Flat Space Holography Overview

• Flat space holography is a framework that generalizes the holographic principle to asymptotically flat spacetimes, encoding gravitational dynamics in a lower-dimensional nonlocal field theory.
• It reveals that entanglement entropy follows a volume law, indicating maximal mixing and nonlocal correlations that lead to trivial boundary correlators.
• The approach is supported by its consistency with Schwarzschild black hole entropy, suggesting that nonlocal boundary degrees of freedom capture essential features of flat space quantum gravity.

Flat space holography is a program that seeks to generalize the holographic principle—best understood in asymptotically anti–de Sitter (AdS) spacetimes—to the physically relevant case of asymptotically flat spacetimes, such as Minkowski space. The central question is whether gravitational dynamics in $(d+1)$-dimensional flat spacetime can be encoded in a lower-dimensional non-gravitational field theory; if so, what is the structure of the dual theory and the holographic dictionary? Recent progress has revealed a complex structure involving nonlocal field theories, infinite-dimensional asymptotic symmetries, BMS (Bondi–Metzner–Sachs) invariance, and deep ties to entanglement entropy, black hole thermodynamics, and the maximal entanglement properties of the vacuum. The following sections furnish a detailed account based on (Li et al., 2010).

1. Flat Spacetime Geometry and Holographic Cutoff

The proposal reformulates Euclidean flat spacetime via polar coordinates: $$ds^2_{\mathbb{R}^{d+1}} = d\rho^2 + \rho^2 ds^2_{S^d}$$ where the holographic dual theory is placed on the $d$-sphere $S^d$ at a large but finite cutoff radius $\rho = \rho_\infty$.

The partition function of gravity on this flat background is conjectured to be dual to that of a boundary theory on $S^d$, paralleling the AdS/CFT paradigm. However, in contrast to AdS, the dual field theory is found to be inherently nonlocal. The holographic dictionary is modified: instead of a boundary at spatial infinity with conformal symmetry, one considers a boundary at finite cutoff radius endowed with boundary conditions compatible with the more global structure of Minkowski space. This geometric setting is essential for analyzing the interplay between entanglement, correlation functions, and black hole entropy in the flat space context.

2. Holographic Entanglement Entropy: Maximally Mixed and Volume Law

Applying the Ryu–Takayanagi prescription adapted to this flat space setup, the entanglement entropy $S_A^{(\text{hol})}$ of a boundary subsystem $A \subset S^d$ is computed as

$$S_A^{(\text{hol})} = \frac{\mathrm{Area}(\gamma_A)}{4G_N^{(d+1)}}$$

where $\gamma_A$ is the bulk minimal surface anchored on $\partial A$. The crucial and new observation is that, contrary to the area law characteristic of local boundary field theories and ground states, in flat space this prescription yields a volume law:

• For a small boundary region $A$ (small opening angle $\alpha \ll 1$), $S_A$ is proportional to the volume of $A$, not its area.
• More generally, the entanglement entropy is extensive on the boundary. Any infinitesimal region $A$ is maximally entangled with its complement, such that

$S_A = \log \dim \mathcal{H}_A$

and the reduced density matrix $\rho_A$ is maximally mixed.

This behavior signals the presence of nonlocal correlations and is in stark contrast to the usual holographic duals of AdS spacetimes where entanglement entropy reflects locality in the dual conformal field theory. Here, the boundary dual must be highly nonlocal to accommodate this maximal entanglement structure.

3. Nonlocal Scalar Field Theory Toy Model

To account for the volume law, a nonlocal boundary field theory is constructed as a free (Gaussian) scalar on $S^d$ with action

$$S_{\text{boundary}} = \int d\Omega_d\, \phi \, f(-\Delta) \, \phi$$

where $\Delta$ is the Laplacian on $S^d$. The function $f(x)$ is chosen to engineer desired nonlocal behavior. For the standard choice $f(x) = x^p$, the entanglement entropy follows the familiar area law. To achieve a volume law, the paper proposes

$f(x) = e^{x^q}$

with $q = 1/2$ yielding

$S_A \propto (L/a)^{d-1}$

where $L$ is the radius of $S^d$ and $a$ is a UV cutoff, manifestly a volume law for entanglement. In particular, the choice

$$S_{\text{boundary}} = \int d\Omega_d\, \phi \, e^{-\Delta} \, \phi$$

acts as a canonical example. The volume law persists for generic nonlocal modifications—even when interactions are added. This toy model captures both the necessary nonlocality and the resulting maximal boundary entanglement.

$f(x)$ functional form	Entropy scaling	Locality
$x^p$	Area law, $(L/a)^{d-2+2p}$	Local
$e^{x^q}$	Volume law for $q=1/2$	Nonlocal

4. Triviality and Vanishing of Correlation Functions

A direct consequence of the maximal entanglement and nonlocality is the trivialization of boundary correlation functions. The argument proceeds as follows:

• The standard bulk-to-boundary process yields boundary $n$-point functions.
• After renormalization (involving adding nonlocal counterterms), all $n$-point connected correlators vanish or reduce to contact terms (`$\delta$-functions and derivatives thereof) on the boundary.
• For any finite set of boundary points $\{x_i\}$, the region $A = \{x_1,\ldots,x_n\}$ is maximally entangled and $\rho_A$ factorizes as

$\rho_A = \bigotimes_{i=1}^n \rho_{x_i}$

with each $\rho_{x_i}$ being maximally mixed.

Thus,

$$\langle O(x_1) \ldots O(x_n) \rangle = \operatorname{Tr}[\rho_A O(x_1)\ldots O(x_n)] = 0$$

except for contact terms, reflecting the effective "infinite" local temperature of all boundary sectors as in a maximally mixed or infinite-temperature state.

This structure is further elucidated by analogy to random (maximally entangled) spin chains, where all local correlators are washed out by maximal mixing among spins.

5. Consistency with Schwarzschild Black Hole Entropy

A major consistency check for the proposed flat space holographic correspondence is its compatibility with the entropy of Schwarzschild black holes in asymptotically flat spacetimes. For the Lorentzian metric

$$ds^2 = d\rho^2 + \rho^2 ( -dt^2 + \cosh^2 t \, d\Omega^2_{d-1} )$$

a static observer at $\rho = \rho_0$ experiences an Unruh temperature $T_U = 1/(2\pi \rho_0)$. The entanglement entropy for a half-boundary ($A$ comprising half of $S^d$) scales as

$S_A \propto \rho_\infty^{d-1} T_U^{d-1}$

This scaling exactly matches the Bekenstein–Hawking entropy for a Schwarzschild black hole, $S_{BH} \propto T_{BH}^{d-1}$, up to numerical constants. Therefore, the nonlocal maximally entangled dual theory reproduces the bulk entropy bound, providing strong support for the identification of nonlocal boundary degrees of freedom as black hole microstates.

6. Summary and Implications

Flat space holography, as formulated in (Li et al., 2010), proposes a holographic dual to flat spacetime on $S^d$ consisting of a nonlocal field theory characterized by:

• Extensive, volume-law entanglement entropy, $S_A \propto$ (volume of $A$), even for infinitesimal $A$ (i.e., all subsystems are maximally entangled with their complement).
• Maximally mixed reduced density matrices, with $S_A = \log(\dim \mathcal{H}_A)$.
• Trivial or vanishing boundary correlators, after renormalization, reflecting maximal decoherence and the absence of local structure.
• A toy model constructed from a nonlocal kinetic term in a scalar field theory, $$S_{\text{boundary}} = \int d\Omega_d \, \phi \, e^{-\Delta} \phi$$, capturing these features explicitly.
• Agreement with the Bekenstein–Hawking black hole entropy, showing that the entropy produced by the dual matches bulk gravitational predictions and thus supports this highly nonlocal theory as a candidate encoding of flat space quantum gravity.

This framework highlights a radical departure from the AdS/CFT paradigm: the dual theory is not local and not conformal, but rather maximally entangled and trivial under local measurements, signifying a "maximal mixing" phase and providing a consistent accounting of black hole entropy in flat spacetimes. The mechanism outlined serves as a concrete proposal for flat space holography, setting a reference point for both future extensions and modifications that incorporate additional bulk fields, more general boundary conditions, or higher curvature structures.