⚛️ quantum physics

Quantum theory over dual-complex numbers

This paper proposes a unified framework for continuous and discrete quantum physics by extending quantum theory to dual-complex numbers, demonstrating that the resulting "dual quantum theory" remains mathematically consistent without requiring division by infinitesimals and successfully applying this formalism to unify the Dirac equation with the Dirac Quantum Walk.

Original authors: P. Arrighi, D. Bakircioglu, N. L. Houyet

Published 2026-03-19

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: P. Arrighi, D. Bakircioglu, N. L. Houyet

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 you are trying to describe the movement of a particle in the universe. Physicists have two main ways of doing this:

The Continuous View: Like a smooth river flowing. This is the traditional way, using complex numbers (math with imaginary parts) to describe waves and smooth changes.
The Discrete View: Like a video game character stepping from one pixel to the next. This is the "Quantum Walk" approach, where things happen in tiny, distinct jumps.

For a long time, these two views have been like oil and water. They speak different mathematical languages. When physicists try to turn the "pixelated" video game into a "smooth river," they often have to do messy, approximate math, and sometimes the beautiful symmetries of the universe (like how physics looks the same to everyone, no matter how fast they are moving) get broken in the process.

This paper introduces a new mathematical "translator" called Dual-Complex Numbers.

Here is the simple breakdown of what the authors did:

1. The Magic Ingredient: The "Almost Zero" Number

Standard complex numbers use $i$ (where $i^2 = -1$ ). The authors added a new ingredient: a number called $\epsilon$ (epsilon).

$\epsilon$ is so small it's practically zero.
But here's the magic trick: $\epsilon^2 = 0$ .

Think of $\epsilon$ as a "microscopic step." If you take one step, you move. If you take two steps at the exact same time, you cancel out and stay still. This allows the math to capture "tiny changes" (infinitesimals) without getting bogged down in complicated higher-order math.

2. The Problem They Solved

The authors faced two big objections from skeptics:

"You can't divide by zero!" Since $\epsilon$ is so small, dividing by it usually breaks math.
"Does the math still make sense?" In quantum mechanics, things must stay "unitary" (meaning probability is always conserved, nothing disappears). Skeptics thought adding this weird $\epsilon$ would break this rule.

The Solution: The authors proved that even with this weird number, the math holds up.

They showed you can still do the necessary calculations without dividing by zero.
They proved that "probability" is still preserved. The system doesn't lose or gain energy; it just shifts slightly.

3. The "Universal Translator"

The coolest part of this paper is what this new math does. It acts as a bridge.

Imagine you have a video game character (the Discrete model) and a real-life person walking (the Continuous model). Usually, to see how the game character becomes a real person, you have to manually tweak the code, frame by frame, and hope it looks right.

With Dual-Complex Quantum Theory, the authors showed that you can write one single equation that describes both the pixelated steps and the smooth flow simultaneously.

The "real" part of the number describes the main movement.
The " $\epsilon$ " part automatically calculates the tiny corrections needed for the next step.

It's like having a GPS that doesn't just tell you where you are, but also instantly calculates the tiny adjustments needed for the road ahead, all in one go.

4. The Real-World Win: The Dirac Equation

To prove this works, they applied it to the Dirac Equation, which describes how electrons (fermions) move.

The Old Way: You have the "Dirac Equation" (smooth) and the "Dirac Quantum Walk" (discrete). They are cousins, but they don't perfectly match. When you try to force the discrete one to look continuous, tiny errors (like $O(\epsilon^2)$ ) creep in and break the "Lorentz Covariance" (the rule that physics looks the same to everyone).
The New Way: By using Dual-Complex numbers, the "tiny errors" vanish because $\epsilon^2 = 0$ $ϵ^{2} = 0$ .
- The discrete steps and the smooth flow become identical.
- The symmetry is restored perfectly. The "pixelated" world now respects the same laws of physics as the "smooth" world, exactly.

The Big Picture Analogy

Think of the universe as a movie.

Traditional Physics tries to describe the movie as a smooth, continuous film reel.
Quantum Computing tries to describe it as a series of individual frames.
This Paper says: "What if we describe the movie as a frame plus a tiny hint of the next frame?"

By adding that tiny hint ( $\epsilon$ ), the math automatically knows how to transition from one frame to the next without any glitches. It unifies the "frame-by-frame" view with the "smooth movie" view, proving that they are actually two sides of the same coin.

In short: The authors built a new mathematical language that lets us treat the "jumpy" quantum world and the "smooth" classical world as the same thing, fixing long-standing errors and making the math cleaner, more consistent, and more powerful.

1. Problem Statement

The paper addresses the fundamental tension between continuous and discrete models in quantum physics.

The Challenge: Continuous models (e.g., the Dirac equation) are mathematically elegant and preserve symmetries like Lorentz covariance, but they are often difficult to simulate computationally. Discrete models (e.g., Quantum Walks) are computationally convenient but often break continuous symmetries when taking the continuum limit, requiring ad-hoc corrections or failing to be exactly covariant at higher orders.
The Gap: There is a lack of a unified mathematical language that treats infinitesimal variations (discrete steps) and continuous evolution on equal footing without resorting to informal "neglect of higher-order terms."
Specific Objections: Extending quantum mechanics from the complex field $\mathbb{C}$ $C$ to the ring of dual-complex numbers $\mathbb{C}[\varepsilon]$ $C [ε]$ (where $\varepsilon^2=0$ $ε^{2} = 0$ ) faces two major theoretical hurdles:
1. $\mathbb{C}[\varepsilon]$ is a ring, not a field, meaning division by infinitesimals is undefined, yet quantum mechanics relies on normalization (division).
2. It is unclear if the concept of unitarity (preservation of probability/norm) remains consistent when $\varepsilon^2=0$ .

2. Methodology

The authors propose replacing the complex scalar field in the postulates of quantum mechanics with the dual-complex ring $\mathbb{DC} = \mathbb{C}[\varepsilon]/\langle\varepsilon^2\rangle$ .

Mathematical Foundation:
- Dual-Complex Numbers: Elements are of the form $z + t\varepsilon$ , where $z, t \in \mathbb{C}$ and $\varepsilon^2=0$ . This structure naturally encodes first-order Taylor expansions (automatic differentiation).
- Linear Algebra Extension: The authors rigorously define dual-complex vector spaces, inner products, and adjoints. They introduce a lexicographic ordering to define valid probabilities and establish that dual-complex Hermitian and Unitary operators are diagonalizable.
- Postulates: They reformulate the four standard postulates of quantum mechanics (States, Evolution, Measurement, Composite Systems) using dual-complex numbers.
  - States: $|\psi_\varepsilon\rangle = |\psi\rangle + \varepsilon|\phi\rangle$ , where $|\psi\rangle$ is the significant part and $|\phi\rangle$ is the infinitesimal variation.
  - Evolution: Defined by dual-complex unitary operators $U_\varepsilon = U + \varepsilon V$ .
  - Measurement: Defined by a family of operators satisfying the completeness relation $\sum M_\varepsilon^\dagger M_\varepsilon = I$ .
Consistency Proofs:
- The authors prove that dual-complex unitary operators preserve the norm and that probabilities remain "appreciably positive" (non-negative and non-infinitesimal) or zero.
- They demonstrate that renormalization never requires dividing by an infinitesimal, resolving the first objection.
- They establish an equivalence between the discrete-time unitary formulation and the continuous-time Schrödinger equation in this new framework.
Translation Mechanism:
- The paper introduces a method to map between conventional parametrized quantum operations (parameterized by a real step size $h$ ) and their dual-complex extensions.
- A dual-complex operator $U_\varepsilon$ corresponds to the first-order Taylor expansion of a conventional operator $U_h$ .
- Conversely, any dual-complex unitary can be "corrected" by adding $O(h^2)$ terms to recover a strictly conventional unitary operator.

3. Key Contributions

Formalization of Dual-Complex Quantum Theory: A self-consistent set of postulates extending quantum mechanics to $\mathbb{C}[\varepsilon]$ , proving that unitarity, norm preservation, and probability conservation hold despite the ring structure.
New Linear Algebra Results:
- Proof that dual-complex unitary operators are diagonalizable.
- Establishment of the spectral theorem for dual-complex unitary operators.
- Proof that the exponential of a dual-complex anti-Hermitian operator is unitary, and the logarithm of a unitary is anti-Hermitian.
Unified Framework for Discrete/Continuous Physics: The framework provides a rigorous way to treat discrete steps as infinitesimal variations of continuous parameters, automating the continuum limit.
Exact Discrete Lorentz Covariance: Application to the Dirac Quantum Walk (QW) demonstrates that the dual-complex formulation restores exact discrete Lorentz covariance, a property usually broken in standard discrete models at second order.

4. Results

Consistency: The theory is internally consistent. Norms are preserved at all times, and renormalization is well-defined because the significant part of the probability is always non-zero for valid states.
Equivalence: A dual-complex evolution $U_\varepsilon$ is equivalent to the first-order variation of a parametrized conventional evolution $U_h$ . Specifically, $U_\varepsilon = U_0 + \varepsilon \frac{dU_h}{dh}|_{h=0}$ .
Application to Dirac Equation:
- The authors applied the framework to the Dirac Quantum Walk.
- By setting the step size $\Delta x = \Delta t = \varepsilon$ , the discrete update rule of the QW automatically yields the Dirac equation (a continuous PDE) when the dual-complex algebra is applied.
- The discrete evolution operator $U_\varepsilon$ and the continuous generator are shown to be the same object in this formalism.
Restoration of Symmetry:
- In standard Quantum Walks, discrete Lorentz covariance holds only to first order; $O(\varepsilon^2)$ terms break the symmetry.
- In the dual-complex formulation, these second-order terms vanish identically because $\varepsilon^2=0$ . This results in exact discrete Lorentz covariance for the QW, matching the continuous theory perfectly.

5. Significance

Foundational Impact: The work challenges the debate on whether quantum mechanics requires a field (like $\mathbb{C}$ ) or if a ring structure suffices. It suggests that dual-complex numbers offer a natural bridge between discrete computational models and continuous physical laws.
Computational Utility: It provides a rigorous tool for automatic differentiation within quantum algorithms. Instead of manually deriving continuum limits or perturbation expansions, the dual-complex formalism automates the extraction of first-order variations.
Symmetry Preservation: The restoration of exact discrete Lorentz covariance in the Dirac QW is a significant breakthrough for lattice field theory and quantum simulation, potentially solving long-standing issues regarding symmetry breaking in discrete spacetime models.
Future Directions: The authors suggest this framework could be extended to higher-order dual numbers, quaternions, or Grassmann numbers (for fermions), offering a generalized language for quantum foundations and quantum gravity research.

In summary, the paper successfully demonstrates that extending quantum theory to dual-complex numbers is not only mathematically consistent but also a powerful unifying tool that resolves the tension between discrete and continuous physics, specifically by automating the continuum limit and preserving fundamental symmetries.