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Are the basic postulates of QM, such as complex Hilbert space, unitary evolution, Hermitian operator observables, projection hypothesis etc., the unique and only set of postulates that gives rise to a semi-deterministic and semi-probabilistic theory, in which the time evolution is non-degenerate? By non-degenerate, I mean different initial states never produce the same final state probabilities, which in QM is guaranteed by unitarity.

Phrased in another way, is it possible to prove from some general principles, such as semi-determinism, semi-reversibility (not for collapse), causality, existence of non-compatible observables etc., that a physical theory must satisfy these postulates? In particular, is it possible to prove that complex numbers, or a mathematical equivalent, must be fundamental to the theory?

I haven't studied anything about foundational issues of QM, so feel free to point out if I'm being a crackpot. I suppose this question may be similar to something like "can you prove that gravity must be a metric theory entirely from the equivalence principle?", whose answer is no, but I'll be glad if it turns out to be otherwise.

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The mere fact that you used the phrase "feel free to point out if I'm being a crackpot" probably disqualifies you from being a crackpot :-P Crackpottery isn't a matter of just making a mistake, it entails stubbornly clinging to a pet theory in the face of overwhelming evidence against it. Not an accusation we throw around lightly. This is a decent question, anyway. The only issue I foresee is possible confusion over exactly which postulates constitute "the postulates of QM." –  David Z Apr 10 '11 at 5:59
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This reminds of a previous question. While I go search for it, check out the article by John Baez. His punchline is that complex numbers isn't the only choice. Reals and quaternions are in the game too. But we know reals aren't good enough. On the other hand, quaternions certainly are. –  Marek Apr 10 '11 at 6:51
    
if you allow the collapse to be a "real event that changes something" and if you allow genuine non-locality associated with such a collapse, then you may design lots of non-QM theories that satisfy your conditions. All of them will be pathological at a theoretical level and inconsistent with observations. There are lots of attempts to "model" the quantum collapse by easily "visualizable", spiritually classical processes. All of these things are wrong. QM is the unique solution to a set of sensible conditions but you haven't described those conditions accurately. –  Luboš Motl Apr 10 '11 at 8:13
    
I'm curious why you think that non-degeneracy is so crucial that it should be emphasized over the other conditions you mention before it. –  Peter Morgan Apr 10 '11 at 13:26
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There are some recent efforts in trying to derive the mathematical structure of quantum mechanics from some reasonable and/or operational axioms. You may want to give a look, for example, at http://arxiv.org/abs/1011.6451 and references therein.

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I dont know if this is an answer to the Question, anyway, from
Informational derivation of Quantum Theory

Quantum theory can be derived from purely informational principles. Five elementary axioms-causality, perfect distinguishability, ideal compression, local distinguishability, and pure conditioning-define a broad class of theories of information-processing that can be regarded as a standard. One postulate-purification-singles out quantum theory within this class. The main structures of quantum theory, such as the representation of mixed states as convex combinations of perfectly distinguishable pure states, are derived directly from the principles without using the Hilbert space framework.

Complex numbers, tensors, matrix and vector algebra, etc, can be substituded by a simpler analog in Geometric Algebra framework.

To the problem: can a different set of principles achieve the same as QM? In principle yes.

QM, has any other theory, is a framework of principles and conclusions based on that. We validade the conclusions but we can not prove that, in some future, other theory using a diferent set of principles can not arrive to the same conclusions. The history has proved that all theories are temporary.

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No, in fact our current postulates allow certain ambiguity in the description of the same physical system (or at least of the possible set of measurements we can extract from them), suggesting that there might be a more concise underlying theory (or set of postulates) that groups such descriptions under a equivalence class

look at the answer to this question for an example.

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