Researchers Restore Hidden Phase Space in Real-Valued Quantum Theory

Researchers at Bergische Universität Wuppertal and Universität Leipzig have revisited a surprising chapter in quantum history: a real-valued Schrödinger equation initially proposed, and then abandoned, by Erwin Schrödinger in 1926. While the standard Schrödinger equation relies on complex numbers, the team demonstrates that it can be rewritten as a second-order equation using only real-valued scalar fields. This simplification obscures fundamental elements of quantum mechanics, including the Born density and the derivation of the uncertainty relation. The work argues these aren’t failures of real variables, but indications of an incomplete local representation needing restoration of the underlying Hamiltonian phase-space structure, where the missing component reappears as a canonical partner. Thus the symbol can be removed, but the symplectic and complex structure it encodes cannot.

Erwin Schrödinger proposed a real-valued Schrödinger equation in 1926, a surprising detail given his later association with the complex-valued formulation now considered standard. This early exploration, detailed in his fourth communication that year, initially presented a real-valued wave equation, a second-order equation in time and fourth-order in space, which Schrödinger ultimately abandoned, partly due to difficulties extending it to non-conservative systems. Robert Chen later revisited this concept in the late 1980s, demonstrating a generalization applicable to time-dependent potentials. Oliver Passon and Bernd Rosenow have recently revisited Chen’s work and published a paper on the topic. The resulting equation, a single real-valued field governed by a second-order differential equation, presents mathematical challenges. They argue that while the complex symbol can be removed, the symplectic and complex structure it encodes cannot.

The pursuit of a real-valued formulation of quantum mechanics, though ultimately eclipsed by the standard complex approach, occupied some of the field’s earliest and most prominent thinkers. Complex numbers are now integral to the Schrödinger equation, but their necessity wasn’t immediately accepted, and alternatives were actively explored in the mid-1920s and beyond. This restoration reveals an internal gauge structure when considering minimal magnetic coupling.

The work builds on Stueckelberg’s formulation, equipped with a distinguished operator.

Current investigations into the foundations of quantum mechanics are revealing surprising connections to its earliest formulations. This isn’t simply historical curiosity; a detailed analysis demonstrates that rewriting the Schrödinger equation as a single real-valued field, while seemingly simplifying the mathematics, obscures fundamental elements of quantum mechanics.

Although Schrödinger abandoned this approach fairly quickly, in part because he was unable to extend it to non-conservative systems, modern analysis reveals a more nuanced picture. This simplification isn’t a triumph of reduction, but rather an obscuring of fundamental elements.

Erwin Schrödinger’s initial foray into real-valued quantum equations, proposed in 1926, represents a surprising historical detail given his later association with the complex Schrödinger equation. This early work, though quickly abandoned due to difficulties extending it to non-conservative systems, has seen renewed interest, prompting a re-evaluation of the necessity of complex numbers in foundational quantum mechanics. Notably, the work corrects a Green-function argument from a 1991 paper by Robert Chen, and provides an explicitly real construction of Kramers degeneracy. The researchers also observe that the reconstruction map fails for composite systems, bearing on the recent no-go debate.

Erwin Schrödinger’s exploration of real-valued quantum mechanics predates the widespread adoption of his now-ubiquitous complex-valued equation; he initially proposed a real formulation in 1926 before shifting his focus. This early work, largely forgotten, challenges the assumption that complex numbers are intrinsic to the foundations of quantum theory. The team’s analysis reveals that reducing the equation obscures crucial elements, including the Born density and current, and complicates deriving the uncertainty relation. They demonstrate that this reduced form requires “restoring the underlying Hamiltonian phase-space structure,” effectively reintroducing a missing component. This restoration isn’t merely a mathematical trick but a necessity for a complete local description of quantum phenomena. The researchers emphasize this isn’t a replacement for the Hilbert-space postulates, but a different way to express existing quantum dynamics.

Despite the pervasive use of complex numbers in quantum mechanics, the standard Schrödinger equation, the researchers demonstrate, can be recast as a second-order equation governing a single real-valued scalar field, a simplification that obscures crucial quantum elements.