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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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There is a fundamental result already conjectured by von Neumann but proved just at the end 20'th century by Solèr (in addition to a partial result already obtained 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 $L^2$. The pure states are here unit vectors up to signs instead of up to phases. It is therefore evident that decomposing the complex wavefunction into real and complex part is not a real QM since in that case pure states are unit vectors up to $SO(3)$ rotations.

The second possibility is the one considered "standard" nowadays, a Hilbert space over the field of complex numbers and the pure states are unit vectors up to phases as is well known.

The third, quite exotic possibility, is a Hilbert space whose scalars are quaternions. Now pure states are unit vectors up to quaternionic phases (quaternionic factors with unitary norm).

This third possibility has been investigated by several authors (see the book by Adler for instance).

As a matter of fact we only know physical systems described in complex Hilbert spaces, is there a fundamental reason to rule out the other two cases?

It seems possible to prove that the first possibility is only theoretical for physical reasons under some hypotheses. When one deals with physical systems described in real Hilbert spaces CCRs induce a complex structure and, in fact it is equivalent to deal with a complex Hilbert space.

CCRs of $X$ and $P$ can enter the game through two different ways. You can suppose that your theory admits position and momentum as fundamental operators, or you may assume that your system is covariant under the action of a unitary irreducible representation of the (extended) Galileo group. I find both possibilities not very satisfactory. On the one hand position is not so relevant (massles particles do not admit any notion of position operator), on the other hand Galileo's group is not fundamental in modern physics. Together with a PhD student of mine, M. Oppio, I recently discovered a much more satisfactory argument (http://xxx.lanl.gov/abs/1611.09029 published in Reviews in Mathematical Physics 29 n.6, (2017) 1750021 DOI: 10.1142/S0129055X17500210) to introduce a natural unique (Poincare' invariant) complex structure in a quantum theory initially formulated in a real Hilbert space, assuming to deal with an elementary system of relativistic nature, supporting an irreducible representation of Poincaré group in terms of automorphisms of the lattice of elementary propositions and admitting an irreducible von Neumann algebra of observables generated by the representation itself (within a refinement of Wigner's idea of elementary relativistic system).

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  • $\begingroup$ 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 :) $\endgroup$ – Tom W Mar 27 '14 at 13:34
  • $\begingroup$ Do you have a ref for the Soler paper? 1900 seems a tad early. Thanks! $\endgroup$ – Art Brown Jun 24 '15 at 3:45
  • $\begingroup$ 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 $\endgroup$ – Valter Moretti Jun 24 '15 at 8:37
  • $\begingroup$ @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. $\endgroup$ – Nikolaj-K Jun 24 '15 at 11:29
  • $\begingroup$ You are right I was really too sloppy! Thank you very much for having pointed out this problem. $\endgroup$ – Valter Moretti Jun 24 '15 at 11:41
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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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