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Eur. J . Phys.

S

(1984)

25-32.

Printed n Northern Ireland

25

The quaternion group and modern

P

R Girardt

RCs.

Al be r t ler , BBtiment

A,

54150 Br iey , F rance

Received

9

D e c e m b e r

1983

Abstract

The pa per shows how various physical

covariance groups: S0 3), he Lorentz group, the

genera l theory of relativity group, the Clifford algebra ,

SU(2) and the conformal group can easily be related to

the quaternion group. The quaternion calculus is intro-

duced and several physical applications: crystallography,

the k inematics of rigid body motion , the Thom as

precession, the special theory

of

relativity, classical

electromagnetism, the equation of motion of the

general theory of relativity, and Dirac’s relativistic

wave equation are discussed.

1

Introduction

Severa l au thor s have , in the pages of this journal,

advoca tedhentroduct ion of groupheoryn

physics teaching. Yet , s imple presentat ions together

withphysicalappl icat ions eem obe carce. n

the fol lowing, I shal l show that severalof t h e m a j o r

covar iance groups

of

physics can easi ly be rela ted

t o a f in i te group, namely, he abs tract quaternion

group .Mo st (if no t all)

of

thehys icalnd

mathemat ica l r esu l t s p resen ted in th i s paper can be

found sca t te red in the l i t e ra tu re over

a

t ime inter-

val reaching into the 19th century. After in troduc-

ing theu a t e r n i o nr o u pn dh e u a t e r n i o n

algebra, the paper analyses the connec t ion of vari-

ousmajorcovar iancegroups

of

physicswith he

qua te rn ion g roup toge ther w i th impor tan t phys ica l

applications.

2. The quaternion group and the quaternion

algebra

T he abs t r ac t qua te rn ion g roup (deno ted Q) which

wasiscovered by Wil l iamowanami l ton

( H a n k i n s 1 9 8 0 , C r o w e 1 9 6 7 a , H a m i l t o n 1 9 3 1 - 6 7 ,

Grav es 1882-9) in 184 3, is cons t i tu ted by the eight

e lem ents * l , * i,

* j ,

*k, satisfying the relations

t

Permanent address: Universitv of Wisconsin-Madison.

Dept

of

the History

of

Science; M adison, Wisconsin

53706,

USA

R6som6 L’article montre comment plusieurs groupes

de c ovariance de la physique:

S0 3),

e groupe de

Lorentz, le groupe de la relativitt gtn tral e, I’algkbre de

Clifford,

SU(2)

et le groupe conforme peuvent facile-

ment etre relits au groupe des quaternions. Le calcul

quaternionien est introduit et plusieurs applications

physiques: la cristallograp hie, la cintm atique du mouve-

ment d es corps rigides, la prtcession Thomas, la thto rie

de la relativitt restreinte, l’tquation du m ouvement de

la relativitt gtntrale et l’tquation relativiste de Dirac

sont traittes.

i Z = j Z = k Z = i j k = - l

or , mo re expl ici tly

i 2 = j z = k z = - 1

i j = -ji = k

j k = - k j = i

k i = - i k = j

1

i s the un i t e lement .

The

subgroups of

Q are:

(1)

(1, -1)

(1, -1, i, -i)

(1, -1,

j ,

-j)

(1 , -1, k, -k).

To ob ta in he qua te rn ion a lgebra (deno ted

H ,

cons ider he vec to r space

of

ordered sets of four

numbersreal r omplex) :

a,

,

b,

. . called

quaternions$

a = ao l+

a l i +

a2j+

a,k

= (ao,

al

az,ad

= (ao,

a).

‘t

For

a

comp lete bibliography on works on quatern ions up

to1912,seeMacfarlane 1904)andtheBulletin 1900-13)

0143-0807/84/05OW25+08 2.25

984 The Institute of Physics

Br

the European Physical Society



26

P

R Girard

This vector space is transformed into an associative

quaternion algebra via the multiplication xule

ab = ( aobo l b l - a2b2- a3b3)

+(aobl+albo+a2b3-a3b2)i

+(aob2+a2bo+a3bl-a,b3)j

+

(aob3

+ a3b0

+

alb2-

a2bl)k

which is obtained from the multiplication table

of

Q.

Theabove multiplication rule can easily be

remembered by expressing it in terms of the ordi-

nary dot and vector products

a b = ( a o b o - a ~ b , a o b + b o a + a x b )

if a

x bf

0 , the quaternionproduct is noncommuta-

tive. The presence of the dot and vector products n

thequaternion product is no coincidence since,

historically, W J Gibbs (Crowe 1967b, Bork 1966)

introducedhese two products by splitting the

quaternion product and by taking

a, = bo= 0.

It is

to be noticed that the quaternion algebra is distinct

from the algebra of the quaternion group which is

eight-dimensional. The real quaternion algebra

H ( R ) ,which contains R and C as particular cases,

has a unique place among all associative algebras

over R since it is the only finite-dimensional divi-

sion algebra (Dickson 1914) over

R

esides

R

and

C. I shall use both H(R) and H ( C ) .The latte r is

still an associative algebraut not a division

algebra.

Considering a quaternion q = qo

+

qli

+

q2j

+

q3k,

I shall use the following terminology (Hamilton

1969a):

sq = 40

is the scalar part of q ;

Vq =

qli+ qd q&

is the vector part of q ;

qc=q0-q1i-q2j-q3k

is the quaternion conjugate?

of

q ;

q*=q$+qTi+q:j+qzk

is the complex conjugate of

q

Nq = qqc = qcq

=q;+q:+q;+q:

is the norm

of

q. Among frequently used relations

are:

(48)

=

(SC)* (q*h

4-1 = %/WC

(qlq2)c= 42fq1c

S(q1q2) = S(q2q1)

S(q1q2q3)

=

S(q24391)

=

S(q3q1q2)

and the important relation

t

Hamilton wrote K q instead of qc .

The differential (Hamilton 1969b)

of

a quaternion

product is given by the relation

d(qlq2)

=

( d q h

+

ql(dq2)

where the order of the terms has to be respected.

Finally, its tobe kept in mind that a scalar

commutes with any quaternion.

3 Representations

of a group

By definition, a representation

of

a group G is a

mapping of

G

onto a group T of operators (acting

on a vector space) which preserve the group law,

i.e., are such that

T(glIT(g2)

=

%,g,)

for any

g,

of

G.

The representation is said to be

linear if the operators are linear.

The theory

of

representations of groups allows

the precise definition

of

the concept of a physical

quantity (Kustaanheimo 1955).

To

this effect, con-

sider an abstract group

G

and a representation T

of it acting on a vector space V.

If

T is a transitive

transformation group (Iyanaga and Kawada 1977)

of

V, then the elements

of

V

are

called physical

quantities with respect to G. The group

T

is called

the covariance group. As illustrations, consider the

following examples: if G is the abstract three-

dimensional rotation group and

T

a representation

of

it, the elements

of

V

are called vectors (in the

physical sense); if G is the general linear group, the

elements of V are called tensors; if G is the

Clifford group, the elements of

V

are called

spinors. Once a physical quantity is defined, physi-

cal equations are obtained by combining physical

quantities with respect to the same group and rep-

resentation.

As

an example of a quaternion representation

(Du Val 1964b), consider the transformations T

given by

q’ = aqb

where a,

b,

q, q’ are complex quaternions

(of

norm

f0). These transformations constitute a group, the

product of two transformations T2T1being defined

by

q”

=

azq’b2= a2alqblb2.

The inverse transformation is giveny q

=

a-’q‘b-’. Take for

G

the group H x H of ordered

sets

of

twoquaternions (of norm

f0)

with the

multiplication rule

(az,

2)x

(al,

bl)

=

blb2).

Since

T2(a2,

2)T1(alr

l )

= T(a2a1, lb2)

it follows that

T

is a representation

of

H

xH.

This



The quaternion group

27

representation yields several of theajor

covariance groups of physics as particular cases, as

we shall now see, ogether with various physical

applications of the quaternion formalism.

4. ‘Ihe rotation

group

SO(3)

The three-dimensional rotation group SO(3) is the

group of transformations which leave the quantity

x * + y 2 + z2 invariant. Any such transformation can

be expressed in the quaternion form (Brand 1947)

q’

=

rqr,

where

r

is a quaternion such that

rr =

1, and where

q =xi+yj+zk ,’=x’i+y‘j+z‘k. Reciprocally,

any transformation q’

=

rqr, (with

rr

= 1) is a rota-

tion. The invariance of the norm of q results from

the relation

4 ’ 4

=

(rqrC)(vJc qqc

= x 2 + y 2 + z 2 .

The inverse transformation is given by q = rcq’r;

the composition of two rotations isgivenby the

rule

4’

=

r ~ ( r ~ q r ~ ~ ) r ~ ~(r2rl)dr2rl)c.

Hence, SO(3)onstitutesepresentataion of

H ( R )X H ( R ) .Since, generally, r2rl rlr2, it follows

that the composition

of

three-rotations is generally

noncommutative. If

r

is written in the form

r =

(cos B , sin )

where

e

is a unit vector (i.e.

Ne

=

l) , then the

transformation q ’= rqr, corresponds to a conical

rotation

of

Vq y 28around he vector

e.

The

scalar part of q remains invariant under a rotation

since

Sq’= Sr,rq

=

Sq.

As physical applications of

S0(3),

we shall consider

crystallography and the kinematics of rigid body

motion.

4.

.

Crystallography

If

r

takes successively all the values of a finite

subgroup

of

real quaternions, the transformations

q ’= rqr, will constitute a subgroup of SO(3)

( r

and

- r leading to he same rotation) . The finite sub-

groups

of

real quaternions (Stringham 1881, Shaw

1907a) are of five types:

(i) cyclic groups (order of the group: N = r )

m4”Ir= (cos 2.rr/r, sin 27r/re)”

= (cos 2.rrn/r, sin 2.rrnlre)

with

n = 1 . . r

and where

e

is

any unit vector;

(ii) double-dihedral groups

( N

=

4 r )

4n/r h

P

with S(pm)

=

0, p2

=

-1, and n

= 1 . r, h = 1 . .

4 ;

(iii) double-te trahedral group

(N =

24)

*l, *i, *j, *k

f(*l*i*j*k);

(iv) double-octahedral group

( N =

48)

*l , hi , *j, *k

i(*l*i*j*k),2-1’2(*l*i)

2-”’(*1* j), 2-”’(*l

*

)

2-”’(fi* j) , 2-’I2(*j

*

k)

2-1’2(*k*i);

(v)

double-icosahedral groupt

(N =

120)

*k2n/5, *jk2n/5

k2””(i ~ k ) k ~ ’ ’ ~

*

1

w 2 )

1 02 1/2

k2”/’(i ok)jk2”’

with n,

S = 1 . .5 ,

o = 2 cos 72”=

; -l + 5 l I 2 ) .

These groups, together with the rotation transfor-

mation q ’ = rqr, and the inversion transformation

q‘ = qc, yield all 32 classes of crystals (Shaw 1907b,

1922).

4.2.

Kinematics

of

rigid body motion

The most general motion of a rigid body being a

combination

of

a translation and a rotation, it can

thereforebe represented by the transformation

(Tait 1890a, Laisant 1877)

q’

=

S

+

rqr,

1)

with

rr = 1

and where

S

corresponds to the transla-

tion

( r

and

S

are functions

of

the time

t) .

By taking

the derivative with respect to the ime, one obtains

dq‘ldt

=

[ds/dt

+

(dr/dt)qr,

+

rq(dr,/dt)] r(dq/dt)r,

or

dq’/dt

=

[ds/dt +f(wrqrc- rqr,w)]+ r(dq/d t)r,

= [ds/dt

+

V(orqr,)]+ r(dq/dt)rc

where o = 2(dr/d t)rc is the instantaneous angular

velocity quaternion

6.1

is a vector since So = 0,

because of the relation rr = 1). If q, for example,

corresponds to he coordinates with respect to a

system rigidly attached to the body and q’ to the

coordinates with respect to a fixed inertial system,

then the quantity in square brackets corresponds to

the velocity of a fixed point of the rigid body and

the second term to the relative velocity.

A

second

differentiation of equation 1)yields

t

Compare

with

Klein

(1913).



28 P

R

Girard

d2q‘/dt2= [d2s/dt2+ d2r/dt2)qrc+ rq(d2rJdt2)

+2(dr/dt)q(drC/dt)l

+2[(dr /dt)(dq/dt)rc+ r(dqldt)(dr,/dt)l

+

r(d2q/dt2)rc.

The quantity in the first square bracketscorres-

ponds to he acceleration of a fixed point of the

rigid body; he second ermcorresponds to he

Coriolis acceleration, and the last te rm to the rela-

tive acceleration. tseems difficult to conceive a

simpler derivation of this useful relation.

5.

The Lorentz

group

The Lorentz group is the group of homogeneous

transformations which conserve the quantity cZt2-

x2 -y 2- z2 . If there is

no

time reversal and

n o

space reflection, the transformations are said to be

orthochronous proper. Any orthochronous proper

Lorentz transformation can be xpressed by a

quaternionransformation (Synge 1972b, Cailler

1917) of the type

q’

=

aqar (2)

where a is a complex quaternion such that aa,

=

1,

and where q

=

(ct, ix, iy, iz) , q’= (ct’, ix‘, iy‘, iz ’)t . I

shall refer to quaternions such that qf = q as min-

quats (minkowskian quaternions)$. The above

transformationhas the usual six parametersand

conserves the norm since

4‘4:

=

(aqaf)(asaf)c= qqc.

It also preserves the minquat character of q since

4’: = (aqa$)f = aqa$= q’.

The composition of two Lorentz transformations

follows the rule

q’

=

a2[alq(al)Tl(a2)f = (a2al)q(a2alE

=

a3q(aJf

with

a3=

azal.

Since the transformations 2) do

form a group, and from the considerations above, it

follows that he Lorentz groupconstitutesarep-

resentation of

H ( C )x H ( C ) .

A pureLorentz transformation is a Lorentz

transformation which contains

no

rotation and is

of

the type

q’ = bqbf

t

From now on, and in the rest of this paper, bold face

type will be used for the elements i,

1, k

of the abstract

quaternion group; i in ordinary roman type represents

the complex imaginary

-1)’”.

$ The term is used by Synge (1972a). In contradistinc-

tion to Synge, I shall adopt the form q = (ct, ix, iy, iz)

instead

of

q

=

(ict, x,

y,

z ) ,

in order to have ds2

=

dq dq, >

0.

where

b

is a minquat such that bb, =

1.

A general

Lorentz transformation is obtained by combining a

pureLorentz transformation bqbfwith a three-

dimensional rotation? rqrc. This can be done in two

ways since q’ = b(rqr,)bf is generally different from

q’

=

r(bqbf)r,.

Reciprocally, any general Lorentz transformation

q’

=

aqaf can be decomposed into a pure Lorentz

transformation and a rotation. The problem simply

consists in finding two quaternions b and r such

that a = br (a = r’b’) where b (b’) is a unit minquat

(i.e. a minquat of norm equal to

1)

and

r (r‘)

a real

unit quaternion (rr,

1, r’r;

=

1).

The equation a =

br (and similarly a

=

r’b’) with the above conditions

is solved as follows (Conway 1948, 1953b). Since

a f = rfbf

=

r,b

one has aaf

=

b2, which is an equation of the type

d = b2, whered is a unit minquat.

To

solve this

equation, write

2b2= 2d

b2d

=

d 2

b2d,

=

1

add, and obtain

b2(2

+

d

+

d,)

= 1+

d)’.

The solution is thus given by

b=*(l+d)/ lN(l+d)l ’”

whered

=

aa: in the case here considered, and

where the vertical bars stand for the absolute value.

Finally, one verifies that this is indeed a solution.

The quaternion

r

of the rotation is given by

r =

b,a

= * ( a

+

a*)/\N(l+ aaf)l”2.

The problem

of

the reduction of a Lorentz trans-

formation into a pure Lorentz transformation and a

rotation is thus solved in the most general case. As

an immediate application, consider a Lorentz trans-

formationobtained rom the composition of two

pure Lorentz transformations. Since a =

b l bz

will in

general be a complex quaternion (not a minquat),

one will generally have

a

=

br. Hence, the resulting

Lorentz transformation will generally contain a ro-

tation; this is the essence

of

the Thomas precession

effect (Thomas 1926,1927). As further applica-

tions

of theLorentz group, we shall now briefly

discuss the special theory

of

relativity and classical

electromagnetism.

5.1. The

special theory

of

relativity

Consider the pure Lorentz transformation

q’ = bqbf

t

The previous results concerning rotations are not

affected by the introduction of minquats.




with

b = (cosh 4 i sinh

)

q = (ct, ix, iy, iz)

q’

=

(ct, ix’, iy’, iz’)

and where e is a real unit vector.The four-veclocity

transforms according to the relation

U’

=

bubr

where

U =

dq/ds,

U‘

= dq’/ds, and ds’= dq dq,. If

U =

1,

one has

U ’= bbr = (cosh4, i sinh )’

= (cosh

29,

i sinh

2 ).

29

If, furthermore,

cosh

24

=

(1

U’/C’)-~’’=

y

tanh 24

=

-u/c

then

U’

y,

iyu/ce).

Thus, the above transformation transforms a point

at rest into a point moving with a speed U in the

direction -e; it corresponds to a standard Lorentz

transformation (from a passive point

of view) in the

e direction.

The four-momentum of a particle of rest mass m

is defined by p = mu and satisfies the relation ppc

=

m’ since uu,= I. The four-angular momentum of a

material point can be defined by

L

= V(qp,)

= [0, r

x

p+ (Er/c -c@)].

Under an arbitrary Lorentz transformation, L

transforms according to the rule

L’ = V(q‘p:)

=

V(aqaTa*p,a,)

= V{a[S(qpJ +V(qpJlaJ

=

V(aLa,)

= aLa,

where the last equation follows from the relation

S(aLa, )=O. Hence, he transformation ule

of

L

yields the relativistic invariant

LL, = (r x p)’- (Er/c -c@)*.

5.2.

Classical electromagnetism

Consider the four vector potential A = (4/c, iA),

the relativistic operator

and the quaternion conjugate operator

D,=

(h,

V).

Under a Lorentz transformation, D transforms?

like

q = (ct, ix, iy, z)

D‘= aD( )a:.

The quaternion

F

=

D,A

is equal to

F

= (0, -B E/c)

if one adopts the Lorentz gauge S(D,A) O and

the usual definitions of the electric and magnetic

fields E and B , respectively. Under a Lorentz

transformation,

F

transforms according to F ’

=

a*FaT, which yields the relativistic invariant

F F ;

=

m,

= B’- E’lc’)

+

2iB E/c.

Furthermore, since

DD,= D,D

=

a’ a’ a’ a’

2 a t Z ax2

ay’

a 2

Maxwell’s eight equations for the vacuum are given

by one quaternion equation Silberstein

1914,

Cail-

ler

1917,

Hermann

1974,

Imaeda

1976,

Edmonds

1978), namely

or

where J = p c , ipu) is the four current density. Max-

well’s equations can also be written in the form

DADAc)=

p o J C

where DA, = 0,

B

iE/c).

one obtains

If one applies the operator D, to equation 3)

D J D D A )= poDJ.

Since S(DJ)= 0 expresses the conservation of

electric charge, it follows that the gauge condition

has to satisfy the equation

SD, OA)

= SO(D,A)= U(SD,A)

=

0

which is less restrictive than heLorentz gauge

S(D,A)=

0. If

DJ =

0, one has the relation

D,(DF)= [0,U(B

+

E/c)]

=

0.

t

To

prove this,

use

the relation

a a ax-

ax’. ax-x’*

write Out q’

= aqa:,

q

=

a,q’a*, nd compare the coeffi-

cients.

- = _ . _



30

P R

Girard

Theequation

of

motion of a particle with an

electric charge E in an electromagnetic field is given

by

dp/ds = &[(DA,)u- u(D,A)]

where = m u is the four-momentum

of

the

particle. The relativistic covariance

of

the equation

of motion is manifest since a quaternion of

the type

fgch transforms like q.

6 The general theory

of

relativity GTR) roup

The

GTR

groupt is simply the generalised Lorentz

group

U = auar (4)

where

a

is an arbitrary function such that au, =

1

and where

W

is a minquat. At any given space-time

point, the

GTR

group reduces to the Lorentzgroup.

By differentiating relation (4), one obtains

du'/ds

=

(da/ds)uat+ au(dar/ds) a(du/ds)a$

(5)

where ds2= dq dq, and dq

=

c dt, i dx, i dy, iz).

As an application, consider U tobe he four-

velocity

of

a particle in its instantaneous rest frame

(thus du'/ds = 0) and U to be the four-velocity of

the particle with respect to a fixed inertial system.

After rearrangement (and taking into account the

relation

a,

d a

=

-da,a, since aa,=

l ) ,

elation

(5)

yields

du/ds = -[(a, da/ds)w u(a, da/ds)*].

This equation corresponds to the equation

of

mo-

tion of the general theory of relativity. If the func-

tion a,da/ds is known, the motion is determined.

Newton's law of gravitation, for example, is ob-

tained by considering the nonrelativistic limit and

by taking

2a, dalds= [0,cr2) (L+ iA/c)]

where L = r X v is the angular momentum per unit

mass, and where

A = ( v x L - G M r / r )

is the misnamed Runge-Lenz vectorHamilton

1969c, Tait 1890b, Goldstein 1975, 1976) per unit

mass of the particle.

7 . Tbe

clifford algebra

The Clifford algebra (Shaw 1907c, Clifford 1878) is

the algebra with n generators

e , . . . ,,,

such that

ef = -1, eiei+ eiei

= 0

with i j = 1 . . n i f j ) , and

ei (eiek =

(eei

ek

with i j ,

k =

1 . .n. The order

of

the algebra is 2 .

t For a historical analysis, see Gira rd 1981

The Clifford algebra is directly linked to quatern-

ions via the following theorem (Shaw 1907c,

Clifford 1968b).

Theorem: If

n

= 2m (m integer), the Clifford

algebra

(of

order

2

=

4 ') is the direct product

of

m

quaternion algebras. If n = 2 m 1, the Clifford

algebra (of order 2

=

2' '+') is the direct product

of m quaternion algebras and the algebra

0 ; = 0 0 = 0 1

WOW, =

W 1 0 0

= W,.

As illustrations of the theorem, we shall consider

the cases for n

=

2,

3,

4.

n

= 2: The Clifford algebracontains the four

elements -1, e , , e2, e,e2, and is isomorphic to the

quaternionalgebra

1, i ,

j,

k

over R or C, the

elements

e , , e2

being respectively associated with

i

and

j.

n

=

3:

The Clifford algebra is isomorphic to the

algebra given by the direct product

1,

,

j,

k) (o0,

the generators e , , e2, e3

of

the Clifford algebra

being respectively associated with w0, io o,

kWl.

The general element

of

this algebra can be written

in the form q +up , where q and p

are

quaternions

and where W satisfies the relation W' = 1 and com-

mutes with any quaternion. Clifford was to call the

element q + o p a iquaternion (Clifford 1968a,

1968c)$.

n = 4: This Clifford algebra is isomorphic to the

algebra obtained by forming the direct product

El , i t , k1)

032,

i2

jz,k2)

where E stands for the nit element of the quatern-

ion group; the four generators

e , ,

e2, e3 , e4

of

the

Clifford algebra correspond respectively to

ilE2,

jlE2, i2E1, 2El.As is well known, Dirac's matrices

constitute a representation of this Clifford algebra.

Expressed in their standard form (Fliigge 1974a),

the Dirac matrices

y1 Y Z , ~ 3 ,

4

YS

=

YlYZY3Y4

are

respectively equivalent to the quaternion operators

(Conway 1953a, Synge 1 97 2 ~):

i(

) i ,

(

)k,

)j,

k

)k,

( )k

operating on the complex function

+ = 4 0 + + 1 i + + 2 j + + 3 k ;

the composition of two operators, say

j )k

fol-

lowed by ( ) i , isgivenby the usual rule: ij ( ) k i .

Dirac's relativistic wave equation (FXigge 1974b)

can thus be written in the quaternion form

(iDl)dd +W Z ) + ~ (iQ)+j + ( k D d + k +x4 = 0

where:

$ Clifford's biquaternion s are distinct from Hamilton's

biquaternions; the lat ter

are

simply com plex quatern-

ions.




31

D p

=

(ic/hc)Ap

a

ax

A. = i4

e 4 =

V

Do

=

-(i/c)(a/at).

More symmetrical formsre possible (Conway

1953c,Edmonds 974). The function

J,

is an

example of a pinor in four dimensions. More

generally, the Clifford algebra allows the definition

of spinors in n dimensions (Chevalley 1954, Waer-

den 1974).

8.

Other groups

As

further and final applications of quaternions to

physics, let us briefly mention SU(2) and the con-

formal group.

8.1. The unitary group

SU(2)

The unitary groupSU(2) is the group of 2 x 2

unitary matricest of determinant 1. This group is

isomorphic to the group of real quaternions ,via the

correspondence (Du Val 1964a)

with det A= Nq = 1.

8.2.

The

conformal

group

It is well-known that Maxwell's equations in

vacuum are covariant not only with respect to the

Lorentz group but also with respect to the confor-

mal group. This 15 (real) parameter group can be

defined (Bessel-Hagen 1921) as the group of trans-

formations such that if N(dq) =0, with dq =

c dt,idx,dy,dz), then N(dq')=O. Conse-

quently, the group contains in particular the space-

time translations, the Lorentz transformations, and

the dilatations q = Aq, where is a constant. The

transformations relative tohe four remaining

parameters can be expressed in quaternion

form

by

the formulaet:

is easy to show that the conformal transformations

given by equation (6) forma group, the inverse

transformation being given by

q

=

1 q'uJ1q'.

Furthermore, one has the relations

Q

q a d

=

q'a,q

N(dq')

=

(Nq'/Nq)'N(dq)

Nq' = Nq/N(l+

qa,)

dq' = (1 +qa,) dq( 1+

a,q) .

From the second relation, it follows that the trans-

formationshere considered are indeed conformal

transformations.

9

Conclusion

The paper has related major covariance groups of

physics to he quaterniongroup ndhas given

several physical applications, most of them at

undergraduate level. It is hoped that the treatment

of the representations of the quaternion group will

have shown that quaternion group theory is quite

accessible to undergraduate tudents and might

further their comprehension of physics by exhibit-

ing the deep unity of physical phenomena.

Rderencea

Bessel-HagenE 1921 Math. Annal. 84 269

Bork

A

M 1966

A m .

J.

Phys.

34

202

Brand L 1947 Vector and Tensor Analysis (New York:

Bulletin of the International Association

for

Promoting

Wiley) pp. 417-21

the Study of Quaternions and Allied Systems

of

Mathematics 1900-13 (Lancaster, Pennsylvania) bib-

liography in the 1905, 1908-10, 1912-3 issues

Cailler C 1917

Archs. Sci.

Phys.

Nat.

ser. 4, 44 p. 237

Chevalley C C 1954 The Algebraic Theory

of

Spinors

(New York: Columbia University Press) pp. 55-8

Clifford W K 1878 A m . J. Math.

1

350

968a Mathematical Papers ed.

R

Tucker (1882;

reprinted 1968 New York: Chelsea) pp. 181-200

968b ibid. pp. 266-76

968c ibid. pp. 385-96

Conway

A

W 1948

Pont. Acad.

S c i

Acta

l2

259

953a Selected Pawrs ed.

J

McConnell (Dublin:

q

=

(1

+

qa,) q

=

q(1+ a,q)

Dublin Institute for Advanced Studies) pp. 204-22

953b ibid. pp. 206-8

= (q + qq, )/[ 1+2 S(q ac)+ aa,qq,] (6)

953c ibid. p. 212

~~

..

Crowe M 1967-A History

of

Vector Analy sis: The

where

q 4 are

minquats

and

where

is a Evolution of t h Idea

of a

Vectorial System (Notre

minquat. The aboveormulae can

also

be written

London: university

f

N~~~~

q y

= q - l+

a,.

(7) E k s o n L E 1914 Linear Algebras (Cambridge: Cam-

1967b ibid. pp.

150-6

By using the quaternion formula (ab)

=

b a it bridgeUniversity Press) pp. 10-2

A-' =

(A')*,

where

(A')*

is the complex conjugate mat- To derive the second equation, differentiate equation

F A

unitary matrix is a complex matrix A such that

rix transpose

of

A . (7) and use the relation d(q ) = -q-'(dq)q which

fol-

i

Compare with Giirsey et al 1980.

lows from the differentiation of the relation q q

=

qq-1= 1.



32

P Girard

Du Val P 1964a

Homographies, Quaternions and Rota-

964b

ibid.

pp. 42-3

Edmonds

J

D Jr 1974

Am. J. Phys. 42

222

978

Am. J. Phys. 46

430

Fliigge S 1974a

Practical Quantum Mechanics

(New

York: Springer) vol. 2 p. 167

974b

ibid.

p. 189

Girard, P

R

1981

The Conceptual Development of Ein-

stein’s General Theory of Relativity PhD Thesis

Uni-

versity of Wisconsin-Madison), published by Univer-

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tions

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H

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Am.

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1931-67

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Elements of Quaternions

2 vols (1899-1901;

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969b

ibid.

vol. 1 ch. 2

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Sir William Rowan Hamilton

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R

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Spinors, Cliflord and Cayley Algebras

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S

and Kawada Y (eds) 1977

Encyclopaedic

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vol. 2 pp. 299-301

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Lectures

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the Icosahedron and the Solu-

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of

Equations

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G

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PP

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ser. 3,

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325-400

Macfarlane A 1904

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Synopsis of Linear Associatiue Algebra;

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(Washington DC:

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907b

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Macmillan) ch. 8

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~ 1972c

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pp. 48-64

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Quaternions

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L H

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the quaternion group and modern physics girard

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