Why Kenneth Wilson’s discoveries matter so much to quantum gravity
In 1982, Kenneth Wilson won the Nobel Prize in Physics for an unusual achievement. He had developed a theory about scale — size in lay…
In 1982, Kenneth Wilson won the Nobel Prize in Physics for an unusual achievement. He had developed a theory about scale — size in lay parlance. In particular, Wilson had shown how the rigid barriers that appear to exist between different scales break down close to what are called critical points — place where objects transition from one phase to another. For example, as ice gets closer to melting suddenly what is going on at the molecular level affects what is going on at the macroscopic level and vice versa. The isolation between sizes breaks down, and all scales contribute to the transition.
Wilson’s work had far reaching implications. For the recently completed Standard Model of quantum theory, Wilson showed how scale affected what fundamental constants of nature you would measure and, more importantly, why. Up until Wilson’s work, physicists had believed they were playing some kind of mathematical game with their equations to get them to give sensible results. Wilson showed that was not true. Rather, it all came down to scale. Constants like the universal gravitational constant and the fine structure constant are not constant at all but depend very much on the scale at which they are measured.
He showed, in particular, that their equations stopped working at a particular scale but that under certain circumstances you could ignore what was going on at smaller scales where you didn’t understand what was happening. Instead, you could relate one scale to a slightly smaller, neighboring scale and come up with equations for how the constants of nature had to change from scale to scale to scale as your experimental apparatus probed to smaller and smaller lengths. The mathematical technique, nonsensically called “renormalization group theory” for historical reasons, changed quantum physics forever and is one of the most important mathematical achievements in physics in the last 50 years.
That isn’t the only strange thing going on at the microscopic level. Scientists have proved that if you go to small enough scales, time can actually move backwards for short periods. Machines made of only a handful of molecules may benefit from this odd temporal distortion by essentially resetting themselves, periodically, after expending their energy; thus, recharging. The reason is because the 2nd law of thermodynamics, which governs the direction we perceive time to flow, is a statistical law and, like all statistical laws, it gets violated when the sample size becomes small. At small sizes, violations become more and more likely and time can, indeed, has been observed to run backwards.
In the world of technology, while airplanes use fixed wings similar to large birds, at small scales, that will not do. Smaller scales mean more turbulence, tiny eddies in air, that demand a more insect like wing motion. At even smaller sizes, whip-like flagella move in a strange corkscrew like pattern to move single celled organisms along, including sperm cells that enable human reproduction.
Meanwhile, at the largest scales, the universe becomes like a smooth fluid. Every galaxy and supercluster is like a molecule amid billions. While the universe, even at the largest scale, has some web-like structure to it, it is largely smooth in how it moves and expands. Yet, at one point billions of years ago, the largest structure we know, the 46 billion light year observable universe, was smaller than the nucleus of an atom. At this scale, the universe collided with the tiniest scale, and we do not know what happens at that point. We only know what happens when the universe is a bit larger and the ordinary physics we know emerged.
The funny thing about scale is that none of this was even visible until the telescope and microscope were invented and turned into scientific instruments. Galileo and Antonie van Leeuwenhoek both built businesses around these instruments becoming near monopolies for a time. Yet, they also used them to make significant expansions of our knowledge of what is going on at the largest scales in Galileo’s case and the smallest in Leeuwenhoek’s case. From the moons of Jupiter to a drop of pond water, that was when our world started to expand in two directions.
If you measure size in a logarithmic scale, with each number on the scale representing not a measure of distance but the power of ten of that measure, you can see human progress expanding in both directions with ever finer measurements producing better and better understandings of those scales. Here is an xkcd example for height. And one for depth. You can enjoy the complete tour by watching this video:
The key is that for different scales we use different physics ordinarily. At the scale of galaxies and the universe we use Einstein’s field equations. There is nothing in there about all the diversity of life on Earth. Atoms aren’t there. None of that matters.
Likewise, at the smallest scales we have quantum physics. Supposedly, all physics is quantum, yet we don’t use quantum physics to talk about how oceans behave. And, while there have been a lot of attempts, very little had been achieved in applying quantum physics to the scale of stars or the universe. The only real exception might be some of Stephen Hawking’s work on Black Hole thermodynamics, entropy, and Hawking radiation which marries the two up. Still, most of that is unconfirmed. Quantum effects are largely invisible for large, stellar black holes. We don’t have the instruments to create tiny black holes that we can study to confirm most of those predictions. And we have yet to find any naturally occurring primordial black holes that might be small enough to observe them.
The fact is that every scale has its own physics and cross-scale physics is fairly rare. The one exception might be electromagnetism, but even that will break down at the smallest scales as it combines with the other forces.
That is what makes Wilson’s work both fascinating and also worrying because he showed that at these critical points, all scales matter. That means that if large scale structures in the universe experience critical points now or in the past we may not be able to understand them depending on how far down the scale goes unless we have an overarching theory that combines the large and the small.
Black holes and singular structures are probably where these critical points play out. In those cases, we are using Einstein’s physics but Einstein’s physics ignores the tiny quantum scale. We have to have a physics that includes both Einstein’s and quantum scale physics in order to understand these critical phenomena. We can’t ignore either.
The straightforward solution was to “quantize” Einstein’s equations but that turned out to be hard and no one has been able to figure out how to do it. Others said, well of course you can’t do that, because Einstein’s is just a large scale “effective” theory that doesn’t apply at any other scales. We need to find the “real” theory of gravity that works at all the scales.
The suggestion was that this had already been invented by a guy named Polyakov. Polyakov’s equations did work at all scales. Polyakov’s equations also had something very similar to Einstein’s gravitational field in them. Recently, mathematicians have indeed proved that this program works in two dimensions. It had taken them forty years to prove it and just for 2D. If they could do it in 4D, they might have a working theory. None other than string theory.
Another suggestion was that gravity had its own smallest scale and that there was literally nothing smaller than that. After all, Einstein’s equations did tell us how long things are. It is a “metric” field theory, metric meaning “measure”. Gravity changes how long and wide and deep things are. If it can decide those things, then why would it not simply tell us how small things can be?
In computers we always introduce a smallest size when we are simulating anything because computers are finite machines. But for most things that smallest size is superimposed on some background that is not assumed to have a smallest size. With gravity, if you do assume a smallest size, it isn’t superimposed on anything. Every point in space and time has no “location” because to have a location you have to have some background reality to compare it to. For example, I can say that I am at a particular latitude and longitude on the Earth’s surface, but that is relative to a coordinate system superimposed on the Earth. Suppose I want to tell you where a particular latitude and longitude is? I can’t give a latitude and longitude for it! It is just wherever it is. But maybe I can give its location by explaining what points are near it?
It is similar with this idea. You have to define every point by its relationships with every other point. In fact, that is the only defining feature of any given point in this theory of space and time. It is connected to adjacent points that have some smallest distance. This is (part of) loop quantum gravity.
There are other theories that attempt in one way or another to do away with the scale problem — for it is a scale problem. My own theory does away with it by quantizing gravity in a 5th dimension. That quantization doesn’t tell you what is going on at the smallest scales — it only tells you that those scales should be manageable because gravity evolves deterministically albeit chaotically at the smallest scales. It also fundamentally challenges quantum physics by incorporating it into Einstein’s theory of general relativity.
Whatever the answer is, it has to address this one question: what happens when you take gravity to the smallest scales? Does gravity change? Does quantum physics change? Does reality just stop? And if we do happen upon the right theory, we still have a lot of work to prove that it matches what we see at the largest scales. That has proved to be non-trivial for the top contenders.
Recently, I saw somebody on twitter ask if we could just only have rational numbers as measures of position and time. Would that help this problem? Rational numbers are ratios of integers, and mathematicians have known for centuries that not all numbers are rational numbers. In fact, there are infinitely more numbers that are not rational than there are rational numbers. Yet, every number you have ever encountered in your life is a rational number. The reason is because we simply cannot process numbers that have infinite numbers of non-repeating digits, as all non-rational numbers such as pi must. (To see why, just think about what a number with a finite number of digits after the decimal place is. It is just a ratio of a whole number to some power of ten.) The question is: would that solve the scale problem?
Oddly enough, it would not solve it. Rational numbers can represent any scale you choose. People get bent out of shape over continuous numbers, but they are rarely ever the real problem in physics. It is easy enough to control any problems continuous numbers introduce by using limits and scaling. I think non-mathematicians are just afraid of them because when you go into continuous numbers you lose the cozy intuitiveness of discrete numbers. You have to think about functional spaces and infinite dimensional operators. It gets dizzying. But the real problem isn’t continuity, it is scale. That is a problem with the infinity of numbers not the continuity of them. Introduce a finite universe and you might have it solved but you have to answer the question then: how big or small is the universe? So you trade one question for another impossible to answer one.
Without being able to measure what happens with black hole singularities yet, it is impossible to know what the real answer is. Perhaps gravitational wave detectors will one day tell us. Another possibility is that a new theory will be introduced that explains some other phenomena such as dark matter and dark energy and doesn’t have the same scale problems as Einstein’s. Whatever the solution is will need to work on vast and tiny scales.
Wilson, Kenneth G. “The renormalization group: Critical phenomena and the Kondo problem.” Reviews of modern physics 47.4 (1975): 773.
Wilson, Kenneth G. “Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture.” Physical review B 4.9 (1971): 3174.
Polyakov, A. “Fine structure of strings.” Nuclear Physics B 268.2 (1986): 406–412.
Rovelli, Carlo. “Loop quantum gravity.” Living reviews in relativity 11.1 (2008): 1–69.