Is there a smallest length?

Good ideas start with a question. Great ideas start with a question that comes back to you. One such question that has haunted scientists and philosophers since thousands of years is whether there is a smallest unit of length, a shortest distance below which we cannot resolve structures. Can we look closer and always closer into space, time, and matter? Or is there a limit, and if so, what is the limit?

I picture our foreign ancestors sitting in their cave watching the world in amazement, wondering what the stones, the trees and they themselves are made of – and starving to death. Luckily, those smart enough to hunt down the occasional bear eventually gave rise to human civilization sheltered enough from the harshness of life to let the survivors get back to watching and wondering what we are made of. Science and philosophy in earnest is only a few thousand years old, but the question whether there is smallest unit has always been a driving force in our studies of the natural world.

The old Greeks invented atomism, the idea that there is an ultimate and smallest element of matter that everything is made of. Zeno’s famous paradoxa sought to shed light on the possibility of infinite divisibility. The question came back with the advent of quantum mechanics, with Heisenberg’s uncertainty principle that fundamentally limits the precision by which we can measure. It became only more pressing with the divergences in quantum field theory that are due to the inclusion of infinitely short distances.

It was in fact Heisenberg who first suggested that divergences in quantum field theory might be cured by the existence of a fundamentally minimal length, and he introduced it by making position operators non-commuting among themselves. Like the non-commutativity of momentum and position operators leads to an uncertainty principle, so does the non-commutativity of position operators limits how well distances can be measured.

Heisenberg’s main worry, which the minimal length was supposed to deal with, was the non-renormalizability of Fermi’s theory of beta-decay. This theory however turned out to be only an approximation to the renormalizable electro-weak interaction, so he had to worry no more. Heisenberg’s idea was forgotten for some decades, then picked up again and eventually grew into the area of non-commutative geometries. Meanwhile, the problem of quantizing gravity appeared on stage and with it, again, non-renormalizability.

In the mid 1960s Mead  reinvestigated Heisenberg’s microscope, the argument that lead to the uncertainty principle, with (unquantized) gravity taken into account. He showed that gravity amplifies the uncertainty so that it becomes impossible to measure distances below the Planck length, about 10^-33 cm. Mead’s argument was forgotten, then rediscovered in the 1990s by string theorists who had noticed using strings to prevent divergences by avoiding point-interactions also implies a finite resolution, if in a technically somewhat different way than Mead’s.

Since then the idea that the Planck length may be a fundamental length beyond which there is nothing new to find, ever, appeared in other approaches towards quantum gravity, such as Loop Quantum Gravity or Asymptotically Safe Gravity. It has also been studied as an effective theory by modifying quantum field theory to include a minimal length from scratch, and often runs under the name “generalized uncertainty”.

One of the main difficulties with these theories is that a minimal length, if interpreted as the length of a ruler, is not invariant under Lorentz-transformations due to length contraction. This problem is easy to overcome in momentum space, where it is a maximal energy that has to be made Lorentz-invariant, because momentum space is not translationally invariant. In position space one either has to break Lorentz-invariance or deform it and give up locality, which has observable consequences, and not always desired ones. Personally, I think it is a mistake to interpret the minimal length as the length of a ruler (a component of a Lorentz-vector), and it should instead be interpreted as a Lorentz-invariant scalar to begin with, but opinions on that matter differ.

The science and history of the minimal length has now been covered in a recent book by Amit Hagar:

Discrete or Continuous? The Quest for Fundamental Length inModern Physics
Amit Hagar
Cambridge University Press (2014)

Amit is a philosopher but he certainly knows his math and physics. Indeed, I suspect the book would be quite hard to understand for a reader without at least some background knowledge in math and physics. Amit has made a considerable effort to address the topic of a fundamental length from as many perspectives as possible, and he covers a lot of scientific history and philosophical considerations that I had not previously been aware of. The book is also noteworthy for including a chapter on quantum gravity phenomenology.

My only complaint about the book is its title because the question of discrete vs continuous is not the same as the question of finite vs infinite resolution. One can have a continuous structure and yet be unable to resolve it beyond some limit (this is the case when the limit makes itself noticeable as a blur rather than a discretization). On the other hand, one can have a discrete structure that does not prevent arbitrarily sharp resolution (which can happen when localization on a single base-point of the discrete structure is possible).

(Amit’s book is admittedly quite pricey, so let me add that he said should sales numbers reach 500 Cambridge University Press will put a considerably less expensive paperback version on offer. So tell your library to get a copy and let’s hope we’ll make it to 500 so it becomes affordable for more of the interested readers.)

Every once in a while I think that there maybe is no fundamentally smallest unit of length, that all these arguments for its existence are wrong. I like to think that we can look infinitely close into structures and will never find a final theory, turtles upon turtles, or that structures are ultimately self-similar and repeat. Alas, it is hard to make sense of the romantic idea of universes in universes in universes mathematically, not that I didn’t try, and so the minimal length keeps coming back to me.

Many if not most endeavors to find observational evidence for quantum gravity today look for manifestations of a minimal length in one way or the other, such as modifications of the dispersion relation, modifications of the commutation-relations, or Bekenstein’s tabletop search for quantum gravity. The properties of these theories are today a very active research area. We’ve come a long way, but we’re still out to answer the same questions that people asked themselves thousands of years ago.

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This post first appeared on Starts With a Bang with the title "The Smallest Possible Scale in the Universe" on August 12, 2014.