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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 '14 at 8:27

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up vote 9 down vote accepted

There is a fundamental theorem already conjectured by von Neumann but proved just at the end 20'th century (corrected) 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 CCR induce 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 '14 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/… –  Valter Moretti Mar 27 '14 at 18:48
    
Do you have a ref for the Soler paper? 1900 seems a tad early. Thanks! –  Art Brown Jun 24 at 3:45
    
The end of 1900! See SAMUEL S. HOLLAND, Jr: "ORTHOMODULARITY IN INFINITE DIMENSIONS; A THEOREM OF M. SOLER", BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 32, Number 2, April 1995 –  Valter Moretti Jun 24 at 8:37
    
@ValterMoretti: I read your answer without reading the comments and was also confused at first. "End of 1900" should be the end of the year 1900, while you mean the end of the 20'th century. I see people discuss an associated issue here. –  NikolajK Jun 24 at 11:29

The question is begging a simple answer:
Because nature can be represented as resonators that respond in a delayed way we adopt to study it using complex numbers.

When an electronic circuit is excited by a regular periodic source of tension (measured as a real function - $V_0\cos(\omega\cdot\ t)$ for instance) the most common answer is a variation in the current that is not in phase with the excitation : $I_0\cos(\omega\cdot\ t-\phi_0)$ for instance. This is a real (i.e. not complex) answer. The capacitors and inductances act as energy stores and imply a delayed and real valued responses. Example: a capacitor integrates the current, i.e. it has memory of the past.

The best way to study the real valued response of the electronic circuits is with the Complex formalism, invented 100's of years in advance by mathematicians to solve problems in algebra (starting with the roots of a cubic equation).

The excitation/response can be represented as a pure complex number:
$V_0e^{j\omega t}$ / $I_0e^{j\omega t-\phi_0}=ke^{\phi_0}$ .
Where the tension and the current are the projections of rotating vectors in the Argand's plane on the real axis.

The application of the 'complex' formalism in QM is so natural as it's use on electronics, as long as there are resonators that respond in a delayed way.

Those subjects can be treated with other formalisms, without complex numbers:
Quaternions (by Hamilton)
GA - Geometric Algebra see also Hestenes: Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics

side notes:
delayed ... is a deep consequence of relativity. There is nothing special on the 'complex nature' of the wavefunction and there is nothing to be assumed.
In Physics we use the language of Mathematics, but they are distinct domains, and we can not coerce Nature to assume some mathematical formalism.
Everything that can be done with complex algebra can be represented with quaternions. The use of the GA formalism is much more interesting as a representation of the physical world in comparison to the usual complex representation.

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