Physics – Quantum Physics
Scientific paper
2009-07-06
Phys. Rev. A 81, 022109 (2010)
Physics
Quantum Physics
v2: Clarifications, and minor corrections and modifications. Results unchanged. v3: Minor changes to introduction and conclusi
Scientific paper
10.1103/PhysRevA.81.022109
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.
Goyal Philip
Knuth Kevin H.
Skilling John
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