Wilsonian Cutoffs and Integrating-Out Proposals

Cutting an asymptotically AdS spacetime at a finite radius gives an exact factorization of the bulk path integral, but it does not by itself give a Wilsonian effective action with a sharp boundary momentum cutoff. The precise statement is more useful: the ultraviolet radial region prepares a cutoff wavefunctional whose Hamilton–Jacobi evolution generates momentum-dependent multi-trace couplings. A Wilsonian interpretation is controlled only after the state, foliation, regulator map, and locality range have been specified.

Required background. Radial Cutoffs and Hamilton–Jacobi Flow supplies the radial canonical evolution used below. Wilsonian Coarse Graining and Theory Space supplies the field-theory meaning of integrating out modes.

Helpful background. Hamiltonian Constraints and Radial Canonical Transformations explains why gravitational radial evolution is constrained rather than ordinary time evolution.

First application. Integrate out a free scalar region between the AdS boundary and a finite radius and derive the induced quadratic multi-trace kernel.

Splitting the bulk path integral

Use Euclidean Poincaré AdS for the cleanest construction,

$$ds^2=\frac{L^2}{z^2}\left(dz^2+dx^i dx^i\right), \qquad 0<z<\infty ,$$

and introduce a surface $z=\ell$. Holding the field value $\varphi_\ell(x)$ fixed on that surface factorizes the path integral as

$$Z[\alpha] =\int \mathcal D\varphi_\ell\, \Psi_{\mathrm{UV}}[\alpha,\varphi_\ell;\ell]\, \Psi_{\mathrm{IR}}[\varphi_\ell;\ell].$$

$\Psi_{\mathrm{UV}}$ integrates over $\epsilon<z<\ell$ with asymptotic source $\alpha$; $\Psi_{\mathrm{IR}}$ integrates over $z>\ell$ with whatever regularity, state, or interior boundary condition defines the problem. The equality is a gluing identity. Its interpretation as renormalization requires additional input because a radial slice retains all boundary momenta and because changing the foliation changes the split.

Writing $\Psi_{\mathrm{UV}}=e^{-S_{\mathrm{UV}}}$ turns radial Schrödinger evolution into a functional Hamilton–Jacobi equation at leading order in the bulk loop expansion,

$$\partial_\ell S_{\mathrm{UV}} +H_\ell\!\left[\varphi_\ell, \frac{\delta S_{\mathrm{UV}}}{\delta\varphi_\ell}\right]=0.$$

This equation is exact classically and receives functional second-derivative terms quantum mechanically. Local counterterms remove the $\epsilon\to0$ divergences; the remaining finite functional depends on the chosen subtraction and on the UV boundary condition.

Free scalar and the induced double-trace kernel

For a free scalar, translation invariance makes the cutoff functional Gaussian. At vanishing asymptotic source write

$$S_{\mathrm{UV}}^{(2)} =\frac12\int\!\frac{d^dk}{(2\pi)^d}\, \varphi_\ell(k)\,K_\ell(k)\,\varphi_\ell(-k).$$

Choose the outward-normal sign so that the renormalized canonical momentum at the inner edge of the UV region is $\Pi_\ell(k)=K_\ell(k)\varphi_\ell(k)$. If $u_k(z)$ solves the radial Klein–Gordon equation with the selected UV boundary condition, then

$$K_\ell(k) =\left. \frac{\sqrt g\,g^{zz}\partial_z u_k}{u_k} \right|_{z=\ell} +K_{\mathrm{ct}}(k,\ell).$$

Substitution into the Hamilton–Jacobi equation gives a Riccati flow for $K_\ell$. Equivalently, the ratio above converts the second-order bulk equation into a first-order equation. For $k\ell\ll1$, its analytic part has a derivative expansion,

$$K_\ell(k)=c_0(\ell)+c_2(\ell)k^2+c_4(\ell)k^4+\cdots ,$$

while nonanalytic terms carry long-distance response. Re-expressing the Gaussian integral in boundary variables turns this kernel into a scale-dependent quadratic multi-trace coupling. This is the concrete sense in which integrating over a radial region induces Wilsonian-looking interactions, as developed by Heemskerk and Polchinski 2011, §§2–4 and Faulkner, Liu, and Rangamani 2011, §§2–3.

The construction has three independent choices that must not be conflated:

• local counterterms choose renormalized canonical coordinates;
• the surface $z=\ell$ chooses a radial factorization;
• a boundary coarse-graining prescription chooses which field-theory modes are removed.

Only the last item is Wilsonian by definition.

Why radius is not a sharp momentum cutoff

The redshift relation suggests a scale $\mu\sim L^{-1}e^{A}$, or $\mu\sim1/\ell$ in Poincaré coordinates. It correctly organizes scale covariance and many derivative expansions. It does not project out all modes with $\lvert k\rvert>\mu$: the field $\varphi_\ell(k)$ exists for every $k$, and the kernel becomes strongly momentum dependent when $k\ell\gtrsim1$.

A literal Wilson action can be recovered only after specifying a boundary regulator and a map from bulk data to the regulated boundary variables. Such maps may be nonlocal, and different maps can yield actions related by field redefinitions or by scheme transformations rather than identical functionals. The radial construction therefore supports a family of Wilsonian proposals, not a regulator-independent identity between position and energy.

Gauge fields and gravity sharpen the distinction. Gauss and diffeomorphism constraints couple data across the cutoff surface, edge modes may be needed for factorization, and gauge-invariant boundary operators are not obtained by independently integrating each local bulk field. The radial wavefunctional must satisfy the constraints before it can be compared with a gauge-invariant boundary effective action.

State, foliation, and real-time dependence

In Euclidean vacuum preparations, regularity in the interior often selects a convenient $\Psi_{\mathrm{IR}}$. A thermal circle, an excited-state insertion, or a Lorentzian Schwinger–Keldysh contour changes the state functional without changing the coordinate value of $\ell$. The same radial surface can therefore correspond to different effective density matrices or influence functionals.

Likewise, two bulk foliations that approach the same boundary scale need not induce the same intermediate canonical variables. After transforming observables and counterterms consistently they can agree on physical boundary correlators, but their cutoff actions need not agree term by term. A proposed exact radial/Wilsonian map must state how it behaves under these changes.

Validity test: hold the boundary scale fixed

An effective diagnostic is to compare two preparations with the same asymptotic theory and nominal boundary scale:

1. compute $K_\ell(k)$ in the Euclidean vacuum;
2. repeat with a thermal or excited-state interior condition;
3. change to a foliation whose induced metric gives the same local redshift scale;
4. compare separated-point observables after applying the same boundary regulator.

The kernels may differ because a cutoff functional contains state and canonical-coordinate information. Agreement of renormalized long-distance observables is the invariant test. If a proposal predicts a unique, state-independent local cutoff action, these examples falsify that stronger statement; they do not falsify radial factorization or Hamilton–Jacobi evolution.

What the construction licenses

The calculation establishes that integrating over a radial region produces an evolving boundary functional and, for a free scalar, an explicit momentum-dependent multi-trace kernel. It supports a Wilsonian interpretation in a declared low-momentum derivative expansion with a fixed regulator, state, and foliation. It does not establish a universal sharp map $z=1/\mu$, nor does it show that geometric radial evolution alone integrates out boundary degrees of freedom.

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

• Faulkner, Thomas, Hong Liu, and Mukund Rangamani. “Integrating Out Geometry: Holographic Wilsonian RG and the Membrane Paradigm.” Journal of High Energy Physics 2011, 51 (2011). DOI; arXiv:1010.4036.
• Heemskerk, Idse, and Joseph Polchinski. “Holographic and Wilsonian Renormalization Groups.” Journal of High Energy Physics 2011, 31 (2011). DOI; arXiv:1010.1264.
• Papadimitriou, Ioannis. “Holographic Renormalization as a Canonical Transformation.” Journal of High Energy Physics 2010, 014 (2010). DOI; arXiv:1007.4592.