GENERAL ARTICLE Algebra and Geometry of Hamilton’s Quaternions ‘Well, Papa, Can You Multiply Triplets?’ Govind S Krishnaswami and Sonakshi Sachdev Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three-dimensional geometry. Un- able to multiply and divide triples, he invented a non-commutative division algebra of quadru- ples, in what he considered his most significant (left) Govind Krishnaswami is on the faculty of the work, generalizing the real and complex number Chennai Mathematical systems. We give a motivated introduction to Institute. His research in quaternions and discuss how they are related to theoretical physics spans Pauli matrices, rotations in three dimensions, the various topics three sphere, the group SU(2) and the celebrated from quantum field theory and particle physics to fluid, Hopf fibrations. plasma and non-linear dynamics. 1. Introduction (right) Sonakshi Sachdev is Every morning in the early part of the above-cited month a PhD student at the [October 1843], on my coming down to breakfast, your Chennai Mathematical (then) little brother William Edwin, and yourself, used Institute. She has worked on to ask me, “Well, Papa, can you multiply triplets”? the dynamics of rigid bodies, Whereto I was always obliged to reply, with a sad shake fluids and plasmas. of the head: “No, I can only add and subtract them.” W R Hamilton in a letter dated August 5, 1865 to his son A H Hamilton [1]. Imaginary and complex numbers arose in looking for ‘impossible’ solutions to polynomial equations such as x2 + 1 = 0. Even a sceptic might be convinced of their utility upon seeing the remarkable Ferro–Tartaglia–Car- dano–Bombelli formula for a real root of cubic equa- tions such as x3 =20x + 25, expressed as sums of cube roots of complex numbers! No such general expression in Keywords terms of real radicals is possible. Some of the mystery Quaternions, division algebra, surrounding complex numbers was removed once they rotations, spheres, Hopf map. RESONANCE ⎜ June 2016 529 GENERAL ARTICLE could be visualized as vectors in the plane, following the work of Euler, Wessel, Argand and Gauss. Hamilton entered the scene in 1830 by defining complex numbers z = x + yi as ordered pairs (or ‘couples’) (x, y)ofreal numbers. He abstracted rules for adding, multiplying and dividing ordered pairs obeying the commutative, 1 1 Commutative means z1 z2 = z2 associative and distributive laws of arithmetic .Thus z , associative means z (z z ) = 1 1 2 3 the complex numbers C constitute a ‘field’ just like the (z z ) z while z (z + z ) = z z 1 2 3 1 2 3 1 2 reals R. The manner in which the geometry of the plane + z z is distributivity. 1 3 was encoded in the algebra of complex numbers2 made 2 The parallelogram law for add- a deep impression on Hamilton. The problem of finding ing vectors is just the sum of an algebra of triples (α, β, γ) to describe the geometry complex numbers, a rotation cor- of vectors in three-dimensional (3D) space haunted him responds to multiplication by a for at least fifteen years. While he knew how to add complex number of unit magni- tude, the cosine of the angle and subtract triples, the problem of multiplying them between vectors is given in terms and dividing by non-zero triples seemed insurmountable. of the scalar product of the cor- What is more, by analogy with the complex numbers, responding complex numbers, he wanted the Euclidean length (square-root of the sum etc. of squares of the components) of the product of a pair 3 The cross product of vectors of triples to equal the product of their lengths3.Incur- had not yet been discovered. rent language, Hamilton was looking for a real, three- Besides being non-associative dimensional, normed, associative, division algebra; we a × (b × c) ≠ (a × b) × c, it does not satisfy the condition | a × b| now know that such an algebra does not exist. = |a||b|. Cross products were invented by Grassmann within a In 1843, Hamilton found an ingenious way around this year of Hamilton's discovery of problem, combining two major innovations. The first quaternions. was to drop the commutative law for multiplication. He wrote a triple as t = α+βi+γj and assumed by analogy with the complex numbers, that i2 = j2 = −1. It follows that t2 = α2 − β2 − γ2 +2αβi +2αγj + βγ(ij + ji). (1) Moreover, he wanted the ‘law of moduli’ |t|2 = |t2| to hold. Now |t|2 = α2 +β2 +γ2 would equal |t2| if ij+ji = 4 The quotations in this para- 0, since graph are from a letter dated October 17, 1843 from Hamilton (α2 −β2 −γ2)2 +(2αβ)2 +(2αγ)2 =(α2 +β2 +γ2)2. (2) to J T Graves, which was re- printed in Vol.xxv, pp.489–495 Hamilton was “tempted for a moment to fancy that” of [2]. ij = −ji =04. But this “seemed uncomfortable” and 530 RESONANCE ⎜ June 2016 GENERAL ARTICLE he settled on the “less harsh” condition ij = −ji = k, reserving to himself to “inquire whether k was 0 or not”. The second innovation, was to “admit in some sense a fourth dimension of space for the purpose of calculating 5 k with triples” . In other words, he assumed that was 5 In taking this step, Hamilton linearly independent of 1, i and j. Applying the asso- was partly motivated by specu- ciative law, he inferred that k2 = ijij = −1, ki = j lations on how time and space and jk = i. Hamilton vividly describes his elation upon may be “girdled” together, as 6 well as some vague notions discovering these multiplication rules: “They started (based on Kantian philosophy) into life, or light, full grown, on the 16th of October, of geometry and arithmetic be- 1843, as I was walking with Lady Hamilton to Dublin, ing the sciences of space and and came up to Brougham Bridge, which my boys have time respectively [3]. since called the Quaternion Bridge. That is to say, I 6 The article by N Mukunda in then and there felt the galvanic circuit of thought close; this issue of Resonance places and the sparks which fell from it were the fundamental Hamilton's discovery of quater- equations in i, j, k exactly such as I have used them ever nions in the wider context of his life and work. See also the ar- since.” Extract from a letter dated October 15, 1858 ticle by R Nityananda devoted to from Hamilton to P G Tait [1]. Hamilton's work in optics. In more recent language [4, 5], quaternions are quadru- ples of real numbers q =(α, β, γ, δ) forming a four- dimensional real vector space H (named after Hamilton). In terms of the canonical basis: Figure 1. Plaque on Broome Bridge, Dublin, commemorat- ing Hamilton’s discovery of quaternions. Photograph by Brian Dolan, In- genious Ireland (http://ingeniousireland.ie), re- printed with permission RESONANCE ⎜ June 2016 531 GENERAL ARTICLE e =(1, 0, 0, 0),i=(0, 1, 0, 0),j=(0, 0, 1, 0) and k =(0, 0, 0, 1), (3) q = αe + βi+ γj + δk. It is natural to write q = αe + u, where αe is called the ‘real’ or ‘scalar’ part of q.The other part u = βi+γj+δk may be regarded as a vector in three-dimensional space and is called the ‘imaginary’ or ‘vectorial’ part of q. In fact, the word vector first appears in Hamilton’s writings of 1845. Quaternions derive Quaternions derive their power from the fact that they their power from the can be multiplied with each other, so that they form a fact that real algebra (a vector space where vectors can be mul- they can be multiplied tiplied to get other vectors). Since any quaternion is a linear combination of the basis quaternions e, i, j and k, with each other, so it suffices to specify their products. First, e is taken to that they form a real be the multiplicative identity (sometimes denoted 1), algebra (a vector so eq = qe = q for any quaternion q. In addition, space where vectors i2 = j2 = k2 = −e, while can be multiplied to get other ij = −ji = k, jk = −kj = i and ki = −ik = j. (4) vectors). Notice the analogy with cross products of unit vectors ˆi × jˆ = kˆ in 3D Euclidean space. The formula for the product of two quaternions q = (α, u = βi + γj + δk)andq =(α, u = βi + γj + δk) can be computed using the quaternion multiplication table (Table 1) and written compactly in terms of the dot and cross products of their vectorial parts: e i j k e e i j k i i −e k −j j j −k −e i Table 1. Hamilton’s quater- k k j −i −e nion relations. 532 RESONANCE ⎜ June 2016 GENERAL ARTICLE qq =(αα + ββ i2 + γγ j2 + δδ k2, Two quaternions (βα + αβ )i +(γα + αγ )j +(δα + αδ )k commute if and + βγ (ij)+βδ (ik)+γβ (ji) only if their vector + γδ (jk)+δβ (ki)+δγ (kj)) parts are collinear. =(αα − u · u,αu + αu +(βγ − γβ)k +(δβ − βδ)j +(γδ − δγ)i) =(αα − u · u,αu + αu + u × u) . (5) Two quaternions commute qq = qq iff their vector parts are collinear: u × u = 0. Moreover, the two- sided multiplicative inverse (reciprocal) of q is q−1 = (α2 + u2)−1(α, −u). Quaternion multiplication allows us to define polynomials in a quaternion variable. Inter- estingly, a polynomial of the nth degree may have more than n quaternion roots, see Box 1 for an example.