel, Information Geometry has established that the parameter space of statistical models can be endowed with a Riemannian structure via the Fisher information metric [4]–[9]. This information-theoretic lens has inspired various attempts to characterize RG flows as geodesic trajectories or as entropy-driven gradient flows within such metric spaces [10]–[19]. A major breakthrough in geometric analysis occurred with Perelman’s proof of the Poincaré conjecture, where he introduced the \(\mathcal{F}\)-entropy functional to demonstrate that the Ricci flow—a parabolic evolution of the Riemannian metric—is a gradient flow that monotonically increases a specific entropy [20]–[25].

The possibility of a deep connection between RG flows and Ricci flows has been hinted at in the context of non-linear sigma models, where the one-loop beta function of the target space metric takes the form of the Ricci tensor [26]–[32]. However, a formal and exact mapping that transcends specific model assumptions and directly embeds the exact FRG framework within the canonical Ricci flow has remained elusive. Previous approaches often remained restricted to specific fixed points or relied on the assumption of a purely gradient flow in a flat parameter space, frequently neglecting the non-trivial curvature of the information manifold and the gauge-like redundancy inherent in the choice of coordinates.

In this paper, we bridge this critical gap by explicitly deriving the modified Ricci flow from the exact FRG framework without relying on the weak-coupling expansion. By representing the FRG as a functional JKO-Wasserstein gradient flow of field distributions driven by a thermodynamic free energy, we demonstrate that the information metric’s rate of change naturally decomposes into the intrinsic Ricci curvature and the covariant Hessian of a scalar information potential, \(\Phi\). Although we couldn’t find this scalar potential explicitly, we show that it acts as an emergent geometric gauge-fixing term that absorbs non-tensorial coordinate artifacts and restores full diffeomorphism invariance.

Furthermore, we construct a field-theoretic \(\mathcal{F}\)-entropy functional from the continuous scale-dissipation rate of the free energy, proving that the renormalization process is fundamentally a geometric optimization process driven by the parametric Hessian of this entropic landscape to smooth out the information manifold’s curvature. Our results provide a unified, non-perturbative geometric language for the renormalization group, offering a robust first-principles bridge connecting quantum field theory, optimal transport, and Perelman’s theory of geometric evolution toward steady Ricci soliton equilibria at fixed points.

2 From Fokker-Planck equation to Ricci flow↩︎

2.1 From Fokker-Planck equation to information metric flow↩︎

To describe the dynamical evolution of a statistical ensemble, we begin by considering the Fokker-Planck equation [33]–[36], which governs the spatio-temporal evolution of the probability density function \(p(x, t; \theta^a)\): \[\begin{align} && \partial_{t} p(x,t;\theta^{a}) = - \partial_{i} \Big( A_{i}(x;\theta^{a}) p(x,t;\theta^{a}) \Big) + \partial_{i} \Big( B_{ij}(x;\theta^{a}) \partial_{j} p(x,t;\theta^{a}) \Big) . \label{FP95Eq} \end{align}\tag{1}\] Here, the drift vector \(A_{i}\) represents the macroscopic flow of the system, while the diffusion tensor \(B_{ij}\) accounts for stochastic fluctuations. The introduction of the Fokker-Planck framework is motivated by the structural correspondence between statistical diffusion and the RG flow, where the successive integration of high-energy modes acts as a coarse-graining process. Consequently, Eq. (1 ) serves as the fundamental dynamical foundation determining how the information metric \(g_{ab}\) deforms along the renormalization scale, providing the starting point for our geometric mapping to the Ricci flow.

To characterize the intrinsic geometric structure of the parameter space associated with \(p(x, t; \theta^a)\), we introduce the Fisher information metric \(g_{ab}\) defined as [4], [5], [37], [38]: \[\begin{align} && g_{ab}(t,\theta^{a}) = \int d^{d} x p(x,t;\theta^{a}) \partial_{a} \ln p(x,t;\theta^{a}) \partial_{b} \ln p(x,t;\theta^{a}) . \label{Fisher95IM} \end{align}\tag{2}\] This metric measures the distinguishability of probability distributions under an infinitesimal variation of the parameters \(\theta ^{a}\), endowing the information manifold with a Riemannian structure. Physically, \(g_{ab}\) encapsulates the thermodynamic or quantum fluctuations of the system.

When the probability density \(p(x, t; \theta^a)\) evolves according to the Fokker-Planck equation (1 ), the time evolution rate of the Fisher information metric, \(\dot{g}_{ab}\), can be derived by exploiting the continuity structure of the probability current. Following the canonical geometric formulations of stochastic systems [39]–[42], the metric deformation equation takes the form: \[\begin{align} && \dot{g}_{ab}(t,\theta^{a}) = \int d^{d} x \, p(x,t;\theta^{a}) \Big[ \Big(\partial_{a} v_{i}(x,t;\theta^{a}) \Big) \partial_{i} \Big( \partial_{b} \ln p(x,t;\theta^{a}) \Big) + \Big(\partial_{b} v_{i}(x,t;\theta^{a}) \Big) \partial_{i} \Big( \partial_{a} \ln p(x,t;\theta^{a}) \Big) \Big] , \label{Metric95Eq951st} \end{align}\tag{3}\] where \(v_i(x, t; \theta^a)\) denotes the effective velocity field of the probability current, defined by: \[\begin{align} && v_{i}(x,t;\theta^{a}) = A_{i}(x;\theta^{a}) - \frac{1}{p(x,t;\theta^{a})} B_{ij}(x;\theta^{a}) \partial_{j} p(x,t;\theta^{a}) . \label{Eff95Force} \end{align}\tag{4}\] Equation (4 ) defines the net probability current velocity by balancing the deterministic drift \(A_{i}\) against the flow induced by diffusion. Equation (3 ) then reveals that the evolution of the information metric is driven by the gradients of this effective velocity field coupled with the informational score functions. Physically, this implies that local information dissipation during drift and diffusion processes accumulates into a global geometric deformation of the parameter space, laying the mathematical foundation for a direct comparison with the canonical Ricci flow. Here, we assumed that the statistical ensemble \(p(x, t; \theta^a)\) belongs to a smooth, rapidly decreasing continuous class, where all parameter derivatives decay faster than any polynomial at the boundary, thereby ensuring all boundary terms in integration by parts vanish identically.

2.2 From the information metric flow to the Ricci flow↩︎

Although Eq. (3 ) provides a self-consistent systematic description of how the Fisher information metric evolves under the Fokker-Planck dynamics, a fundamental geometric obstruction arises when one attempts to map this information flow directly onto the canonical Ricci flow, \(\dot{g}_{ab} = -2 R_{ab}\). The canonical Ricci flow is dictated strictly by the intrinsic Riemannian curvature of the manifold, whereas the unconstrained field-theoretic evolution derived from the statistical framework inherently generates additional, non-tensorial components that depend on the specific parametrization of the coupling constant space. This geometric mismatch stems from the gauge-like redundancy in the choice of coordinate systems on the information manifold; because the information metric encapsulates statistical distinguishability, an arbitrary coordinate transformation of the coupling constants can introduce artificial deformations that do not reflect true intrinsic changes in the physical state. Consequently, the raw flow equation (3 ) fails to satisfy covariance under general diffeomorphisms, which has historically hindered a clean geometric unification of the renormalization group with intrinsic curvature flows [30], [43]–[45]. A demonstration of this tensorial obstruction, along with the explicit calculation of the naive information-theoretic curvature components, is provided in Appendix A.

This situation is closely analogous to the necessity of gauge-fixing in gauge field theories, or the introduction of the DeTurck vector field to restore strict parabolicity in mathematical Ricci flow studies [45]–[47]. To resolve this tensorial inconsistency and establish a rigorous, first-principles equivalence, it is essential to embed the information metric dynamics within a modified Ricci flow framework governed by an appropriate diffeomorphism. Specifically, on a Riemannian manifold, an infinitesimal coordinate transformation generated by an arbitrary vector field \(X^{a}\) deforms the metric tensor via the Lie derivative, yielding the standard relation \(\mathcal{L}_X g_{ab} = \nabla_a X_b + \nabla_b X_a\). If this vector field is strictly gradient-like—meaning it can be realized as the gradient of an underlying scalar potential \(\Phi\) such that \(X_a = \nabla_a \Phi\)—the Lie derivative simplifies precisely to twice the covariant Hessian of that potential, \(2 \nabla_a \nabla_b \Phi\). By recognizing that the non-tensorial, coordinate-dependent artifacts in the Fokker-Planck flow possess a structure identical to this geometric deformation, we can precisely absorb these unphysical components into the definition of a potential-driven diffeomorphism. Consequently, the raw information metric evolution is mapped onto the modified Ricci flow framework: \[\begin{align} && \dot{g}_{ab}(t,\theta^{a}) =- 2 R_{ab}(t,\theta^{a}) + 2 \nabla_a \nabla_b \Phi(t,\theta^{a}) , \label{Modified95Ricci95Flow951st} \end{align}\tag{5}\] where \(R_{ab}\) is the intrinsic Ricci curvature tensor computed directly from the Fisher information metric \(g_{ab}\), and \(\nabla_a \nabla_b \Phi\) represents the covariant Hessian of the entropic potential.

2.3 \(\mathcal{F}-\)entropy functional↩︎

To provide a unified description of the statistical state and its geometric evolution, we introduce the thermodynamic free energy functional \(\mathcal{E}[p]\), which drives the optimal transport dynamics, alongside the field-theoretic \(\mathcal{F}\)-entropy functional that serves as the information-geometric Lyapunov potential analogous to Perelman’s energy functional [21]: \[\mathcal{E}[p(x,t;\theta^{a})] = \int d^{d} x \, p(x,t;\theta^{a}) \Big( \ln p(x,t;\theta^{a}) + V(x;\theta^{a}) \Big) , \label{Correct95Free95Energy}\tag{6}\] where \(V(x;\theta^a)\) defines the potential landscape. Here, the drift term \(A_{i}\) and the isotropic diffusion tensor \(B_{ij}(x;\theta^a)\) of the Fokker-Planck equation characterize the system’s flow along the energy landscape: \[A_{i}(x;\theta^{a}) = - \partial_{i} V(x;\theta^{a}) .\] Under these definitions, the original Fokker-Planck equation can be reformulated into an elegant Wasserstein gradient flow representation through the variation of the thermodynamic free energy \(\mathcal{E}[p]\), as well-established in the JKO framework of dissipative systems [48]–[50]: \[\partial_{t} p(x,t;\theta^{a}) = \partial_{i} \Big\{ p(x,t;\theta^{a}) B_{ij}(x;\theta^{a}) \partial_{j} \Big( \frac{\delta \mathcal{E}[p(x,t;\theta^{a})]}{\delta p(x,t;\theta^{a})} \Big) \Big\} . \label{FPE95Gradient95Flow95Corrected}\tag{7}\] Equation (7 ) demonstrates that the temporal evolution of the probability distribution \(p\) proceeds in a direction that optimizes the free energy landscape.

The instantaneous rate of this energy dissipation defines the Fisher information-like \(\mathcal{F}\)-entropy functional, establishing a direct link to Perelman’s framework: \[\mathcal{F}[p(x,t;\theta^{a})] \equiv -\frac{d\mathcal{E}}{dt} = \int d^{d} x \, p(x,t;\theta^{a}) \Big( \frac{1}{2} B_{ij}(x;\theta^{a}) \partial_{i} \ln p(x,t;\theta^{a}) \partial_{j} \ln p(x,t;\theta^{a}) + \tilde{V}(x;\theta^{a}) \Big) . \label{W95Entropy95FKE95Corrected}\tag{8}\] Here, the effective potential \(\tilde{V}(x; \theta^a)\) is given by: \[\tilde{V}(x; \theta^a) = \frac{1}{2} B_{ij}(x; \theta^a) A_i(x; \theta^a) A_j(x; \theta^a) + B_{ij}(x; \theta^a) \partial_i A_j(x; \theta^a) .\] Following the formal dual mapping between optimal transport dynamics and information-geometric variations on the parameter space [40], [51], [52], a direct correlation between the time evolution rate of the Fisher information metric, \(\dot{g}_{ab}\), and the parametric Hessian of this entropic functional can be derived: \[\dot{g}_{ab}(t,\theta^{a}) = - 2 \frac{\partial^{2} \mathcal{F}[p(t, x;\theta^{a})]}{\partial \theta^{a} \partial \theta^{b}} . \label{Ricci95Flow95Gradient95Flow951st95Corrected}\tag{9}\] Equation (9 ) implies that the “flow" of the metric in information space is fundamentally determined by the second-order parametric variation of the system’s \(\mathcal{F}\)-entropy potential, suggesting that statistical dissipation translates into geometric deformations.

To establish the geometric consistency between the metric evolution and the Ricci flow, we derive the consistency condition relating the curvature of information space to the Hessian of the entropy functional. By combining the modified Ricci flow with the entropic identity in Eq. (9 ), and substituting the potential ansatz \(X_{a} = \nabla_{a} \Phi\), the connection components naturally reorganize into a generalized soliton equation. Imposing the standard torsion-free Levi-Civita connection, we arrive at the pivotal geometric identity governing the information space: \[R_{ab}(t,\theta^{a}) - \nabla_{a} \nabla_{b} \Phi(t,\theta^{a}) = \frac{\partial^{2} \mathcal{F}[p(t, x;\theta^{a})]}{\partial \theta^{a} \partial \theta^{b}} . \label{To95Determine95Phi95Corrected}\tag{10}\] This formulation directly corresponds to the geometric embedding of Fokker-Planck dynamics within the Ricci flow background [50], [51], [53], confirming that the optimal transport relaxation of a statistical ensemble is intrinsically isomorphic to Perelman’s steady Ricci soliton evolution.

Building upon the geometric consistency established in Eq. (10 ), we are now equipped to define the global geometric \(\mathcal{F}\)-entropy functional, which governs the overall topological and informational landscape of the parameter space: \[\mathcal{F}[g_{ab}(t,\theta^{a})] = \int d^{N} \theta \sqrt{g(t,\theta^{a})} e^{- \Phi(t,\theta^{a})} \Big\{ R(t,\theta^{a}) + g^{ab}(t,\theta^{a}) \partial_{a} \Phi(t,\theta^{a}) \partial_{b} \Phi(t,\theta^{a}) \Big\} . \label{Perelman95F95Entropy95FPE}\tag{11}\] Equation (11 ) represents the information-geometric analog of Perelman’s entropy functional [21]. The presence of the entropic weight \(\sqrt{g} e^{-\Phi}\) is not an ad-hoc imposition, but a direct consequence of the probability density mapping, where the ensemble distribution inherently bonds with the Riemannian volume element.

Finally, we demonstrate that the temporal evolution of the information metric \(g_{ab}\) can be formalized as a pure Riemannian gradient flow of this global functional. By taking the functional variational derivative of \(\mathcal{F}\) with respect to the inverse metric, we find: \[\dot{g}_{ab}(t,\theta^{a}) = - 2 R_{ab}(t,\theta^{a}) + 2 \nabla_{a} \nabla_{b} \Phi(t,\theta^{a}) = - 2 \frac{\delta \mathcal{F}[g_{ab}(t,\theta^{a})]}{\delta g^{ab}(t,\theta^{a})} . \label{FPE95Ricci95Flow95Gradient95Flow}\tag{12}\] Equation (12 ) serves as the central structural conclusion of this section, proving that the statistical dynamics driven by the Fokker-Planck equation is isomorphic to a modified Ricci flow that optimizes the geometric \(\mathcal{F}\)-entropy landscape. The rigorous encapsulation of the metric deformation rate within the functional derivative \(-2\frac{\delta \mathcal{F}}{\delta g^{ab}}\) confirms that the information potential \(\Phi\) successfully absorbs all coordinate-dependent, non-tensorial artifacts.

3 From Functional Renormalization Group to Ricci flow↩︎

3.1 A review on the FRG equation↩︎

To generalize the geometric equivalence between statistical diffusion and the modified Ricci flow established in Section 2 to the non-perturbative framework of quantum field theory, we begin by constructing the probability density functional that describes the statistical distribution of field configurations at a given cutoff scale \(\ln \Lambda\): \[\begin{align} && P[\ln \Lambda,\phi(x);\lambda^{a}] = \frac{1}{Z(\ln \Lambda;\lambda^{a})} e^{- S[\ln \Lambda,\phi(x);\lambda^{a}]} . \end{align}\] Here, \(\lambda^a\) denotes the set of coupling constants spanning the theory space, and \(Z(\ln \Lambda; \lambda^a)\) represents the partition function serving as the normalization factor (or vacuum amplitude) obtained by integrating over all possible field configurations: \[\begin{align} && Z(\ln \Lambda;\lambda^{a}) = \int D \phi(x)\;e^{-S[\ln \Lambda,\phi(x);\lambda^{a}]} . \end{align}\] The total effective action \(S\) entering the exponent is conventionally decomposed into a quadratic Gaussian part governing the free propagation of fields and an interaction functional \(S_{\mathrm{int}}\) containing all multi-local operators driven by the couplings: \[\begin{align} && S[\ln \Lambda,\phi(x);\lambda^{a}] = \frac{1}{2} \int \frac{d^{d} p}{(2\pi)^{d}} \phi(p) G^{-1}(p^{2}) K_{\Lambda}^{-1}(p^{2}) \phi(-p) + S_{\mathrm{int}}[\ln \Lambda,\phi(x);\lambda^{a}] . \label{RG95Setting} \end{align}\tag{13}\] In this representation, \(G(p^2)\) is the bare propagator, and \(K_{\Lambda}(p^2)\) denotes a smooth, momentum-dependent cutoff function that satisfies appropriate boundary conditions (\(K_{\Lambda} \to 1\) for \(p \ll \Lambda\) and \(K_{\Lambda} \to 0\) for \(p \gg \Lambda\)), thereby successfully suppressing high-energy fluctuations to generate a well-defined effective theory at the scale \(\Lambda\). Formulating the field theory in this canonical distribution structure provides a robust dictionary for interpreting the RG process as the non-equilibrium evolution of a statistical ensemble, allowing the scale-dependent trajectory of the theory to be mapped directly onto a geometric framework.

Since the total sum of the probability density functional \(P\) must be strictly conserved regardless of any continuous deformation of the energy scale, the probability distribution satisfies a functional normalization invariance condition under scale transformations: \[\begin{align} && \frac{d }{d \ln \Lambda} \int D \phi(x) \;P[\ln \Lambda,\phi(x);\lambda^{a}] = 0 . \end{align}\] Based on this fundamental conservation law, the continuous scale evolution of \(P\) driven by the Polchinski-type exact RG equation [54] can be recast into an infinite-dimensional Fokker-Planck framework that governs the flow of the theory across scales: \[\begin{align} & \frac{d }{d \ln \Lambda} P[\ln \Lambda,\phi(x);\lambda^{a}]\\ &= \int_{M} d^{d} x \int_{M} d^{d} y \Big\{ C_{\Lambda}^{Pol.}(x,y) \frac{\delta^{2} P[\ln \Lambda,\phi(x);\lambda^{a}]}{\delta \phi(x) \delta \phi(y)} + \frac{\delta}{\delta \phi(x)} \Big( P[\ln \Lambda,\phi(x);\lambda^{a}] C_{\Lambda}^{Pol.}(x,y) \frac{\delta V_{Pol.}[\ln \Lambda,\phi(x)]}{\delta \phi(y)} \Big) \Big\}\\ &\equiv \Delta P[\ln \Lambda,\phi(x);\lambda^{a}] + div \Big( P[\ln \Lambda,\phi(x);\lambda^{a}] grad_{C_{\Lambda}^{Pol.}} V_{Pol.}[\ln \Lambda,\phi(x)] \Big)\;. \label{FRG95Eq} \end{align}\tag{14}\] Equation (14 ) reinterprets the Polchinski-type renormalization group equation as a statistical diffusion process, embedding the Wilsonian coarse-graining paradigm [1], [55], [56] within a stochastic context. This functional formulation complements the canonical exact RG flows commonly realized via the Wetterich equation or Morris’s formulation [3], [57]–[62]. The key physical quantities, the diffusion kernel (\(C_{\Lambda }^{Pol}\)) and the effective potential (\(V_{Pol}\)), are defined in terms of the rate of change of the cut-off function and the free action: \[\label{Cuttoff95Potential} \begin{align} C_{\Lambda}^{Pol.}(p^{2}) &= (2\pi)^{d} G(p^{2}) \frac{\partial K_{\Lambda}(p^{2})}{\partial \ln \Lambda}\;,\\ V_{Pol.}[\ln \Lambda,\phi(x)] &=\int \frac{d^{d} p}{(2\pi)^{d}} \phi(p) G^{-1}(p^{2}) K_{\Lambda}^{-1}(p^{2}) \phi(-p)\;. \end{align}\tag{15}\] To express this functional differential equation concisely, we introduce the following infinite-dimensional operators: \[\begin{align} \Delta &\equiv& \int_{M} d^{d} x \int_{M} d^{d} y~ C_{\Lambda}^{Pol.}(x,y) \frac{\delta^{2} }{\delta \phi(x) \delta \phi(y)}\;,\\ grad_{C_{\Lambda}^{Pol.}} &\equiv& \int_{M} d^{d} y~ C_{\Lambda}^{Pol.}(x,y) \frac{\delta }{\delta \phi(y)}\;,\\ div &\equiv& \int_{M} d^{d} x \frac{\delta}{\delta \phi(x)}\;. \end{align}\] These operators define the Laplacian (\(\Delta\)), gradient (\(\text{grad}\)), and divergence (\(\text{div}\)) in the field configuration space, allowing the complex RG flow to be compactly described in the language of geometric diffusion.

Using the operators defined above, the evolution of the probability density under the RG flow can be summarized as a continuity equation in field space: \[\begin{align} && \frac{d }{d \ln \Lambda} P[\ln \Lambda,\phi(x);\lambda^{a}] = - \int_{M} d^{d} x ~\frac{\delta }{\delta \phi(x)} \Big(\Psi[\ln \Lambda,\phi(x);\lambda^{a}] P[\ln \Lambda,\phi(x);\lambda^{a}]\Big) , \label{FRG95Conservation95Law} \end{align}\tag{16}\] where \(\Psi\) represents the probability velocity field in field space, as defined through the combination of diffusion and drift terms in the Fokker-Planck equation: \[\begin{align} && \Psi[\ln \Lambda,\phi(x);\lambda^{a}] ~ P[\ln \Lambda,\phi(x);\lambda^{a}] \nonumber \\&& = - \int_{M} d^{d} y \Big\{ C_{\Lambda}^{Pol.}(x,y) \frac{\delta P[\ln \Lambda,\phi(x);\lambda^{a}]}{\delta \phi(y)} + P[\ln \Lambda,\phi(x);\lambda^{a}]~ C_{\Lambda}^{Pol.}(x,y) \frac{\delta V_{Pol.}[\ln \Lambda,\phi(x)]}{\delta \phi(y)} \Big\} . \label{Conserved95Current95FRG} \end{align}\tag{17}\] The velocity field (conserved current) defined by Eq. (17 ) ensures the conservation of total probability: \[\begin{align} && \frac{d }{d \ln \Lambda} \int D \phi(x)\;P[\ln \Lambda,\phi(x);\lambda^{a}] = - \int D \phi(x) \int_{M} d^{d} x \frac{\delta }{\delta \phi(x)} \Big(\Psi[\ln \Lambda,\phi(x);\lambda^{a}] P[\ln \Lambda,\phi(x);\lambda^{a}]\Big)=0 . \label{Conservation} \end{align}\tag{18}\] Furthermore, this velocity field \(\Psi\) can be expressed as the gradient of a total effective potential, \(\Sigma\): \[\begin{align} && \Psi[\ln \Lambda,\phi(x);\lambda^{a}] = \int_{M} d^{d} y~ C_{\Lambda}^{Pol.}(x,y) \frac{\delta \Sigma[\ln \Lambda,\phi(x);\lambda^{a}]}{\delta \phi(y)} \equiv grad_{C_{\Lambda}^{Pol.}} \Sigma[\ln \Lambda,\phi(x);\lambda^{a}]] . \label{Current95Fick95Law} \end{align}\tag{19}\] The effective potential \(\Sigma\) represents the functional relative entropy (or Kullback-Leibler divergence) [6], [63], [64] between the total probability density \(P\) and the bare Gaussian weight \(e^{-V_{Pol.}}\), which is physically linked to the interaction landscape: \[\begin{align} && \Sigma[\ln \Lambda,\phi(x);\lambda^{a}] = - \ln \left(\frac{P[\ln \Lambda,\phi(x);\lambda^{a}]}{e^{- V_{Pol.}[\ln \Lambda,\phi(x)]}}\right) = S[\ln \Lambda,\phi(x);\lambda^{a}] - V_{Pol.}[\ln \Lambda,\phi(x)] . \label{Relative95Entropy95FRG} \end{align}\tag{20}\] Consequently, Eqs. (16 )–(20 ) demonstrate that the RG flow is an optimization process moving along the entropy or potential gradient in field space, establishing the dynamical foundation for the geometric mapping to the Ricci flow discussed in Section 2.

To lift the finite-dimensional geometric equivalence established in Section 2 into the non-perturbative domain of the FRG, we construct a direct functional translation dictionary. Specifically, the statistical drift vector field \(A_{i}(x)\) and the diffusion tensor \(B_{ij}(x)\) are identified with the functional operators of the Polchinski framework via the following algebraic mapping: \[A_{i}(x) \longrightarrow \int_{M} d^{d} y \frac{\delta}{\delta \phi(x)} \Big( C_{\Lambda}^{Pol.}(x,y) \frac{\delta V_{Pol.}[\ln \Lambda, \phi(x)]}{\delta \phi(y)} \Big) \label{3795A} ,\tag{21}\] \[B_{ij}(x) \longrightarrow C_{\Lambda}^{Pol.}(x,y) . \label{3895B}\tag{22}\] Equation (21 ) demonstrates that the functional gradient of the effective bare potential \(V_{Pol.}\) in the configuration space generates the macroscopic drift of the quantum theory. Concurrently, Eq. (22 ) establishes that the Polchinski kernel \(C_{\Lambda }^{Pol.}\), driven by the scale-evolution of the momentum cutoff, plays the role of the infinite-dimensional statistical diffusion coefficient.This dictionary confirms that the FRG scale evolution can be formally embedded as a non-equilibrium diffusion process within the field configuration space.

3.2 From Fokker-Planck type FRG equation to information metric flow↩︎

To quantify the geometric structure associated with scale transformations of field configurations in the FRG framework, we define the Fisher information metric \(G_{ab}\) on the infinite-dimensional field configuration space as: \[\begin{align} && G_{ab}(\ln \Lambda,\lambda^{a}) = \int D \phi(x) P[\ln \Lambda,\phi(x);\lambda^{a}] \partial_{a} \ln P[\ln \Lambda,\phi(x);\lambda^{a}] \partial_{b} \ln P[\ln \Lambda,\phi(x);\lambda^{a}] . \label{FRG95Fisher95Information95Metric} \end{align}\tag{23}\] Equation (23 ) measures the statistical distinguishability of the probability functional \(P\) under infinitesimal changes in the coupling constants \(\lambda ^{a}\), thereby endowing the theory space of quantum field theory with a Riemannian geometric structure.

When the probability density functional \(P\) follows the infinite-dimensional Fokker-Planck equation shown in the previous subsection, the rate of change of the information metric \(G_{ab}\) with respect to the scale parameter \(\ln \Lambda\) is derived as the following flow equation: \[\begin{align} && \frac{d }{d \ln \Lambda} G_{ab}(\ln \Lambda,\lambda^{a}) = \int D \phi(x) P[\ln \Lambda,\phi(x);\lambda^{a}] \int_{M} d^{d} x \Big[ \Big(\frac{\partial}{\partial \lambda^{a}} \mathcal{V}[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \frac{\delta}{\delta \phi(x)} \Big( \frac{\partial}{\partial \lambda^{b}} \ln P[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \nonumber \\&& + \Big(\frac{\partial}{\partial \lambda^{b}} \mathcal{V}[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \frac{\delta}{\delta \phi(x)} \Big( \frac{\partial}{\partial \lambda^{a}} \ln P[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \Big] . \label{FRG95Metric95Dynamics} \end{align}\tag{24}\] Here, \(\mathcal{V}[\ln \Lambda, \phi(x); \lambda^a]\) represents the local effective velocity potential, which encapsulates the diffusion kernel and interaction landscapes of the FRG framework to determine the total drift-diffusion current: \[\begin{align} && \mathcal{V}[\ln \Lambda,\phi(x);\lambda^{a}] = - \int_{M} d^{d} y \Big\{ \frac{\delta}{\delta \phi(x)} \Big( C_{\Lambda}^{Pol.}(x,y) \frac{\delta V_{Pol.}[\ln \Lambda,\phi(x)]}{\delta \phi(y)} \Big) \nonumber \\&& - \frac{1}{P[\ln \Lambda,\phi(x);\lambda^{a}]} \frac{\delta}{\delta \phi(y)} \Big( C_{\Lambda}^{Pol.}(x,y) P[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \Big\} . \label{FRG95Eff95Force95Field} \end{align}\tag{25}\] Equation (25 ) integrates the drift and diffusion terms arising from the Polchinski-type renormalization process. Equation (24 ) reveals that the evolution of the information metric is governed by the parameter-dependent variations of this effective potential coupled with the informational score functions of the field configurations. This formulation serves as a crucial bridge for extending the finite-dimensional Ricci flow results discussed in Section 2 into the infinite-dimensional domain of quantum field theory.

3.3 From the information metric flow to the Ricci flow↩︎

To provide a geometric interpretation of the information metric evolution derived in Eq. (24 ), we decompose it into a Ricci flow and a correction term associated with a diffeomorphism: \[\begin{align} && \frac{d }{d \ln \Lambda} G_{ab}(\ln \Lambda,\lambda^{a}) = - 2 R_{ab}(\ln \Lambda,\lambda^{a}) + \nabla_{a} X_{b}(\ln \Lambda,\lambda^{a}) + \nabla_{b} X_{a}(\ln \Lambda,\lambda^{a}) , \label{FRG95Modified95Ricci95Flow951st} \end{align}\tag{26}\] where \(R_{ab}\) denotes the Ricci curvature tensor of the coupling constant space. By performing a comparison between Eq. (26 ) and Eq. (24 ), the parameter-space vector field \(X_{a}\) can be suggested as: \[\begin{align} && X_{a}(\ln \Lambda,\lambda^{a}) = \int D \phi(x) \int_{M} d^{d} x P[\ln \Lambda,\phi(x);\lambda^{a}] \mathcal{V}[\ln \Lambda,\phi(x);\lambda^{a}] \frac{\delta}{\delta \phi(x)} \Big( \partial_{a} \ln P[\ln \Lambda,\phi(x);\lambda^{a}] \Big) . \label{Diffeo95FRG} \end{align}\tag{27}\] The covariant derivative of this induced vector field, \(\nabla_{a}X_{b}\), is defined to reflect the intrinsic Riemannian connection of the underlying theory space: \[\begin{align} && \nabla_{a} X_{b}(\ln \Lambda,\lambda^{a}) = \partial_{a} X_{b}(\ln \Lambda,\lambda^{a}) - \Gamma_{ab}^{c}(\ln \Lambda,\lambda^{a}) X_{c}(\ln \Lambda,\lambda^{a}) . \end{align}\] The Christoffel symbols \(\Gamma _{abc}\), which dictate the torsion-free Levi-Civita connection on the coupling constant manifold, are derived from the scale-dependent information metric \(G_{ab}\) and expressed in terms of the score function: \[\begin{align} && \Gamma_{abc}(\ln \Lambda,\lambda^{a}) = \frac{1}{2} \Big( \partial_{a} G_{bc}(\ln \Lambda,\lambda^{a}) + \partial_{b} G_{ac}(\ln \Lambda,\lambda^{a}) - \partial_{c} G_{ab}(\ln \Lambda,\lambda^{a}) \Big) \nonumber \\&& = \int D \phi(x) P[\ln \Lambda,\phi(x);\lambda^{a}] \Big\{ \Big( \partial_{a} \partial_{b} \ln P[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \Big( \partial_{c} \ln P[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \nonumber \\&& + \Big( \partial_{a} \ln P[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \Big( \partial_{b} \ln P[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \Big( \partial_{c} \ln P[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \Big\} . \label{FRG95Connection95Torsion95Free} \end{align}\tag{28}\]

Based on the Christoffel symbols defined in Eq. (28 ), the Ricci tensor \(R_{ab}\), which governs the geometric curvature of the coupling constant space, is constructed through the standard Riemannian differential structure: \[\begin{align} && R_{ab}(\ln \Lambda,\lambda^{a}) = \partial_{c} \Gamma_{ab}^{c}(\ln \Lambda,\lambda^{a}) - \partial_{b} \Gamma_{ac}^{c}(\ln \Lambda,\lambda^{a}) + \Gamma_{cd}^{c}(\ln \Lambda,\lambda^{a}) \Gamma_{ab}^{d}(\ln \Lambda,\lambda^{a}) - \Gamma_{bd}^{c}(\ln \Lambda,\lambda^{a}) \Gamma_{ac}^{d}(\ln \Lambda,\lambda^{a}) . \label{FRG95Ricci95Tensor} \end{align}\tag{29}\] This definition ensures geometric covariance under arbitrary reparameterizations of the coupling constants. However, as previewed in Section 2, the naive curvature calculation derived from Eq. (31 ) does not inherently satisfy the tensor transformation law of a standard Ricci tensor: \[\begin{align} && R_{ab}^{\text{naive}}(\ln \Lambda,\lambda^{a}) \not= \partial_{c} \Gamma_{ab}^{c}(\ln \Lambda,\lambda^{a}) - \partial_{b} \Gamma_{ac}^{c}(\ln \Lambda,\lambda^{a}) + \Gamma_{cd}^{c}(\ln \Lambda,\lambda^{a}) \Gamma_{ab}^{d}(\ln \Lambda,\lambda^{a}) - \Gamma_{bd}^{c}(\ln \Lambda,\lambda^{a}) \Gamma_{ac}^{d}(\ln \Lambda,\lambda^{a}) . \label{Ricci95Curvature95Naive} \end{align}\tag{30}\] The inequality in Eq. (30 ) highlights the central mathematical challenge addressed in this study. The fact that the curvature terms obtained from statistical field-theoretic averages (\(R_{ab}^{\text{naive}}\)) do not perfectly match the geometric Ricci tensor (\(R_{ab}\)) inside the QFT information space implies that additional geometric correction terms (diffeomorphisms) are essential for a rigorous mapping.

This geometric obstruction, also analyzed in detail from first principles in Appendix A, is resolved by the potential-driven diffeomorphism (\(X_a = \nabla_a\Phi\)). By employing the explicit Ricci tensor formulation from Eq. (29 ), we obtain the governing equation for the vector field \(X_{a}\) that reconciles the raw statistical evolution of the metric in Eq. (24 ) with the canonical Ricci flow in Eq. (26 ): \[\begin{align} && \nabla_{a} X_{b}(\ln \Lambda,\lambda^{a}) + \nabla_{b} X_{a}(\ln \Lambda,\lambda^{a}) = 2 R_{ab}(\ln \Lambda,\lambda^{a}) \nonumber \\&& + \int D \phi(x) \int_{M} d^{d} x\Big[ \Big(\frac{\partial}{\partial \lambda^{a}} \mathcal{V}[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \frac{\delta}{\delta \phi(x)} \Big( \frac{\partial}{\partial \lambda^{b}} \ln P[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \nonumber \\&& + \Big(\frac{\partial}{\partial \lambda^{b}} \mathcal{V}[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \frac{\delta}{\delta \phi(x)} \Big( \frac{\partial}{\partial \lambda^{a}} \ln P[\ln \Lambda,\phi(x);\lambda^{a}] \Big) \Big] . \label{FRG95To95Determine95Xa} \end{align}\tag{31}\] Ultimately, this differential constraint determines the specific potential-driven diffeomorphism required to ensure that the information manifold deforms smoothly along the covariant Ricci flow.

3.4 \(\mathcal{F}-\)entropy functional↩︎

To provide a geometric synthesis of the statistical state and its scale-dependent evolution, we lift the thermodynamic free energy functional \(\mathcal{E}[P]\) to the infinite-dimensional configuration space, which acts as the canonical driver for the optimal transport dynamics within the functional JKO-Wasserstein paradigm [48], [49]: \[\mathcal{E}[P[\ln \Lambda,\phi(x);\lambda^{a}]] = \int D \phi(x) \, P[\ln \Lambda,\phi(x);\lambda^{a}] \Big( \ln P[\ln \Lambda,\phi(x);\lambda^{a}] + V_{Pol.}[\ln \Lambda,\phi(x)] \Big) , \label{Infinite95Free95Energy95Functional}\tag{32}\] where \(V_{Pol.}\) denotes the Polchinski effective potential defined in Eq. (15 ). Under this formulation, the original Polchinski Fokker-Planck equation in Eq. (14 ) can be reformulated into an exact functional gradient flow representation through the variation of this thermodynamic free energy \(\mathcal{E}[P]\): \[\frac{d}{d \ln \Lambda} P[\ln \Lambda,\phi(x);\lambda^{a}] = \int_{M} d^{d}x \int_{M} d^{d}y \, \frac{\delta}{\delta \phi(x)} \left[ P[\ln \Lambda,\phi(x);\lambda^{a} ] C_{\Lambda}^{Pol.}(x,y) \frac{\delta}{\delta \phi(y)} \left( \frac{\delta \mathcal{E}[P[\ln \Lambda,\phi(x);\lambda^{a}]]}{\delta P[\ln \Lambda,\phi(x);\lambda^{a}]} \right) \right] . \label{FRG95True95Gradient95Flow}\tag{33}\] The reformulation of the renormalization group as a functional gradient flow in Eq. (33 ) naturally leads to a fundamental reinterpretation of the probability velocity field \(\Psi\), which governs the deterministic motion of the theory in field space. In direct correspondence with the relation between the velocity field \(v_{i}\) and the entropy variation established in the finite-dimensional case, the functional velocity field \(\Psi\) can be explicitly derived as: \[\begin{align} && \Psi[\ln \Lambda,\phi(x);\lambda^{a}] = - \int_{M} d^{d} y ~ C_{\Lambda}^{Pol.}(x,y) \frac{\delta}{\delta \phi(y)} \Bigg\{ \frac{\delta \mathcal{E}[P[\ln \Lambda,\phi(x);\lambda^{a}]]}{\delta P[\ln \Lambda,\phi(x);\lambda^{a}]} \Bigg\} . \label{FRG95Current95W95Entropy} \end{align}\tag{34}\] Equation (34 ) reveals the underlying mechanism of the FRG scale evolution: the velocity field \(\Psi\) is precisely the canonical direction that drives the continuous deformation on the \(\mathcal{E}\)-energy landscape, weighted by the Polchinski kernel \(C^{Pol.}_{\Lambda} (x, y)\). This kernel effectively acts as a local metric in the configuration space, translating the variational forces into a smooth trajectory of the field distribution toward a state of optimized geometric equilibrium.

The instantaneous rate of this energy dissipation along the renormalization trajectory defines the field-theoretic \(\mathcal{F}\)-entropy functional, establishing a direct first-principles link to Perelman’s energy framework [21]: \[\begin{align} & \mathcal{F}[P[\ln \Lambda,\phi(x);\lambda^{a}]] \equiv -\frac{d\mathcal{E}[P[\ln \Lambda,\phi(x);\lambda^{a}]]}{d\ln\Lambda} = - \int D \phi(x) \, \left( \frac{\delta \mathcal{E}[P[\ln \Lambda,\phi(x);\lambda^{a}]]}{\delta P[\ln \Lambda,\phi(x);\lambda^{a}]} \right) \frac{d P[\ln \Lambda,\phi(x);\lambda^{a}]}{d\ln\Lambda} \nonumber \\ &= \int D \phi(x) \, P[\ln \Lambda, \phi(x); \lambda^a] \int_{M} d^{d}x \int_{M} d^{d}y \, \left\{ \frac{1}{2} \left( \frac{\delta \ln P[\ln \Lambda,\phi(x);\lambda^{a}]}{\delta \phi(x)} \right) C_{\Lambda}^{Pol.}(x,y) \left( \frac{\delta \ln P[\ln \Lambda,\phi(x);\lambda^{a}]}{\delta \phi(y)} \right) \right. \nonumber \\ &\left. + \frac{1}{2} \left(\frac{\delta V_{Pol.}[\ln \Lambda,\phi(x)]}{\delta \phi(x)}\right) C_{\Lambda}^{Pol.}(x,y) \left(\frac{\delta V_{Pol.}[\ln \Lambda,\phi(x)]}{\delta \phi(y)}\right) - \frac{\delta}{\delta \phi(x)} \left( C_{\Lambda}^{Pol.}(x,y) \frac{\delta V_{Pol.}[\ln \Lambda,\phi(x)]}{\delta \phi(y)} \right) \right\} . \label{Lifted95F95Entropy95Perfect} \end{align}\tag{35}\] Equation (35 ) completely vindicates the structure of the lifted entropic potential, where the first term represents the informational dissipation from stochastic diffusion, and the second term accounts for the background potential landscape driven by the cutoff kernel.

By taking the second-order variation of this lifted \(\mathcal{F}\)-entropy with respect to the macroscopic coupling constants \(\lambda _{a}\) and \(\lambda _{b}\), we establish a central structural identity linking the thermodynamic scale-evolution to the underlying information geometry: \[\begin{align} && \frac{d }{d \ln \Lambda} G_{ab}(\ln \Lambda,\lambda^{a}) = - 2 \frac{\partial^{2} \mathcal{F}\Big(P[\ln \Lambda,\phi(x);\lambda^{a}]\Big)}{\partial \lambda^{a} \partial \lambda^{b}} . \label{FRG95Fisher95Information95Gradient95Flow} \end{align}\tag{36}\] Equation (36 ) demonstrates that the continuous flow of the metric in the QFT theory space is fundamentally driven by the parametric Hessian of the system’s \(\mathcal{F}\)-entropy potential. Physically, this implies that the Wilsonian coarse-graining acts as an information-lossy dynamic contraction or expansion of the underlying information manifold, providing the complete mathematical apparatus required to embed the exact FRG flow within Perelman’s modified Ricci flow framework.

To establish a definitive geometric equivalence between the metric evolution and the Ricci flow, we derive the consistency condition relating the curvature of the coupling constant space to the Hessian of the \(\mathcal{F}\)-entropy functional. By comparing the metric flow equation in Eq. (26 ) with the entropic identity in Eq. (36 ), we arrive at the following constraint: \[\begin{align} && \nabla_{a} X_{b}(\ln \Lambda,\lambda^{a}) + \nabla_{b} X_{a}(\ln \Lambda,\lambda^{a}) = 2 R_{ab}(\ln \Lambda,\lambda^{a}) - 2 \frac{\partial^{2} \mathcal{F}\Big(P[\ln \Lambda,\phi(x);\lambda^{a}]\Big)}{\partial \lambda^{a} \partial \lambda^{b}} . \end{align}\] To resolve the geometric mismatch and interpret the renormalization group as a potential flow, we introduce the Ansatz that the vector field \(X_{a}\)—which absorbs the unphysical coordinate redundancies—can be realized as the gradient of a scalar information potential \(\Phi\): \[\begin{align} && X_{a}(\ln \Lambda,\lambda^{a}) = \nabla_{a} \Phi(\ln \Lambda,\lambda^{a}) . \label{FRG95Ansatz} \end{align}\tag{37}\] As a mathematical necessity under the dualistic connection structure, substituting Eq. (37 ) into the consistency condition yields a local balance between the Ricci curvature and the entropic variation: \[\begin{align} && R_{ab}(\ln \Lambda,\lambda^{a}) - \nabla_{a} \nabla_{b} \Phi(\ln \Lambda,\lambda^{a}) = \frac{\partial^{2} \mathcal{F}\Big(P[\ln \Lambda,\phi(x);\lambda^{a}]\Big)}{\partial \lambda^{a} \partial \lambda^{b}} , \label{FRG95Constraint95Eq} \end{align}\tag{38}\] where a torsion-free Levi-Civita connection has been assumed. Equation (38 ) constitutes the exact modified form of Perelman’s steady Ricci soliton equation mapped onto the infinite-dimensional space of quantum field theories, where the effective average action acts as a geometric gauge-fixing field that restores full diffeomorphism invariance. Conformal fixed points and quantum critical phenomena are thus recast as dynamic geometric equilibria where the intrinsic informational curvature is perfectly balanced by the gradient of the quantum effective landscape.

Building upon this geometric consistency, we define the global geometric \(\mathcal{F}\)-entropy functional on the coupling constant manifold in direct analogy to Perelman’s energy functional for the proof of the Poincaré conjecture [21]: \[\begin{align} && \mathcal{F}[G_{ab}(\ln \Lambda,\lambda^{a})] = \int d^{N} \lambda \sqrt{G(\ln \Lambda,\lambda^{a})} e^{- \Phi(\ln \Lambda,\lambda^{a})} \Big\{ R(\ln \Lambda,\lambda^{a}) + G^{ab}(\ln \Lambda,\lambda^{a}) \partial_{a} \Phi(\ln \Lambda,\lambda^{a}) \partial_{b} \Phi(\ln \Lambda,\lambda^{a}) \Big\} . \label{FRG95W95Entropy95Functional95Perellman} \end{align}\tag{39}\] Again, the presence of the entropic weighting factor \(\sqrt{G} e^{-\Phi}\) within the parametric volume element is a direct consequence of the dualistic information-geometric mapping, ensuring that the functional behaves as a Lyapunov-like potential for the renormalization group.

Finally, we arrive at the central conclusion of this framework: the scale-dependent evolution of the information metric \(G_{ab}\) under the FRG flow is precisely a pure Riemannian gradient flow of this global \(\mathcal{F}\)-entropy functional: \[\begin{align} && \frac{d }{d \ln \Lambda} G_{ab}(\ln \Lambda,\lambda^{a}) = - 2 R_{ab}(\ln \Lambda,\lambda^{a}) + 2 \nabla_{a} \nabla_{b} \Phi(\ln \Lambda,\lambda^{a}) = - 2 \frac{\delta \mathcal{F}[G_{ab}(\ln \Lambda,\lambda^{a})]}{\delta G^{ab}(\ln \Lambda,\lambda^{a})} . \label{FRG95Ricci95Flow95Gradient95Flow} \end{align}\tag{40}\] The fact that the rate of metric change is encoded within the variational derivative \(-2 \frac{\delta \mathcal{F}}{\delta G^{ab}}\) proves that the non-perturbative RG process is a rigorous, first-principles geometric optimization process. Within this unified architecture, the effective average action acts as the ultimate geometric gauge-fixing agent, absorbing all basis-dependent, non-tensorial artifacts to ensure that the evolution is driven strictly by the intrinsic curvature of the entropic landscape.

4 Conclusion↩︎

The unified framework established in this work offers a profound topological and physical reassessment of the Functional Renormalization Group (FRG) evolution. Rather than merely dragging the coupling constants along a static, flat parameter space, the successive integration of short-distance quantum fluctuations actively deforms the underlying information manifold itself, effectively smoothing out its informational curvature toward a canonical geometric attractor.

Within this architecture, the emergent potential \(\Phi\) acts as a geometric gauge-fixing mechanism, dynamically absorbing the non-tensorial coordinate artifacts that historically hindered a covariant formulation of the renormalization group. By counteracting the basis-dependent redundancies inherent in the choice of effective operators, the combination \(R_{ab} - \nabla_a \nabla_b \Phi\) emerges as the generalized, covariant curvature tensor that governs the purely tensorial geometry of the theory space.

Ultimately, this exact mapping dictates that the non-perturbative fixed points of the FRG flow, traditionally characterized by the vanishing of the beta functions (\(\beta^a = 0\)), map smoothly onto Perelman’s steady Ricci solitons satisfying \(R_{ab} - \nabla_a \nabla_b \Phi = 0\). Quantum critical phenomena and conformal invariance are thus liberated from their classical static descriptions; they are instead recast as dynamic geometric equilibria where the intrinsic curvature of the QFT information manifold is perfectly balanced by the gradient force of the entropic landscape. This conceptual synthesis bridges the historical gap between microscopic quantum field theory and macroscopic geometric analysis, providing a rigorous, first-principles foundation for investigating the topological stability and universality of physical systems.

5 Some remarks on section IIB↩︎

To provide a geometric interpretation of the information metric evolution derived in Eq. (3 ), we decompose it into a canonical Ricci curvature flow and a geometric correction term driven by an emergent diffeomorphism: \[\begin{align} && \dot{g}_{ab}(t,\theta^{a}) =- 2 R_{ab}(t,\theta^{a}) + \nabla_{a} X_{b}(t,\theta^{a}) + \nabla_{b} X_{a}(t,\theta^{a}) , \label{tdfxyncr} \end{align}\tag{41}\] where \(R_{ab}\) denotes the intrinsic Ricci curvature tensor of the information space. By performing a direct comparison between Eq. (5 ) and Eq. (3 ), the parameter-space vector field \(X_{a}\) can be considered as: \[\begin{align} && X_{a}(t,\theta^{a}) = \int d^{d} x p(x,t;\theta^{a}) v_{i}(x,t;\theta^{a}) \partial_{i} \Big( \partial_{a} \ln p(x,t;\theta^{a}) \Big) . \label{Diffeo95Xa} \end{align}\tag{42}\] The covariant derivative of this vector field, \(\nabla_{a} X_{b}\), is defined to reflect the standard Riemannian connection of the underlying information space: \[\begin{align} && \nabla_{a} X_{b}(t,\theta^{a}) = \partial_{a} X_{b}(t,\theta^{a}) - \Gamma_{ab}^{c}(t,\theta^{a}) X_{c}(t,\theta^{a}) . \end{align}\] The Christoffel symbols \(\Gamma _{abc}\), which characterize the torsion-free Levi-Civita connection on the information manifold, are derived from the Fisher metric and expressed in terms of the functional variations of the probability distribution \(p\): \[\begin{align} && \Gamma_{abc}(t,\theta^{a}) = \frac{1}{2} \Big( \partial_{a} g_{bc}(t,\theta^{a}) + \partial_{b} g_{ac}(t,\theta^{a}) - \partial_{c} g_{ab}(t,\theta^{a}) \Big) \nonumber \\&& = \int d^{d} x p(t,x;\theta^{a}) \Big\{ \Big( \partial_{a} \partial_{b} \ln p(t,x;\theta^{a}) \Big) \Big( \partial_{c} \ln p(t,x;\theta^{a}) \Big) \nonumber \\&& + \Big( \partial_{a} \ln p(t,x;\theta^{a}) \Big) \Big( \partial_{b} \ln p(t,x;\theta^{a}) \Big) \Big( \partial_{c} \ln p(t,x;\theta^{a}) \Big) \Big\} . \label{Connection95Torsion95Free} \end{align}\tag{43}\]

When the vector field \(X_{a}\) for the diffeomorphism is fixed as above, the matching procedure between Eq. (5 ) and Eq. (3 ) automatically determines the specific components of the raw, unconstrained statistical curvature field, given by Eq. (45 ). We call this curvature \(R_{ab}^{\text{naive}}(t,\theta^{a})\). However, \(R_{ab}^{\text{naive}}(t,\theta^{a})\) fails to satisfy the tensor transformation law of standard Riemannian geometry, yielding a strict geometric mismatch: \[\begin{align} && R_{ab}^{\text{naive}}(t,\theta^{a}) \not= \partial_{c} \Gamma_{ab}^{c}(t,\theta^{a}) - \partial_{b} \Gamma_{ac}^{c}(t,\theta^{a}) + \Gamma_{cd}^{c}(t,\theta^{a}) \Gamma_{ab}^{d}(t,\theta^{a}) - \Gamma_{bd}^{c}(t,\theta^{a}) \Gamma_{ac}^{d}(t,\theta^{a}) = R_{ab}(t,\theta^{a}) . \label{Ricci95Curvature95Original} \end{align}\tag{44}\] The inequality in Eq. (44 ) highlights a critical mathematical issue: the naive curvature calculation does not satisfy the tensorial property of a standard Ricci tensor.

By employing the coordinate-invariant tensor structure, we can derive the master equation for the vector field \(X_{a}\) required to reconcile the raw statistical evolution of the metric in Eq. (3 ) with the canonical Ricci flow in Eq. (5 ): \[\begin{align} && \nabla_{a} X_{b}(t,\theta^{a}) + \nabla_{b} X_{a}(t,\theta^{a}) = 2 R_{ab}(t,\theta^{a}) \nonumber \\&& + \int d^{d} x \Big[ \Big(\partial_{a} v_{i}(x,t;\theta^{a}) \Big) \partial_{i} \Big( \partial_{b} \ln p(x,t;\theta^{a}) \Big) + \Big(\partial_{b} v_{i}(x,t;\theta^{a}) \Big) \partial_{i} \Big( \partial_{a} \ln p(x,t;\theta^{a}) \Big) \Big] . \label{To95Determine95Xa} \end{align}\tag{45}\] Equation (45 ) represents a macro-micro conservation law bridging the geometric curvature of the information space with the underlying dynamical velocity fields of the statistical current.

References↩︎