[Paper Review] A Brief History of Quaternions and of the Theory of Holomorphic Functions of Quaternionic Variables

Amy Buchmann|arXiv (Cornell University)|Nov 25, 2011

Algebraic and Geometric Analysis8 references9 citations

TL;DR

This paper traces the historical development of quaternions from Hamilton's 1843 discovery and the failure of his triplet theory, then explores modern efforts to define holomorphic functions on quaternions through slice-regularity—where a function is holomorphic on every complex slice through the origin. The key contribution is establishing a consistent theory of quaternionic holomorphic functions using this slice-regularity condition, enabling holomorphic behavior for polynomials and power series in quaternions.

ABSTRACT

In this paper I will give a brief history of the discovery (Hamilton, 1843) of quaternions. I will address the issue of why a theory of triplets (the original goal of Hamilton) could not be developed. Finally, I will discuss briefly the history of various attempts to define holomorphic functions on quaternionic variables.

Motivation & Objective

• To trace the historical origins of quaternions, particularly Hamilton's discovery on October 16, 1843, and the failed attempt to construct a theory of triplets.
• To explain why a consistent algebraic structure for triplets (three imaginary units) could not be formed, leading to the need for four-dimensional quaternions.
• To examine the evolution of holomorphic function theory in the context of quaternions, culminating in the modern concept of slice-regular functions.
• To establish the foundation for a non-commutative analog of complex analysis by defining holomorphic-like behavior on quaternionic domains.

Proposed method

• Using historical accounts and mathematical definitions, the paper reconstructs Hamilton’s path to discovering quaternions via the non-commutative algebraic relations: $i^2 = j^2 = k^2 = ijk = -1$.
• Defining the skew field $\mathbb{H}$ as a non-commutative extension of $\mathbb{C}$, with elements $q = a + bi + cj + dk$, where $a,b,c,d \in \mathbb{R}$, and multiplication governed by the quaternion multiplication rules.
• Analyzing the impossibility of constructing a consistent algebraic structure for triplets by showing that such a system cannot satisfy the properties of a division algebra.
• Introducing the concept of slice-regularity: a function $f: \Omega \to \mathbb{H}$ is slice-regular if its restriction to each complex line $L_I = \mathbb{R} + \mathbb{R}I$ is holomorphic.
• Applying the definition to show that monomials $aq^n$ and power series $\sum_{n=0}^\infty a_n q^n$ are slice-regular where convergent.
• Using Hamilton’s own letter to the reader as a primary source to anchor the historical narrative and provide personal insight into the discovery process.

Experimental results

Research questions

• RQ1 Why was it impossible to construct a consistent algebraic system of triplets with properties analogous to complex numbers?
• RQ2 What motivated Hamilton’s shift from triplets to quaternions, and how did the discovery occur on October 16, 1843?
• RQ3 How can the concept of holomorphic functions be generalized to the non-commutative setting of quaternions?
• RQ4 What properties must a quaternionic function satisfy to be considered holomorphic in a meaningful way?
• RQ5 Can a theory of holomorphic functions on quaternions be constructed that preserves key features of complex analysis, such as analyticity and power series representation?

Key findings

• Hamilton discovered quaternions on October 16, 1843, while walking along the Royal Canal in Dublin, and carved the fundamental identity $i^2 = j^2 = k^2 = ijk = -1$ onto Brougham Bridge.
• The theory of triplets failed because no consistent algebraic structure with three imaginary units could satisfy the required field properties, particularly associativity and division.
• The concept of slice-regularity provides a viable generalization of holomorphic functions to quaternions by requiring holomorphicity on every complex slice $L_I = \mathbb{R} + \mathbb{R}I$.
• All polynomials $f(q) = \sum_{n=0}^n a_n q^n$ with quaternion coefficients are slice-regular, and so are power series $\sum_{n=0}^\infty a_n q^n$ where convergent.
• The theory of slice-regular functions is a promising, emerging field with foundational results already established, suggesting that the mathematical development of quaternions remains active and evolving.
• Hamilton’s personal letter confirms the emotional and intellectual intensity of the discovery moment, with the 'spark' of insight occurring during a walk with his family.