Nature of spacetime in the holographic boundary

Question

My understanding from popular science articles is that the boundary is a field theory that has no gravity and has one less spatial dimension than the bulk.

However, I am not sure I understood this picture correctly. I just read a recent article at quanta magazine that states:

"A solar system in the central anti-de Sitter region, for instance, can be described as a collection of particles scattered around the boundary that obey only quantum theory and have no sense of gravity or space-time at all".

So my question is, did the quote mean no curved spacetime, or there is not even a minkowski spacetime associated with the physics in the boundary? I dont even understand how it is possible to have a quantum theory of particles without either space or time.

Answer 1

The AdS/CFT correspondence is a conjecture where every (quantum)-gravitational theory in $D$-dimensional space-time with AdS asymptotics is proposed to be a dual of some (quantized) conformal field theory (CFT) in $D-1$ dimensional space-time conformally equivalent to Minkowski space-time. In other words, the CFT lives in a metric of the form $$ds^2 = \Omega^2 (t,x^i)\left[-d t^2 + \sum_{i =1}^{D-2} (d x^i)^2 \right]\,,$$ where $\Omega$ is any smooth nonvanishing function on space-time.

The reason for this freedom is that the CFT is not sensitive towards this rescaling, it only cares about the lightcone (causal) structure of the space-time, not about the absolute magnitudes of time or space intervals. Funnily enough, Minkowski space-time can conformally map to huge patches of other space-times with different curvature, such as dS and AdS space-times, so the CFT is really quite insensitive to the local space-time geometry. This is also the reason why people are not particularly eager to specify which sort of geometry the CFT lives in, since this is not particularly meaningful. I believe this is what the author of the Quanta magazine article was trying to hint at.

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P.S.: CFTs are funny beasts in various different ways. The Quanta article talks about particle in the CFT at the boundary, but what exactly should one call a particle in a CFT is up to debate.

Answer 2

Under normal circumstances, the boundary theory in a holographic duality definitely inhabits a space-time of its own. So one might suppose that this was a journalistic error.

However, there are a number of avantgarde ideas, popular in the milieu of elite physicists that Quanta Magazine reports on, which could give rise to the otherwise unusual claim that the boundary theory "obey[s] only quantum theory and [has] no sense of gravity or space-time at all".

First of all, a lot of recent work on holography has taken place in AdS2. This includes work on obtaining the Page curve of Hawking radiation, and the controversial claim to have constructed a traversable wormhole by entangling qubits in a quantum computer.

AdS2 has one space and one time dimension; its boundary has one time dimension and no spatial dimension at all. The best-known proposed duality in AdS2 is that JT gravity on AdS2, is dual to the SYK model. The SYK model is just a bunch of entangled fermions living on a single point.

Also, it's actually routine for quantum field theory in 0+1 dimensions to be referred to as quantum mechanics. Search this site for "0+1 dimensions" and you will find a number of discussions as to why.

Finally, it's possible that the journalist was thinking of work that models AdS/CFT as a quantum error-correcting code. In some versions of this, the boundary space is treated simply as a tesselation with one qubit per tile, and the bulk is (for example) a branching tree-graph with more qubits at each vertex.

My guess is that the quote by the journalist, reflects the views of physicists working on topics like these.

Answer 3

The holographic principle is a idea that suggests that the information content of a region of space can be encoded on a lower-dimensional boundary, such as a horizon. This means that the physics in the bulk (the higher-dimensional space) can be described by a quantum field theory on the boundary (the lower-dimensional surface) that has no gravity.

The quote you mentioned means that there is no spacetime at all on the boundary, not even a flat Minkowski spacetime. The quantum theory on the boundary is defined on a fixed background that does not change with time or space. The particles on the boundary are not really particles, but operators that act on quantum states.