Schr\"odinger's real-valued equation revisited

The Schrödinger equation can be rewritten as a second-order equation for a single real-valued scalar field.

Taken at face value, however, this one-field reduction obscures several local structures of quantum mechanics: the Born density and current, the role of momentum as a generator, and the usual derivation of the uncertainty relation.

We argue that these difficulties do not show that real variables fail.

They show that the reduced equation is not, by itself, a complete local representation of the theory.

Its coherent real form is obtained by restoring the underlying Hamiltonian phase-space structure, where the missing component reappears as a canonical partner.

In this real phase-space formulation, the standard structures reappear locally, and minimal magnetic coupling reveals an internal \(SO(2)\) gauge structure.

Thus, the symbol \(i\) can be removed, but the symplectic and complex structure it encodes cannot.