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Hidden Supersymmetry in Wigner-Yang Quantum Mechanics

This paper generalizes the Wigner-Yang quantum mechanical framework on the real line to reveal a hidden supersymmetric structure where the reflection operator serves as the grading operator, demonstrating that non-interacting systems correspond to Witten's SUSY model and establishing a Hooke-Newton duality between harmonic and Coulomb-like potentials within this deformed Heisenberg algebra.

Original authors: Georg Junker

Published 2026-08-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Georg Junker

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

Imagine the universe as a giant, cosmic game of billiards. For centuries, physicists have believed they knew the exact rules of the game: how balls (particles) bounce, how they move, and how they talk to each other. These rules are written in a language called quantum mechanics, which is a bit like a rulebook for the very small, where things can be in two places at once or act like waves. A key part of this rulebook is the "commutation relation," a fancy way of saying that if you try to measure a particle's position and its speed at the same time, the order in which you do it matters. It's like trying to put on your socks and shoes; doing it in the wrong order leaves you in a very awkward state.

But what if the rulebook isn't as rigid as we thought? What if the universe allows for a few extra moves, a hidden flexibility in how particles interact? This is the question that has fascinated scientists for decades. In 1950, a physicist named Eugene Wigner asked a simple but profound question: "Do the equations that tell us how things move actually force us to use these specific, rigid rules?" He suspected that maybe, just maybe, there was more than one way to write the rules of the game. Later, another physicist named Yang found a way to write these rules down using a special kind of math that included a "mirror" operator—a mathematical way of flipping things over, like looking in a mirror. This led to a new, "deformed" version of quantum mechanics where the usual rules are slightly bent, but the physics still makes sense. Why does this matter? Because sometimes, when you bend the rules just right, you discover hidden symmetries—secret patterns in the universe that link different types of particles together, almost like a cosmic dance where partners swap places without anyone noticing.

This paper takes a fresh look at those bent rules, exploring a system called "Wigner-Yang quantum mechanics." The author, Georg Junker, decides to play a game of "one step at a time." First, he ignores the second rule of motion (how position changes) and focuses only on the first (how momentum changes). By doing this, he creates a "generalized" version of the theory. He discovers that in this simplified world, the "mirror" operator acts like a referee that sorts particles into two teams: "even" and "odd." This sorting mechanism turns out to be the key to a hidden structure called "supersymmetry" (SUSY). Think of SUSY as a magical pair of gloves where if you have a left-handed particle, there's a hidden right-handed partner waiting to be found. The paper shows that in this generalized system, the universe naturally organizes itself into these supersymmetric pairs, provided the "mirror" function behaves in a specific way. If the function acts one way, the symmetry is perfect and unbroken; if it acts another, the symmetry breaks, and the partners go their separate ways.

Next, the author brings back the second rule of motion to return to the original, full theory proposed by Wigner and Yang. He tests this theory on two classic scenarios: a particle bouncing back and forth in a box (the harmonic oscillator) and a particle being pulled by a force that gets stronger the closer it gets to the center (a Coulomb-like potential, similar to how gravity or electricity works). He finds that the harmonic oscillator doesn't quite show off this hidden supersymmetry in the way we might hope. However, the Coulomb-like system is a different story. Here, the hidden symmetry shines brightly. The paper proves that the energy levels of this system are perfectly paired up, just like the gloves in the supersymmetric dance.

Perhaps the most playful discovery in the paper is a "duality" between these two systems. The author shows that you can mathematically transform the problem of a particle being pulled by a Coulomb force into the problem of a particle bouncing in a harmonic oscillator, and vice versa. It's like having a magical map that turns a steep mountain climb into a gentle walk on a flat plain, yet the destination (the energy levels) remains the same. This transformation isn't just a simple stretch; it involves a clever change of variables that mixes the particle's position with a "mirror" flip. The paper suggests that this connection is a deeper, more complex version of a relationship known since the days of Newton and Hooke, but now updated with these new, bent rules of quantum mechanics.

Ultimately, the paper doesn't claim to have rewritten the entire rulebook of the universe, but it does show that within this specific, deformed version of quantum mechanics, nature loves to hide symmetries. It suggests that by looking at the equations of motion in a slightly different order, we can uncover a supersymmetric structure that was previously hidden. The author establishes that for certain potentials, like the Coulomb-like one, this hidden symmetry is real and unbroken, offering a new way to understand how particles interact. While the math is heavy, the core idea is a delightful reminder that even in the rigid world of quantum physics, there is room for mirrors, hidden partners, and surprising connections between the forces that shape our world.

Technical Summary: Hidden Supersymmetry in Wigner-Yang Quantum Mechanics

Problem Statement
This work revisits Eugene Wigner's 1950 inquiry regarding whether the equations of motion uniquely determine quantum mechanical commutation relations. Specifically, it addresses the Wigner-Yang quantum mechanics (WYQM) framework, where the canonical commutation relations are deformed by a reflection operator $R$ (acting as $(Rg)(x) = g(-x)$). The paper seeks to generalize Yang's explicit coordinate representation of the momentum operator to arbitrary symmetric potentials $V(x) = V(-x)$ and to investigate the conditions under which these systems exhibit a hidden supersymmetric (SUSY) structure. While previous studies focused on the harmonic oscillator or specific half-line problems, this work aims to establish a comprehensive SUSY framework for general symmetric potentials on the full real line and to explore the duality between Hooke's (harmonic) and Newton's (Coulomb) laws within this deformed algebra.

Methodology
The author employs a two-step approach to reconstruct the quantum system:

Generalised Wigner-Yang Quantum Mechanics (gWYQM):
The author first implements only the first equation of motion (EoM), $\dot{p} = -V'(x)$ , which corresponds to Newton's law. They introduce a generalized momentum operator $P = -i\hbar[\partial_x - f(x)R]$ , where $f(x)$ is an arbitrary odd function. This leads to a generalized derivative $D_x = \partial_x - f(x)R$ and a deformed Hamiltonian $H = -\frac{\hbar^2}{2m}D_x^2 + V(x)$ . The Hilbert space is decomposed into even ( $H_+$ ) and odd ( $H_-$ ) subspaces based on the eigenvalues of the reflection operator $R$ , which acts as a grading operator.
Standard WYQM with Interaction:
The second EoM, $\dot{x} = p$ , is then imposed. This constraint forces the function $f(x)$ to take the specific form $\nu/x$ , recovering the original Wigner-Yang deformation parameter $\nu$ . The author analyzes the resulting Hamiltonian for general symmetric potentials, reducing the eigenvalue problem on the full line to radial problems on the half-line ( $r \geq 0$ ) with specific boundary conditions (Neumann for $H_+$ , Dirichlet for $H_-$ ).
SUSY Analysis and Duality:
The author constructs SUSY charges using ladder operators derived from the generalized derivative. They examine the spectrum and eigenfunctions for two specific potentials: the harmonic oscillator and the attractive Coulomb-like potential $V(x) = -\gamma/|x|$ . Finally, they establish a "modified Hooke-Newton duality" by mapping the Coulomb-like eigenvalue problem onto the harmonic oscillator problem via a coordinate transformation $x = z|z|$ and a wavefunction rescaling, accounting for the deformation parameters.

Key Contributions and Results

Generalized Representation: The paper extends Yang's representation to a general class of symmetric potentials, defining a "generalised Wigner-Yang quantum mechanics" (gWYQM) where the deformation function $f(x)$ is arbitrary (subject to being odd).
Hidden SUSY in Free Systems: For the free case ( $V=0$ ), the author demonstrates that gWYQM is structurally identical to Witten's model of Supersymmetric Quantum Mechanics (SUSY QM). The reflection operator $R$ serves as the Witten parity operator. The paper clarifies that unbroken SUSY requires the ground state to be an even function, imposing specific asymptotic conditions on $f(x)$ (specifically $\lim_{x\to-\infty} f(x) > 0 > \lim_{x\to+\infty} f(x)$ ).
SUSY in Interacting Systems:
- Harmonic Oscillator: For the standard WY harmonic oscillator, the even and odd sectors possess disjoint spectra, preventing a direct SUSY structure unless a non-local potential term involving $R$ is introduced.
- Coulomb-like Potential: For the attractive potential $V(x) = -\gamma/|x|$ , the author identifies a hidden SUSY structure. The spectra of the even ( $H_+$ ) and odd ( $H_-$ ) sectors exhibit a degeneracy $E^+_{n+1} = E^-_n$ , characteristic of unbroken SUSY. The author explicitly constructs the supercharges and shows that the SUSY Hamiltonian includes a non-local interaction term $\delta(x)R$ , which vanishes for physical eigenfunctions that are zero at the origin.
Hooke-Newton Duality: The paper establishes a duality mapping the Coulomb-like WYQM system to the harmonic oscillator WYQM system. Unlike standard mappings, this transformation involves a modification of the deformation parameter. The relation between the deformation parameters $\nu$ (Coulomb) and $\mu$ (Harmonic) is given by $\mu = 2\nu - \frac{1}{2}R$ . When restricted to parity subspaces, this yields $\mu_{\pm} = 2\nu \mp \frac{1}{2}$ . This mapping successfully reproduces the known energy spectra and eigenfunctions of the Coulomb system from those of the harmonic oscillator.

Significance and Claims
The paper claims to provide a unified framework for understanding the supersymmetric nature of Wigner-Yang quantum mechanics. By treating the reflection operator as a grading operator, the author shows that WYQM naturally embeds Witten's SUSY QM model. The work extends previous findings (such as those by Beckers, Debergh, and Plyushchay) by:

Analyzing the full real line rather than restricting to the positive half-line, thereby addressing boundary conditions and the behavior of wavefunctions at the origin.
Providing explicit expressions for SUSY transformations in the presence of a Coulomb-like potential.
Generalizing the Hooke-Newton duality to the deformed algebra, showing that the duality holds but requires a specific, operator-dependent scaling of the angular momentum-like parameters (a "Langer-type modification").

The author concludes that whenever the effective radial problem yields eigenvalues dependent only on the sum of the radial quantum number and the deformation parameter ( $n + \nu$ ), a hidden SUSY structure exists. They suggest this framework could be extended to other conditionally exactly solvable partners of the hydrogen atom and applied to WKB approximations in WYQM, though these are noted as subjects for future investigation rather than results presented here.

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