Are the basic postulates of QM the only set of postulates that can give rise to a sensible semi-probabilistic physical theory?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.