Why Quantum Theory is Complex

          @misc{ pirsa_PIRSA:09040062,
            doi = {10.48660/09040062},
            url = {https://pirsa.org/09040062},
            author = {Goyal, Philip},
            keywords = {Quantum Foundations},
            language = {en},
            title = {Why Quantum Theory is Complex},
            publisher = {Perimeter Institute},
            year = {2009},
            month = {aug},
            note = {PIRSA:09040062 see, \url{https://pirsa.org}}
          }
          

Abstract

Complex numbers are an intrinsic part of the mathematical formalism of quantum theory, and are perhaps its most mysterious feature. We show that it is possible to derive the complex nature of the quantum formalism directly from the assumption that a pair of real numbers is associated to each sequence of measurement outcomes, and that the probability of this sequence is a real-valued function of this number pair. By making use of elementary symmetry and consistency 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. Reference: arXiv:0907.0909 (http://arxiv.org/abs/0907.0909)