The Quaternions

The sphere S3 and the quaternions

BK

If the sphere S3 resembles the circle S1 more than the sphere S2, it is because both correspond to the group of units of a number field: the field of complex numbers for S1 and the field of quaternions for S3. But this does not work with S2 because R3 is not a number field.

To understand these structures, I will recall how one makes R2 a number field by embedding it in the algebra of 2×2 square matrices with real coefficients. We will then see that a similar construction allows R4 to be endowed with a field structure (the field of quaternions) by embedding it in the algebra of 2×2 square matrices with complex coefficients.

1. The field of complex numbers

Let M2(R) be the space of 2×2 matrices with real coefficients. It is the space of matrices

ö ÷ ÷ ø | d |
|---|

where a,b,c,d are real numbers. We know how to:

1. add two matrices of M2(R),

2. multiply a matrix of M2(R) by a real number,

3. multiply two matrices of M2(R) together.

To summarize these three properties, we say that M2(R) is an algebra. It is of dimension 4 because 4 parameters are needed to describe it.

To every vector (x,y) of the plane R2, one can associate the square matrix with real coefficients

ö ÷ ÷ ø | x |
|---|

We then notice that

1. the sum of 2 such matrices is still of the same form

(y

1

• y

2

)

ö ÷ ÷ ø ,| x | 1 | + x | 2 |
|---|---|---|---|

2. the product by a real number l of such a matrix z is still of this type

ö ÷ ÷ ø ,| l | x |
|---|---|

3. the matrix product of 2 such matrices as well

x

1

x

2

y

1

y

2

y

1

x

2

• x

1

y

2

• (y 1 x 2 + x 1 y 2 )

x

1

x

2

y

1

y

2

x

1

x

2

y

1

y

2

ö ÷ ÷ ø .

x

1

x

2

y

1

y

2

We have therefore constructed a subspace of M2(R), which we will denote C, stable under addition, multiplication by a scalar and matrix multiplication. We say that C is a subalgebra of M2(R). It is of dimension 2. One can notice an important property of this algebra: it is commutative

z

1

z

2

= z

2

z

1

.

z

1

z

2

= z

2

z

1

.

The determinant of a matrix z Î C is written

ê ê ê ê = x2 + y2 ,| x |
|---|

consequently, every non-zero matrix z in C is invertible and its inverse

--- ****

æ ç ç è

| - | y |
|---|

ö ÷ ÷ ø | x |
|---|

also belongs to C. We say that C is a field. We have just rediscovered the field of complex numbers.

The matrix corresponding to the vector (1,0) is the identity matrix ****

ö ÷ ÷ ø | 1 |
|---|

The matrix corresponding to the vector (0,1) is denoted i ****

ö ÷ ÷ ø | 0 |
|---|

It possesses the remarkable property

i

2

=

1

i

2

=

1

With these notations, we see that every complex number is written uniquely as

where x and y are real numbers.

The conjugate of an element z in C is defined as the transpose of the matrix z

| - | y |
|---|

ö ÷ ÷ ø | x |
|---|

In the field of complex numbers there are two remarkable subspaces. On the one hand there are the multiples of the identity matrix 1, that is to say the set of matrices of the form

ö ÷ ÷ ø | x |
|---|

which is stable under addition and multiplication. It is the real space, identical to the field of real numbers and which appears as a subfield of C. It is also defined by the equation

There is also the subspace of matrices z with zero trace (the trace of a matrix is the sum of its diagonal elements)

ö ÷ ÷ ø | 0 |
|---|

It is called the imaginary space. It is stable under addition but not under multiplication. We notice that it is also the space of 2×2 matrices that are antisymmetric.

With these definitions, the ordinary scalar product of two vectors of R2 is easily calculated. If z1 is the matrix corresponding to (x1,y1) and z2 is the matrix corresponding to (x2,y2), then --- ****

(x 1 ,y 1 )·(x 2 ,y 2 ) = 1 2 Tr (z 1 - z 2 )

(x 1 ,y 1 )·(x 2 ,y 2 ) = 1 2 Tr (z 1 - z 2 )

The norm of a vector (x,y) corresponding to z is written

**** --- ****

\ \ (x,y) \ \ 2 = x 2 + y 2 = Det (z) = 1 2 Tr (z - z )

\ \ (x,y) \ \ 2 = x 2 + y 2 = Det (z) = 1 2 Tr (z - z )

It is also called the modulus or the norm of the complex number z and is denoted | z|. We have of course | z1z2| = | z1|| z2|.

The subgroup of units of the field C, which we denote U(1), is the set of elements z of C that satisfy

\ z\ 2 = z - z = x 2 + y 2 = 1 .

\ z\ 2 = z - z = x 2 + y 2 = 1 .

This set is stable under multiplication. The element 1 belongs to U(1). Every element of U(1) is invertible and its inverse is still in U(1). We say that U(1) is a group. Note that the sum of two elements of U(1) is not in U(1) in general. The preceding construction has therefore brought out the circle S1 as a group, the group of unit elements of C. The tangent to the circle S1 at the point 1, which is called the Lie algebra of the group U(1), is the imaginary line defined by

2 The field of quaternions

Definition.

Let us take up the preceding construction to endow R4 with a field structure. Let M2(C) be the space of 2×2 matrices with complex coefficients. To every pair of complex numbers (a,b) one can associate the square matrix with complex coefficients

ö ÷ ÷ ÷ ø | - | a |
|---|

| - | a |
|---|

We then notice that:

1. the sum of 2 such matrices is still of the same form

(b

1

• b

2

)

a

1

a

2

a

1

a

2

ö ÷ ÷ ÷ ø ,

a

1

a

2

2. the product by a real number l of such a matrix q is still of this type

ö ÷ ÷ ÷ ø ,| l | - | a |
|---|---|---|

3. the matrix product of 2 such matrices as well

a 1 a 2 - b 1 - b 2

a

1

b

2

• b

1

a

2

• a 1 - b 2 - - b 2 - a 1

• a 1 - a 2 - - b 1 - b 2

• a 1 - a 2 - - b 1 - b 2

ö ÷ ÷ ÷ ø .

• a 1 - a 2 - - b 1 - b 2

We have therefore constructed a subalgebra of M2(C) which we will denote K. But unlike C, this algebra is not commutative. Indeed, we have for example

ö ÷ ÷ ø æ ç ç è

ö ÷ ÷ ø = æ ç ç è

ö ÷ ÷ ø | 0 |
|---|

whereas

ö ÷ ÷ ø æ ç ç è

ö ÷ ÷ ø = æ ç ç è

ö ÷ ÷ ø | 0 |
|---|

The determinant of a matrix q Î K is written

ê ê ê ê ê = | a|2 + | b|2 ,| - | a |
|---|

| - | a |
|---|

consequently, every non-zero matrix q in K is invertible and its inverse

--- ****

æ ç ç ç è

| - | a |
|---|

| - | b |
|---|

| - | b |
|---|

ö ÷ ÷ ÷ ø | a |
|---|

also belongs to K. We call K the field of quaternions. It is also a real vector space of dimension 4. Every quaternion q can be written in the form

q = t

1

• x

I

• y

J

• z

K

q = t

1

• x

I

• y

J

• z

K

where

ö ÷ ÷ ø I = æ ç ç è

ö ÷ ÷ ø J = æ ç ç è

ö ÷ ÷ ø K = æ ç ç è

ö ÷ ÷ ø | 0 |
|---|

One easily verifies the following relations:

I2 = J2 = K2 = - 1,

IJ = - JI = K , ** JK** = - KJ = I , KI = - IK = J.

The ordinary scalar product of two vectors in R4 represented by the quaternions

q

1

= t

1

1

• x

1

I

• y

1

J

• z

1

K

q 1 = t 1 1 + x 1 I + y 1 J + z 1 K

and

q

2

= t

2

1

• x

2

I

• y

2

J

• z

2

K

q 2 = t 2 1 + x 2 I + y 2 J + z 2 K

is written

--- ****

(t 1 ,x 1 ,y 1 ,z 1 )·(t 2 ,x 2 ,y 2 ,z 2 ) = 1 2 Tr (q 1 - q 2 )

(t 1 ,x 1 ,y 1 ,z 1 )·(t 2 ,x 2 ,y 2 ,z 2 ) = 1 2 Tr (q 1 - q 2 )

The norm of a vector (t,x,y,z) represented by q is written

**** --- ****

\ \ (t,x,y,z) \ \ 2 = t 2 + x 2 + y 2 + z 2 = Det (q) = 1 2 Tr (q - q )

\ \ (t,x,y,z) \ \ 2 = t 2 + x 2 + y 2 + z 2 = Det (q) = 1 2 Tr (q - q )

It is also called the modulus or the norm of the quaternion q and is denoted | q|. We have of course | q1q2| = | q1|| q2|.

Real and imaginary quaternions.

The conjugate of an element q in K is defined as the transpose of the conjugate matrix of q

| - | a |
|---|

| - | b |
|---|

| - | b |
|---|

ö ÷ ÷ ÷ ø | a |
|---|

A quaternion q is said to be real if it is of the form

ö ÷ ÷ ø | t |
|---|

where t is a real number. This allows R to be identified with a subfield of K. The subspace of real quaternions is defined by the equation

| q = | - | q |
|---|

One also defines the space of imaginary quaternions H by the equation

| Tr | (q) = 0 . |
|---|

It is a space of dimension 3, generated by I,J,K. It is also defined as the space of matrices M of M2(C), antihermitian

M = t`M = - M*

(M* is the transpose of the matrix of complex conjugate elements) with zero trace

| Tr | (M) =0 |
|---|

Remark 1: * In the case of the field C, we had observed that the imaginary space consisted of the antisymmetric real 2×2 matrices. But the trace of an antisymmetric matrix *

tM = - M * is necessarily zero, which is not the case for an antihermitian matrix *

M = t`M = - M .*

*The group of units. * * *

The subgroup of units of the field K, which we denote SU(2), is the set of elements q of K that satisfy

\ q\ 2 = z - z = t 2 + x 2 + y 2 + z 2 = 1 .

\ q\ 2 = z - z = t 2 + x 2 + y 2 + z 2 = 1 .

It is the set of points belonging to the sphere S3, which thus appears endowed with a group structure, the group SU(2). The tangent space (of dimension 3) to the sphere S3, which is called the Lie algebra of the group SU(2), is the imaginary space defined by

| Tr | (q) = 0 . |
|---|

One can verify that the Lie bracket of two elements of H, X and Y

| [X,Y] = XY | - | YX |
|---|

still belongs to H.

The tangent space to the sphere S3 at any point q is obtained by left-multiplying by q the elements of the imaginary space H

T

q

S

3

= q H.

T

q

S

3

= q H.

Each geodesic of the sphere S3 starting from the point 1 in the direction X (where X belongs to H and | X| = 1) is defined by

| q(s) = exp(sX) = cos(s) | 1 | + sin(s) X |
|---|

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