Flat-Space Holographic Dictionary

• Flat-Space Holographic Dictionary is a systematic framework relating Minkowski boundary quantum field theory correlators to bulk dynamics through adapted holographic renormalization.
• It employs the Hamilton–Jacobi formalism with asymptotic expansion and a Riccati-type differential equation to compute counterterms and renormalize the on-shell action.
• The method bridges scattering theory and Carrollian symmetry, aligning Dirichlet boundary conditions with S-matrix elements and classical field observables.

The flat-space holographic dictionary encompasses the systematic relation between correlators or observables in a quantum field theory (QFT) defined on the boundary of flat (Minkowski) spacetime and the classical dynamics of fields, typically scalar or gravitational, in the corresponding bulk geometry. Unlike the AdS/CFT correspondence, where the spatial infinity is conformal and the dictionary is underpinned by fixed boundary geometries, the flat-space dictionary must reconcile the fundamentally different IR and boundary conditions of Minkowski spacetime. Core to the construction is the adaptation of holographic renormalization: the flat-space Hamilton–Jacobi formalism is used to define the renormalized on-shell action, the necessary boundary counterterms, and the matching of sources/VEVs between bulk fields and dual operators. Recent work extends systematic treatments from scalar fields to more general field content, and aligns with older constructions in scattering theory.

1. Hamilton–Jacobi Formalism and Flat-Space Slicing

The foundational step is to foliate $(d+1)$-dimensional Minkowski spacetime in radial slices, with $r$ as the coordinate normal to future null infinity or spatial infinity. The canonical ADM decomposition gives

$$ds^2 = N^2(r)\,dr^2 + h_{ab}(r, x)\,(dX^a + N^a dr)\,(dX^b + N^b dr)$$

with typical gauge choices such as $N=1,\ N^a=0,\ h_{tt}=-1,\ h_{ij}=r^2 \hat g_{ij}$ yielding

$ds^2 = dr^2 - dt^2 + r^2 d\Omega_{d-1}^2$

for calculations in spherical/tortoise coordinates.

For a real scalar $\Phi$, the action is

$$S[\Phi] = \frac{\mathfrak{s}}{2} \int d^{d+1}x\,\sqrt{|g|} \left( g^{\mu\nu} \partial_\mu \Phi \partial_\nu \Phi - m^2 \Phi^2 \right)$$

with $\mathfrak{s}=\pm 1$ according to signature. The canonical momentum conjugate to $\Phi$ is

$$\Pi(r,x) = \frac{\delta L}{\delta(\partial_r \Phi)} = \mathfrak{s} \sqrt{|h|} \partial_r \Phi$$

and the Hamilton–Jacobi (HJ) principle imposes that for the on-shell action $S[\Phi, r]$,

$$\Pi(r,x) = \frac{\delta S[\Phi(\cdot), r]}{\delta \Phi(r, x)}$$

The bulk Hamiltonian constraint $\mathscr{H}$ then generates the HJ equation for $S$ as a functional partial differential equation:

$$H\left[ \Phi, \Pi \right] + \partial_r S[\Phi, r] = 0$$

where $H$ is an integral over $N \mathscr{H}$.

2. Asymptotic Expansion and Renormalization

The classical solution for $\Phi(r, t, \Omega)$ near $r \to \infty$ admits a Thomé-type expansion:

$$\Phi(r, t, \Omega) = \frac{1}{r^{(d-1)/2}} \int \frac{d\omega}{2\pi} e^{-i\omega t} \sum_{l, I} Y_{lI}(\Omega) \Big[ e^{+i|\omega| r} \tilde{\phi}^{(I)}(\omega, l, I) + e^{-i|\omega| r} \tilde{\phi}^{(II)}(\omega, l, I) \Big] + \cdots$$

The regulated on-shell action is dominated by exponentially growing terms as $r \to \infty$, necessitating a boundary counterterm action of the form

$$S_{\rm ct} = -\frac{1}{2} \int_{r=r_0} d^d x \sqrt{|h|} \Phi f(-\partial_t^2, \Delta_{\gamma}; r) \Phi$$

where the scalar function $f$ must solve a Riccati-type ODE derived from the asymptotic HJ equation:

$$\partial_r f + \frac{d-1}{r} f + f^2 + \big( -\partial_t^2 + \frac{1}{r^2} \Delta_\Omega - m^2 \big) = 0$$

with physical (regular, minus-branch) solutions typically $f \sim -i|\omega| + \frac{d-1}{2r} + \dots$. This summing of local differential-operator counterterms is a distinctive feature of the flat-space case, contrasting with the power-law expansion in the AdS setting (Ammon et al., 16 Dec 2025).

After counterterm subtraction, the renormalized on-shell action is defined by

$$S^{\rm ren}[\phi^{(II)}] = \lim_{r_0 \to \infty} \left( S^{\rm os}_{\rm reg}(r_0) - S_{\rm ct}(r_0) \right)$$

and is functionally finite.

3. Operator Identification and the Flat-Space Dictionary

The flat-space holographic dictionary is formulated by the rule:

• Source: The Dirichlet boundary value $\phi_s(x)$, identified with $\phi^{(II)}(x)$, encodes the scattering data entering the Arefeva–Faddeev–Slavnov (AFS) generating functional for QFT S-matrix elements.
• Expectation Value: The renormalized momentum

$$\Pi_{\rm ren}(x) = \lim_{r_0 \to \infty} (\Pi(r_0, x) - \delta S_{\rm ct}/\delta\Phi(r_0, x)) = \frac{\delta S^{\rm ren}}{\delta \phi^{(II)}(x)}$$

defines the dual operator's expectation value

$\langle \mathcal{O}(x) \rangle = \Pi_{\rm ren}(x)$

In frequency space, this yields the familiar "Carrollian" (null-infinity) two-point correlators, e.g., for massless fields,

$$\langle \mathcal{O}(x) \mathcal{O}(x') \rangle \propto \frac{1}{(\Delta x)^2 - i \epsilon} \delta^{d-1}(\Omega - \Omega')$$

The full dictionary mirrors the GKPW rules but is adapted so that all correlation functions are consistent with Carrollian symmetry and the underlying causal structure of the conformal boundary of Minkowski spacetime (Ammon et al., 16 Dec 2025).

4. Boundary Conditions, Scattering Data, and Variational Problem

Requiring a well-posed variational problem at $r \to \infty$ singles out the Dirichlet data on $\phi^{(II)}(x)$. In Lorentzian signature, with the standard $i\epsilon$ prescription, $\phi^{(II)}(\omega, \hat{x})$ encompasses positive-frequency data on past null infinity $\mathscr{I}^-$ and negative-frequency data on future null infinity. Imposing the rule

$$\phi^{(II)}(\omega, \hat{x}) = \phi_s(\omega, \hat{x})$$

with the identification $\hat{x} \to \pm \hat{x}$ for $\omega \gtrless 0$ exactly reproduces the AFS scattering boundary conditions. This ensures the bulk–boundary map is compatible with QFT S-matrix elements and S-matrix analyticity.

The boundary limit is defined for the growing solution branch, and the expectation value of the operator is determined by the leading normalizable term, paralleling the standard AdS/CFT structure but with essential adjustments for Minkowski asymptotics (Ammon et al., 16 Dec 2025).

5. Flat Holographic Renormalization Workflow

The method for constructing the flat-space holographic dictionary is summarized as:

1. ADM Decomposition: Write bulk Minkowski metric in radial gauge.
2. Canonical Momenta: Define $\Pi(r, x)$ as functional derivatives of the action with respect to $\Phi$.
3. Hamilton–Jacobi Equation: Obtain the HJ equation for the on-shell action functional $S$.
4. Asymptotic Expansion: Expand bulk fields and $S$ in terms exhibiting exponential divergences at $r \to \infty$.
5. Counterterms: Construct $S_{\rm ct}$ by solving the Riccati equation so as to cancel all divergences.
6. Renormalized Action: Define $S^{\rm ren}$ as the subtracted on-shell action in the $r \to \infty$ limit.
7. Operator Map: The source $\phi_s$ and expectation $\langle \mathcal{O} \rangle$ are respectively $\phi^{(II)}$ and $\delta S^{\rm ren}/\delta \phi^{(II)}$.

Higher-point correlators and Witten diagrams are constructed using flat-space adapted bulk-to-boundary and bulk-to-bulk propagators, with the $i\epsilon$ prescription, generalizing the AdS technology (Ammon et al., 16 Dec 2025).

6. Comparisons and Extensions

Relationship to AdS/CFT

The flat-space holographic dictionary is a limiting case of the AdS/CFT dictionary, but with crucial adjustments due to the absence of natural scale separation, conformal boundary, and the presence of power-law versus exponential divergences. The necessity of resumming an infinite tower of local differential-operator counterterms with a Riccati equation is a technical novelty compared to the polynomial-order truncation in AdS (Ammon et al., 16 Dec 2025).

Scattering Theory and Carrollian Limit

This formalism aligns naturally with the Carrollian limit and BMS symmetry algebras at null infinity: the correlation functions reconstructed via the flat holographic method exhibit the required invariance properties and encode physical scattering data directly, distinguishing them from the conventional Euclidean CFT correlators of AdS/CFT. The map is consistent with modern S-matrix theory and celestial holography.

Generalizations and Outlook

The current flat-space dictionary, as formulated via HJ renormalization, is constructed for scalar fields but is in principle extendable to gauge, gravitational, and higher-spin fields, provided appropriate boundary terms and gauge-fixing are incorporated (Ammon et al., 16 Dec 2025). For gravitational duals, the matching of BMS symmetry at null infinity and the holographic reconstruction of stress tensors remain active topics.

Step	Bulk Side	Boundary (Flat QFT)
Canonical momentum	$\Pi(r, x)$	$\delta S^{\rm ren}/\delta \phi^{(II)}(x)$
Source	$\phi_s(x) = \phi^{(II)}(x)$	Scattering data
Renormalized action	$S^{\rm ren}[\phi^{(II)}]$	AFS or Carrollian generating functional
Counterterms	Riccati solution for $S_{\rm ct}$	Removal of exponential divergences

The flat-space holographic dictionary thus systematizes the extraction of finite, physical QFT correlators from divergent bulk actions in asymptotically flat geometries, generalizing the powerful techniques of AdS holography to the domain relevant for S-matrix theory, Carrollian physics, and celestial holography (Ammon et al., 16 Dec 2025).