Reading boundary correlations from a bulk field

Choose a chart and state what it covers

The global map places the boundary on a circle and makes arrival times easy to compare. A second chart is convenient for a field close to the boundary. It flattens the boundary and labels radial position by height above it. These are Poincaré coordinates.

In Lorentz signature, the boundary has $d$ spacetime directions and the bulk has one extra coordinate $z>0$. Both $x^\mu$ and $z$ have length units. The curvature radius is $L$, the boundary approaches $z=0$, and $\eta_{\mu\nu}$ is the flat metric with one negative direction.

In Lorentz signature this chart covers only part of global AdS. Infinite radial height approaches a Poincaré horizon, which is not the end of the global spacetime. Changing charts changes the drawing, coordinate clocks and the chosen boundary conformal representative. Causal reachability still follows from the same metric.

Source: João Penedones

Turn a mass into a radial exponent

Place a free scalar field $\Phi$ on this fixed geometry. A scalar carries no vector direction under a coordinate change. In natural units $c=\hbar=1$, its mass $m$ has inverse-length units. It obeys the Klein–Gordon equation. For the boundary expansion, continue the boundary time to Euclidean signature, so its metric is positive.

Insert the metric into the curved-space Laplacian. At fixed boundary wave number, boundary derivatives have an extra factor $z^2$ and become subleading near the boundary. Try a power $\Phi\sim z^\alpha$. Each radial derivative lowers the exponent; its coordinate prefactor restores it. All leading terms then have the same power.

Source: João Penedones

Fix a source and read a response

A second-order equation needs boundary conditions. In standard quantization the coefficient $J(x)$ of the slower branch is a prescribed source. It specifies how the boundary drives the bulk field. The other coefficient, $A(x)$, carries state or response information. Interior regularity, a quantum state and, in Lorentz signature, temporal conditions complete the problem.

The schematic expansion below suppresses local terms determined by derivatives of the source. Some integer exponent differences also introduce logarithms. After renormalization, $A$ is proportional to $\langle\mathcal O\rangle_J$, possibly with additional local source terms. Its proportionality constant depends on field and operator normalization and cannot be read from the powers alone.

A boundary operator specifies a quantum observable. Its two-point function describes joint fluctuations in a specified state. Varying the source in a generating functional yields an expectation value; another variation gives the corresponding two-point function. Euclidean, time-ordered and retarded definitions require their own prescriptions.

At zero source, a normalized limit of $z^{-\Delta}\Phi$ can instead define the boundary operator directly. Penedones section 3.2, equations (77)–(78), constructs correlations by this limiting procedure. Equations (93)–(99) in section 3.2.2 relate the same correlations to a boundary source and generating functional.

Source: João Penedones · Xi Dong

Work through a massless example

Take AdS3, so $d=2$, and set the scalar mass to zero. The exponent equation becomes $\alpha(\alpha-2)=0$. The radial powers are zero and two. Standard quantization assigns the boundary operator $\Delta=2$. Boundary conditions specify how the two roots enter the physical problem.

First turn off the source. The leading field term falls as $z^2$, so halving radial height reduces that term to one quarter. Multiplying by $z^{-2}$ leaves a finite boundary quantity. A position-dependent source in this massless example generally also needs a local $z^2\log(z/L)$ term. The source-free example avoids that term without removing it from the general expansion.

Now compare two separated scalar primary operators in the flat Euclidean boundary vacuum. Translation, rotation and scale symmetry fix the separation dependence. Doubling the distance from $r$ to $2r$ reduces the correlation by a factor of sixteen. Operator normalization cancels from this ratio.

Source: João Penedones

Add the assumptions needed for gravity

Fixed-background field theory already produces boundary correlations with conformal properties. A gravitational correspondence also makes the metric dynamical. Changes of the metric supply the boundary stress-energy tensor; one free scalar alone does not supply this ingredient.

Semiclassical gravity needs a curvature radius much larger than the quantum-gravity length. A boundary large-number or large-$N$ expansion describes weak connected corrections to factorized correlators of suitably normalized operators. The stress-tensor two-point strength $C_T$ grows with $L^{d-1}/G_N$.

A further approximation by Einstein gravity and a few low-energy fields requires a sparse light spectrum and a large gap to additional higher-spin single-particle excitations. Otherwise many extra fields remain light and the simple low-energy equations omit them. A large number of degrees of freedom, strong interactions and a sparse spectrum play distinct roles; a proposed dual theory must state the conditions it actually satisfies.

Source: João Penedones

Ask a causal question with a causal response

Distant measurements can be correlated. Sending a message requires changing a drive at one location and observing whether another outcome responds. Linear response uses a retarded correlator built from an operator commutator. Spacelike commutators vanish in a local boundary theory even when a two-point expectation value is nonzero.

The static extremal curve in the next chapter connects boundary endpoints at one time. It calculates a spatial length and has no emission time or retarded response. Use null trajectories to determine causal reachability, then use extremal curves to study entanglement. Each question has its own calculation.