Shut up and Calculate?
One often reads about how successful quantum theory is, in that it can predict the results of experiments with superb accuracy. If there is a problem with quantum theory then it is not a problem of being able to do calculations, we are told, it is a problem with what it all means. I don't accept this argument. After all, one can take any subject and question what it actually means. But somehow it only happens with quantum theory. My claim is that far from being able to "Shut up and Calculate", in fact the calculations are in general so difficult as to be impracticable, and that this is what is at the root of the interpretational problems of quantum theory.
In general relativity, it was gradually realised that for some objects the tendency to collapse under gravity couldn't be balanced by any possible force. Such an object would continue its collapse down to a singularity, forming what is now known as a black hole. However, there was considerable resistance to this idea, and it was thought possible that black holes wouldn't form in realistic situations - maybe the examples being considered were artificial. Then in the 1960's Penrose and others proved that this wasn't the case, and that if you started with a massive enough body then you would get a singularity. This type of general result is critical to the development of a subject. However, in quantum theory such results are much harder to find - we tend to have to make do with approximations, which always leave some doubt as to whether they are really applicable. This is why people constantly argue about the meaning of quantum mechanics, while they don't for general relativity. (Well maybe I do, but that's a different matter).
So what has been calculated?
So we know that matter will collapse under gravity at high densities. Such results have also been obtained using quantum mechanics. However, what about normal matter? Here we have huge numbers of atoms, each consisting of a nucleus surrounded by electrons, and the weak, strong and electromagnetic forces and the exclusion principle mean that all of these particles end up in a certain configuration. What if some other configuration resulted in a lower energy? Can we show that matter won't suddenly collapse into such a configuration? This is called the stability of matter problem, and is surprisingly difficult. Many results have been obtained, in particular by Elliot Lieb and co-workers. However there is no overall result - for example moving from non-relativistic to relativistic quantum mechanics means starting again from scratch.
The question of what happens when a measurement is made is at the centre of interpretational issues in quantum mechanics. Around 1930 papers were written describing a model of a particle being absorbed by an atom, thus changing its state. This was taken as a simple model of a measuring device. Now my description of a measuring device would include several things, such as that it starts in a metastable state, it amplifies the incoming signal, and it loses heat to the environment. You might expect that these would become included in the model of a measuring device, as the subject progressed. Surprisingly this didn't happen - the 1930 idea of a quantum measuring device held sway for about 50 years. Clearly during this time some people, such as Willis Lamb, tried to take the calculations further, but their ideas didn't seem to enter the mainstream of the subject . More recently ideas such as decoherence have been introduced but even this seems to have become embroiled in interpretational issues.
So what happens now?
My belief is that in quantum theory that hand-waving arguments have somehow become more important than trying to deal with the substantial computational problems which remain. This leads to a situation where one can never be quite sure what the 'standard' interpretation actually says. For example Penrose has proposed a theory of 'Objective reduction of the wave function', and has devised an experiment to test this. Likewise Deutsch has thought up an experiment to test the possibility of the many-worlds interpretation. However, we are also told that different interpretations are physically equivalent. How can this be? I have a suspicion that if the experiments were carried out then whatever the result then some people would say 'Yes that agrees with the Copenhagen Interpretation'. I would like to see more work done on clarifying what standard quantum theory actually predicts in these cases. If that means that much more effort has to be put into doing the calculations, and in attacking difficult mathematics, then I for one would see that as a good thing.