Square roots of complexified quaternions

This paper investigates the square roots of complexified quaternions (including Hamilton quaternions, coquaternions, nectorines, and conectorines) by utilizing their isomorphism with 3-dimensional multivectors of Clifford algebras, demonstrating that such roots can be discrete, continuous, or non-existent.

The Gist

Imagine you are a master builder working in a universe where numbers aren't just points on a line, but full-blown 3D objects that can spin, flip, and twist. This is the world of Quaternions.

You've probably heard of regular numbers (like 5 or -3) and complex numbers (which involve the square root of -1, often written as $i$). Quaternions are like the "big brothers" of complex numbers. They are used by engineers to help rockets fly smoothly and robots move their arms without getting dizzy.

But here's the puzzle this paper solves: If you have a Quaternion, how do you find its "square root"?

In normal math, the square root of 4 is 2 (because $2 \times 2 = 4$). But in the wild world of Quaternions, things get messy. Sometimes there are two answers, sometimes four, sometimes a whole infinite circle of answers, and sometimes... no answer at all.

Here is the simple breakdown of what the authors, Adolfas and Artūras, discovered.

1. The Four Flavors of Quaternions

The paper starts by saying there isn't just one type of Quaternion. There are four distinct "flavors":

• Hamilton Quaternions: The classic, standard kind used in physics and spaceflight.
• Coquaternions, Conectorines, and Nectorines: These are the "cousins." They look similar but have different rules for how they multiply. Think of them like different dialects of the same language; they sound alike but have different grammar rules.

The authors wanted to know: If I give you a number from any of these four families, what number, when multiplied by itself, gives you back the original?

2. The Magic Translator (Clifford Algebras)

Solving this directly in the Quaternion world is like trying to untangle a knot while wearing thick gloves. It's hard to see what's going on.

The authors' brilliant trick was to use a translator. They realized that all these Quaternion "dialects" are actually isomorphic (mathematically identical) to something called Clifford Algebras.

The Analogy:
Imagine Quaternions are written in a secret code that is hard to crack. The authors found a dictionary that translates this code into a different language (Clifford Algebras) where the rules are much clearer and the "knots" are easier to untangle.

Once they translated the Quaternion problem into the Clifford language, they could use a known "algorithm" (a step-by-step recipe) to find the roots. Then, they translated the answer back into Quaternion language.

3. The Surprising Results

When they applied this method, they found that the "Square Root" in this world is much more complicated than in school math.

• The "Multiple Choice" Surprise: In normal math, $\sqrt{4}$ is just $\pm 2$. In the Quaternion world, a single number might have four different square roots.
• The "Infinite Circle" Surprise: Sometimes, the answer isn't just a few numbers. It can be a continuous line or a whole sphere of answers. Imagine if the square root of 4 wasn't just 2, but any number on a specific circle. That's what happens here.
• The "Ghost" Surprise: Sometimes, a number simply has no square root. It's like asking for the square root of a negative number in regular math (before complex numbers were invented). In the Quaternion world, some numbers are just "rootless."

4. Real vs. Complex Quaternions

The paper also looked at "Complex Quaternions."

• Real Quaternions: The coefficients are normal numbers.
• Complex Quaternions: The coefficients are complex numbers (involving $i$).

The authors showed that even these "super-complex" Quaternions can be translated into a 3D Clifford algebra (specifically $Cl_{3,0}$). This means the same "translator" works for both simple and super-complex versions, revealing that the roots can be incredibly elaborate structures.

The Big Picture

Why does this matter?

Think of Quaternions as the GPS system for 3D rotation. If you want to program a robot arm to move from Point A to Point B, or a spaceship to reorient itself in zero gravity, you need to do math with Quaternions.

If you need to reverse a process (like "undo" a rotation), you need to find the square root. This paper provides the ultimate instruction manual for finding those roots. It tells engineers and physicists:

1. Which "flavor" of Quaternion you are using.
2. How to translate it to find the answer.
3. Whether you will get one answer, many answers, a circle of answers, or no answer at all.

In short: The authors built a universal key that unlocks the "square root" mystery for all types of Quaternions, showing us that the universe of numbers is far richer, stranger, and more colorful than we ever imagined.

Technical Summary

1. Problem Statement

The paper addresses the mathematical challenge of finding square roots for various types of quaternions, specifically:

• Real Quaternions: Including the standard Hamilton quaternion ($H$), the split quaternion/coquaternion ($H_{coq}$), the conectorine ($H_{con}$), and the nectorine ($H_{nec}$).
• Complex Quaternions (Biquaternions): The complexified versions of the four algebras listed above, where coefficients are complex numbers.

Key Issues:

• Unlike complex numbers, where the square root generally yields two values ($\pm$), quaternionic square roots can be discrete, continuous (forming manifolds like spheres or hyperbolic surfaces), or non-existent.
• Existing literature often focuses on polynomial roots or specific cases (like $\sqrt{-1}$ in biquaternions), but a comprehensive algorithm for finding square roots across all four non-commutative quaternion types (both real and complex) was lacking.
• The authors aim to determine the conditions under which roots exist and to provide explicit formulas for calculating them.

2. Methodology

The core methodology relies on the isomorphism between quaternion algebras and Clifford algebras (Geometric Algebras). Instead of deriving root-finding algorithms directly from quaternion multiplication rules, the authors map quaternion problems into the framework of Clifford algebras, where robust root-finding algorithms already exist.

Step-by-Step Approach:

1. Isomorphism Mapping:

   • Real Quaternions: The authors map the four real quaternion types to 2D Clifford algebras:
     • Hamilton ($H$) $\cong Cl_{0,2}$
     • Coquaternion, Conectorine, Nectorine $\cong Cl_{2,0}$ (or $Cl_{1,1}$ depending on the basis choice, though the paper standardizes on $Cl_{2,0}$ for complexification).
   • Complex Quaternions: The authors demonstrate that all four complexified quaternion algebras are isomorphic to the 3D Euclidean Clifford algebra, $Cl_{3,0}$. This is achieved by treating the complex unit $I$ as an additional basis vector in the real algebra.

2. Root Extraction in Clifford Algebras:

   • The problem is transformed: finding $\sqrt{q}$ becomes finding $\sqrt{Q}$ where $Q$ is a multivector in the corresponding Clifford algebra.
   • The authors utilize a previously developed algorithm (referenced as [18]) for calculating square roots of multivectors in $Cl_{3,0}$. This algorithm handles:
     • Discrete roots: Specific isolated solutions.
     • Continuous roots: Solutions involving free parameters (e.g., when the determinant is zero, leading to spherical or hyperbolic solution sets).
     • Non-existence: Cases where no real solution exists (e.g., negative determinants in specific signatures).

3. Inverse Mapping:

   • Once the roots are calculated in the Clifford algebra domain, they are mapped back to the specific quaternion basis using the established isomorphism tables.

3. Key Contributions

• Unified Framework: The paper establishes a unified method to handle square roots for all four non-commutative quaternion types (Hamilton, coquaternion, conectorine, nectorine) by reducing them to Clifford algebra problems.
• Complex Quaternion Isomorphism: A significant theoretical contribution is the proof that all complexified quaternion algebras are isomorphic to the single real algebra $Cl_{3,0}$. This simplifies the analysis of biquaternions significantly.
• Comprehensive Root Classification: The paper categorizes the nature of the roots, showing that they can be:
  • Multiple discrete roots (e.g., 2 or 4 distinct roots).
  • Continuous families of roots (infinite solutions parameterized by free variables).
  • Non-existent (no solution).
• Explicit Algorithms and Examples: The authors provide explicit formulas and a step-by-step algorithm (Algorithm 1) for computing these roots, accompanied by concrete numerical examples for every quaternion type.

4. Results

The paper presents specific findings through examples:

• Real Quaternions:

  • For $q = 1+k$, the Hamilton and Conectorine quaternions yield roots involving irrational coefficients, while Coquaternions and Nectorines yield simpler rational roots.
  • For $q = 1+2i+3j+4k$, the Hamilton and Conectorine algebras yield roots, but the Coquaternion and Nectorine algebras yield no roots because their determinants are negative ($D < 0$). This highlights that the existence of a root depends heavily on the specific algebraic signature.

• Complex Quaternions (Biquaternions):

  • The authors demonstrate that complex quaternions can have four distinct roots (e.g., for $B = -(2+I)k$).
  • They show cases where roots are nilpotent (squared to zero) or zero divisors (product of non-zero elements is zero), properties not present in standard Hamilton quaternions.
  • Example: $\sqrt{-Ik}$ in the complex Hamilton algebra yields four distinct roots derived from the $Cl_{3,0}$ basis vectors.

• Algorithm Validation:

  • The provided Algorithm 1 successfully computes both isolated roots and continuous root sets for multivectors in $Cl_{3,0}$, which are then translated back to quaternion forms.

5. Significance

• Theoretical Physics and Engineering: Quaternions are critical in robotics, aerospace (attitude control), and quantum physics. Understanding the full spectrum of their roots (including continuous sets) is vital for solving nonlinear equations, such as those found in the nonlinear Schrödinger equation or optimal trajectory planning.
• Computational Efficiency: By leveraging the isomorphism with Clifford algebras, the paper offers a computationally efficient path to solving quaternionic equations using existing Geometric Algebra software libraries, avoiding the need to derive complex quaternion-specific solvers from scratch.
• Mathematical Insight: The work clarifies the structural differences between the four quaternion types, demonstrating that while they may look similar, their root structures (existence and multiplicity) differ fundamentally due to the signs of their basis vector squares.
• Foundation for Future Work: The paper sets the stage for analyzing higher-order roots and polynomial equations in these algebras, extending the Fundamental Theorem of Algebra to these non-commutative, complexified systems.

In summary, Dargys and Acus provide a rigorous, algorithmic solution to the square root problem for a broad class of quaternions by exploiting the deep structural connections between quaternion algebras and Clifford algebras, revealing a rich landscape of discrete, continuous, and non-existent solutions.