Radial Cutoffs and Hamilton–Jacobi Flow

The radial Hamilton–Jacobi equation turns holographic renormalization into a local asymptotic recursion. Hamilton’s principal function is the regulated on-shell action at a cutoff surface; replacing its canonical momenta by functional derivatives converts the bulk Hamiltonian constraint into equations for divergent counterterms. The finite, nonlocal remainder carries response and state data that the ultraviolet recursion cannot determine.

Required background. Fefferman–Graham expansions identify local and normalizable radial coefficients. Helpful background. Local counterterms and subdivergences explain why ultraviolet divergences are local, Wilsonian coarse graining supplies the comparison that must not be assumed, and the 1PI effective action fixes functional-derivative conventions.

First application. Insert a local scalar ansatz for the cutoff action into the radial Hamilton-Jacobi equation and solve its leading divergent coefficients.

Radial phase space and the Hamiltonian constraint

Choose an outward radial coordinate $r$ and decompose a Euclidean bulk metric as

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

The induced fields $(\gamma_{ij},\phi)$ are coordinates on radial phase space. Their momenta are

$$\pi^{ij}=\frac{\delta S_r}{\delta\gamma_{ij}}, \qquad \pi_\phi=\frac{\delta S_r}{\delta\phi},$$

when evaluated on a classical solution ending on the cutoff surface. Lapse and shift are Lagrange multipliers, so the principal function $S_r[\gamma,\phi]$ obeys Hamiltonian and momentum constraints rather than an unconstrained evolution equation:

$$\mathcal H\!\left(\gamma,\phi; \frac{\delta S_r}{\delta\gamma}, \frac{\delta S_r}{\delta\phi}\right)=0, \qquad \mathcal H_i\!\left(\gamma,\phi; \frac{\delta S_r}{\delta\gamma}, \frac{\delta S_r}{\delta\phi}\right)=0.$$

For Einstein gravity with the site curvature convention, the precise signs depend on whether the unit normal points toward increasing or decreasing $r$. A derivation must state that choice together with the Gibbons–Hawking term. Constraint identities are a safer convention check than comparing an isolated sign in $\pi^{ij}$.

Local derivative recursion

Near an AdS boundary, split

$$S_r=S_{\mathrm{loc}}+\Gamma_r,$$

where $S_{\mathrm{loc}}$ is a covariant derivative expansion,

$$S_{\mathrm{loc}} =\int\mathrm d^d x\sqrt\gamma \left[U(\phi)+\Phi(\phi)R[\gamma] +\frac12M(\phi)(\nabla\phi)^2+\cdots\right],$$

and $\Gamma_r$ contains finite nonlocal data. Substitution in the Hamilton–Jacobi constraint orders terms by dilatation weight. Algebraic equations determine $U$, then $\Phi$ and $M$, and then higher-derivative coefficients. A vanishing recursion denominator signals a logarithmic term and a conformal anomaly, not an arbitrary failure of the method Papadimitriou and Skenderis 2004, §§2–4.

The split is not unique. Adding a finite local functional shifts contact terms and canonical momenta while preserving the symplectic form. Changing a boundary condition or performing a Legendre transform, by contrast, can define a different theory or ensemble.

Free-scalar calculation at a cutoff

Consider a Euclidean scalar on fixed Poincaré AdS,

$$S=\frac12\int_{z\ge\epsilon}\!\mathrm d^{d+1}x\sqrt G \left(G^{MN}\partial_M\phi\partial_N\phi+m^2\phi^2\right).$$

The outward normal to the regulated region at $z=\epsilon$ is $n=-(z/L)\partial_z$. On shell,

$$S_{\mathrm{reg}} =\frac12\int_{z=\epsilon}\!\mathrm d^d x\sqrt\gamma\, \phi\,n\!\cdot\!\partial\phi.$$

With $\phi\sim z^{d-\Delta}\phi_{(0)}$, its leading term is

$$S_{\mathrm{reg}} \sim-\frac{d-\Delta}{2L} \int\mathrm d^d x\sqrt\gamma\,\phi^2.$$

The local Hamilton–Jacobi ansatz therefore begins with

$$S_{\mathrm{ct}}^{(0)} =\frac{d-\Delta}{2L} \int\mathrm d^d x\sqrt\gamma\,\phi^2,$$

for this normal and Euclidean action convention. Reversing the normal reverses both the regulated boundary term and the counterterm sign; the renormalized variational problem is unchanged after a consistent translation.

At the next derivative order, the Hamilton–Jacobi equation produces the same denominator $2\Delta-d-2$ that appears in the Fefferman–Graham recursion. Thus the asymptotic field equation and the canonical method independently locate the logarithmic resonance.

What the flow determines

The divergent terms of $S_r$ are local because they arise at finite dilatation weight. The finite functional $\Gamma=\lim_{r\to\infty}\Gamma_r$ determines renormalized connected correlators after a state and interior condition are supplied. Its functional derivatives satisfy the finite Ward identities inherited from $\mathcal H_i$ and any gauge constraints.

The radial equation is not automatically a Wilsonian beta function. A surface at one radial position does not impose a sharp, state-independent boundary momentum cutoff, and integrating out a radial region generally generates momentum-dependent multi-trace terms. The correct statement here is narrower: the Hamilton–Jacobi constraint evolves cutoff data and recursively isolates local divergences.

Adversarial and invariant checks

1. Reverse the normal and verify that $S_{\mathrm{reg}}+S_{\mathrm{ct}}$ is unchanged after translating both signs.
2. Insert the counterterm momentum into the Hamiltonian constraint and check cancellation at each dilatation weight.
3. At a resonance, replace the singular coefficient by a logarithmic term and recover the anomaly.
4. Apply a finite local canonical transformation and verify preservation of $\int\delta\pi\wedge\delta q$.
5. Vary $\Gamma$ under a boundary diffeomorphism and recover the momentum Ward identity.

A procedure that cancels the action divergence but violates a constraint or the variational principle has not completed holographic renormalization.

Exercises

Why can the Hamilton–Jacobi recursion determine divergent counterterms without determining the state-dependent one-point function?

Solution

Divergent coefficients occur at fixed local dilatation weights and are fixed algebraically by the near-boundary constraint. The normalizable coefficient belongs to the finite nonlocal solution $\Gamma$ and depends on interior regularity, causal conditions, or the chosen state. Ultraviolet locality cannot determine that global datum.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• de Boer, J., Verlinde, E., and Verlinde, H. “On the Holographic Renormalization Group.” Journal of High Energy Physics 2000, 003 (2000). DOI. arXiv.
• Papadimitriou, I., and Skenderis, K. “AdS/CFT Correspondence and Geometry.” In IRMA Lectures in Mathematics and Theoretical Physics 8, 73–101 (2005). DOI. arXiv.
• Papadimitriou, I., and Skenderis, K. “Correlation Functions in Holographic RG Flows.” Journal of High Energy Physics 2004, 075 (2004). DOI. arXiv.
• Skenderis, K. “Lecture Notes on Holographic Renormalization.” Classical and Quantum Gravity 19 (2002): 5849–5876. DOI. arXiv.