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I want to use the 2d wave equation ($\frac{\partial^2u}{\partial t^2}=\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}$) with periodic boundary conditions as a simple toy model of a quantum mechanical system. A formulation in the form $i\hbar\!\frac{\partial}{\partial t}\left|\psi\right\rangle=H\left|\psi\right\rangle$ with a suitable Hamiltonian $H$ in a suitable Hilbert space would be nice.

For this I could use Dirac's trick of turning the second order in time Klein-Gordon-equation into the first order in time Dirac-equation. This will turn my second order equation in one real component into a first order equation in 4 complex components. I guess I won't get away with less than 2 complex components, but 4 components seem a bit excessive. Is there is a simple Hamiltonian that gets away with 2 complex components?

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