Patrick r girard quaternions, clifford algebras and relativistic physics birkhäuser (2007)

Patrick R. Girard

Quaternions,
Clifford Algebras and
Relativistic Physics

Birkhäuser
Basel . Boston . Berlin


Author:
Patrick R. Girard
INSA de Lyon
Département Premier Cycle
20, avenue Albert Einstein
F-69621 Villeurbanne Cedex
France
e-mail:

Igor Ya. Subbotin
Department of Mathematics and Natural Sciences
National University
Los Angeles Campus
3DFL¿F&RQFRXUVH'ULYH
Los Angeles, CA 90045
USA
e-mail:

2000 Mathematical Subject Classification: 15A66, 20G20, 30G35, 35Q75, 78A25, 83A05, 83C05, 83C10

Library of Congress Control Number: 2006939566
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at <>.

ISBN 978-3-7643-7790-8 Birkhäuser Verlag AG, Basel – Boston – Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use
permission of the copyright owner must be obtained.
Originally published in French under the title “Quaternions, algèbre de Clifford et physique relativiste”.
© 2004 Presses polytechniques et universitaires romandes, Lausanne
All rights reserved
© 2007 Birkhäuser Verlag AG, P.O. Box 133, CH-4010 Basel, Switzerland
Part of Springer Science+Business Media
Printed on acid-free paper produced from chlorine-free pulp. TCF f
f
Printed in Germany
ISBN-10: 3-7643-7790-8
e-ISBN-10: 3-7643-7791-7
ISBN-13: 978-3-7643-7790-8
e-ISBN-13: 978-3-7643-7791-5
987654321

www.birkhauser.ch


To Isabelle, my wife, and to our children: Claire, B´eatrice, Thomas and Benoˆıt


Foreword

The use of Clifford algebra in mathematical physics and engineering has grown
rapidly in recent years. Clifford had shown in 1878 the equivalence of two approaches to Clifford algebras: a geometrical one based on the work of Grassmann
and an algebraic one using tensor products of quaternion algebras H. Recent developments have favored the geometric approach (geometric algebra) leading to an
algebra (space-time algebra) complexified by the algebra H ⊗ H presented below
and thus distinct from it. The book proposes to use the algebraic approach and
to define the Clifford algebra intrinsically, independently of any particular matrix
representation, as a tensor product of quaternion algebras or as a subalgebra of
such a product. The quaternion group thus appears as a fundamental structure of
physics.
One of the main objectives of the book is to provide a pedagogical introduction to this new calculus, starting from the quaternion group, with applications to
physics. The volume is intended for professors, researchers and students in physics
and engineering, interested in the use of this new quaternionic Clifford calculus.
The book presents the main concepts in the domain of, in particular, the
quaternion algebra H, complex quaternions H(C), the Clifford algebra H ⊗ H
real and complex, the multivector calculus and the symmetry groups: SO(3),
the Lorentz group, the unitary group SU(4) and the symplectic unitary group
USp(2, H). Among the applications in physics, we examine in particular, special
relativity, classical electromagnetism and general relativity.
I want to thank G. Casanova for having confirmed the validity of the interior
and exterior products used in this book, F. Sommen for a confirmation of the
Clifford theorem and A. Solomon for having attracted my attention, many years
ago, to the quaternion formulation of the symplectic unitary group.
Further thanks go to Professor Bernard Balland for reading the manuscript,
the Docinsa library, the computer center and my colleagues: M.-P. Noutary for
advice concerning Mathematica, G. Travin and A. Valentin for their help in Latex.
For having initiated the project of this book in a conversation, I want to
thank the Presses Polytechniques et Universitaires Romandes, in particular, P.-F.
Pittet and O. Babel.

viii

Foreword

Finally, for the publication of the english translation, I want to thank Thomas
Hemping at Birkhă
auser.
Lyon, June 2006
Patrick R. Girard


Contents

Introduction

1

1 Quaternions
1.1 Group structure . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Finite groups of order n ≤ 8 . . . . . . . . . . . . . . . . .
1.3 Quaternion group . . . . . . . . . . . . . . . . . . . . . . .
1.4 Quaternion algebra H . . . . . . . . . . . . . . . . . . . .
1.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Polar form . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Square root and nth root . . . . . . . . . . . . . .
1.4.4 Other functions and representations of quaternions
1.5 Classical vector calculus . . . . . . . . . . . . . . . . . . .
1.5.1 Scalar product and vector product . . . . . . . . .
1.5.2 Triple scalar and double vector products . . . . . .

1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3
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11
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14
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16
17

2 Rotation groups SO(4) and SO(3)
2.1 Orthogonal groups O(4) and SO(4) . . . . . . .
2.2 Orthogonal groups O(3) and SO(3) . . . . . . .
2.3 Crystallographic groups . . . . . . . . . . . . .
2.3.1 Double cyclic groups Cn (order N = 2n)
2.3.2 Double dihedral groups Dn (N = 4n) .
2.3.3 Double tetrahedral group (N = 24) . . .
2.3.4 Double octahedral group (N = 48) . . .
2.3.5 Double icosahedral group (N = 120) . .

2.4 Infinitesimal transformations of SO(4) . . . . .
2.5 Symmetries and invariants: Kepler’s problem .
2.6 Exercises . . . . . . . . . . . . . . . . . . . . .

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19
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27
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32
34

3 Complex quaternions
3.1 Algebra of complex quaternions H(C) . . . . . . . . . . . . . . . .
3.2 Lorentz groups O(1, 3) and SO(1, 3) . . . . . . . . . . . . . . . . .
3.2.1 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37
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38
38

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x

Contents
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38
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41

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47
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48
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50
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54

4 Clifford algebra
4.1 Clifford algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Clifford algebra H ⊗ H over R . . . . . . . . . . . . . . . . .
4.2 Multivector calculus within H ⊗ H . . . . . . . . . . . . . . . . . .
4.2.1 Exterior and interior products with a vector . . . . . . . . .
4.2.2 Products of two multivectors . . . . . . . . . . . . . . . . .
4.2.3 General formulas . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Classical vector calculus . . . . . . . . . . . . . . . . . . . .
4.3 Multivector geometry . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Analytic geometry . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Orthogonal projections . . . . . . . . . . . . . . . . . . . . .
4.4 Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Infinitesimal elements of curves, surfaces and hypersurfaces
4.4.3 General theorems . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

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59
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61
62
64
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64
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69
69
69
71
72

5 Symmetry groups
5.1 Pseudo-orthogonal groups O(1, 3) and SO(1, 3) . . . .
5.1.1 Metric . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Symmetry with respect to a hyperplane . . . .
5.1.3 Pseudo-orthogonal groups O(1, 3) and SO(1, 3)
5.2 Proper orthochronous Lorentz group . . . . . . . . . .
5.2.1 Rotation group SO(3) . . . . . . . . . . . . . .
5.2.2 Pure Lorentz transformation . . . . . . . . . .
5.2.3 General Lorentz transformation . . . . . . . . .
5.3 Group of conformal transformations . . . . . . . . . .
5.3.1 Definitions . . . . . . . . . . . . . . . . . . . .
5.3.2 Properties of conformal transformations . . . .

75
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75
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78
79
81
82
82
83

3.3

3.4
3.5

3.6
3.7
3.8

3.2.2 Plane symmetry . . . . . . . . . . . . .
3.2.3 Groups O(1, 3) and SO(1, 3) . . . . . . .
Orthochronous, proper Lorentz group . . . . .
3.3.1 Properties . . . . . . . . . . . . . . . . .
3.3.2 Infinitesimal transformations of SO(1, 3)
Four-vectors and multivectors in H(C) . . . . .
Relativistic kinematics via H(C) . . . . . . . .
3.5.1 Special Lorentz transformation . . . . .

3.5.2 General pure Lorentz transformation . .
3.5.3 Composition of velocities . . . . . . . .
Maxwell’s equations . . . . . . . . . . . . . . .
Group of conformal transformations . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . .

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Contents

xi
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groups
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91
. 91
. 91
. 92
. 94
. 94
. 94
. 97
. 99
. 99
. 100
. 103

7 Classical electromagnetism
7.1 Electromagnetic quantities . . . . . . . . . . . . . .
7.1.1 Four-current density and four-potential . .
7.1.2 Electromagnetic field bivector . . . . . . . .
7.2 Maxwell’s equations . . . . . . . . . . . . . . . . .
7.2.1 Differential formulation . . . . . . . . . . .
7.2.2 Integral formulation . . . . . . . . . . . . .
7.2.3 Lorentz force . . . . . . . . . . . . . . . . .
7.3 Electromagnetic waves . . . . . . . . . . . . . . . .
7.3.1 Electromagnetic waves in vacuum . . . . . .
7.3.2 Electromagnetic waves in a conductor . . .

7.3.3 Electromagnetic waves in a perfect medium
7.4 Relativistic optics . . . . . . . . . . . . . . . . . . .
7.4.1 Fizeau experiment (1851) . . . . . . . . . .
7.4.2 Doppler effect . . . . . . . . . . . . . . . . .
7.4.3 Aberration of distant stars . . . . . . . . .
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . .

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105
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123
124
125

8 General relativity
8.1 Riemannian space . . . . . .

8.2 Einstein’s equations . . . . .
8.3 Equation of motion . . . . . .
8.4 Applications . . . . . . . . . .
8.4.1 Schwarzschild metric .
8.4.2 Linear approximation

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127
127
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129
130
130
133

5.4

5.5

5.3.3 Transformation of multivectors
Dirac algebra . . . . . . . . . . . . . .
5.4.1 Dirac equation . . . . . . . . .
5.4.2 Unitary and symplectic unitary
Exercises . . . . . . . . . . . . . . . .

6 Special relativity
6.1 Lorentz transformation . . . . . . . . .
6.1.1 Special Lorentz transformation

6.1.2 Physical consequences . . . . .
6.1.3 General Lorentz transformation
6.2 Relativistic kinematics . . . . . . . . .
6.2.1 Four-vectors . . . . . . . . . . .
6.2.2 Addition of velocities . . . . . .
6.3 Relativistic dynamics of a point mass .
6.3.1 Four-momentum . . . . . . . .
6.3.2 Four-force . . . . . . . . . . . .
6.4 Exercises . . . . . . . . . . . . . . . .

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84
85
85
86
88


xii

Contents

Conclusion

135

A Solutions

137

B Formulary: multivector products within H(C)

153

C Formulary: multivector products within H ⊗ H (over R)

157

D Formulary: four-nabla operator ∇ within H ⊗ H (over R)

161

E Work-sheet: H(C) (Mathematica)

163

F Work-sheet H ⊗ H over R (Mathematica)

165

G Work-sheet: matrices M2 (H) (Mathematica)

167

H Clifford algebras: isomorphisms

169

I

171

Clifford algebras: synoptic table

Bibliography

173

Index

177


Introduction
If one examines the mathematical tools used in physics, one finds essentially three
calculi: the classical vector calculus, the tensor calculus and the spinor calculus.
The three-dimensional vector calculus is used in nonrelativistic physics and also
in classical electromagnetism which is a relativistic theory. This calculus, however,
cannot describe the unity of the electromagnetic field and its relativistic features.
As an example, a phenomenon as simple as the creation of a magnetic induction by a wire with a current is in fact a purely relativistic effect. A satisfactory
treatment of classical electromagnetism, special relativity and general relativity
is given by the tensor calculus. Yet, the tensor calculus does not allow a double
representation of the Lorentz group and thus seems incompatible with relativistic
quantum mechanics. A third calculus is then introduced, the spinor calculus, to
formulate relativistic quantum mechanics. The set of mathematical tools used in
physics thus appears as a succession of more or less coherent formalisms. Is it
possible to introduce more coherence and unity in this set? The answer seems to
reside in the use of Clifford algebra. One of the major benefits of Clifford algebras is that they yield a simple representation of the main covariance groups of
physics: the rotation group SO(3), the Lorentz group, the unitary and symplectic
unitary groups. Concerning SO(3), this is well known, since the quaternion algebra
H which is a Clifford algebra (with two generators) allows an excellent representation of that group . The Clifford algebra H ⊗ H, the elements of which are simply

quaternions having quaternions as coefficients, allows us to do the same for the
Lorentz group. One shall notice that H ⊗ H is defined intrinsically independently
of any particular matrix representation. By taking H ⊗ H (over C), one obtains the
Dirac algebra and a simple representation of SU(4) and USp(2, H). Computations
within these algebras have become straightforward with software like Mathematica which allows us to perform extended algebraic computations and to simplify
them. One will find as appendices, worksheets which allow easy programming of
the algebraic (or numerical) calculi presented here. One of the main objectives of
this book is to show the interest in the use of Clifford algebra H ⊗ H in relativistic
physics with applications such as classical electromagnetism, special relativity and
general relativity.


Chapter 1

Quaternions
The abstract quaternion group, discovered by William Rowan Hamilton in 1843, is
an illustration of group structure. After having defined this fundamental concept
of physics, the chapter examines as examples the finite groups of order n ≤ 8 and
in particular, the quaternion group. Then the quaternion algebra and the classical
vector calculus are treated as an application.

1.1 Group structure
A set G of elements is a group if there exists an internal composition law ∗ defined
for all elements and satisfying the following properties:
1. the law is associative
(a ∗ b) ∗ c = a ∗ (b ∗ c),

∀a, b, c ∈ G,

2. the law admits an identity element e

a ∗ e = e ∗ a = a,

∀a ∈ G,

3. any element a of G has an inverse a
a ∗ a = a ∗ a = e.
Let F and G be two groups. A composition law on F × G is defined by
(f1 , g1 )(f2 , g2 ) = (f1 f2 , g1 g2 ),

(fi ∈ F, gi ∈ G, i = 1, 2);

the group F × G is called the direct product of the groups F and G.
Examples.

1. Cyclic group Cn of order n the elements of which are
(b, b2 , b3 , . . . , bn = e)

and where b represents, for example, a rotation of 2π/n around an axis.


4

Chapter 1. Quaternions
2. Dihedral group Dn of order 2n generated by two elements a and b such that
a2 = bn = (ab)2 = e.
One has in particular b−h a = abh (h = 1 · · · n); indeed, since
(ab)−1 = b−1 a−1 = b−1 a = ab,
one has

b−1 (b−1 a)b = b−2 ab = b−1 ab2 = ab3

and thus b−2 a = ab2 ; by proceeding similarly by recurrence, one establishes
the above formula.

1.2 Finite groups of order n ≤ 8
The finite groups of order n ≤ 8, except the quaternion group which will be treated
separately, are the following.
1. n = 1, there exists only one group (1 = e).
2. n = 2, only one group exists, the cyclic group C2 consisting of the elements
(b, b2 = e).
Examples.

(a) the group constituted by the elements (−1, 1);

(b) the group having the elements (b : rotation of ±π around an axis,
b2 = e).
3. n = 3, only one group is possible: the cyclic group C3 of elements (b, b2 ,
b3 = e) where b, b2 are elements of order 3.
4. n = 4, two groups exist:
(a) the cyclic group C4 constituted by the elements (b, b2 , b3 , b4 = e) where
the element b2 is of order 2, and where (b, b3 ) are elements of order 4;
(b) the Klein four-group defined by
I 2 = J 2 = (IJ)2 = 1
or, equivalently I 2 = J 2 = K 2 = IJK = 1 with K = IJ and the
multiplication table

1
I
J
K

1
1
I
J
K

I
I
1
K
J

J
J
K
1
I

K
K
J
I
1

.


1.2. Finite groups of order n ≤ 8

5

The Klein four-group is isomorphic to the direct product of two cyclic
groups C2 ,
(−1, 1) × (b, b2 = e)
= 1 ≡ (1, b2 ), I ≡ (1, b), J ≡ (−1, b), K ≡ (−1, b2 ) .
Example. The group constituted by the elements (I: rotation of π
around the axis Ox, J: rotation of π around the axis Oy, K = IJ:
rotation of π around the axis Oz).
5. n = 5, there exists only one group, the cyclic group C5 having the elements
(b, b2 , b3 , b4 , b5 = e).
6. n = 6, two groups are possible:
(a) the cyclic group C6 (b, b2 , b3 , b4 , b5 , b6 = e);
(b) the dihedral group D3 defined by the relations
a2 = b3 = (ab)2 = e,
leading to the multiplication table

b
b2
b3 = e
a
ab
ba

b
b2
e
b
ab
ba

b

b2
e
b
b2
ba
a
ab

b3 = e
b
b2
e
a
ab
ba

a
ba
ab
a
e
b2
b

ab
a
ba
ab

b
e
b2

ba
ab
a
ba
b2
b
e

with b−h a = abh (h = 1, 2, 3). This group is the first noncommutative
group of the series.
Example. The symmetry group of the equilateral triangle (see Fig. 1.1).
7. n = 7, there exists only one group, the cyclic group C7 of elements (b, b2 , b3 ,
b4 , b5 , b6 , b7 = e).
8. n = 8, there exist five groups, among them the quaternion group which will
be treated separately.
(a) The cyclic group C8 of elements (b, b2 , b3 , b4 , b5 , b6 , b7 , b8 = e).


6

Chapter 1. Quaternions

B
b(M )

y

a(M )

D
x

Oz

ba(M )

C

M

b2 (M )

ab(M )

A

Figure 1.1: Symmetry group of the equilateral triangle; b represents a rotation of
the point M by 2π/3 around the axis Oz, a a symmetry of M in the plane ABC
with respect to the mediatrice CD and M an arbitrary point of the triangle.
(b) The group S2×2×2 , direct product of the Klein four-group with C2 ,
(1, I, J, K) × (1, −1) = (±1, ±I, ±J, ±K)
with 1 = (1, 1), −1 = (1, −1), ±I = (I, ±1), ±J = (J, ±1), ±K =
(K, ±1) ; the multiplication table is given by
1
−1
I

−I
J
−J
K
−K

1
1
−1
I
−I
J
−J
K
−K

−1
−1
1
−I
I
−J
J
−K
K

I
I
−I
1

−1
K
−K
J
−J

−I
−I
I
−1
1
−K
K
−J
J

J
J
−J
K
−K
1
−1
I
−I

−J
−J
J
−K

K
−1
1
−I
I

K
K
−K
J
−J
I
−I
1
−1

−K
−K
K
−J
;
J
−I
I
−1
1

the group is commutative.
(c) the group S4×2 , direct product of C4 with C2 and constituted by the
elements

(b, b2 , b3 , b4 = e) × (1, −1) = (±b, ±b2 , ±b3 , ±1);
it is a commutative group.


1.3. Quaternion group

7

(d) The group D4 (noncommutative) defined by
a2 = b4 = (ab)2 = e
with the multiplication table

b
b2
b3
b4 = e
a
ab
ba
ab2

b
b2
b3
e
b
ab
ab2
a
ab

b2
b3
e
b
b2
ab2
ba
ab
a

b3
e
b
b2
b3
ba
a
b2
ab

b4 = e
b
b2
b3
e
a
ab
ba
ab2

a
ba
ab2
ab
a
e
b3
b
b2

ab
a
ba
ab2
ab
b
e
b2
b3

ba
ab2
ab
a
ba
b3
b2
e
b

ab2
ab
a
ba
ab2
b2
b
b3
e

and b−h a = abh (h = 1, 2, 3, 4).
Example. The symmetry group of the square (see Fig. 1.2).
C

y

ba(M )

b(M )

b2 (M )

a(M )

Oz

x

ab2 (M )

D

B

M

b3 (M )

ab(M )

A

Figure 1.2: Symmetry group of the square; b is a rotation of π/2 around the axis
Oz, a a symmetry with respect to the axis Ox in the plane ABCD and M an
arbitrary point of the square.

1.3 Quaternion group
The quaternion group (denoted Q) was discovered by William Rowan Hamilton
in 1843 and is constituted by the eight elements ±1, ±i, ±j, ±k satisfying the


8

Chapter 1. Quaternions

relations
i2 = j 2 = k 2 = ijk = −1,
ij = −ji = k,
jk = −kj = i,
ki = −ik = j,

with the multiplication table

1
−1
i
−i
j
−j
k
−k

1
1
−1
i
−i
j
−j
k
−k

−1
−1
1
−i
i
−j
j
−k
k

i
i
−i
−1
1
−k
k
j
−j

−i
−i
i
1
−1
k
−k
−j
j

j
j
−j
k
−k
−1
1
−i
i

−j
−j
j
−k
k
1
−1
i
−i

k
k
−k
−j
j
i
−i
−1
1

−k
−k
k
j
−j
−i
i
1
−1

the element of the first column being the first element to be multiplied and 1 being
the identity element. The element −1 is of order 2 (i.e., its square is equal to 1)
and the elements (±i, ±j, ±k) are of order 4. The subgroups of Q are
(1)
(1, −1)
(1, −1, i, −i)
(1, −1, j, −j)
(1, −1, k, −k).

1.4 Quaternion algebra H
1.4.1 Definitions
Consider the vector space of numbers called quaternions a, b, . . . constituted by
four real numbers
a = a0 + a1 i + a2 j + a3 k
= (a0 , a1 , a2 , a3 )
= (a0 , a) = (a0 , a)
where S(a) = a0 is the scalar part and V (a) = a = a the vectorial part. This


1.4. Quaternion algebra H

9

vector space is transformed into the associative algebra of quaternions (denoted
H) via the multiplication
ab = (a0 b0 − a1 b1 − a2 b2 − a3 b3 )
+ (a0 b1 + a1 b0 + a2 b3 − a3 b2 )i
+ (a0 b2 + a2 b0 + a3 b1 − a1 b3 )j
+ (a0 b3 + a3 b0 + a1 b2 − a2 b1 )k

and in a more condensed form
ab = (a0 b0 − a · b, a0 b + b0 a + a × b)
where a · b = (a1 b1 + a2 b2 + a3 b3 ) and a × b = (a2 b3 − a3 b2 )i + (a3 b1 − a1 b3 )j +
(a1 b2 − a2 b1 )k are respectively the usual scalar and vector products. Historically,
these two products were obtained by W. J. Gibbs [17] by taking a0 = b0 = 0 and
by separating the quaternion product in two parts.
The quaternion algebra constitutes a noncommutative field (without divisors
of zero) containing R and C as particular cases. Let a = a0 + a1 i + a2 j + a3 k be a
quaternion, the conjugate of a, the square of its norm and its norm are respectively
ac = a0 − a1 i − a2 j − a3 k,
2

|a| = aac = a20 + a21 + a22 + a23 ,
|a| =

|a|2 ,

with the following properties
(ab)c = bc ac ,
|ab|2 = |a|2 |b|2 ,
the last relation deriving from (ab)c ab = bc ac ab = (aac )(bbc ) ; furthermore,
S(ab) = S(ba)
thus S [a (bc)] = S [(bc) a] = S [b (ca)] = S [(ca) b] and therefore
S (abc) = S (bca) = S (cab) ,

a−1

2

(a1 a2 · · · an )−1 =

=

a−1 =

ac
,
aac

ac
aac

a
aac

(a1 a2 · · · an )c
|a1 a2 · · · an |2

=

1

2,

|a|

−1
−1
= a−1
n an−1 · · · a1 .

10

Chapter 1. Quaternions

To divide a quaternion a by the quaternion b (= 0), one simply has to resolve the
equation
xb = a
or
by = a
with the respective solutions
x = ab−1 = a
y = b−1 a =
and the relation |x| = |y| =

bc
|b|2

,

bc a
2

|b|

|a|
|b| .

Examples. Consider the quaternions a = 2 + 4i − 3j + k and b = 5 − 2i + j − 3k;

1. the vectorial parts a, b and the conjugates ac , bc are
a = 4i − 3j + k,
ac = 2 − 4i + 3j − k,
2. the norms are given by

b = −2i + j − 3k,
bc = 5 + 2i − j + 3k;
√
√
aac = 30,
√
bbc = 39;

|a| =
|b| =
3. the inverses are
a−1

=

b−1

=

ac
2

|a|
bc

2

|b|

=

2 − 4i + 3j − k
,
30

=

5 + 2i − j + 3k
;
39

4. one can realize the following operations
a+b
a−b

= 7 + 2i − 2j − 2k,
= −3 + 6i − 4j + 4k,

ab

= 24 + 24i − 3j − 3k,

ba

= 24 + 8i − 23j + k,

(ab)−1 =

(ab)c
|ab|

S(x)

= b−1 a−1 =

4
4i
j
k
−
+
+
,
195 195 390 390

8i
9j
13k
−2
−
+
−
),
15
15 10

30
3k
−2 16i 7j
−
+
−
),
= b,
y = a−1 b = (
15
15
30 10
13
= S(y),
|x| = |y| =
.
10

xa = b,
ay

2

√
1 170,
√
|ba| = 1 170,

|ab| =

x = ba−1 = (


1.4. Quaternion algebra H

11

1.4.2 Polar form
Any nonzero quaternion can be written
a = a0 + a1 i + a2 j + a3 k
0 ≤ θ ≤ 2π

= r(cos θ + u sin θ),
with r = |a| =

a20 + a21 + a22 + a23 being the norm of a and
a0
,
r
a0
cot θ = ± ,
|a|

cos θ =

±

a21 + a22 + a23
,
r

|a|
tan θ = ± ;
a0
sin θ =

the unit vector u (uuc = 1) is given by
u=

±(a1 i + a2 j + a3 k)
a21 + a22 + a23

with a21 + a22 + a23 = 0. Since u2 = −1, one has via the De Moivre theorem
an = rn (cos nθ + u sin nθ).
Example. Consider the quaternion a; let us determine its polar form
a

=

|a| =
tan θ

=

Answer: a

=

3 + i + j + k,
√
√

√
12 = 2 3,
|a| = 3,
|a|
1
θ = 30◦ ,
= √ ,
a0
3
√
i+j+k
√
2 3 cos 30◦ +
sin 30◦ .
3

1.4.3 Square root and nth root
Square root
The square root of a quaternion a = a0 + a1 i + a2 j + a3 k can be obtained algebraically as follows. The equation b2 = a with b = b0 + b1 i + b2 j + b3 k leads to the
following equations
b20 − b21 − b22 − b23 = a0

(1.1)

2b0 b1 = a1 ,
2b0 b2 = a2 ,

sgn(b0 b1 ) = sgn(a1 ),
sgn(b0 b2 ) = sgn(a2 ),

(1.2)
(1.3)

2b0 b3 = a3 ,

sgn(b0 b3 ) = sgn(a3 ).

(1.4)


12

Chapter 1. Quaternions

Writing t = b20 the above equation (1.1) leads to
t−

a21 + a22 + a23
= a0
4t

or
t2 − a 0 t −

a21 + a22 + a23
= 0.
4

One obtains
a20 + a21 + a22 + a23

≥0
2
a0 − a20 + a21 + a22 + a23
−(b21 + b22 + b23 ) = a0 − b20 =
,
2
t = b20 =

a0 +

hence
ε
b0 = √
2

(ε = ±1).

a20 + a21 + a22 + a23 + a0

The equations (1.2), (1.3), (1.4) lead to
−a21
,
4
−(a22 + a23 )
b20 (−b22 − b23 ) =
,
4
b20 (−b21 ) =

(1.5)

(1.6)

with
−b22 − b23 = b21 + a0 − b20
= b21 +

a0 −

a20 + a21 + a22 + a23
.
2

Equation (1.6) then becomes with t = b21 and using (1.5)
a21
4t

t+

a0 −

a20 + a21 + a22 + a23
2

=−

a22 + a23
4

thus
t=

a21
2

a20 + a21 + a22 + a23 − a0
a21 + a22 + a23

,


1.4. Quaternion algebra H

13

hence b1 (one proceeds similarly for b2 , b3 ); finally, one obtains
ε
b0 = √
2

a20 + a21 + a22 + a23 + a0
a1

ε
b1 = √
2

a21

ε
b2 = √

2

a21 + a22 + a23

ε
b2 = √
2

a21 + a22 + a23

+ a22 + a23
a2
a3

(ε = ±1),

a20 + a21 + a22 + a23 − a0 ,
a20 + a21 + a22 + a23 − a0 ,
a20 + a21 + a22 + a23 − a0 .

Example. Consider the quaternion a = 1 + i + j + k ; find its square root b
b0

=

Answer: b =

ε √
ε
√ 3,

b1 = b2 = b3 = √ ,
2
6
j
k
1 √
i
±√
3+ √ + √ + √
.
2
3
3
3

nth roots
The nth root of a quaternion a = r(cos ϕ + u sin ϕ), where ϕ can always be chosen
within the interval [0, π] with an appropriate choice of u, is obtained as follows [9].
1. Supposing sin ϕ = 0, the equation bn = a with b = R(cos θ+e sin θ), θ ∈ [0, π],
leads to
Rn = r, cos nθ = cos ϕ, sin nθ = sin ϕ, e = u
and thus to
1

R = rn,

θ=

(ϕ + 2kπ)
n

(k = 0, 1, . . . , n − 1);

finally, one has
1

b = r n cos

(ϕ + 2kπ)
(ϕ + 2kπ)
+ u sin
n
n

(k = 0, 1, . . . , n − 1).

2. When sin ϕ = 0, the vector e in b is arbitrary. If a > 0, one has ϕ = 0 and
thus θ = 2πm
n (m
√= 0, 1, . . . , n − 1). For n = 2, one obtains θ = 0, π and thus
the real roots ± a. With n > 2, certain values of θ (= 0 or π) give nonreal
roots, the vector e being arbitrary. With a < 0, one has ϕ = π, θ = (2m+1)π
n
(m = 0, 1, . . . , n − 1), certain values of θ = π give nonreal roots b , the vector
e being arbitrary.


14

Chapter 1. Quaternions

Example. Find the cubic root of
a

= 3+i+j+k
√
(i + j + k)
√
sin 30◦ ;
= 2 3 cos 30◦ +
3

Answer: b =
θ

=

√
2 3

1
3

cos θ +

(i + j + k)
√
sin θ ,
3

10◦ , 130◦ , 250◦.

1.4.4 Other functions and representations of quaternions
The exponential ea is defined by [30]
a
a2
a3
+
+
+ ···
1!
2!
3!
where a is an arbitrary quaternion. Furthermore, one defines
ea = 1 +

a2
a4
a2p
ea + e−a
=1+
+
+ ···+
,
2
2!
4!
(2p)!
ea − e−a
a

a3
a5
a2p+1
sinh a =
=
+
+
+ ···+
,
2
1!
3!
5!
(2p + 1))!

cosh a =

and thus ea = cosh a + sinh a. For an arbitrary quaternion a, let U (V (a)) = u
(u2 = −1) be a unit vector, and therefore one has ua = au ; consequently, one
can define the trigonometric functions
a2
a4
eua + e−ua
=1−
+
+ ··· ,
2
2!
4!
eua − e−ua

a
a3
a5
sin a =
=
−
+
+ ··· .
2
1!
3!
5!
Example. Let a = uθ be a quaternion without a scalar part with u ∈ Vec H,
u2 = −1 and θ real; one has
cos a =

cosh uθ
sinh uθ

=
=

cos θ,
u sin θ,

euθ

=

cos θ + u sin θ,

cos uθ
sin uθ

=
=

cosh θ,
u sinh θ.

In particular, if a = iθ,
cosh iθ = cos θ,

sinh iθ = i sin θ,

cos iθ = cosh θ,

sin iθ = i sinh θ.


1.5. Classical vector calculus

15

One can represent a quaternion a = a0 + a1 i + a2 j + a3 k by a 2 × 2 complex
matrix (with i being the usual complex imaginary)
a0 + i a3
−i a1 − a2

A=

−i a1 + a2
a0 − i a3

or by a 4 × 4 real matrix
⎡

a0
⎢ a1
A=⎢
⎣ a2
a3

−a1
a0
a3
−a2

−a2
−a3
a0
a1

⎤
−a3
a2 ⎥
⎥.
−a1 ⎦
a0

The differential of a product of quaternions is given by
d(ab) = (da)b + a(db),

a, b ∈ H,

the order of the factors having to be respected.

1.5 Classical vector calculus
1.5.1 Scalar product and vector product
Let a, b, c ∈ Vec H, be three quaternions without a scalar part, a = a1 i + a2 j + a3 k,
b = b1 i + b2 j + b3 k, c = c1 i + c2 j + c3 k. The norm of a is
|a| =

√
aac =

a21 + a22 + a23

and
ab + ba ab − ba
+
2
2
= (−a · b, a × b).

ab =

One defines respectively the scalar product and the vector product of two vectors
a, b by
(ab + ba)

= a1 b 1 + a2 b 2 + a3 b 3 ,
2
(ab − ba)
a×b≡a×b=
2
= (a2 b3 − a3 b2 )i + (a3 b1 − a1 b3 )j + (a1 b2 − a2 b1 )k,

(a, b) ≡ a · b = −

with ab = −(a, b) + a × b. Geometrically, one has
(a, b) = |a| |b| cos α,
|a × b| = |a| |b| sin α,


16

Chapter 1. Quaternions

α being the angle between the two vectors a and b. Furthermore,
2

2

2

|ab| = |a| |b| = (ab)(ab)c
= [−(a, b) + a × b] [−(a, b) − a × b]
= (a, b)2 − (a × b)2
2

= (a, b)2 + |a × b|

which is coherent with the above geometrical expressions. One has the properties
(a, b) = (b, a),
(λa, b) = λ(a, b),

λ ∈ R,

(a, b + c) = (a, b) + (a, c),
a × b = −b × a,
a × λb = λ(a × b),
a × (b + c) = a × b + a × c.

1.5.2 Triple scalar and double vector products
The triple scalar product is defined by
[a(b × c) + (b × c)a]
2
[a(bc − cb) + (bc − cb)a]
=−
2

(a, b × c) = −

and satisfies the relations
(a, b × c) = (b, c × a) = (c, a × b),
(a, b × c) = −(c, b × a)
which are established using the relations
S(abc) = S(bca) = S(cab)
and
S(abc) = S(abc)c = −S(cba) = −S(acb),

S(abc) = −S(bac).
In particular
(a × b, a) = (a × b, b) = 0
which shows that a × b is orthogonal to a and b.