How to define the boundary of an infinite space in the holographic principle?

In the holographic principle, an analogy is made between the information of a black hole being encoded on the event horizon, and the information of the universe being encoded onto a two-dimensional boundary of the universe. However, one of the possibilities for the extent of the physical universe is that it is infinite. In this case, how does one define the boundary?

Answer

How to define the boundary of an infinite space in the holographic principle?

The answer is at least relatively well-understood in spacetimes with a negative cosmological constant, but the real world has a positive cosmological constant, and the answer in that case is not yet known. The rest of this post elaborates on these statements.

The one case where the holographic principle is relatively well-understood (emphasis on relatively) is in Anti-de Sitter (AdS) spacetime, or spacetimes that are asymptotically AdS. This is the type of spacetime that would be expected if the cosmological constant were negative.

AdS (or asymptotically AdS) spacetime does not have a "boundary" in the usual sense. In fact, pure AdS is homogeneous, meaning that there are isometries (distance-preserving transformations) that can transform any given point to any other given point. A homogeneous spacetime cannot have a boundary in the usual sense, because no point can be "closer to the boundary" than any other point if the spacetime is homogeneous.

However, AdS spacetime does have a special property that is often loosely described using the word "boundary." Just like in flat spacetime, an inertial observer can send out a pulse of light (or anything that travels at the speed of light) so that it reflects from a faraway mirror and returns back to the observer. The special thing about AdS spacetime is that in the limit as the mirror is moved farther and farther away from the observer, the round-trip time for the light pulse approaches a finite value. (In flat spacetime, the round-trip time would grow without bound.) For this reason, AdS spacetime is often described as having a "boundary" at spacelike infinity. For more discussion about this, see "Anti De Sitter Space And Holography" (https://arxiv.org/abs/hep-th/9802150v2).

The holographic principle in asymptotically AdS spacetime is realized by the AdS/CFT correspondence, where CFT stands for Conformal Field Theory. A CFT is a special kind of quantum field theory that is symmetric with respect to angle-preserving transformations, including transformations that are not distance-preserving. In particular, it is symmetric with respect to changes of the overall scale, and this symmetry can act as though it were a "translation" symmetry in an extra spatial dimension. In fact, a CFT defined in $D$-dimensional spacetime has the same spacetime symmetry group as $D+1$-dimensional AdS spacetime. Over the past 20 years, physicists have accumulated compelling mathematical that the $D$-dimensional CFT is actually equivalent to string theory in a higher-dimensional AdS spacetime. This is the AdS/CFT correspondence, also called gauge-gravity duality. According to page 16 in "Gauge/gravity duality" (https://arxiv.org/abs/gr-qc/0602037),

...this duality is itself our most precise definition of string theory...

So the AdS/CFT correspondence is an explicit example of the holographic principle at work in a spacetime that does not have any boundary in the usual sense, although it does have a boundary-like property with respect to the propagation of light rays, as described above. However, despite the common language, the lower-dimensional CFT is not "located on the boundary". According to page 14 in "AdS Dynamics from Conformal Field Theory" (https://arxiv.org/abs/hep-th/9808016),

the CFT does not really `live' on the boundary of the AdS spacetime; rather, it fills the bulk. ... much as worldsheet CFT is a representation of perturbative string dynamics. Indeed, the perturbative string also fills spacetime due to its quantum fluctuations; pointlike UV perturbations -- the vertex operators -- represent perturbations at the asymptotic boundary of spacetime, yet one would not say that the string worldsheet resides at the (conformal) boundary of Minkowski space.

AdS spacetime corresponds to a negative cosmological constant. But in the real world, the cosmological constant is positive, as it is in de Sitter (dS) spacetime. In contrast to asymptotically AdS spacetimes, asymptotically dS spacetimes do not have anything resembling a "boundary" at spacelike infinity, and the holographic principle is not well-understood in dS spacetime. According to page 9 in "The Holographic Bound in Anti-de Sitter Space" (https://arxiv.org/abs/hep-th/9805114),

It remains to ask whether one can build a similarly sharpened holographic hypothesis for theories with zero (or even positive) cosmological constant. The answer will require some new ideas, since Minkowski space (or de Sitter space) has no obvious close analog of the `boundary at spatial infinity' by which holography is realized when the cosmological constant is negative.

Pages 34-35 in "Lectures on AdS/CFT from the Bottom Up" (http://sites.krieger.jhu.edu/jared-kaplan/files/2016/05/AdSCFTCourseNotesCurrentPublic.pdf) says it like this:

Let us make a couple of comments about cosmology, which naturally transpires (during the early epoch of inflation) in quasi-deSitter spacetime. ... Now we have a metric that has an obvious Penrose diagram. Note that spatially there is no boundary, and the spacetime only ends in time... This is the origin of the usual statement that deSitter space only has a boundary in the infinite past and infinite future. The absence of any notion of time on the boundary is one (mild) reason to worry about attempts at holography in deSitter spacetime...