pbs | 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.
