I finally have found some time to blog a bit and one thing that has been on my mind is the quantum-classical contrast. One of the points commonly made when attempting to distinguish between classical systems and quantum systems is that the latter are probabilistic. This is often then (somewhat carelessly) extrapolated to mean that classical physics is exact while quantum physics is not. Not only is that an oversimplification, it is misleading.
First, while it is true that quantum physics is probabilistic, it does not mean that there aren’t quantum systems for which the probability of a given state is precisely 100%. In fact, suppose you have system in a superposition of states. Once it’s been measured, it will then be in a specific eigenstate. If the system is isolated then it is guaranteed to remain in that eigenstate and thus all subsequent measurements will find it in that state (J.J. Sakurai has an excellent discussion of this in his book Modern Quantum Mechanics). In addition, it is often assumed that the uncertainty principle prevents precise measurements. In fact all the uncertainty principle says is that it is impossible to simultaneously measure non-commuting observables to arbitrary accuracy. Nothing prevents us from measuring a single observable (assuming the measuring apparatus doesn’t include a “hidden” operator that fails to commute with the observable) and nothing prevents us from simultaneously measuring commuting observables precisely. In fact, the most accurate physical theory – known to match experiment to better than ten parts in one-billion () – is quantum electrodynamics (QED).
With that said, isn’t classical physics exact? If I calculate some force via Newton’s laws, won’t the result match experiment perfectly? Actually, it depends on how literally we take the word “perfect” to be. How many decimal places are required for perfection? Well, technically (if we take the dictionary’s definition of the word) all of them! Take a anything that measures weight (the gravitational force), for instance, e.g. a bathroom scale. Usually we’re happy if our bathroom scale is accurate to within a pound or so (note to the rest of world: scales should not measure kilograms since that is a measure of mass!). But suppose for some perverse reason we wanted better accuracy. In order to do that, we’d need to develop a more sensitive instrument. We could go on doing this in an attempt to measure our weight more and more accurately, but at some point, we’re going to get down to counting the atoms in our body which are quantum objects!
In other words, in classical physics, accuracy is what we make it. Inevitably, continually seeking to increase one’s accuracy in measurements will lead one into the quantum realm. So maybe the growing trend to view the universe as essentially quantum, á là Bob Griffiths, is a good one. If so, then in attempts to unify quantum mechanics and gravity (general relativity) it stands to reason that general relativity is the one that needs altering. But general relativity is a phenomenally successful theory. We rely on it daily for GPS coordinates (yes, general relativity is used). While it may not be as accurate as QED, it’s still quite accurate. Considering how different in form it is from quantum mechanics, this suggests that something about it is right (or, rather, it can’t be complete crap). In particular, it predicts that time is not absolute (and that there are technically three different kinds of time: coordinate time, proper time, and spacetime intervals). Since this has been experimentally proven and (non-relativistic) quantum mechanics does not have a mechanism with which to deal with this (it treats time as absolute), this clearly marks a failing of quantum mechanics.
Now, QED is technically the merger of quantum mechanics and special relativity which, in theory, should solve the time problem. Since it has served as the model (directly or indirectly) for most subsequent quantum field theories (including string theory), it would stand to reason that it would be a good model for quantum gravity – and it well may turn out that way. But, to date, experimental verification has eluded physicists. In addition, it suffers from complexity. But perhaps it’s greatest problem may be what it shares with most classical theories: it’s a field theory.
Classical physical theories are all field theories at their core. While this is purely conjecture, it might be that this desire to “classicize” things motivated the stubborn insistance on field theoretic techniques. Most field theorists, however, will tell you that the main reason field theory is needed is because there are no multi-particle solutions to the Klein-Gordon equation for photons unless one treats them as a field (those damned, pesky photons!). Everything – literally everything – in quantum field theory has followed from this. First, this assumes the Klein-Gordon equation is a good model, but we know it doesn’t include spin which is a major failing. So it is far from perfect. But, in addition, fields are not discrete and we assume the universe is discrete (thanks to quantum mechanics). But is it?
In fact, we may never know. It is possible that the uncertainty principle, instead of telling us that the universe is discrete, simply limits our knowledge about the universe and it is that knowledge that is discrete. As such it is conceivable that we will never find a single, universally accepted theory of quantum gravity. Yet we stubbornly cling to our belief in a universe of physical fields rather than questioning whether our underlying assumptions were biased to begin with. Don’t get me wrong. Field theory is very accurate and quite useful. But don’t assume it represents reality. It’s simply a model and there may exist a better model that we have hitherto ignored.