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The sound waves are real, and they can interfere, so corresponding apparat may be used in quantum mechanics. We also may use the time dependence in a form of orthogonal matrix multiplying the initial constant vector.

So why do we must postulate that the wavefunction is initially complex?

This question is very similar to the corresponding questions at the site, but there I didn't find the answer.

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Possible duplicates: physics.stackexchange.com/q/8062/2451 , physics.stackexchange.com/q/17168/2451 and links therein. –  Qmechanic Mar 27 at 8:27
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1 Answer 1

up vote 6 down vote accepted

There is a fundamental theorem already conjectured by von Neumann but proved just at the end of 1900 by Solèr (in addition to a partial result already obtained by by Piron in the sixties) which establishes (relying on the theory of orthomodular lattices and projective geometry) that the general phenomenology of Quantum Mechanics can be described only by means of three types of Hilbert spaces. (All fundamental theorems of quantum theory like, say, Wigner theorem, can be proved in these three cases.) One is a Hilbert space over the field of real numbers. In this case wavefunctions can be taken as real valued functions if the system is described in terms of a $L^2$ space. The second possibility is the one considered "standard" nowadays, a Hilbert space over the field of complex numbers. The third, quite exotic possibility, is a Hilbert space whose scalars are quaternions. This third possibility has been investigated by several authors (see the book by Adler for instance).

It is possible to prove, however, that the first possibility is only theoretical for physical reasons. In fact when one deals with physical systems described in real Hilbert spaces, invariant under time-reversal, the time reversal operation induces a complex structure and, in fact it is equivalent to deal with a complex Hilbert space.

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What are the 'physical reasons' that disqualify the first case? Could you provide a worked example of the time-reversal operation? I think a little more explicit mathematics would improve the answer :) –  Tom W Mar 27 at 13:34
Sorry I am a bit busy now. In the meantime, see for instance the third answer to physics.stackexchange.com/questions/32422/… –  V. Moretti Mar 27 at 18:48
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