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This question is specifically about the robustness of mathematical models.

Special relativity can be derived from very basic principles. Assuming that space is homogeneous and isotropic and that relative velocity is measured the same by two observers is enough to show that the ransformation from one frame to another must have the form of a standard Lorentz transformation with some universal constant c, which is infinity in Galileian relativity.

Thus, the simple assumptions outlined above plus an estimate for a universal bound on speed are enough mathematically to derive all of special relativity. This shows that, in some sense, special relativity is the only theory that works in flat spacetime.

What about quantum mechanics? Dirac's book derives QM from general principles but throws in a few 'jumps' in logic that work great but seem to leave the door open to other mathematical theories. My question is,

Is there a small set of basic assumptions and experimental evidence which require quantum mechanics to have the mathematical form that it does?

For instance, does linear superposition of states (a simple assumption given experimental evidence) imply that observables must be linear operators (instead of other functions of states)? Is there any other inequivalent mathematical framework that satisfies linear superposition and the correspondence principle?

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This is more an extended comment than an answer.

Given the principle of relativity (along with a handful of almost axiomatic assumptions), it is indeed possible to derive a general coordinate transformation that involves an invariant speed.

Which leaves the conjecture that the invariant speed is the measured speed of light, c, a matter of empirical verification.

It seems, then, that you're asking if there is an analogous line of reasoning that involves a quantum of action leaving the value of said quantum a matter of empirical verification?

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This is exactly what I am asking. –  Brian Rushton Feb 25 at 2:21
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@BrianRushton, great question! –  Alfred Centauri Feb 25 at 2:30
    
@BrianRushton Another good reference for Alfred's link is the "from group postulates" on the Wikipedia page for Derivations of the Lorentz transformation. The language of group theory makes for a highly succinct retelling of this derivation. I suggest, as background, answers to this Physics SE question, especially Joseph f Johnson's. –  WetSavannaAnimal aka Rod Vance Feb 25 at 2:53
    
I saw the argument about special relativity as an exercise in that short relativity book everyone uses in freshman year (I think Wheeler was an author). –  Brian Rushton Feb 25 at 3:00
    
I found this paper:journals.aps.org/prd/pdf/10.1103/PhysRevD.31.1341 It has a section on uniqueness of quantum mechanics, but I don't understand the paper well enough to distill the axiomatic assumptions that are used (in particular, I do not understand what they mean by a complementary structure). –  Brian Rushton Feb 26 at 19:16
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