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Quantum mechanics, the measurement problem, and the nature of reality
3.0 Look, if you sort of squint and defocus a bit... yes! I can see it, it's a dinosaur, isn't it?
3.1 Copenhagen Interpretation
The CI is the most orthodox interpretation of QM. It contains what we usually regard as the basic postulates of QM, but what are just parts of a philosophical interpretation of the wavefunction that have a physical use.
The CI acknowledges Heisenberg's uncertainty principal, and the statistical approach of Born. It is this part of the interpretation that means the average behaviour of a large number of quantum events converges on classical law, as is absolutely necessary (otherwise it simply wouldn't agree with reality). Born also gives rise to the probability law, where P=(psi).(psi)*.
It also says that the state vector - that is, the representation of the wavefunction - holds everything that we know about the system. The state vector contains all the information there is, and nothing else. This is why, when our knowledge changes (ie we make a measurement), the wavefunction collapses.
As well as this, the CI sensibly does not make non-observable interpretations on the meaning of quantum mechanics, as many other interpretations do. In a way, this is wise because it leaves the orthodox way open and stays with only what we know works. However, we saw earlier that sometimes it can be useful to look at a theory in a different way: the reality doesn't change, but the metaphor may lead us to great understanding. Remember, there is nothing intrinsically 'correct' or 'real' about the formulation, it's just a convenient way of working. It's just a representation of the universe.
3.2.0 Hidden Variable theories and the Bell Inequality
As far as representations of the universe go, some believe that even the CI has gone too far with the interpretation that the wavefunction contains all our knowledge of the system. Many, Einstein among them, believed that there was an objective universe underneath, we just couldn't see it. These are the so-called 'Hidden Variable' theories.
3.2.1 Why hidden variable theories are attractive
There is actually an experimental setup which illustrates a facet of QM which people regarded to be unwanted, and wanted to find a way to circumvent.
Einstein, Podolsky and Rosen (commonly known as EPR) presented what they regarded as a paradox. Simply, consider a system where two particles are 'entangled' such that they have, say, opposite spin (an example put forward by Bohm and Aharonov). Moving these particles a long way apart such that measuring one will in no way affect the other, then measuring one will tell us already the spin of the other without actually observing it (its spin will be opposite). This would imply either:
Nonlocal interaction, which is undesirable. Much of science is based in isolating a system and being able to treat it as independent from the rest of the universe. Now suddenly it can be affected from outside by a seeming completely different quantum measurement - EPR found this an indication that QM was an incomplete theory because the alternative made more sense;
or, a hidden variable which carried the information of the predetermination.
3.2.2 Locality and non-locality
What EPR based their argument that QM was an incomplete theory on was that the theory should be local, and to be local there had to be hidden variables. Famously Bell showed that the local hidden-variable theories were inconsistent with quantum mechanics (the results of which are empirically borne out).
It remains to be seen whether it is possible to formulate a hidden-variable (that is, where there is actually more information there, we just can't see it), even a non-local one, that is consistent with the predictions of QM. Personally I would rather have a theory which seems to go against common-sense but still be simple instead of a theory which satisfies our minds but whose components are not fully justified with what we can definitely call objective reality.
3.3.4 Other hidden-variable models
Early thinking, even by Heisenberg, adhered to the idea that there were actual quantities of even conjugate variables, but the act of measuring one disturbed the other. This has been shown to be inadequate - particles do actually behave as if they are in a superposition of a number of states, not just being disturbed from one value to another.
The question of observation also comes up here. Given there is already a hidden value for a quantity, why is the wavefunction a superposition of states of this quantity?
Einstein believed that the state vector represented only the observer's knowledge of the particle, and that there were objective physical processes that constituted observations - we just didn't understand what these processes were.
I find this view difficult; there is nothing intrinsic about us as observers, and no way we can draw a line across a set of events and say that these ones are observations and these ones are not. The universe does not decide when a wavefunction collapses any more than it predetermines what a value is going to be before said value is forced to be discrete. The wavefunction should either collapse to begin with (which it doesn't), or at the last possible moment. That there is no distinct time that it collapses is the interpretation given in the transactional approach.
3.3 Transactional approach
Cramer's Transactional Interpretation makes no physical predictions - it's rather a way of interpreting the wavefunction such that questions of how and when it collapses become meaningless.
The wavefunction (psi) is taken to be an actual physical wave as opposed to a representation of the probability. When a quantum event takes place between two quantum objects (the emitter E and the absorber A), an 'offer wave' of the state vector is sent out in all directions by E. It is important to realise that no observables are carried by this wave initially.
When this wave reaches A, A sends back an echo or 'confirmation wave' of (psi)*, such that the typical real wave is set up as a standing wave in space-time. It is along this wave that momentum, energy, and other quantities that need to be conserved are transferred. This wave remains until the transaction is complete.
The transactional approach renders a number of the problems of QM meaningless: there is no difficulty about when the wavefunction collapse occurs because the gradual change of wavefunction occurs over a space-time interval. This also means that there is no discontinuity.
Also, despite being a hidden-variable theory (it postulates the existence of a real unobservable wave), it avoids the trap of Bell's Inequality which renders local theories false by being explicitly non-local. Problems with it remain though given that it doesn't make any new predictions and places a limit on how much we can actually know about the fundamental nature of the universe - quantities are hidden, after all. However, it is a relatively new interpretation (only 1980) and there hasn't been much work on it since so it is possible that there are deeper properties to find.
3.4 In other worlds, this section might be different.
There are other interpretations that are worth mentioning, but I won't spend much time on them. Occum's Razor is a useful principle to apply to unfalsifiable and complex theories, however much 'better' they make us feel about QM.
One such theory is the Many Worlds Interpretation (MWI). It declares that the wavefunction never actually collapses - the universe branches off into separate realities for each possibility. Unnervingly this means that areas of the universe incredibly different will have multiplying realities due to a quantum event which is likely to never (measurably) affect them. This brings in some more relativistic problems. Another difficulty (apart from the obvious conceptual problem of having many equally valid realities - and versions of yourself) is that there is still a very distinct arrow of time. The universe is splitting in the future direction, but not into the past.
Since however the state vector is an actual real quantity, Everett's paper declares that the EPR locality problem now becomes meaningless. This seems like answering one question at the cost of asking a multitude of others.
3.5 The micro/macroscopic border
It's not just interpretations that can resolve some of the paradoxes of QM, but clarifying what we actually mean by quantum systems and measurement. By investigating where exactly the interface is between quantum effects and the classical world it is possible to gain much understanding in what is happening to the wavefunction.
A semi-quantitative measure called 'disconnectivity' was introduced by Leggett to tackle the Schrodinger's cat problem. It characterises the degree to which a system is isolated from effects which otherwise would hide any quantum phenomena. He has found that in the Schrodinger's Cat system, there is a small amount of disconnectivity which places the cat in a classical domain - and this isn't just an arbitrary evaluation. Some of Leggett's predictions have been borne out experimentally in macroscopic quantum tunnelling - experimental techniques in this field have probed so close to the quantum world that the issue of collapse is becoming very unclear.
Although investigating into where the 'collapse trigger' actually is because the micro- and macroscopic worlds doesn't yield a full interpretation of quantum mechanics, it does provide insight into the nature of collapse and may move on to more fundamental issues.
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