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.

The Interpretation of Quantum Mechanics

So far on this blog, I have argued that quantum mechanics should be most aptly seen as a generalization of probability theory, necessary to account for complementary propositions (propositions which can't jointly be known exactly). Quantum mechanics can then be seen to emerge either as a generalization (more accurately, a deformation) of statistical mechanics on phase space, or, more abstractly (but cleaner in a conceptual sense) as deriving from quantum logic in the same way classical probability derives from classical, i.e. Boolean, logic.

Using this picture, we've had a look at how it helps explain two of quantum mechanics' most prominent, and at the same time, most mysterious consequences -- the phenomena of interference and entanglement, both of which are often thought to lie at the heart of quantum mechanics.

In this post, I want to have a look at the interpretation of quantum mechanics, and how the previously developed picture helps to make sense of the theory. But first, we need to take a look at what, exactly, an interpretation of quantum mechanics is supposed to accomplish -- and whether we in fact need one (because if we find that we don't, I could save myself a lot of writing).

Untangling Entanglement

What to Feynman was interference (see the previous post), to Erwin Schrödinger (he of the cat) was the phenomenon known as entanglement: the 'essence' of quantum mechanics. Entanglement is often portrayed as one of the most outlandish features of quantum mechanics: the seemingly preposterous notion that the outcome of a measurement conducted over here can instantaneously influence the outcome of a measurement carried out way over there.
Indeed, Albert Einstein himself was so taken aback by this consequence of quantum mechanics (a theory which, after all, he helped to create), that he derided it as 'spooky' action at a distance, and never fully accepted it in his lifetime.
However, viewing quantum mechanics as a simple generalization of probability theory, which we adopt in order to deal with complementary propositions that arise when not all possible properties of a system are simultaneously decidable, quantum entanglement may be unmasked as not really that strange after all, but in fact a natural consequence of the limited information content of quantum systems. In brief, quantum entanglement does not qualitatively differ from classical correlation; however, the amount of information carried by the correlation exceeds the bounds imposed by classical probability theory.

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.

The Origin Of The Quantum, Part I: An Incomplete Phase Space Picture

In the last post, we have familiarized ourselves with some basic notions of algorithmic information theory. Most notably, we have seen how randomness emerges when formal systems or computers are pushed to the edges of incompleteness and uncomputability.
In this post, we'll take a look at what happens if we apply these results to the idea that, like computers or formal systems, the physical world is just another example of a universal system -- i.e. a system in which universal computation can be implemented (at least in the limit).
First, recall the idea that information enters the description of the physical world through viewing it as a question-answering process: any physical object can be uniquely identified by the properties it has (and those it doesn't have); any two physical objects that have all the same, and only the same, properties are indistinguishable, and thus identified. We can thus imagine any object as being described by the string of bits giving the answers to the set of questions 'Does the object have property x?' for all properties x; note that absent an enumeration of all possible properties an object may have, this is a rather ill-defined set, but it'll serve as a conceptual guide.
In particular, this means that we can view any 'large' object as being composed of a certain number of 'microscopic', elementary objects, which are those systems that are completely described by the presence or absence of one single property, that may be in either of two states -- having or not having that particular property. Such a system might, for instance, be a ball that may be either red or green, or, perhaps more to the point, either red or not-red. These are the systems that can be used to represent exactly one bit of information, say red = 1, not-red = 0. Call such a system a two-level system or, for short, and putting up with a little ontological inaccuracy, simply a bit.