What is the holographic correspondence?

One of the hardest things to describe in theoretical physics is what happens when lots of particles interact with each other. Essentially, it is impossible to solve this problem exactly, and so the approaches that are currently used rely on several types of approximation.

What I want to describe is how, maybe, approaches in String Theory might be used to solve some of these really important “hard” problems. There’s no way that I can explain all the details (honestly, I don’t understand them!) but hopefully this will be a picture of how weird, esoteric, and very mathematical concepts can be say something useful about reality.

This approach is generically called “holography” for reasons that will become clear(er) later.

One of the approximate approaches to describing interacting particles that has been used to great effect is called “perturbation theory”. This applies when the interactions between the particles are relatively weak. How it works could be a whole post in itself, but perhaps for now it is enough to say that the existence of perturbation theory makes some problems with weak interactions “easy” in the sense that they can be approximately solved.

Crucially, it turns out that many of the complicated string theories that try to describe how quantum gravity works have interactions between particles which can be treated in perturbation theory.

The point of holography is that it might be possible to discover a dictionary or a way of translating between the “easy” string theory and a “hard” theory with strong interactions. Using this dictionary, it is possible to start from the “hard” theory, translate the calculation into the “easy” gravity analogue, do the calculation, and translate the results back to the “hard” context.

2017-10-04-BulkBoundary
The “hard” theory with strong interactions lives on the (red) boundary of a space with a (green) black hole at the centre.

The diagram above is a sketch of how to visualise this process. The “easy” gravity theory exists in a bulk with a certain number of dimensions, whereas the “hard” theory lives in a space which is one dimension smaller, at the edge (or “boundary”). This is where the term holography comes from: The physical theory is a hologram which is projected from the bulk like R2D2’s message from Princess Leia.

Most intriguingly, when the “hard” theory has a temperature above absolute zero (which all physical materials must have) the gravity theory contains a black hole at its centre which has an event horizon.

So, the calculation for the complicated experimental quantity that you are interested in on the boundary can be translated through the bulk to the event horizon of the black hole. There, the properties of the theory on the boundary get converted into the properties of space-time near the black hole. This is what he dictionary does. Perturbation theory can then be used to get an approximate answer in that context. Finally, the answer is moved back through the bulk to the boundary where it can be interpreted in the original context.

Of  course, the technical details of how to actually do this in mathematics is very complicated, but there is one well-understood example of this process.

Quarks are fundamental particles and can be glued together to make protons and neutrons. The particles which do the glueing are called gluons. The gluons and the quarks are strongly interacting and so they fall into the category of “hard” theories. But, there is a well-defined correspondence between a supersymmetric particle theory which lives in eight spatial dimensions and one time dimension (so, nine in total) and an “easy” string theory which lives in ten dimensions. This correspondence has been used to derive results which would otherwise not be possible.

One of the current questions for people who work on holography is whether this is just a fortuitous specific case, or whether these correspondences are more general.

In condensed matter, there are also strongly interacting materials which theorists find very difficult to describe. One really important example is the high temperature superconductor materials.

The question is whether a holographic correspondence can be found for a theory that can make predictions about these materials? To put that another way, is there a higher-dimensional, gravity-like theory which gives a theory for a superconductor as its hologram?

A lot of people are looking at this question at the moment.

There are some encouraging things which have been done already. For example, the materials which go superconducting at low temperature also have weird behaviour at higher temperatures where they don’t superconduct. These properties have been calculated within the gravity theory, and shows some similar features to those seen in experiments.

But there is also a lot that is not known yet. For example, it is very difficult to include effects of the underlying material crystal, or include the existence of the quantum-mechanical spin of the particles. Both of these details will be important to design new materials which sustain superconductivity at even higher temperature.

This is really a field which is still in its infancy, but the underlying idea behind it is intriguing: if the theorists working on it can progress to the point where it can make predictions, it would be very exciting indeed.

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