The Origin of Force

 

As I have expanded upon lengthily in this previous post, interference is a key phenomenon in quantum theory. In this post, we will see how it can be used to explain the existence of forces between certain objects, using the example of the electromagnetic force in particular. 

The usual popular account of the quantum origin of forces rests on the notion of virtual particles. Basically, two charged particles are depicted as 'ice skaters' on a frictionless plane; they exchange momentum via appropriate virtual particles, i.e. one skater throws a ball over to the other, and both receive an equal amount of momentum imparted in opposite directions. This nicely explains repulsive forces, i.e. the case in which both skaters are equally charged. In order to explain attraction, as well, the virtual particles have to be endowed with a negative momentum, causing both parties to experience a momentum change in the direction towards the other. Sometimes, this is accompanied by some waffle about how this is OK for virtual particles, since they are not 'on-shell' (which is true, but a highly nontrivial concept to appeal to for a 'popular level' explanation).

In this post, after the introduction, I will not talk about virtual particles anymore. The reason for this is twofold: first, the picture one gets through the 'ice-skater' analogy is irreducibly classical and thus, obfuscates the true quantum nature of the process, leaving the reader with an at best misleading, at worst simply wrong impression. Second, and a bit more technically, virtual particles are artifacts of what is called a perturbation expansion. Roughly, this denotes an approximation to an actual physical process by means of taking into account all possible ways the process can occur, and then summing them to derive the full amplitude -- if you're somewhat versed in mathematical terminology, it's similar to approximating a function by means of a Taylor series. The crucial point is that the virtual particles are present in any term of this expansion, but the physical process does not correspond to any of those terms, but rather, to their totality. So the virtual-particles analogy can't give you the full picture.

What is Quantum Mechanics?

So far, I've told you a little about where I believe quantum theory comes from. To briefly recap, information-theoretic incompleteness, a feature of every universal system (where 'universal' is to be understood in the sense of 'computationally universal'), introduces the notion of complementarity. This can be interpreted as the impossibility for any physical system to answer more than finitely many questions about its state -- i.e. it furnishes an absolute restriction on the amount of information contained within any given system. From this, one gets to quantum theory via either a deformation of statistical mechanics (more accurately, Liouville mechanics, i.e. statistical mechanics in phase space), or, more abstractly, via introducing the possibility of complementary propositions into logic. In both cases, quantum mechanics emerges as a generalization of ordinary probability theory. Both points of view have their advantages -- the former is more intuitive, relying on little more than an understanding of the notions of position and momentum; while the abstractness of the latter, and especially its independence from the concepts of classical mechanics, highlights the fundamental nature of the theory: it is not merely an empirically adequate description of nature, but a necessary consequence of dealing with arbitrary systems of limited information content. For a third way of telling the story of quantum mechanics as a generalized probability theory see this lecture by Scott Aaronson, writer of the always-interesting Shtetl-Optimized.
But now, it's high time I tell you a little something about what, actually, this generalized theory of probability is, how it works, and what it tells us about the world we're living in. First, however, I'll tell you a little about the mathematics of waves, the concept of phase, and the phenomenon of interference.