The problem with point particles

Quantum mechanics is often viewed as a “weird” theory, with all sorts of non-intuitive predictions. However, there are more serious conceptual problems with classical mechanics, at least in its simpler formulations. One of these is what happens near point particles — in fact, point particles in classical mechanics lead to all sorts of infinities. I’ll say a little about this below, as well as talk about what quantum mechanics and string theory have to say about these short-distance infinities.

First, the terminology: classical mechanics is physics without discretisation — that is, anything that isn’t quantum. The cornerstones of classical mechanics are Maxwell’s electromagnetism, and Einstein’s Special and General Relativity. Quantum mechanics, or quantum theories in general, are theories which include the ideas (all related) of wave/particle duality; uncertainty relations between position and momentum; and discretisation of quantities such as momentum and energy. Point particles are particles with no spacial extent, ie. infintely small; all fundamental particles are either points or very, very close to point-size.

Let’s start with one of the simplest classical systems: an isolated, charged point particle. Coulomb’s law,

[tex]E(r) = \frac{Q}{r^2}[/tex]

expresses the electric field due to the particle of charge Q, at a distance r. By considering the energy stored in a capacitor, it’s easy to show that the energy of an electric field is proportional to E². But the integral of this over all space is infinity! Yes, simple classical mechanics predicts that all isolated charged point particles have infinite energy. Add relativity, and the equivalence of mass and energy, and we should measure infinite mass for all point particles. Ooops – this is a serious problem! The infinity comes from extending the integral all the way to zero radius, so is called an ultraviolet divergence — ultraviolet light having a short wavelength.

Having shown a problem with classical mechanics, we can start adding quantum mechanics. Initially, things seem worse: by the uncertainty principle, probing very small lengths means a great uncertainty in the momentum of the particles doing the probing, and being probed. Since they cannot have velocity greater than the speed of light, this means great uncertainty in their mass. In fact, probe distances smaller than the Plank length, 1.616×10^-35, and a black hole develops — destroying what you were attempting to probe.

Catastrophic as this seems, it allows us to say that distances less than the Planck length make no sense — that is, there is a smallest definable distance. Point particles are equivalent to particles of this size, as we can’t tell them apart. Equivalently, we define an ultraviolet cutoff — a highest energy that a particle can have. This seems arbitrary, but it allows us to avoid the infinities we saw in the classical case, in a mathematically rigorous manner termed renormalisation, for all forces other than gravity. This is something we cannot do in classical mechanics, as classical mechanics has no “natural length scale”.

Some artifacts, however, remain. The energy of the field surrounding a particle (and so the observed particle energy) is now dependent on the exact length cutoff we choose. This means that physical masses for “bare” particles, without their surrounding fields, are very hard to define. So physics deals with observed particle masses.

The exception to the above is gravity. For various reasons, gravity is not renormalisable. One way to think about it is to note that gravity affects the curvature of spacetime, and so gravity changes the meaning of lengths near gravitational sources. We saw that probing small distances implies encountering strong gravitational fields, till eventually it’s hard to define what we mean by a standard length cutoff. This failure of renormalisability is means that we have no theory of gravity at short distances. So relativity and quantum mechanics are not yet compatible — we need a quantum theory of gravity.

String theory seems to be the best candidate at the moment. It is far from understood, but seems to include, even require gravity, in order to be consistently defined. One way to think about this MIGHT be the following (no promises that this is the correct picture): the problems above result from infinities extremely close to charges. If the particle is an extended object, like a string, then there is only a charge per unit length. Get sufficiently close, so that the string looks long in comparison, and the charge per unit length approaches zero. The infinite field strengths don’t cause the same problems.

What is the weirdest part of this story, for me? Around every single particle is a sea of particles which we can’t define properly. When we look closely, we see particles at the maximum possible energies we can define — if not even tiny black holes. We can’t even say what the actual particle under all this mess is, or what its mass is. But this all doesn’t matter to the rest of physics, science, even life. Everywhere other than the surface of black holes, we can completely ignore the complexity at these short lengths, and treat particles as nice, well behaved points. This seems to me to be an entirely non-trivial statement about the organisation of our universe.

Exercise for the reader: Do NOT, whatever you do, look too closely at anything. The universe might decide to show you what’s REALLY happening, and then we’re all in trouble.