Editors' note: Last year the Foundational Questions Institute's third essay contest posed the following question to physicists and philosophers: “Is Reality Digital or Analog?” The organizers expected entrants to come down on the side of digital. After all, the word “quantum” in quantum physics connotes “discrete” —hence, “digital”. Many of the best essays held, however, that the world is analog. Among them was the entry by David Tong, who shared the second-place prize. The article here is a version of his essay.
In the late 1800s the famous german mathematician Leopold Kronecker proclaimed, “God made the integers, all else is the work of man.” He believed that whole numbers play a fundamental role in mathematics. For today's physicists, the quote has a different resonance. It ties in with a belief that has become increasingly common over the past several decades: that nature is, at heart, discrete—that the building blocks of matter and of spacetime can be counted out, one by one. This idea goes back to the ancient Greek atomists but has extra potency in the digital age. Many physicists have come to think of the natural world as a vast computer described by discrete bits of information, with the laws of physics an algorithm, like the green digital rain seen by Neo at the end of the 1999 film The Matrix.
Yet is that really the way the laws of physics work? Although it might seem to contradict the spirit of the times, I, among many others, think that reality is ultimately analog rather than digital. In this view, the world is a true continuum. No matter how closely you zoom in, you will not find irreducible building blocks. Physical quantities are not integers but real numbers—continuous numbers, with an infinite number of digits after the decimal point. The known laws of physics, Matrix fans will be disappointed to learn, have features that no one knows how to simulate on a computer, no matter how many bytes its memory has. Appreciating this aspect of these laws is essential to developing a fully unified theory of physics.
An Ancient Enigma
The debate between digital and analog is one of the oldest in physics. Whereas the atomists conceived of reality as discrete, other Greek philosophers such as Aristotle thought of it as a continuum. In Isaac Newton's day, which spanned the 17th and 18th centuries, natural philosophers were torn between particle (discrete) theories and wave (continuous) theories. By Kronecker's time, advocates of atomism, such as John Dalton, James Clerk Maxwell and Ludwig Boltzmann, were able to derive the laws of chemistry, thermodynamics and gases. But many scientists remained unconvinced.
Wilhelm Ostwald, winner of the 1909 Nobel Prize in Chemistry, pointed out that the laws of thermodynamics refer only to continuous quantities such as energy. Similarly, Maxwell's theory of electromagnetism describes electric and magnetic fields as continuous. Max Planck, who would later pioneer quantum mechanics, finished an influential paper in 1882 with the words: “Despite the great success that the atomic theory has so far enjoyed, ultimately it will have to be abandoned in favor of the assumption of continuous matter.”
One of the most powerful arguments of the continuous camp was the seeming arbitrariness of discreteness. As an example: How many planets are there in the solar system? I was told at school that there are nine. In 2006 astronomers officially demoted Pluto from the planetary A-list, leaving just eight. At the same time, they introduced a B-list of dwarf planets. If you include these, the number increases to 13. In short, the only honest answer to the question of the number of planets is that it depends on how you count. The Kuiper belt beyond Neptune contains objects in size ranging from mere microns to a few thousand kilometers. You can count the number of planets only if you make a fairly arbitrary distinction between what is a planet, what is a dwarf planet, and what is just a lump of rock or ice.
Quantum mechanics ultimately transformed the digital-analog debate. Whereas the definition of a planet may be arbitrary, the definition of an atom or an elementary particle is not. The integers labeling chemical elements—which, we now know, count the number of protons in their constituent atoms—are objective. Regardless of what developments occur in physics, I will happily take bets that we will never observe an element with √500 protons that sits between titanium and vanadium. The integers in atomic physics are here to stay.
Another example occurs in spectroscopy, the study of light emitted and absorbed by matter. An atom of a particular type can emit only very specific colors of light, resulting in a distinctive fingerprint for each atom. Unlike human fingerprints, the spectra of atoms obey fixed mathematical rules. And these rules are governed by integers. The early attempts to understand quantum theory, most notably by Danish physicist Niels Bohr, placed discreteness at its heart.
But bohr's was not the final word. Erwin Schrödinger developed an alternative approach to quantum theory based on the idea of waves in 1925. The equation that he formulated to describe how these waves evolve contains only continuous quantities—no integers. Yet when you solve the Schrödinger equation for a specific system, a little bit of mathematical magic happens. Take the hydrogen atom: the electron orbits the proton at very specific distances. These fixed orbits translate into the spectrum of the atom. The atom is analogous to an organ pipe, which produces a discrete series of notes even though the air movement is continuous. At least as far as the atom is concerned, the lesson is clear: God did not make the integers. He made continuous numbers, and the rest is the work of the Schrödinger equation.
In other words, integers are not inputs of the theory, as Bohr thought. They are outputs. The integers are an example of what physicists call an emergent quantity. In this view, the term “quantum mechanics” is a misnomer. Deep down, the theory is not quantum. In systems such as the hydrogen atom, the processes described by the theory mold discreteness from underlying continuity.
Perhaps more surprisingly, the existence of atoms, or indeed of any elementary particle, is also not an input of our theories. Physicists routinely teach that the building blocks of nature are discrete particles such as the electron or quark. That is a lie. The building blocks of our theories are not particles but fields: continuous, fluidlike objects spread throughout space. The electric and magnetic fields are familiar examples, but there are also an electron field, a quark field, a Higgs field, and several more. The objects that we call fundamental particles are not fundamental. Instead they are ripples of continuous fields.
A skeptic might say that the laws of physics do contain some integers. For example, these laws describe three kinds of neutrinos, six kinds of quarks (each of which comes in three varieties called colors), and so on. Integers, integers everywhere. Or are there? All these examples are really counting the number of particle species in the Standard Model, a quantity that is famously difficult to make mathematically precise when particles interact with one another. Particles can mutate: a neutron can split into a proton, an electron and a neutrino. Should we count it as one particle or three particles or four particles? The claim that there are three kinds of neutrinos, six kinds of quarks, and so on is an artifact of neglecting the interactions between particles.
Here is another example of an integer in the laws of physics: the number of observed spatial dimensions is three. Or is it? The famous late mathematician Benoît Mandelbrot pointed out that the number of spatial dimensions does not have to be an integer. The coastline of Great Britain, for example, has a dimension of around 1.3. Moreover, in many proposed unified theories of physics, such as string theory, the dimension of space is ambiguous. Spatial dimensions can emerge or dissolve.
I venture to say only one true integer may occur in all of physics. The laws of physics refer to one dimension of time. Without precisely one dimension of time, physics appears to become inconsistent.
Even if our current theories assume reality is continuous, many of my fellow physicists think that a discrete reality still underlies the continuity. They point to examples of how continuity can emerge from discreteness. On the macroscopic scales of everyday experience, the water in a glass appears to be smooth and continuous. It is only when you look much much closer that you see the atomic constituents. Could a mechanism of this type perhaps sit at the root of physics? Maybe if we looked at a deeper level, the smooth quantum fields of the Standard Model, or even spacetime itself, would also reveal an underlying discrete structure.
We do not know the answer to this question, but we can glean a clue from 40 years of efforts to simulate the Standard Model on a computer. To perform such a simulation, one must first take equations expressed in terms of continuous quantities and find a discrete formulation that is compatible with the bits of information in which computers trade. Despite decades of effort, no one has succeeded in doing that. It remains one of the most important, yet rarely mentioned, open problems in theoretical physics.
Physicists have developed a discretized version of quantum fields called lattice field theory. It replaces spacetime with a set of points. Computers evaluate quantities at these points to approximate a continuous field. The technique has limitations, however. The difficulty lies with electrons, quarks and other particles of matter, called fermions. Strangely, if you rotate a fermion by 360 degrees, you do not find the same object that you started with. Instead you have to turn a fermion by 720 degrees to get back to the same object. Fermions resist being put on a lattice. In the 1980s Holger Bech Nielsen of the Niels Bohr Institute in Copenhagen and Masao Ninomiya, now at the Okayama Institute for Quantum Physics in Japan, proved a celebrated theorem that it is impossible to discretize the simplest kind of fermion.
Such theorems are only as strong as their assumptions, and in the 1990s theorists, most notably David Kaplan, now at the University of Washington, and Herbert Neuberger of Rutgers University, introduced various creative methods to place fermions on the lattice. Quantum field theories come in many conceivable varieties, each with different possible types of fermions, and people can now formulate nearly every one on a lattice. There is just a single class of quantum field theory that people do not know how to put on a lattice. Unfortunately, that class includes the Standard Model. We can handle all kinds of hypothetical fermions but not the ones that actually exist.
Fermions in the Standard Model have a very special property. Those that spin in a counterclockwise direction feel the weak nuclear force, and those that spin in a clockwise direction do not. The theory is said to be chiral. A chiral theory is delicate. Subtle effects known as anomalies are always threatening to render it inconsistent. Such theories have so far resisted attempts to be modeled on a computer.
Yet chirality is not a bug of the Standard Model that might go away in a deeper theory; it is a core feature. At first glance, the Standard Model, based on three interlinking forces, seems to be an arbitrary construction. It is only when thinking about the chiral fermions that its true beauty emerges. It is a perfect jigsaw puzzle, with the three pieces locked together in the only manner possible. The chiral nature of fermions in the Standard Model makes everything fit together.
Scientists are not entirely sure what to make of our inability to simulate the Standard Model on a computer. It is difficult to draw strong conclusions from a failure to solve a problem; quite possibly the puzzle is just a very difficult one waiting to be solved with conventional techniques. But aspects of the problem smell deeper than that. The obstacles involved are intimately tied to mathematics of topology and geometry. The difficulty in placing chiral fermions on the lattice may be telling us something important: that the laws of physics are not, at heart, discrete. We are not living inside a computer simulation.