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Minimal length: A source of quantum non-locality

This paper demonstrates that the subtle distinction between Hilbert spaces of canonical and generalized momentum operators in the presence of a minimal length can lead to complex eigenvalues for canonical momentum and enable novel forms of quantum entanglement, thereby enriching the understanding of quantum non-locality.

Original authors: H. Moradpour, S. Jalalzadeh

Published 2026-07-02

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: H. Moradpour, S. Jalalzadeh

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 video game. For decades, physicists have played by a specific set of rules called Quantum Mechanics. One of the most famous rules in this game is the Heisenberg Uncertainty Principle. It basically says: "The more precisely you know where a particle is, the less you know about how fast it's moving, and vice versa." In this classic game, there is no limit to how small a distance can be; you can theoretically zoom in forever.

However, this paper suggests that if we add a new rule to the game—a "Minimum Length" (a smallest possible pixel size for the universe, likely related to gravity)—the rules of the game change in some very strange ways.

Here is a simple breakdown of what the authors discovered:

1. The "Real" vs. The "Dressed" Momentum

Think of Momentum (how much "oomph" a particle has) as a character in our story.

Canonical Momentum ( $\hat{p}$ ): This is the character in their "street clothes." It's the standard momentum we learn about in basic physics.
Generalized Momentum ( $\hat{P}$ ): This is the character wearing a heavy, futuristic suit that accounts for the "Minimum Length" rule.

In the old rules, these two were essentially the same. But with the Minimum Length, they become different. The paper argues that while the "suit-wearing" momentum ( $\hat{P}$ ) is the physical reality we measure, the "street-clothes" momentum ( $\hat{p}$ ) still exists mathematically, but it behaves weirdly.

2. The "Ghost" Numbers (Complex Eigenvalues)

In standard physics, when you measure a property like momentum, you always get a real number (like 5, 10, or -2.5). You never get a "ghost" number.

But the authors found that in this new "Minimum Length" universe, the "street-clothes" momentum ( $\hat{p}$ ) can produce complex numbers.

The Analogy: Imagine you are trying to measure the speed of a car. In our normal world, the speedometer always shows a real number. But in this new world, the speedometer might show a number that includes an "imaginary" part (like $3 + 4i$ ).
Why? Because the "Minimum Length" distorts the mathematical space the particle lives in. The paper shows that for every single "real" measurement of the generalized momentum, there are actually three possible "street-clothes" momenta that could produce it. Two of these three possibilities involve these strange, complex numbers.

3. The Magic Trick: Creating Entanglement from Nothing

The most exciting part of the paper is about Quantum Entanglement.

What is it? Imagine you have two magic coins. If you flip them, they are linked. If one lands on Heads, the other must land on Tails, no matter how far apart they are. This spooky connection is called entanglement. Usually, you need to create these coins in a special way to link them.
The Paper's Claim: The authors show that simply having a "Minimum Length" in the universe is enough to automatically create this entanglement between two particles, even if they weren't specially prepared.

How it works (The Analogy):
Imagine two dancers (particles) moving in opposite directions.

In the old rules, if they move perfectly opposite, they are just two separate dancers.
In the new "Minimum Length" rules, the paper shows that because the "street-clothes" momentum has those three weird possibilities (one real, two complex), the two dancers become mathematically forced to link up.
The state of the system becomes a "superposition" (a mix) of all three possibilities. Because there are multiple possibilities happening at once, the two particles become entangled.

4. The "Spooky" Result

The paper concludes that this new type of entanglement is a source of quantum non-locality.

Non-locality means that what happens to one particle instantly affects the other, even if they are light-years apart.
The authors suggest that the existence of a "Minimum Length" (a smallest possible distance in the universe) is a hidden engine that generates this spooky connection. It's not just a side effect; it's a direct result of the universe having a "pixel size."

Summary

The paper doesn't propose a new machine or a medical cure. Instead, it does a deep mathematical "stress test" on the rules of physics. It finds that if the universe has a smallest possible size (a Minimum Length):

The standard way we calculate momentum gets distorted.
This distortion allows for "imaginary" numbers in our calculations.
This distortion automatically creates a link (entanglement) between particles that wouldn't exist otherwise.

In short: The "pixelation" of the universe might be the secret ingredient that makes the universe "spooky" and connected in ways we didn't fully appreciate before.

Technical Summary: Minimal length: A source of quantum non-locality

Problem Statement
The paper addresses the theoretical implications of a minimal length scale on quantum mechanics, specifically focusing on the transition from the Heisenberg Uncertainty Principle (HUP) to the Generalized Uncertainty Principle (GUP). While the HUP implies no lower bound on position measurement (provided momentum information is lost), various quantum gravity scenarios and Newtonian gravity considerations suggest a non-zero minimal length. This modification alters the commutation relations between position and momentum. The central problem investigated is how this minimal length affects the mathematical structure of the momentum operator, the nature of its eigenvalues, and whether these modifications can serve as a direct source for generating quantum entanglement and non-locality, distinct from previous studies that focused on testing gravity via entanglement or analyzing GUP effects on Bell inequalities.

Methodology
The authors employ a formal operator-based approach within the framework of Quantum Mechanics (QM) modified by GUP.

Operator Definition: They distinguish between the canonical momentum operator ( $\hat{p}$ ) and the generalized (physical) momentum operator ( $\hat{P}$ ). In the presence of minimal length, the relationship is defined as $\hat{P} = \hat{p}(1 + \beta\hat{p}^2)$ to the first order of the GUP parameter $\beta$ .
Hilbert Space Analysis: The study examines the eigenstates and eigenvalues of both $\hat{P}$ and $\hat{p}$ . While $\hat{P}$ is treated as the physical observable with real eigenvalues, the authors analyze the properties of $\hat{p}$ within the minimal length regime. They solve the eigenvalue equation for $\hat{P}$ in the position representation, noting that $\hat{P}$ and $\hat{p}$ commute but possess different Hilbert spaces ( $|P\rangle \neq |p\rangle$ ).
Complex Eigenvalue Investigation: The authors investigate the solutions to the cubic equation relating $\hat{p}$ and $\hat{P}$ . They analyze the boundary conditions required for $\hat{p}$ to remain Hermitian, specifically checking if wave functions vanish at infinity.
Entanglement Construction: To test for non-locality, the authors construct a two-particle system where the total generalized momentum is zero ( $P_1 = -P_2$ ). They express the joint state in the canonical momentum basis, applying a projection operator to enforce canonical momentum conservation ( $p_1 = -p_2$ ) derived from generalized momentum conservation.

Key Results

Complex Eigenvalues for Canonical Momentum: The analysis reveals that the mapping from generalized momentum $P$ to canonical momentum $p$ is a third-order equation, leading to a threefold degeneracy. For a given physical momentum $P$ , there exist three corresponding eigenvalues for $\hat{p}$ ( $P_1, P_2, P_3$ ). Crucially, two of these solutions ( $P_2$ and $P_3$ ) are complex. The corresponding wave functions diverge at infinity, violating the standard boundary condition for Hermiticity. Consequently, the paper concludes that in the presence of minimal length, the canonical momentum operator $\hat{p}$ may possess complex eigenvalues.
Triple Degeneracy and State Expansion: The eigenstate of the generalized momentum $|P\rangle$ is shown to be a superposition of three eigenstates of the canonical momentum: $|P\rangle = \sum_{k=1}^3 \alpha_k |p=P_k\rangle$ .
Generation of Entanglement: For a two-particle system with zero total generalized momentum, the state $|P, -P\rangle$ is expanded in the canonical momentum basis. After applying the projection operator to ensure canonical momentum conservation, the resulting state is a superposition of three terms: $\sum_{i=1}^3 c_i |P_i, -P_i\rangle$ .
Quantum Non-Locality: The Von Neumann entropy of this state is calculated. The entropy is non-zero (indicating entanglement) provided that at least two coefficients $c_i$ are non-zero. The maximum entropy occurs when all three outcomes are equally probable. This demonstrates that the minimal length framework naturally generates entangled states in the canonical momentum space.

Significance and Claims
The paper claims to identify a novel mechanism for quantum non-locality rooted in the existence of a minimal length. The primary significance lies in the finding that the minimal length effect allows the canonical momentum operator to acquire complex eigenvalues, a feature not present in standard QM under typical boundary conditions. Furthermore, the authors report that this framework enables the direct generation of quantum entangled states without relying on external interactions, purely through the structural differences between the Hilbert spaces of canonical and generalized momenta.

The authors position this work as a re-examination of the EPR argument within the GUP framework, suggesting that the replacement of the commutator $[\hat{x}, \hat{p}]$ with $[\hat{x}, \hat{P}]$ reveals a new source of quantum non-locality. They emphasize that while previous works have studied GUP effects on Bell inequalities or used entanglement to test gravity, this study specifically investigates the minimal length's ability to generate non-locality and entangled states featuring complex quantum numbers.

1. The "Real" vs. The "Dressed" Momentum

2. The "Ghost" Numbers (Complex Eigenvalues)

3. The Magic Trick: Creating Entanglement from Nothing

4. The "Spooky" Result

Summary

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