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In Lucien Hardy's influential paper "Quantum Theory from five reasonable axioms," Hardy states in the abstract:

This work provides some insight into the reasons why quantum theory is the way it is. For example, it explains the need for complex numbers and where the trace for- mula comes from.

Can someone explain to me why his axioms explain "the need for complex numbers" ?

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    $\begingroup$ It does not. You can always get away with complex numbers by using the standard 2D real plane. Complex numbers simplify the formalism in many cases though. $\endgroup$ – Wolphram jonny Sep 4 at 12:50
  • $\begingroup$ @Wolphramjonny: could you expand a bit on that comment ? Isn't he claiming that his axioms imply the complex numbers as an underlying algebraic structure ? $\endgroup$ – THC Sep 4 at 14:05
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    $\begingroup$ Do you mind giving part of the argument, as you understand it? $\endgroup$ – Norbert Schuch Sep 5 at 2:14
  • $\begingroup$ @NorbertSchuch: I honestly do not see a clear argument in his paper which implies that the complex numbers "follow" from his axioms. $\endgroup$ – THC Sep 5 at 14:59

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