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Rutherford did say something like this but suggesting a barmaid instead of a six-year-old, while Hilbert suggested the first man on the street. Technically, if we assume the statement correct, then nobody has understood anything about science, ever. But I'd like to have my go at explaining Physics-y stuff as simply as I can, and, perhaps equally importantly, without too much of the "woaaah quantum mechanics is so weird, you just have to accept it like this even though it's completely non-intuitive" crap one often finds in science articles written for the laymen.

Sunday, March 8, 2015

Jigsaw's cat

Quantum mechanics written in terms of the wave function and the Schroedinger equation is quite neat. It is only when we include the 'measurement' component that things usually become messy, and people start getting lost in 'paradoxes'. But measurements are a part of the postulates of the Copenhagen interpretation, thus they play a central, unavoidable role in the theory. The conceptual difficulties that this poses are collectively labeled the 'measurement problem', which represents - I think - the main reason for people to look for interpretations beyond Copenhagen. I'll try to outline the problem here. 
To do that, let's use the good ol' Schroedinger cat. The setup is the following: we take a box and put inside a cat and a vial of poisonous gas. We also put a quantum particle which has a probability of ½ to decay within one hour, as well as a detector and some mechanism that breaks the vial as soon as a decay is detected. 

Thus, quantum mechanics seems to say that, after one hour, the particle is in a superposition state of being decayed and not decayed, and the cat is therefore in a superposition state of being dead and alive. It is at this point important to note that when Schroedinger proposed this famous thought experiment, he did not give it as an illustration of quantum mechanics. Instead, he was demonstrating how there is something wrong with the way we think of it, because of the nonsense zombie-cat result. He was pointing out a problem, not giving a solution. This is very important and from now on you should never let anyone get away with saying that Schroedinger's cat is simultaneously dead and alive, as if that's some strange paradox of QM that one has to accept as true. Point out their mistake! Be that geeky 'technically, you're wrong' guy or gal! Together, we can start a revolution! And make the world a better place! Yes we can!

Uhm, anyway. To better understand the measurement problem, let's take a small modification of the thought experiment: let's put a scientist in the box instead of the cat. Let's also put the particle and the vial of poison inside another box. And let's say the scientist inside the first box is to open the second box one hour from the start of the experiment

Ok, let's say the box will open on its own one hour from the start of the experiment.

And let's say there's another scientist outside the big box.

who's creepily happy about the whole situation

because it's actually a cat controlling a humanoid robot!

It seems that in this situation it is extremely difficult to say at which point of time a measurement occurs and the particle's wave function 'collapses'. Say that the detector measures a decay already at the fifth minute. Has the wave function collapsed already? As far as both scientists are concerned, no it's not, because they haven't 'measured' it yet – they don't know whether the decay has happened. On the other hand, some sort of measurement did take place: the apparatus interacted with the system, right? So it must have also broken the vial of poison. Then, when 55 minutes more pass and the box opens, the second scientist is dead. Was that a 'measurement'? Is the wave-function collapsed now? As far as the cat-scientist is concerned, no it's not, because he hasn't 'measured' it: the big box is still closed. Or, let's have mercy and remove the vial of poison. When the small box opens, the second scientist can simply read out what the detector says

Has the wave function of the particle collapsed then? The detector has now been read out, so maybe yes, but again that's only the case from the point of view of the scientist in the box. As far as the cat outside is concerned, the system is still in a superposition state, and if he didn't know about us removing the vial of poison, the second scientist is himself in a superposition state of being dead and alive (Schroedinger's original nonsense result). Can we say that the wave function has collapsed because a conscious being has read out a detector? That's obviously a very dangerous path to take. What is consciousness anyway? Is a cat a conscious being? What if it's smart enough to build a controllable humanoid robot? 
Bottom line, the measurement postulates present many difficulties on the philosophical level. When does the measurement happen? Does it happen instantaneously? This is a huge problem in cases in which it would imply that an action at one point of space has an instantaneous effect at a different, distant point: that could break the idea that there is a cause-effect relationship in nature. Anyway, how do we even split the world into 'quantum' and 'classical' – isn't everything inherently quantum? That we are able to do such a split is pre-supposed in the Copenhagen postulates, but no rigorous way to do that is given. What if we want to study the whole universe as a quantum system?
In case that hasn't convinced you yet that there is a problem, I'll now return to the original problem as posed by Schrodinger. There is actually more than one way to 'resolve' it within the Copenhagen interpretation. Here I suggested one: do not ascribe any classical properties to the cat before it is observed - it's not a cat, it's a wave function in a superposition state. But it is also possible to think of the cat as a nice, ordinary, classical kitty, which has some probability to be dead, but is not in a superposition state (because that wouldn't make any sense, or would it?) In other words, since there is no rigorous definition of how to split the world into quantum and classical, it's up to us to decide to which world the kitty belongs. What's the difference between the two representations of the situation? I hope this makes it clear:

Different people might prefer one or the other version. I think Schroedinger had the first version in mind when he proposed the problem. In the second version, it's fun to think of the cat as a classical measurement apparatus which is constantly measuring the wave function of the quantum particle... by dying or not dying! In that sense, the wave function cannot be in a superposition state, it's either the wave function of a decayed particle or of an un-decayed one, because at every moment the cat interacts with it and 'measures' which is the case, causing an immediate collapse.

The problem is not with the versions themselves as much as with the fact that they are both allowed. The problem is, in other words, with the supposition - inherent to the Copenhagen postulates - that we can divide the universe into a quantum part, which is described by quantum mechanics, and a classical part, in which we live. Cats, too - or do they? What if instead of a cat we had a bacteria? Should we include it in the quantum system or keep thinking of it classically? What if we had a molecule that would dissolve if exposed to the gas? Is the molecule 'quantum' enough for us to have to include it in the wave function and not in the classical world? And if yes, where do we draw the line between a molecule and a cat... and is this line different for every different setup?

At this point, people who really want to stick to the Copenhagen interpretation will inevitably start talking about decoherence, which is a complex topic, but which, to me personally, always leaves a tang of not really solving the problem. In other words, I think that everybody who is sufficiently bothered by the problem to feel the need to dig deeper has to inevitably leave Copenhagen in search of further answers. And, of course, I'm planning to get to that one day!

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