## Explanation of why QM wavefunction is complex valued

In my previous post titled “QM wavefunction and its many interpretations” I stressed the fact that the quantum mechanical wavefunction is complex valued. This means that $i=\sqrt{-1}$ and the algebra of complex numbers are essential ingredients of the quantum mechanical wavefuntion.

In the post titled “Cayley-Dickson Construction of Complex, Quaternion and Octonion Numbers” I reminded readers that a complex number is really a pair of real numbers. The so-called complex algebra is in fact the algebra of pairs.

In the post titled “Few Comments on the Pair Concept” I mentioned that the electric field $\mathbf{E}$ and the magnetic field $\mathbf{E}$ are dual quantities. The mathematical treatment of dual quantities as complex numbers makes perfect sense because complex algebra manipulates $\mathbf{E}+i\mathbf{B}$  as a whole.

I stressed the fact that we cannot pick two random physical quantities and combine them in the form of a complex number. Such treatment can only be meaningful if the two physical quantities are truly dual.

If QM wavefunction is complex-valued there must a dual pair implicit in the wavefunction. The fundamental question is this: what is the dual pair implicit in the wavefunction? Time and energy? Position and momentum? Some other pair?

While I was thinking along these lines I discovered a physics paper that used the “dual pair” concept to explain why the QM wavefunction (amplitude in the context of Feynman formulation of QM) is complex-valued.

The Origin of Complex Quantum Amplitudes and Feynman’s Rules by Phillip Goyal, Kevin H. Knuth and John Skilling

“Abstract: Complex numbers are an intrinsic part of the mathematical formalism of quantum theory, and are perhaps its most mysterious feature. In this paper, we show that the complex nature of the quantum formalism can be derived directly from the assumption that a pair of real numbers is associated with each sequence of measurement outcomes, with the probability of this sequence being a real-valued function of this number pair. By making use of elementary symmetry conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic. We demonstrate that these complex numbers combine according to Feynman’s sum and product rules, with the modulus-squared yielding the probability of a sequence of outcomes.”