Is there an equivalent to the Schrodinger equation for quantum mechanics over the reals?

Question

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?

Answer 1

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.)

Answer 2

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.