That there is some form of generalized framework from which quantum mechanics follows as a certain limit? Is there any theory that has been discovered that can lead to the same predictions of QM but is not strictly QM?
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2$\begingroup$ If such a thing had been discovered (or even reliably hinted at) it would be big news. But sure, it's possible. Be aware that any such overarching theory would have to reproduce all of quantum mechanics weirdness---that is all very well supported by experiment and is not going away. $\endgroup$– dmckee --- ex-moderator kittenCommented Jul 21, 2012 at 15:23
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$\begingroup$ Can you be more specific? What do you mean by "not strictly QM"? Are you asking about hidden variables, the existence of a GUT, or something else altogether? $\endgroup$– cartomancerCommented Aug 7, 2013 at 23:49
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3 Answers
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The de Broglie-Bohm theory, also known as the Pilot Wave interpretation of quantum mechanics, is one such extension of quantum mechanics (it is in a sense not merely an interpretation), and formulated precisely because the authors doubted that quantum mechanics was fundamental.
The way in which de Broglie-Bohm theory extends quantum mechanics is to postulate that the particles actually do have definite positions and momenta, and that there are phenomenological reasons which prevent us from being able to actually prepare a definite state, which entails that our observations end up being probabilistic in nature. It also posits the existence of a 'pilot wave' which guides the particles along their trajectories, corresponding exactly to the quantum wave function of conventional quantum mechanics; the fact that the pilot wave corresponds exactly to the probability distribution observed of the particles is meant to be a consequence of the theory.
In order to explain the non-local correlations seen in quantum mechanics, the effect of the pilot wave is superluminal in nature, sending influences between particles faster than the speed of light. According to the pilot wave theory, the reasons why we cannot use these influences to signal faster than light is because the ability to signal is conveniently (or inconveniently, if you like) shrouded by the statistical noise of our inability to prepare definite particle states.
The reason why the Born rule holds under this theory — that is, the reason why the pilot wave happens to describe the particle statistics that we see — is meant to be a sort of thermodynamic argument: it is concievable for the distribution of outcomes to be different from the Born rule, but the distribution described by the Born rule is an equilibrium régime of the theory. One conceivable way that we could find definite support for the pilot wave theory, even if we do not anticipate that we will ever be able to observe these conditions, is to find some part of the universe which shows evidence of not being in this "quantum equilibrium".
answered Jul 21, 2012 at 16:00
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1$\begingroup$ But it is unlikely to be more fundasmental. It has big difficulties coping with quantum field theory, which a more fundamental theory would have to fully encompass. $\endgroup$ Commented Jul 21, 2012 at 20:23
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$\begingroup$ @MBN: why wouldn't you? It does clearly feature elements which are not present in quantum mechanics; it just also hypothesizes why the things we observe are limited to what quantum mechanics tells us (in the same way that one can tell from special relativity how it could be that we were "fooled" to believe that Gallilean relativity held). $\endgroup$ Commented Jul 22, 2012 at 10:42
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$\begingroup$ I would called it a different formulation not an extension. As you said one says that particles have definite coordinates at all time, the other says they don't. How can possible one be an extension of the other?! $\endgroup$– MBNCommented Jul 22, 2012 at 10:59
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1$\begingroup$ Special relativity and Gallilean relativity can be experimentally distinguished, not so for Bohmian and Orthodox QM. $\endgroup$– MBNCommented Jul 22, 2012 at 11:02
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I think M-theory is the generalization of quantum mechanics.
Anyway, a more general theory have to include both theory of gravity and quantum mechanics. Currently there is no such theory complete.
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1$\begingroup$ This is not really true--- M-theory doesn't touch the quantum mechanics underneath. You don't need gravity to extend quantum mechanics, I give a trivial example in an answer. $\endgroup$ Commented Jul 22, 2012 at 8:50
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There is a framework which is somewhat more general than quantum mechanics, which is the superoperator formalism of density matrices. It can be regarded as quantum mechanics with fundamental decoherence.
In ordinary quantum mechanics, a density matrix evolves just by unitary evolution
$$ i {\partial\rho\over \partial t } = [H,\rho]$$
Which has opposite sign from the usual evolution of an operator in the Heisenberg picture. To extend quantum mechanics to include fundamentally irrevesible processes, you can just add terms which are not commutators to the right hand side, but which are linear in $\rho$.
$$ i {\partial\rho \over \partial t} = S \rho$$
Or in indices labelling the quantum states:
$$ i {\partial\rho^i_j \over \partial t} = S^{ki}_{lj} \rho^l_k $$ S is a 4 index object, which takes matrices to matrices. The condition that S preserve the trace of $\rho$ is that:
$$ S^{ik}_{jk} = 0 $$
The Hamiltonian evolution is only a special case of the most general evolution. You can add terms to the superoperator to describe decoherence, friction, and all sorts of things, and if you make the new effects small by adjusting a parameter, so that S is approximately given by a Hamiltonian, you reproduce quantum mechanics.
Such a theory is a theory of fundamental decoherence, and it is not likely to be correct in any way. It is useful for quantum computing, and for effective theories of decoherence. Hawking proposed to use such a thing for quantum evolution with black holes, but it is unncessary. But it does answer an old question of Weinberg of whether quantum mechanics admits a reasonable deformation, the answer is yes.
For a fundamental modification of quantum mechanics, something which could plausibly be going on underneath it all, this, in order to be physics, should modify quantum mechanics in a way that reduces the size of the description to a more classical size. I discussed these possibilities here: Consequences of the new theorem in QM? .
