⚛️ quantum physics

Schrödinger's real-valued equation revisited

This paper argues that while the Schrödinger equation can be reformulated as a second-order equation for a single real-valued field, a complete local representation of quantum mechanics requires restoring the underlying Hamiltonian phase-space structure to recover essential features like the Born rule and uncertainty principle, thereby demonstrating that the imaginary unit's role in encoding symplectic structure is indispensable even if the symbol $i$ is removed.

Original authors: Oliver Passon, Bernd Rosenow

Published 2026-07-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Oliver Passon, Bernd Rosenow

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Invisible Thread of Reality

Imagine you are trying to describe a spinning coin. You could say, "It's a flat disk," but that misses the most important part: it's spinning. To fully capture the coin, you need to know not just its position, but also its speed and direction. In the world of quantum mechanics—the physics of the very tiny, where particles like electrons behave like both solid balls and rippling waves—this "spinning" is even more mysterious. The standard way to describe these particles uses a special mathematical tool called a "complex number." Think of a complex number not as a weird, imaginary thing, but as a two-part instruction: one part tells you the particle's height (like a wave going up), and the other tells you its phase (like the wave tilting forward or backward). This two-part system is usually written with a symbol, $i$ , which acts like a magical switch that turns a "height" into a "tilt."

For nearly a century, physicists have wondered: Is this magical switch, $i$ , absolutely necessary? Or is it just a convenient shorthand for something that could be described using only real, everyday numbers? If you could strip away the "imaginary" part, would the universe still make sense? This question isn't just about math; it's about whether the fundamental rules of nature require a hidden dimension of reality that we can't see directly. If the answer is "yes," then the universe is built on something deeper than simple numbers. If the answer is "no," then the "imaginary" part is just a trick of our notation, and the universe is made of real stuff all along.

The Paper's Discovery: The Ghost in the Machine

In this paper, Oliver Passon and Bernd Rosenow revisit a very old idea: that we can rewrite the famous Schrödinger equation (the rulebook for how quantum particles move) using only real numbers, getting rid of the imaginary unit $i$ entirely. They look at a specific attempt to do this, where the complex wave is split into just one real field, $u$ , and its rate of change over time. At first glance, this looks like a win for simplicity. You have one field instead of two, and no imaginary numbers.

However, the authors find that this "one-field" version is like trying to describe a spinning coin by only looking at its shadow on the wall. You can see the shadow move, but you lose the ability to tell which way the coin is spinning, how fast it's going, or even where the probability of finding the particle is located. When they strip away the imaginary part, the beautiful, local rules of quantum mechanics—like the probability density (where the particle is likely to be) and the flow of momentum—become "non-local." This is a fancy way of saying that to know what is happening at one specific spot, you suddenly need to know about the state of the particle everywhere else in the universe at once. It's as if the shadow on the wall only makes sense if you know the exact shape of the entire room, not just the spot where the light hits.

The paper argues that this doesn't mean real numbers fail; it means that the "one-field" reduction is incomplete. The authors show that to fix this, you have to bring back the missing piece, not as a second imaginary number, but as a "canonical partner" (a second real field, $\pi$ ) that acts like the coin's momentum. When you put these two real fields together, they form a "phase space"—a two-dimensional plane where the particle's position and momentum dance together.

Here is the magic trick: In this new real-world view, the imaginary unit $i$ isn't gone; it's just been replaced by a geometric rotation. Imagine the two real fields as coordinates on a piece of paper. The "imaginary" part of quantum mechanics is actually just a rule that says, "Rotate the paper by 90 degrees." The symbol $i$ was just a shorthand for this rotation. The paper proves that you can remove the symbol $i$ from the equations, but you cannot remove the structure it represents: the symplectic geometry (the specific way position and momentum twist around each other).

The authors also tackle a tricky problem: what happens when you add a magnetic field? In the standard complex view, magnetic fields are easy to handle. In the "one-field" real view, they break the equations. But in the authors' "two-field" real phase-space view, magnetic fields are revealed to be something beautiful: they are just a way of twisting the internal rotation of the two fields differently at every point in space. It's like if the "90-degree rotation rule" changed slightly depending on where you were standing. This reveals an internal symmetry (called $SO(2)$) that was hidden in the complex notation.

So, what is the verdict? The paper concludes that while you can rewrite quantum mechanics using only real numbers, you cannot do it with just a single field and its speed. You need a pair of fields that act like position and momentum. The "imaginary" unit $i$ is not a fundamental ingredient of the universe that we can simply delete; it is a compact way of writing down a deep, geometric relationship between two real things. If you try to delete it without replacing it with the right geometric structure, the local rules of physics fall apart, and the universe becomes a confusing, non-local mess. The paper doesn't claim to have discovered a new theory of physics, but rather acts as a diagnostic tool, showing us exactly why the complex numbers we use are so useful: they are the most efficient way to describe a reality that is fundamentally a dance between two real partners.

Technical Summary: Schrödinger's Real-Valued Equation Revisited

Problem Statement
The paper addresses the long-standing question of whether quantum mechanics fundamentally requires complex numbers or if it can be formulated entirely using real-valued variables. While Erwin Schrödinger proposed a real-valued second-order equation in 1926, and Robert Chen later developed a systematic "reduced single-field formulation" (where the wave function $\psi = u + iv$ is reduced to a single real field $u$ ), these approaches face significant conceptual hurdles. Specifically, the reduction obscures local structures essential to standard quantum mechanics: the Born probability density, the probability current, the role of momentum as a generator, and the standard derivation of the uncertainty relation. In the reduced formulation, these quantities often become non-local functionals of the real field $u$ and its time derivative $\dot{u}$ , relying on the inverse Hamiltonian operator $\hat{H}^{-1}$ . The central problem is to determine whether these difficulties indicate a failure of real variables or merely a deficiency in the reduced single-field representation itself.

Methodology
The authors employ a diagnostic approach, first analyzing the limitations of the reduced single-field formulation (Chen's approach) and then reconstructing the theory within a real Hamiltonian phase-space framework.

Analysis of the Reduced Formulation: The paper examines the Schrödinger-Chen reduction, where the imaginary part $v$ is reconstructed from the real part $u$ via $v = \hbar \hat{H}^{-1} \dot{u}$ . The authors demonstrate that while the spectra of the reduced second-order equation match the complex theory, the probabilistic and kinematical structures lose their local character. The probability density $\rho$ and current $j$ become non-local because they depend on $\hat{H}^{-1}$ , an integral operator.
Phase-Space Reconstruction: To resolve the loss of locality, the authors lift the second-order real equation to a first-order Hamiltonian system on a real phase space. They identify the canonical momentum $\pi$ conjugate to the configuration variable $u$ . By requiring the conservation of the Born norm and matching the second-order dynamics, they derive the specific form of the canonical momentum: $\pi = \hbar^2 \hat{H}^{-1} \dot{u}$ .
Geometric Interpretation: The authors interpret the complex structure $i$ geometrically as a symplectic rotation $J$ (satisfying $J^2 = -1$ ) acting on the real phase plane $(u, \pi/\hbar)$ . They analyze how this structure behaves under time reversal, gauge transformations, and magnetic coupling, showing that magnetic fields correspond to the gauging of an internal $SO(2)$ symmetry.

Key Contributions and Results

Diagnosis of Non-Locality: The paper provides a rigorous operator-theoretic diagnosis showing that in the reduced single-field variables $(u, \dot{u})$ , the probability density, current, and momentum are non-local. Crucially, Proposition 2 establishes a state-level non-locality: two distinct admissible sets of Cauchy data $(u, \dot{u})$ can coincide in a neighborhood of a point (including all spatial derivatives) yet yield different Born densities at that point. This proves that the single field $u$ and its time derivative are insufficient for a manifestly local representation of the quantum state.
Restoration of Local Structure: By introducing the canonical partner $\pi$ , the authors show that the theory can be reformulated entirely in real variables while restoring locality. In the real phase-space variables $(u, \pi)$ , the Born density $\rho = u^2 + (\pi/\hbar)^2$ and the probability current become local functionals. The standard uncertainty relation is recovered not as a property of a single field, but as a consequence of the restored local symplectic geometry.
Magnetic Coupling and $SO(2)$ Gauge Structure: The paper extends the analysis to magnetic fields. It demonstrates that the minimal coupling of a charged particle to a vector potential $\mathbf{A}$ corresponds to gauging the internal $SO(2)$ symmetry of the real phase plane. The vector potential acts as the connection required to compare neighboring phase-space planes. This leads to a "gyroscopic" term in the reduced second-order equation, explaining why the naive Chen equation fails for magnetic systems without modification.
Clarification of Time Reversal: The analysis clarifies the nature of time reversal in the real formulation. Time reversal corresponds to a reflection in the $u$ -axis of the phase plane $(u, \pi)$ , which is equivalent to complex conjugation in the standard formulation. This explains why time reversal in quantum mechanics involves complex conjugation: it is necessary to reverse the "velocity" (the second quadrature) in the real phase space.
Comparison with Pauli's Formulation: The paper contrasts Chen's variables (where $u = \text{Re}(\psi)$ ) with Pauli's auxiliary field $U$ . It highlights a dichotomy: Pauli's variables keep the density local but make the field definition dependent on the Hamiltonian (non-kinematical), whereas Chen's variables keep the field kinematical but render the density non-local. Only the full phase-space pair $(u, \pi)$ achieves both locality and Hamiltonian independence.

Significance and Claims
The authors explicitly state that their work does not propose a rival to complex quantum mechanics nor rediscover the known geometric formulation of quantum mechanics (where the Hilbert space is viewed as a real manifold). Instead, they utilize the Schrödinger-Chen reduction as a diagnostic tool.

The paper claims that the difficulties encountered in the reduced single-field formulation do not prove that real variables fail; rather, they demonstrate that the reduced equation is not a complete local representation of the theory. The "missing" component (the imaginary part) is not an independent degree of freedom but a canonical partner required to restore the symplectic structure.

The primary significance lies in the precise identification of what the symbol $i$ encodes: it is a compact notation for an underlying symplectic structure and a complex structure compatible with the metric. The paper concludes that while the symbol $i$ can be removed from the notation, the symplectic and complex structures it encodes cannot be eliminated without losing the local representation of quantum kinematics. The reconstruction of the phase space is necessary to recover the Born rule, momentum, and uncertainty relations in their standard local forms.

The Invisible Thread of Reality

The Paper's Discovery: The Ghost in the Machine

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