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In a deterministic classical theory we describe the positions and velocities of particles using real numbers.

In a non-deterministic classical theory we describe the positions and velocities of particles using real random variables.

In quantum theory we can describe positions and velocities of particles using elements of $C^*$-algebras. But $C^*$-algebras are algebras over complex numbers.

Is there a reason we can't generalize real classical random variables to real quantum random variables?

Qmechanic
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Jagerber48
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    Related: https://physics.stackexchange.com/q/32422/ and questions linked to it – Jules Lamers Dec 31 '23 at 09:44
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    Is switching to matrix notation satisfactory? A+Bi = (A, -B; B, A) – g s Dec 31 '23 at 10:20
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    Self-adjoint operators are an analog of real random variables. They have a real-valued spectrum. Their spectral theory is analogous to the commulative distribution function of a real random variable. The issue I think is why work with Hilbert spaces over $\Bbb{C}$ and not just over $\Bbb{R}$. This has to do with having to use operators that are not self-adjoint in the theory - in particular unitary operators. Once you start doing that you need $\Bbb{C}$. Even orthogonal matrices over $\Bbb{R}$ have an imaginary spectrum. – Chad K Dec 31 '23 at 11:07
  • See Renou, M.O., Trillo, D., Weilenmann, M., Le, T.P., Tavakoli, A., Gisin, N., Acín, A. and Navascués, M., 2021. Quantum theory based on real numbers can be experimentally falsified. Nature, 600(7890), pp.625-629. The paper is open access. – ZeroTheHero Dec 31 '23 at 15:43

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