Friday, February 5, 2010

The problem with Quantum Mechanics

Everyone knows Quantum Mechanics is weird. Many of its principles sound paradoxical.

Matter is both wave and particle. Position and velocity can't be simultaneously specified. Particles have spin even though they can't be spun. Particles carry entanglements across space, allowing a form of teleportation. "Empty" space seethes with activity. Small-scale physics is unpredictable and fundamentally random.

Weird, for sure. But is there any real problem here? Does the theory have some kind of inconsistency or mathematical difficulty, or does it just conflict with our inborn intuitions?

I say mathematical difficulty because that is the only kind of problem that would be a real problem (aside from experimental contradiction). If a theory makes mathematical sense then there's no reason to believe it couldn't represent a universe, no matter how badly it contravenes "common sense". Indeed, mathematics is just extrapolated common sense, so anything that makes mathematical sense can be assimilated into our intuition eventually.

But Quantum Mechanics has resisted this assimilation for almost a century now. The reason for this lies not with any of the oddities cited in the second paragraph; they are all perfectly comprehensible with a bit of study.

The problem with Quantum Mechanics is that it contains no consistent way to say what exists. This is usually referred to as the "measurement problem", because physicists encounter it when studying the measurement process, but in truth virtually everything is a kind of measurement. To even say that something exists, even something as seemingly obvious as a rhinoceros or a planet, is to make a type of measurement.

In Quantum Mechanics the universe consists of the "wave function", &#936. However, &#936 doesn't describe any actual particles, fields, or rhinoceroses, but only the probabilities that they might exist. In order for them to actually exist, there must be a "measurement". But a measurement requires a measurer, and the theory doesn't tell us what or who are the measurers.

Many perfectly valid Quantum Mechanical universes indeed have no actual measurements. Consider a sparsely-filled box of electrons, and imagine that this is the entire universe. Nothing in this universe provides any possibility of measurement, and therefore nothing in this universe really exists. We say the box "contains electrons" because we wrote down our normal &#936 theory for electrons, but beyond this there is nothing - no events, no history, and therefore no real electrons either.

But, one might object, surely in this "=electron universe one can talk about the possible locations and collisions of the electrons, and compute their probabilities? Maybe there's nobody around to see them, but so what - can't they still exist?

Alas, no. In the electron box there are no computable probabilities because the future trajectories of the electrons continue to interfere with each other. It is this interference which is the root of the problem we are discussing. The function of a "measuring device", or "observer", is to wash out the interference of outcomes in the future. Once the interference is washed out, the different outcomes are distinguishable and their probabilities make sense to a high degree - which, for probabilities, means that they very nearly add up to one.

And here is the mathematical crux of the problem: the probabilities never quite add up to one. There's no such thing as a perfect measuring device, because everything is just globs of matter in the first place. A big, complicated thing like a human being does a pretty good job of washing out interference (or to use the more technical lingo, "creating decoherence") but it is never perfect.

So we have "probabilities" that, mathematically speaking, aren't probabilities at all, because they don't add to one. It's like saying there's a 50% chance of flipping heads and a 51% chance of tails; in fact that's exactly what the prediction could be in extreme cases of interference, like the electron box world.

The most accepted solution to this problem, as far as I know, is the one I just alluded to: large blobs of matter create "decoherence" in the things they touch, allowing them to wash out interference to a very high degree. In other words, the probabilities almost add up to one. The come so close that one can argue that the discrepancy can never be noticed in practice.

Well, we don't "notice" the inconsistency between General Relativity and Quantum Mechanics in practice, either - but that hasn't stopped two generations of physicists from trying to resolve it. This problem of probabilities that don't add up to one is equally embarrassing, but gets little attention because nobody has a clue where to start. It is built so deeply into the structure of Quantum Mechanics, and that structure seems so impervious to tinkering, that the effort seems futile.

Personally, I am torn. I believe that anything that exists must rest on a consistent mathematical foundation. The fact that our own universe is built from mathematics suggests to me that this view is right. If it could have been some other way - then why isn't it?

But inconsistent mathematics is as bad as no mathematics at all. Inconsistent mathematics has all the same problems as gods or magic or any of the other non-mathematical fairy tales people have dreamed up over the eons. So, given that the universe clearly uses mathematics, why would it slip in an inconsistency at the very lowest level? Why bother with math at all, in that case?

Yet, I have a feeling that the problems with Quantum Mechanics will not be resolved. The probabilities add up nearly to one in the universe we have right here, so even though it doesn't make any sense in principle, and it wouldn't make sense for some other universes, it's what we will be stuck with - like it or lump it.

For some reason, our universe chooses to exist at the very boundary of conceivability. Perhaps it is a joke of some kind, or perhaps for some reason this is the only kind of existence that is really possible.

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