The Planck length as a minimal length

The best scientific arguments are those that are surprising at first sight, yet at second sight they make perfect sense. The following argument, which goes back to Mead's 1964 paper "Possible Connection Between Gravitation and Fundamental Length," is of this type. Look at the abstract and note that it took more than 5 years from submission to publication of the paper. Clearely, Mead's argument seemed controversial at this time, even though all he did was to study the resolution of a microscope taking into account gravity.

For all practical purposes, the gravitational interaction is far too weak to be of relevance for microscopy. Normally, we can neglect gravity, in which case we can use Heisenberg's argument that I first want to remind you of before adding gravity. In the following, the speed of light c and Planck's constant ℏ are equal to one, unless they are not. If you don't know how natural units work, you should watch this video, or scroll down past the equations and just read the conclusion.

Consider a photon with frequency ω, moving in direction x, which scatters on a particle whose position on the x-axis we want to measure (see image below). The scattered photons that reach the lens (red) of the microscope have to lie within an angle ε to produces an image from which we want to infer the position of the particle.

According to classical optics, the wavelength of the photon sets a limit to the possible resolution Δx But the photon used to measure the position of the particle has a recoil when it scatters and transfers a momentum to the particle. Since one does not know the direction of the photon to better than ε, this results in an uncertainty for the momentum of the particle in direction xTaken together one obtains Heisenberg's uncertainty principle
We know today that Heisenberg's uncertainty principle is more than a limit on the resolution of microscopes; up to a factor of order one, the above inequality is a fundamental principle of quantum mechanics.

Now we repeat this little exercise by taking into account gravity.

Since we know that Heisenberg's uncertainty principle is a fundamental property of nature, it does not make sense, strictly speaking, to speak of the position and momentum of the particle at the same time. Consequently, instead of speaking about the photon scattering off the particle as if that would happen in one particular point, we should speak of the photon having a strong interaction with the particle in some region of size R (shown in the above image).

With gravity, the relevant question now will be what happens with the measured particle due to the gravitational attraction of the test particle.

For any interaction to take place and subsequent measurement to be possible, the time elapsed between the interaction and measurement has to be at least of the order of the time, τ, the photon needs to travel the distance R, so that τ is larger than R. (The blogger editor has an issue with the "larger than" and "smaller than" signs, which is why I avoid using them.) The photon carries an energy that, though in general tiny, exerts a gravitational pull on the particle whose position we wish to measure. The gravitational acceleration acting on the particle is at least of the orderwhere G is Newton's constant which is, in natural units, the square of the Planck length l_Pl. Assuming that the particle is non-relativistic and much slower than the photon, the acceleration lasts about the duration the photon is in the region of strong interaction. From this, the particle acquires a velocity of v ≈ aRThus, in the time R, the aquired velocity allows the particle to travels a distance of L ≈ Gω.

Since the direction of the photon was unknown to within ε, the direction of the acceleration and the motion of the is also unknown. Projection on the x-axis then yields the additional uncertainty ofCombining this with the usual uncertainty (multiply both, then take the square root), one obtainsThus, we find that the distortion of the measured particle by the gravitational field of the particle used for measurement prevents the resolution of arbitrarily small structures. Resolution is bounded by the Planck length, which is about 10^-33cm. The Planck length thus plays the role of a minimal length.

(You might criticize this argument because it makes use of Newtonian gravity rather than general relativity, so let me add that, in his paper, Mead goes on to show that the estimate remains valid also in general relativity.)

As anticipated, this minimal length is far too small to be of relevance for actual microscopes; its relevance is conceptual. Given that Heisenberg's uncertainty turned out to be a fundamental property of quantum mechanics, encoded in the commutation relations, we have to ask then if not this modified uncertainty too should be promoted to fundamental relevance. In fact, in the last 5 decades this simple argument has inspired a great many works that attempted exactly this. But that is a different story and shall be told another time.

To finish this story, let me instead quote from a letter that Mead, the author of the above argument, wrote to Physics Today in 2001. In it, he recalls how little attention his argument originally received:

"[In the 1960s], I read many referee reports on my papers and discussed the matter with every theoretical physicist who was willing to listen; nobody that I contacted recognized the connection with the Planck proposal, and few took seriously the idea of [the Planck length] as a possible fundamental length. The view was nearly unanimous, not just that I had failed to prove my result, but that the Planck length could never play a fundamental role in physics. A minority held that there could be no fundamental length at all, but most were then convinced that a [different] fundamental length..., of the order of the proton Compton wavelength, was the wave of the future. Moreover, the people I contacted seemed to treat this much longer fundamental length as established fact, not speculation, despite the lack of actual evidence for it."