Physicists fly high on the idea that our three-dimensional world is actually two-dimensional, that we live in a hologram, and that we’re all projections on the boundary of space. Or something like this you’ve probably read somewhere. It’s been all over the pop science news ever since string theorists sang the Maldacena. Two weeks ago Scientific American produced this “Instant Egghead” video which is a condensed mashup of all the articles I’ve endured on the topic:
To begin with, physicists haven’t believed this since Minkowski doomed space and time to “fade away into mere shadows”. Moyer in his video apparently refers only to space when he says “reality.” That’s forgiveable. I am more disturbed by the word “reality” that always creeps up in this context. Last year I was at a workshop that mixed physicists with philosophers. Inevitably, upon mentioning the gauge-gravity duality, some philosopher would ask, well, how many dimensions then do we really live in? Really? I have some explanations for you about what this really means.
Q: Do we really live in a hologram?
A: What is “real” anyway?
Q: Having a bad day, yes?
A: Yes. How am I supposed to answer a question when I don’t know what it means?
Q: Let me be more precise then. Do we live in a hologram as really as, say, we live on planet Earth?
A: Thank you, much better. The holographic principle is a conjecture. It has zero experimental evidence. String theorists believe in it because their theory supports a specific version of holography, and in some interpretations black hole thermodynamics hints at it too. Be that as it may, we don’t know whether it is the correct description of nature.
Q: So if the holographic principle was the correct description of nature, would we live in a hologram as really as we live on planet Earth?
A: The holographic principle is a mathematical statement about the theories that describe nature. There’s a several thousand years long debate about whether or not math is as real as that apple tree in your back yard. This isn’t a question about holography in particular, you could also ask that question also in general relativity: Do we really live in a metric manifold of dimension four and Lorentzian signature?
Q: Well, do we?
A: On most days I think of the math of our theories as machinery that allows us to describe nature but is not itself nature. On the remaining days I’m not sure what reality is and have a lot of sympathy for Platonism. Make your pick.
Q: So if the holographic principle was true, would we live in a hologram as really as we previously thought we live in the space-time of Einstein’s theory of General Relativity?
A: A hologram is an image on a 2-dimensional surface that allows one to reconstruct a 3-dimensional image. One shouldn’t take the nomenclature “holographic principle” too seriously. To begin with actual holograms are never 2-dimensional in the mathematical sense; they have a finite width. After all they’re made of atoms and stuff. They also do not perfectly recreate the 3-dimensional image because they have a resolution limit which comes from the wavelength of the light used to take (and reconstruct) the image. A hologram is basically a Fourier transformation. If that doesn’t tell you anything, suffices to say this isn’t the same mathematics as that behind the holographic principle.
Q: I keep hearing that the holographic principle says the information of a volume can be encoded on the boundary. What’s the big deal with that? If I get a parcel with a customs declaration, information about the volume is also encoded on the boundary.
A: That statement about the encoding of information is sloppy wording. You have to take into account the resolution that you want to achieve. You are right of course in that there’s no problem in writing down the information about some volume and printing it on some surface (or a string for that matter). The point is that the larger the volume the smaller you’ll have to print.
Here’s an example. Take a square made out of N2 smaller squares and think of each of them as one bit. They’re either black or white. There are 2N2 different patterns of black and white. In analogy, the square is a box full of matter in our universe and the colors are information about the particles in the inside.
Now you want to encode the information about the pattern of that square on the boundary using pieces of the same length as the sidelength of the smaller squares. See image below for N=3. On the left is the division of the square and the boundary, on the right is one way these could encode information.
There’s 4N of these boundary pieces and 24N different patterns for them. If N is larger than 4, there are more ways the square can be colored than you have different patterns for the boundary. This means you cannot uniquely encode the information about the volume on the boundary.
The holographic principle says that this isn’t so. It says yes, you can always encode the volume on the boundary. Now this means, basically, that some of the patterns for the squares can’t happen.
Q: That’s pretty disturbing. Does this mean I can’t pack a parcel in as many ways as I want to?
A: In principle, yes. In practice the things we deal with, even the smallest ones we can presently handle in laboratories, are still far above the resolution limit. They are very large chunks compared to the little squares I have drawn above. There is thus no problem encoding all that we can do to them on the boundary.
Q: What then is the typical size of these pieces?
A: They’re thought to be at the Planck scale, that’s about 10-33 cm. You should not however take the example with the box too seriously. That is just an illustration to explain the scaling of the number of different configurations with the system size. The theory on the surface looks entirely different than the theory in the volume.
Q: Can you reach this resolution limit with an actual hologram?
A: No you can’t. If you’d use photons with a sufficiently high energy, you’d just blast away the sample of whatever image you wanted to take. However, if you loosely interpret the result of such a high energy blast as a hologram, albeit one that’s very difficult to reconstruct, you would eventually notice these limitations and be able to test the underlying theory.
Q: Let me come back to my question then, do we live in the volume or on the boundary?
A: Well, the holographic principle is quite a vague idea. It has a concrete realization in the gauge-gravity correspondence that was discovered in string theory. In this case one knows very well how the volume is related to the boundary and has theories that describe each. These both descriptions are identical. They are said to be “dual” and both equally “real” if you wish. They are just different ways of describing the same thing. In fact, depending on what system you describe, we are living on the boundary of a higher-dimensional space rather than in a volume with a lower dimensional surface.
Q: If they’re the same why then do we think we live in 3 dimensions and not in 2? Or 4?
A: Depends on what you mean with dimension. One way to measure the dimensionality is, roughly speaking, to count the number of ways a particle can get lost if it moves randomly away from a point. The result then depends on what particle you use for the measurement. The particles we deal with will move in 3 dimensions, at least on the distance scales that we typically measure. That’s why we think, feel, and move like we live in 3 dimensions, and nothing wrong with that. The type of particles (or fields) you would have in the dual theories do not correspond to the ones we are used to. And if you ask a string theorist, we live in 11 dimensions one way or the other.
Q: I can see then why it is confusing to vaguely ask what dimension “reality” has. But what is the most confusing thing about Moyer’s video?
A: The reflection on his glasses.
Q: Still having a bad day?
A: It’s this time of the month.
Q: Okay, then let me summarize what I think I learned here. The holographic principle is an unproved conjecture supported by string theory and black hole physics. It has a concrete theoretical formalization in the gauge-gravity correspondence. There, it identifies a theory in a volume with a theory on the boundary of that volume in a mathematically rigorous way. These theories are both equally real. How “real” that is depends on how real you believe math to be to begin with. It is only surprising that information can always be encoded on the boundary of a volum if you request to maintain the resolution, but then it is quite a mindboggling idea indeed. If one defines the number of dimensions in a suitable way that matches our intuition, we live in 3 spatial dimensions as we always thought we do, though experimental tests in extreme regimes may one day reveal that fundamentally our theories can be rewritten to spaces with different numbers of dimensions. Did I get that right?
A: You’re so awesomely attentive.
Q: Any plans on getting a dog?
A: No, I have interesting conversations with my plants.