Split a mile in half, you get half a mile. Split the half mile, you get a quarter, and on and on, until you’ve carved out a length far smaller than the diameter of an atom. Can this slicing continue indefinitely, or will you eventually reach a limit: a smallest hatch mark on the universal ruler?
The success of some contemporary theories of quantum gravity may hinge on the answer to this question. But the puzzle goes back at least 2500 years, to the paradoxes thought up by the Greek philosopher Zeno of Elea, which remained mysterious from the 5th century BC until the early 1800s. Though the paradoxes have now been solved, the question they posed—is there a smallest unit of length, beyond which you can’t divide any further?—persists.
The most famous of Zeno’s paradoxes is that of Achilles and the Tortoise in a race. The tortoise gets a head start on the faster-running Achilles. Achilles should quickly catch up—at least that’s what would happen in a real-world footrace. But Zeno argued that Achilles will never pass over the tortoise, because in the time it takes for Achilles to reach the tortoise’s starting point, the tortoise too will have moved forward. While Achilles pursues the tortoise to cover this additional distance, the tortoise moves yet another bit. Try as he might, Achilles only ever reaches the tortoise’s position after the animal has already left it, and he never catches up.
Obviously, in real life, Achilles wins the race. So, Zeno argued, the assumptions underlying the scenario must be wrong. Specifically, Zeno believed that space is not indefinitely divisible but has a smallest possible unit of length. This allows Achilles to make a final step surpassing the distance to the tortoise, thereby resolving the paradox.
It took more than two thousand years to develop the necessary mathematics, but today we know that Zeno’s argument was plainly wrong. After mathematicians understood how to sum an infinite number of progressively smaller steps, they calculated the exact moment Achilles surpasses the tortoise, proving that it does not take forever, even if space is indefinitely divisible.
Zeno’s paradox is solved, but the question of whether there is a smallest unit of length hasn’t gone away. Today, some physicists think that the existence of an absolute minimum length could help avoid another kind of logical nonsense; the infinities that arise when physicists make attempts at a quantum version of Einstein’s General Relativity, that is, a theory of “quantum gravity.” When physicists attempted to calculate probabilities in the new theory, the integrals just returned infinity, a result that couldn’t be more useless. In this case, the infinities were not mistakes but demonstrably a consequence of applying the rules of quantum theory to gravity. But by positing a smallest unit of length, just like Zeno did, theorists can reduce the infinities to manageable finite numbers. And one way to get a finite length is to chop up space and time into chunks, thereby making it discrete: Zeno would be pleased.
He would also be confused. While almost all approaches to quantum gravity bring in a minimal length one way or the other, not all approaches do so by means of “discretization”—that is, by “chunking” space and time. In some theories of quantum gravity, the minimal length emerges from a “resolution limit,” without the need of discreteness. Think of studying samples with a microscope, for example. Magnify too much, and you encounter a resolution-limit beyond which images remain blurry. And if you zoom into a digital photo, you eventually see single pixels: further zooming will not reveal any more detail. In both cases there is a limit to resolution, but only in the latter case is it due to discretization.
In these examples the limits could be overcome with better imaging technology; they are not fundamental. But a resolution-limit due to quantum behavior of space-time would be fundamental. It could not be overcome with better technology.
So, a resolution-limit seems necessary to avoid the problem with infinities in the development of quantum gravity. But does space-time remain smooth and continuous even on the shortest distance scales, or does it become coarse and grainy? Researchers cannot agree.
In string theory, for example, resolution is limited by the extension of the strings (roughly speaking, the size of the ball that you could fit the string inside), not because there is anything discrete. In a competing theory called loop quantum gravity, on the other hand, space and time are broken into discrete blocks, which gives rise to a smallest possible length (expressed in units of the Planck length, about 10-35 meters), area and volume of space-time—the fundamental building blocks of our universe. Another approach to quantum gravity, “asymptotically safe gravity,” has a resolution-limit but no discretization. Yet another approach, “causal sets,” explicitly relies on discretization.
And that’s not all. Einstein taught us that space and time are joined in one entity: space-time. Most physicists honor Einstein’s insight, and so most approaches to quantum gravity take space and time to either both be continuous or both be discrete. But some dissidents argue that only space or only time should be discrete.
So how can physicists find out whether space-time is discrete or continuous? Directly measuring the discrete structure is impossible because it is too tiny. But according to some models, the discreteness should affect how particles move through space. It is a miniscule effect, but it adds up for particles that travel over very long distances. If true, this would distort images from far-away stellar objects, either by smearing out the image or by tearing apart the arrival times of particles that were emitted simultaneously and would otherwise arrive on Earth simultaneously. Astrophysicists have looked for both of these signals, but they haven’t found the slightest evidence for graininess.
Even if the direct effects on particle motion are unmeasurable, defects in the discrete structure could still be observable. Think of space-time like a diamond. Even rare imperfections in atomic lattices spoil a crystal’s ability to transport light in an orderly way, which will ruin a diamond’s clarity. And if the price tags at your jewelry store tell you one thing, it’s that perfection is exceedingly rare. It’s the same with space-time. If space-time is discrete, there should be imperfections. And even if rare, these imperfections will affect the passage of light through space. No one has looked for this yet, and I’m planning to start such a search in the coming months.
Next to guiding the development of a theory of quantum gravity, finding evidence for space-time discreteness—or ruling it out!—would also be a big step towards solving a modern-day paradox: the black hole information loss problem, posed by Stephen Hawking in 1974. We know that black holes can only store so much information, which is another indication for a resolution-limit. But we do not know exactly how black holes encode the information of what fell inside. A discrete structure would provide us with elementary storage units.
Black hole information loss is a vexing paradox that Zeno would have appreciated. Let us hope we will not have to wait 2000 years for a solution.
Editor and author’s picks for further reading
arXiv: Minimal Length Scale Scenarios for Quantum Gravity
Sabine Hossenfelder’s detailed look at the past, present, and future of the question, is there a shortest length?
Fermilab Physics in a Nutshell: Planck length, minimal length?
Don Lincoln explains the Planck length and why it’s thought to be the smallest possible length.
The Nature of Reality: Do Black Holes Destroy Information?
A primer on the black hole information paradox.
Physics Today: Questions and answers with Amit Hagar
An interview with philosopher of science Amit Hagar, author of “Discrete or Continuous?: The Quest for Fundamental Length in Modern Physics.”