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Many people have considered alternatives to standard quantum mechanics in which the Hilbert space is over the real instead of the complex numbers - see e.g. here, here, here, here, here, and here. In general, this alternate theory is surprisingly similar to standard quantum mechanics over the complex numbers:

All the greatest hits are still there: interference, entanglement, Bell inequality violations, noncommuting observables, non-unique decompositions of mixed states, universal quantum computing, the Zeno effect, the Gleason and Kochen-Specker theorems.

But I'm not sure how one would generalize the Schrodinger equation to the real setting. In order to get a unitary time-evolution operator, you need to exponentiate an anti-Hermitian operator. The only operator conveniently lying around is the Hamiltonian, which is Hermitian. Fortunately, with the complex numbers there's an easy way to convert a Hermitian operator into an anti-Hermitian one: you just multiply it by $-i$, as is done in the Schrodinger equation. But with the real numbers, I can't think of any natural way to convert the symmetric Hamiltonian operator into an antisymmetric operator which exponentiates to an orthogonal time-evolution operator. How would this work?

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If you have a symmetric operator $S$, you can turn it into an antisymmetric one by embedding it into a complex structure. For example, the analogue of multiplying the self-adjoint $H$ by $i$ would be the tensor product

$$J\otimes S$$

where $J$ is the symplectic form on $\mathbb R^2$. In the embedding, $S$ identifies with $I_2\otimes S$, where $I_2$ is the $2\times2$ identity matrix.

It shouldn't be a surprise that this embedding trick and the complex setting lead to about the same result. In both case you have doubled the number of real dimensions.

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  • $\begingroup$ Fair point, but I'd argue that this trick violates the spirit of real-valued QM by essentially sneaking the complex structure in through the back door. In particular, the "antisymmetrized" operator acts on a different (larger) Hilbert space than the original one. $\endgroup$ – tparker Jul 8 '18 at 1:40
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Quantum mechanics for a real wave function $\psi(\mathbf{r},t)$ has the wave equation ("Schrödinger equation") $$\hbar\frac{\partial}{\partial t}\psi(\mathbf{r},t)=A\psi(\mathbf{r},t),$$ where $A$ is a real anti-Hermitian operator ("Hamiltonian"). A Majorana fermion on the 2D surface of a topological superconductor provides one realization, with a two-component spinor $\psi(x,y,t)$ and a $2\times 2$ Hamiltonian matrix $A$ with elements $$A=v\begin{pmatrix} \partial/\partial y&\partial/\partial x\\ \partial/\partial x&-\partial/\partial y \end{pmatrix}+\begin{pmatrix} 0&V(x,y)\\ -V(x,y)&0\end{pmatrix}. $$ (The coefficient $v$ is the energy-independent velocity of the Majorana fermions and $V$ is a spatially-dependent magnetisation.)

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  • $\begingroup$ A remark - what is a real spinor? Spinors are defined on $\mathbb{C}$ AFAIK, real spinor is just a two component vector, which carries less information than complex spinor? $\endgroup$ – user74200 Jul 10 '18 at 13:36
  • $\begingroup$ Majorana spinors are a real representation of the Clifford algebra $\endgroup$ – Carlo Beenakker Jul 10 '18 at 14:06

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