The real reason the universe may be a hologram Part IV

To Asymptotic Infinity and Beyond

This is the fourth part. If you haven’t read the third part, I encourage you to find it here:

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In the previous posts, we have learned about Anti-de Sitter geometry, Conformal Field Theory, and how the two are connected through propagators, observables, and symmetry. In this post, we will understand what this means for physics and why it is important. Therefore, I want to talk about some examples of how the theory works in practice and what it does for us in quantum gravity.

The problems with quantum gravity are far worse than most people outside the field think. That is why it has remained a problem since the 1950s.

The non-renormalizability of general relativity, Einstein’s theory, runs deep, and as we saw in earlier parts, non-renormalizability isn’t just a failure to figure out the mathematics; it means that the theory is missing something.

What the theory is missing is the subject of countless papers. Still, the Anti-de Sitter space is the only case where Einstein’s theory alone has solutions that resolve some of the deepest issues. It isn’t a full theory of quantum gravity as far as we know, but it is a lot closer.

As Assaf Shomer explains in “A pedagogical explanation for the non-renormalizability of gravity”,

if General Relativity was a renormalizable quantum field theory then its extreme high energy behavior should be that of a conformal field theory in the appropriate number of dimensions. However, our experience with gravity has shown that once enough energy is concentrated in a given region a black hole will form.

This tendency to form black holes at small scales is a pathological feature of general relativity. In flat spacetimes, he demonstrates, black hole entropy does not scale like a conformal field theory (except, as I’ve mentioned, in two dimensions). Therefore, general relativity cannot approach a CFT at small scales since black holes dominate the small-scale dynamics.

A black hole in AdS, however,

gives the correct dependence for a conformal field theory but in one less spacetime dimension. This is a consequence of the AdS/CFT correspondence which claims a complete equivalence, or duality, between quantum gravity in AdS space and a conformal field theory (without gravity)

This does not say we can renormalize general relativity through the AdS/CFT correspondence. In particular, the duality to high-energy, short scale (what physicists call UltraViolet or UV) renormalization of the CFT on the boundary in the AdS bulk is low-energy, long scale (what physicists call InfraRed or IR) renormalization of the AdS fields. Now, general relativity is already IR renormalizable. (In fact, it just becomes a simple wave theory at long distances.) When we say it isn’t renormalizable, we mean it isn’t UV renormalizable. The correspondence doesn’t provide a solution to that.

The method for renormalizing in the IR in AdS is called holographic renormalization. Essentially, the problem that AdS fields face is that they are propagating over an infinite distance to the boundary. Even though we conformally compactify the space to be finite, the closer one gets to the boundary the slower one moves so it is still infinite. Holographic renormalization provides a cutoff in the journey to the boundary so that we can stop our propagators somewhere finite. We then do the usual renormalization song and dance to remove the cutoff and place it in the observable components of the theory.

That is why, when string theorists use the correspondence, they are making an equivalence to string theory in the bulk and a CFT on the boundary, not general relativity. String theory is UV renormalizable, so there is no problem.

In fact, one of the most famous AdS/CFT examples is between a type of string theory called IIB and a particular CFT called Yang-Mills theory. Although that was a great achievement mathematically, neither theory is workable as a theory of gravity.

String theorists continue to work in the AdS/CFT regime and even try to break out of the string theory realm and address “low energy” string theory, which is assumed to be general relativity in the bulk. In this case, one assumes that the small-scale problems are taken care of, and you just need to deal with the moderate to large-scale issues and try to move beyond the pure, “on-shell” form of the bulk theory to what is called “loop order”.

I have my own thoughts on this, which I’ll now dive into.

A new direction

I have pondered all the problems that AdS/CFT faces in becoming a true theory of gravity. I boil them down to these:

• AdS/CFT is dependent on string theory to be a workable theory of quantum gravity. But string theory itself has many of its own problems. It would be helpful to separate the two.

• AdS/CFT provides a connection between a kind of spacetime that doesn’t look like ours and a CFT. Our universe is likely de Sitter in shape, not anti-de Sitter.

• The most successful formulation of AdS/CFT is with AdS in five dimensions, not four. Although it works in any number of dimensions, five has the best examples.

• AdS/CFT does not allow you to build a renormalizable version of general relativity because AdS doesn’t help with UV renormalization, only IR renormalization (also called holographic renormalization).

My idea solves all these problems at once, and it is quite simple. Rather than assuming that you need to have a quantum field theory in the bulk, I have proposed that you can keep a classical, i.e., on-shell, theory in the bulk in five dimensions.

You then take the five-dimensional AdS space, and you start slicing it in such a way that each slice is a four-dimensional de Sitter space.

Each of these de Sitter slices is like its own universe or world.

You then compute the quantum theory as the average of fields over all the de Sitter worlds.

This, surprisingly, is a quantum field theory, but the long-scale dynamics of the five-dimensional theory control the small-scale dynamics of the quantum theory. That means that long-scale, IR renormalization of the five-dimensional classical theory is equivalent to small-scale, UV renormalization of the four-dimensional quantum theory, in the bulk, not on the boundary.

You can imagine why. Suppose I have a tiny pendulum that sits in a room and oscillates randomly, swinging back and forth from thermal currents. Perhaps it has a little fin on it to catch the currents.

If I watch the pendulum over a long period of time, I will see small swings and large swings. Perhaps it will go all the way around a few times from some random kick.

The law of large numbers says, however, that the longer I watch it, the more likely bigger kicks will happen, which will send it swinging around and around for a long time.

If I were to imagine a scenario where I could watch the spring for an infinite length of time, I would have to conclude that there is now a non-zero possibility for a kick of infinite size happening. That is what happens when you encounter infinity. The length of time I watch and the size of the kick are correlated so that if I let one go to infinity, the other will too.

That means that the long-time dynamics of the pendulum are connected to its energy dynamics, which are related to short-time dynamics.

If, therefore, I can limit the amount of time I watch the pendulum, I immediately eliminate the possibility of infinite energy.

The same principle applies to taking averages over the slices of the AdS space. If I cut off the size of the space as one normally does in holographic renormalization, then that cuts off the problems with small-scale energy in the quantum field theory that results from the averaging.

This idea checks all the boxes above. It turns a 5D AdS classical theory into a 4D dS off-shell quantum field theory. It controls small-scale dynamics. And it has nothing to do with string theory, at least not directly.

There are a lot of details to work out still, but it is a worthwhile direction.

Conclusion

The AdS/CFT correspondence seems to be with us to stay. It is hard to ignore something so compelling that connects the two greatest theories of modern physics: general relativity and quantum field theory. And perhaps there is a path forward still, even if string theory remains untestable.

We have seen why AdS is so unique and why CFTs are fundamental to our universe at the smallest scales. We have also seen what connects them through symmetry and propagation.

Does this mean, as the title suggests, that the universe really is a hologram?

I like to think so, but the universe may be much bigger and wilder than we suppose. If my idea is correct and the universe is actually a 5D AdS universe, and we only live in a dS slice of it, then there is an entire dimension to the universe that we perceive only through quantum experiments. That universe is holographic. It makes sense, in fact, that to form a quantum field theory in the bulk, you have to average out one dimension so that it has the same number of dimensions as the conformal field theory on the boundary.

Testing such a theory remains difficult because the realm where small-scale dynamics become important happens at such a small scale that our particle accelerators may never reach it. Nevertheless, there may be other ways to see it. If the universe does have another dimension, it may manifest itself through dark matter or dark energy. Observations in those areas are only getting more precise.

It is only a matter of time before we uncover the truth.