Dear Dr. Bee: Why do some people assume that the Planck length/time are the minimum possible length and time?

“Given that the Planck mass is nothing like a minimum possible mass, why do some people assume that the Planck length/time are the minimum possible length and time? If we have no problem with there being masses that are (much) smaller than the Planck mass, why do we assume that the Planck length and time are the fundamental units of quantisation? Why could there not be smaller distances and times, just as there are smaller masses?”

This question came from Brian Clegg who has written a stack of popular science books and who is also to be found on twitter and uses the same blogger template as I.

Dear Brian,

Before I answer your question, I first have to clarify the relation between the scales of energy and distance. We test structures by studying their interaction with particles, for example in a light microscopes. In this case the shortest structures we can resolve are of the order of the wavelength of the light. This means if we want to probe increasingly shorter distances, we need shorter wavelengths. The wavelength of light is inversely proportional to the light’s energy, so we have to reach higher energies to probe shorter distances. An x-ray microscope for example can resolve finer structures than a microscope that uses light in the visible range.

That the resolution of shorter distances requires higher energies isn’t only so for measurements with light. Quantum mechanics tells us that all particles have a wavelength and the same applies for measurements made by interactions with electrons, protons, and all other particles. Indeed using heavier particles is of advantage to get short wavelengths, which is why electron microscopes reveal more details than light microscopes. Besides this, electrically charged particles are easier to speed up to high energies than neutral ones.

The reason we build particle colliders is because of this simple relation: larger energies resolve shorter distances. Your final question thus should have been “Why could there not be smaller distances [than the Planck length], just as there are larger masses?”

One obtains the Planck mass, length, and time when one estimates the scale at which quantum gravity becomes important, when the quantum uncertainty in the curvature of space-time would noticeably distort the measurement of distances. One then obtains a typical curvature, and a related energy-density, at which quantum gravity becomes relevant. And this scale is given by the Planck scale, that is an energy of about 10¹⁵ times that of the LHC, or 10^-35 meter respectively. This is such a tiny distance/high energy that we cannot reach it with particle colliders, not now and not anytime in the soon future.

“Why do some people assume that the Planck length/time are the minimum possible length and time?”

It isn’t so much assumed as that it is a hint we take from estimates, thought experiments, and existing approaches to quantum gravity. It seems that the very definition of the Planck-scale – it being where quantum gravity becomes relevant – prevents us from resolving structures any shorter than this. Quantum gravity implies that space and time themselves are subject to quantum uncertainty and fluctuations, and once you reach the Planck scale these uncertainties of space-time add to the quantum uncertainties of the matter in space-time. But since gravity becomes stronger at higher energy density, going to even higher energies does no longer improve the resolution, it just causes stronger quantum gravitational effects.

You have probably heard that nobody yet knows how the complete theory of quantum gravity looks like because quantizing it can’t be done the same way as one quantizes the other interactions – this just leads to meaningless infinite results. However, a common way to tame infinities is to cut off integrals or to discretize the space one integrates over. In both cases this can be done by introducing a minimal length or a maximal energy scale. And since the Planck length and Planck energy are the only scales that we know of for what quantum gravity is concerned, it is often assumed that they play the role of this “regulator” that deals with the infinities.

More important than this general expectation though is that indeed almost all existing approaches to quantum gravity introduce a short-distance regulator in one way or the other. String theory resolves short distances by getting rid of point particles and instead deals only with extended objects. In Loop Quantum Gravity, one has found a smallest unit of area and of volume. Causal Dynamical Triangulation is built on the very idea of discretizing space-time in finite units. In Asymptotically Safe Gravity, the resolution of distances below the Planck length seems prohibited because the strength of gravity depends on the resolution. Moreover, the deep connections that have been discovered between thermodynamics and gravity also indicate that Planck length areas are smallest units in which information can be stored.

“Why do we assume that the Planck length and time are the fundamental units of quantisation?”

This isn’t commonly assumed. It is assumed in Loop Quantum Gravity, and a lot of people don’t think this makes sense.

“Why could there not be smaller distances and times, just as there are larger masses?”

The question isn’t so much whether there are distances shorter than (this question might not even be meaningful), but whether you can ever measure anything shorter than. And it’s this resolution of short distances that seems to be spoiled by quantum gravitational effects.

There is very relevant distinction you must draw here, which is that in most cases physicists speak of the Planck length as a “minimum length scale”, not just a “minimum length.” The reason is that not any quantity with a dimension length is a length of something, and not any quantity with a dimension of energy is an energy of something. It is not the distance itself that matters but the space-time curvature, which has the dimension of an inverse distance squared. So if a physicist says “quantum gravity becomes important at the Planck length” what they actually mean is “quantum gravity becomes important when the curvature is of the order of an inverse Planck length squared.” Or “energy densities in the center-of-mass frame are of the order Planck mass to the forth power”. Or at least that’s what they should mean...

And so, that the Planck length acts as a minimal length scale means that particle collisions that cram increasingly more energy into a smaller volumes will not reveal any new substructures once past Planckian curvature. Instead they may produce larger extended objects such as (higher-dimensional) black holes or stringballs and other exotic things. But note that in all these cases it is perfectly possible to exceed Planckian energies; this is referred to as “superplanckian scattering.”

Some models, such as deformed special relativity, do have an actual maximal energy. In this case the maximum energy is that of an elementary particle, which seems okay since we have never seen an elementary particle with an energy exceeding the Planck energy. In these models it is assumed that the upper bound for energies does not apply to composite particles, that you probably had in mind. The Planck energy is a huge energy to have for elementary particles, but its mass equivalent of 10^-5 gram is tiny on every-day scales. Just exactly how it should come about that in these models the maximum energy bound does not apply for composite objects is not well understood and some people, such as I, believe that it isn’t mathematically consistent.

Thanks for an interesting question!