Quantum Theory from Complementarity, and its Implications

          @misc{ pirsa_PIRSA:09100092,
            doi = {10.48660/09100092},
            url = {https://pirsa.org/09100092},
            author = {Goyal, Philip},
            keywords = {Quantum Foundations},
            language = {en},
            title = {Quantum Theory from Complementarity, and its Implications},
            publisher = {Perimeter Institute},
            year = {2009},
            month = {oct},
            note = {PIRSA:09100092 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. In this talk, we show how it is possible to derive the complex nature of the quantum formalism directly from the assumption that a pair of real numbers is associated with 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. We then discuss how complementarity --- the key guiding idea in the derivation --- can be understood as a consequence of the intrinsically relational nature of measurement, and discuss the implications of this for our understanding of the status of the quantum state.