# Is there non-locality in the AdS/CFT?

Many String theorists are trying to restore bulk locality in the AdS/CFT. So does that mean the CFT, the boundary, is non-local? Does matter travel faster than light on the CFT?

You may find a small set of articles discussing non-commutative field theories, or dipole field theories, or field theories in the presence of a magnetic field. These types of field theories will indeed have a certain amount of non-locality built in.

For the most part, however, the "string theorists" you refer to will assume that the conformal field theory is strictly local and causal, with nothing traveling faster than the speed of light. There is then a small puzzle, how a local field theory in $d$-dimensions could arise from a local classical gravity theory in $(d+1)$-dimensions. It seems plausible at first thought that non-local effects in the bulk $(d+1)$-dimensional theory might map to purely local effects in the boundary, after "integrating out" the radial coordinate, or similarly that local effects in the bulk might get smeared out into non-local effects in the boundary conformal field theory.

As it turns out, locality in the bulk is closely associated with a large $N$ limit. These CFTs typically have a parameter $N$ (e.g. a rank of a gauge group) that needs to be very large for the bulk dual to be classical gravity. If $N$ gets small, the gravity theory typically needs to be improved, for example to a string theory. As the fundamental objects of string theory are not points but strings, there is some non-locality built in to string theory that will show up as $1/N$ corrections in the CFT. Even though the bulk becomes slightly non-local when $1/N$ effects are included, the CFT remains completely local. One paper that discusses these issues in greater detail than I can here is this one.

What happens on the CFT side, then? Well here is one funny effect. If one places the CFT on a sphere, then it is possible to perform a scattering experiment where the ingoing and outgoing particles are light like separated with respect to the classical bulk geometry but are not with respect to paths that travel on the sphere. In other words, the particles can take a short cut through the bulk. As a result, one will see fake poles in the scattering amplitude at leading order $N$ that disappear when $1/N$ corrections are added. This paper discusses the effect in more detail than I can here.

This is a very deep question. The $AdS/CFT$ correspondence tells us that the gravitational physics in the $AdS_n$ manifold is equivalent to the conformal field theory $CFT_{n-1}$ identified on the boundary of the anti-de Sitter spacetime $\partial AdS_n$. This boundary is an Einstein spacetime that is Ricci flat. The holographic principle behind this tells us that the field theory on the boundary is equivalent to the gravitational physics in the bulk. This can be seen both as the $CFT_n$ on the boundary is the holographic plate constructing the gravitational physics in the bulk $AdS_n$, or that the bulk $AdS_n$ is the subject of the photograph which produces the $CFT_{n-1}$.

We can first think of the $CFT$ on the boundary without gravity as following the standard construction. Quantum field theories have a local construction. Operators or observables commute with each other if they are on a spatial surface. A spatial surface has at every point a harmonic oscillator field, and observables commute with each other if they have spatial separations. They will have non-trivial commutators if they are separated by a timelike or null direction. This permits dynamics of the sort $$i\frac{\partial{\cal O}}{\partial t}~=~[H,~{\cal O}],$$ and other commutators of field amplitudes which construct propagators. The locality condition, called the Wightman conditions, prevents the occurrence of propagators that are spacelike.

This is related to, but not the same as locality in quantum foundations. Quantum mechanics clearly has nonlocal properties, such as the violation of Bell inequalities. However, QFT formalism restricts the physics in such as way this nonlocality becomes less relevant and to define causal propagation in a way which avoid nonlocality. We know from no-signaling theorems that nonlocality in quantum mechanics does not lead to the communication of information. The Wightman conditions are imposed in order to prevent any signaling by nonlocality when formulating propagators of fields.

This local field theory is identified with the gravitational physics in the interior. String theory requires a background to quantum gravity. One expands the Lagrangian density of gravity as $$S~=~\int d^4r\sqrt{g}{\cal L}~=~\int d^4r\sqrt{g}\left(R~+~\alpha'R_{abcd}R^{abcd}~+~O(\alpha'^2)\right),$$ as modes formed from the background spacetime. We have this expansion according to the string parameter $\alpha'$ which in its full rather unexplored glory means $U(t,\vec r)~=~e^{-S}$ is an element of a enveloping group. We then have a quantum gravity that is defined in an anti-de Sitter spacetime. This does require it be done on a complete patch of the $AdS_4$, for the topologically $AdS_4~\sim~\mathbb S^1\times\mathbb R^3$. Time is a circle, which leads to closed timelike curves. This is a property of spacetimes that violate the Hawking-Penrose energy condition, and the anti-de Sitter spacetime has a Gaussian curvature or cosmological constant $\Lambda~<~0$. As a result we have to specify our physics in a wedge of the spacetime. It illustrate this with a Poincare disk below, where the blue region is a complete region a field may be defined.

Quantum gravity is more nonlocal than quantum field theory. Quantum gravity is the propagation of space or spacetime as a field, and in effect we are propagating in within itself. This is different than the propagator for a standard field, such as a scalar field, Yang-Mill field or a Dirac field, which has spacetime as a fixed background. String theory has this requirement of a classical-like background, which reflects some oddity here with quantum gravity. We are not entirely free to specify quantum gravitation operators without some “anchor” to hang them on. We do though have some ambiguity though, for quantum gravitation to $O(\alpha')$ introduce an uncertainty concerning the identification of a spatial surface. This means that nonlocality of quantum physics, which we broomed away in quantum field theory with commutator conditions pops up with quantum gravity. It is as if the nonlocality removed from CFT emerges in the bulk.

From a physics perspective we do have something a bit odd with this nonlocality. We might ask the question of whether we can eliminate this nonlocality in the gravitational bulk, while at the same time maintaining locality of the CFT on boundary. This is equivalent to asking how can be localize the properties of field amplitudes on a black hole horizon by using a high energy probe, or in effect a Heisenberg microscope. The two slit experiment wave function is a superposition of states through the two slits. An ensemble of experiments produces a wave pattern on the detecting plate. Now consider the resolution of a screen and the Planck length. Suppose a wave emerges from a spherical screen of radius $R$ with pixel size $\Delta x$. The information reaches the center in $T~=~R/c$. The total number of pixels on this screen is $4\pi R^2/\Delta x^2$, which per time $2R/c$ is equal to the Planck rate of transmission $1/T_p$, and so $\Delta x~=~\sqrt{2\pi RcT_p}$, which is very small but much larger than the Planck scale. For the two slit experiment the resolution of the two slits is set by a limit $$\Delta x~=~\sqrt{DcT_p}$$ for $D~\simeq~2\pi R$. For smaller two slits separation the double slit experiment is indistinguishable from a single slit experiment.

The angular uncertainty in the wave is $\Delta\theta~=~\Delta x/D$ $\simeq~ cT_p/D$, and angular uncertainty reduces at larger distance. Direction has a clearer physical meaning clearer at larger scales. The means square uncertainty in distance becomes $\langle x\rangle^2~=~DcT_p/\sqrt{2\pi}$, which is considerably larger than the Planck area. Distance emerges at large scales with increased number of Planck units of information on a screen. If physics could be measured on the Planck scale, the angular uncertainty would be huge. This is the Heisenberg microscope, which illustrates how spatial directions on the Planck scale are not well defined. What is important is the topology induced by the slits a 2D holographic system. The screen and slits define "topological logic gate" information about spacetime. This is a double slit version of how entanglement gives rise to geometry, and entanglement if quantum waves from the $CFT$ have nonlocal properties.

We then have an interesting property; one can squash nonlocality out of the $CFT$ and it enters into the quantum gravitation of the bulk. If one tries to localize quantum gravitational states this necessitates entanglement of quantum states, such as those given by the $CFT$, which have nonlocality. Nonlocality, or equivalently we might say entanglement, is a bit like an incompressible fluid. You can't get rid of it.