Hamilton–Jacobi Holographic Renormalization

• Hamilton–Jacobi holographic renormalization is a variational method that reformulates UV divergence cancellation in bulk gravitational actions as a functional Hamilton–Jacobi problem.
• It employs a local covariant ansatz and dilatation expansion to recursively derive counterterms, effectively handling marginal and relevant deformations.
• The method bridges holographic renormalization with Wilsonian RG flows, providing clear insights into beta-functions, RG trajectories, and the extraction of conformal anomalies.

The Hamilton–Jacobi (HJ) method of holographic renormalization is a first-principles, variational approach to extracting and organizing ultraviolet (UV) divergences of gravitational (and more generally, bulk field theory) actions in spacetimes with boundary—most notably in the context of the AdS/CFT correspondence. By recasting the bulk renormalization problem as a functional HJ equation for the radial evolution of the on-shell action, the method provides a systematic, covariant, and highly generalizable framework for determining the counterterms required to render generating functionals and correlation functions in the dual field theory finite. In contrast to standard techniques such as the Fefferman–Graham (FG) expansion, the HJ approach naturally generates local counterterm ansätze, handles marginal and relevant deformations uniformly, and directly exposes compositional structures such as beta-functions, RG flows, and trace anomalies.

1. Foundations and Hamilton–Jacobi Equation

The core of the HJ method begins with the ADM decomposition of a $(d+1)$-dimensional bulk spacetime into a radial foliation,

$$ds^2 = N^2 dr^2 + \gamma_{ij} (dx^i + N^i dr)(dx^j + N^j dr),$$

where the radial coordinate $r$ plays the role of "Hamiltonian time", and $\gamma_{ij}$ is the induced metric on $r=$ constant slices. Bulk scalar and gauge fields $\Phi^I(r,x)$ and $A_i(r,x)$ are similarly decomposed. The canonical momenta $\pi^{ij}$, $\pi_I$, and (where appropriate) $\pi^i$ are given by functional derivatives of the action with respect to radial derivatives of the fields or metric.

On-shell, the action $S_{\rm os}[\gamma, \Phi]$ (as a functional of the induced fields at the cutoff surface $r=r_c$) must satisfy the radial Hamilton–Jacobi equation, which stems from the Hamiltonian constraint:

$$H\left[\gamma_{ij},\Phi^I;\,\pi^{ij} = \frac{\delta S}{\delta \gamma_{ij}},\, \pi_I = \frac{\delta S}{\delta \Phi^I}\right] = 0,$$

where $H$ is quadratic in the momenta and includes contributions from the boundary Ricci curvature, kinetic terms, and the bulk potential. Variants for massive gravity, asymptotically Lifshitz backgrounds, or higher-derivative gravity follow the same principle, but with more intricate Hamiltonians and local invariants (Elvang et al., 2016, Ma et al., 2022, Chen et al., 2019, Mann et al., 2011).

2. Local Ansatz and Dilatation Expansion

Divergences in the on-shell action arise from near-boundary behavior as $r \to \infty$. The HJ method constructs a local, covariant ansatz for the divergent part of the action, expanded in eigenfunctions of the dilatation operator (which counts powers of the inverse boundary metric or, equivalently, spatial derivatives). For gravity coupled to scalars, the local counterterm action is written as:

$$S_{\rm ct}[\gamma,\Phi] = \int d^d x \sqrt{\gamma} \left[ U_{(0)}(\Phi) + U_{(2)}(\gamma,\Phi) + U_{(4)}(\gamma,\Phi) + \cdots \right],$$

where $U_{(0)}$ is the zero-derivative (potential) term, $U_{(2)}$ contains all two-derivative invariants (such as $R[\gamma]$ and $(\partial \Phi)^2$), and $U_{(4)}$ includes four-derivative terms (curvature squared, derivative operators on the scalar, etc.). Each coefficient is determined recursively by organizing the HJ equation by dilatation weight and solving at each step for all possible covariant invariants at a fixed order (Ma et al., 2022, Elvang et al., 2016, Papadimitriou, 2011, Rajagopal et al., 2015).

This procedure enables direct algebraic determination of all divergent counterterms up to the maximal divergence degree (e.g., $2\lfloor d/2 \rfloor$ derivatives in $d$ boundary dimensions), with logarithmic divergences (i.e., conformal anomalies) manifesting automatically when the dilatation weight vanishes.

3. Recursive Solution and Generated Ansatz Algorithm

Central to efficient implementation is the recursive structure of the HJ equation. For each operator $U_{(2n)}$, the recursion has the general structure:

$$2\partial_r U_{(2n)} + F(U_{(2n)}) + \sum_{m+n'=n} H(U_{(2m)},U_{(2n')}) = 0,$$

where $F$ is linear in $U_{(2n)}$ and carries dilatation weight and mass parameters, while $H(\cdots)$ encapsulates quadratic terms in the lower-order ansatz components. The "generated ansatz" technique systematically constructs the possible set of invariants (the basis $O_n$ at each order), eliminates guesswork, and reduces the recursion for coefficient functions $c_{i a}(r)$ to solvable ODEs (Ma et al., 2022).

In practice, for generic scalar potentials and boundary geometries, the entire set of required counterterms can thus be derived recursively to the necessary order, with explicit formulas available for AdS, flat, and certain anisotropic or curved backgrounds (Kim et al., 2020, Ammon et al., 16 Dec 2025).

4. Extraction of Renormalized Quantities and Holographic Dictionary

After identifying and subtracting $S_{\rm ct}$ from the regulated on-shell action, the renormalized generating functional $S_{\rm ren}$ yields physical observables via functional differentiation:

$$\langle T^{ij}(x) \rangle = \frac{2}{\sqrt{\gamma}} \frac{\delta S_{\rm ren}}{\delta \gamma_{ij}(x)}, \quad \langle O_{\Phi^I}(x) \rangle = \frac{1}{\sqrt{\gamma}} \frac{\delta S_{\rm ren}}{\delta \Phi^I(x)},$$

with analogous expressions for other bulk fields. In the AdS/CFT context, these formulas reproduce the canonical AdS/CFT dictionary, linking boundary values (sources) to dual operator correlators, and allow the computation of conformal/Weyl anomalies directly from the coefficients of logarithmic divergences (Papadimitriou, 2010, Rajagopal et al., 2015).

For marginal and multi-trace deformations, the HJ method naturally encodes the beta-function structure and multi-trace RG flow equations. In the presence of scalar self-interactions, one finds, for marginal $n$-trace couplings $f_n$,

$$\partial_r f_n(r) = \left( d - n\Delta_- \right) f_n(r) + C_n \lambda [f_2(r)]^{n-1} + O(\lambda^2),$$

with logarithmic RG running arising for marginal couplings $n\Delta_-=d$ as

$$f_n(\mu) = f_n(\mu_0) - C_n \lambda \ln \frac{\mu}{\mu_0} + \cdots,$$

precisely matching field-theoretical expectations and boundary-condition approaches (Kim et al., 2021).

5. Generalizations, Applications, and Extensions

The HJ formalism has been adapted and systematically applied to a breadth of settings:

• Thermal states and black holes: At finite temperature, the counterterm structure must account for the broken dilatation symmetry due to compact Euclidean time and the presence of a horizon. The HJ approach yields modified Callan–Symanzik equations and Ward identities involving temperature derivatives and entropy density (Cáceres et al., 11 Oct 2025).
• Non-conformal branes and non-AdS backgrounds: Recursive HJ algorithms have been developed for Einstein–Maxwell–dilaton models and non-conformal D$p$-branes $(p<5)$, with the asymptotic expansion and counterterms adapted to the appropriate scaling exponents and potential forms (Korpas, 2022).
• Massive gravity and higher-derivative theories: By including additional polynomial invariants (in the square root of the metric, for example), the HJ approach accommodates ghosts-free massive gravity and higher-derivative interactions; conformal anomalies in odd dimensions can emerge due to diffeomorphism-breaking terms (Chen et al., 2019, Rajagopal et al., 2015).
• Curved boundaries and mass deformations: On spheres or more general boundary topologies, the HJ characteristic function receives curvature-dependent contributions, and mass deformations generate new terms in the series solution, reflecting correct operator mixing and conformal dimension dependence (Kim et al., 2020).
• Lifshitz and flat-space holography: The HJ method extends to anisotropic Lifshitz backgrounds or flat holography, including adaptations to Carrollian boundaries and null slices (Ammon et al., 16 Dec 2025, Mann et al., 2011).

6. Relation to Wilsonian RG, Functional RG, and Canonical Transformations

A foundational insight underlying the HJ approach is its equivalence, in structure, to Wilsonian renormalization group flows. The radial variable in holography is mapped to the Wilsonian cutoff scale, and the HJ equation becomes the functional RG flow equation for the effective boundary action. For matrix models, large-$N$ gauge theories, or hydrodynamic examples, there is a precise matching between the bulk HJ flow and the Polchinski (or Wetterich) RG equation upon mapping the radial coordinate to the inverse energy scale (Radicevic, 2011, Ogarkov, 2016, Ivanov et al., 2020). This formalism unifies holographic, functional, and canonical RG approaches in a common Hamiltonian structure, with the counterterms interpreted as boundary terms that effect canonical transformations on phase space, rendering the variational problem well-posed and boundary correlators finite (Papadimitriou, 2010).

7. Advantages, Limitations, and Open Problems

Advantages:

• Systematic, local, and covariant identification and removal of divergences.
• Unified treatment of marginal, relevant, and multi-trace deformations, including explicit beta-functions and RG equations.
• Algorithmic and recursive construction of counterterms at arbitrary derivative order and for general field content.
• Direct applicability to curved, anisotropic, and non-AdS backgrounds via appropriate adaptation of the ansatz (Ma et al., 2022, Elvang et al., 2016).

Limitations and Open Problems:

• For theories with many fields or high-derivative corrections, the recursive ODE system becomes increasingly complex.
• Finite counterterms are not fixed by the HJ equation and must be determined by additional physical or symmetry input.
• Extensions to nonlocal counterterms (relevant for flat-space holography) and genuine strongly coupled backgrounds are active topics of research.
• The choice of renormalization scheme, Weyl consistency conditions, and scheme ambiguities in the HJ context require precise characterization (Rajagopal et al., 2015).

The HJ method thus constitutes the modern technical backbone for holographic renormalization, enabling both explicit computations and conceptual insights across a wide array of bulk/boundary systems and serving as the natural interface connecting bulk Hamiltonian dynamics, boundary quantum renormalization, and the global consistency of the holographic dictionary.