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Apeiron, Vol. 12, No. 4, October 2005

371

Quaternions, Maxwell
Equations and Lorentz
Transformations
M. Acevedo M., J. López-Bonilla and M. Sánchez-Meraz
Sección de Estudios de Posgrado e Investigación
Escuela Superior de Ingenieria Mecánica y Eléctrica
Instituto Politécnico Nacional
Edif..Z-4, 3er Piso, Col.Lindavista, 07738 México DF
E-mails: jlopezb@ipn.mx
In this work: a).-We show that the invariance of the Maxwell
equations under duality rotations brings into scene to the




complex vector ( c B + i E ), whose components allow to
construct a quaternionic equation for the electromagnetic field
in vacuo. b).-For any analytic function f of the complex
variable z, it is possible to prove that is a Debye potential for
itself, which permits to reformulate the corresponding
Cauchy-Riemann relations. Here we show that the Fueter
conditions- when z is a quaternion- also accept a similar
reformulation and a very compact quaternionic expression. c).We exhibit how the rotations in three and four dimensions can
be described through a complex matrix relation or
equivalently by a quaternionic formula.

© 2005 C. Roy Keys Inc. — http://redshift.vif.com

Apeiron, Vol. 12, No. 4, October 2005

1.

372

Quaternionic version of the Maxwell
equations.

The Maxwell equations in the source-free case:

∇ • B = 0,

∇ • E = 0,

1 ∂Ε
∂B
,
∇× E = − ,
2
c dt
∂t
are invariant under the duality rotations [1,2]:
∇× B =

Ε ' = ΕCosα + cBSinα ,

(1)

cB ' = − ESinα + cBCosα ,

(2)
in the sense that the fields also satisfy (1) ; the Noether theorem [3-9]
shows [10] that this invariance of the Maxwell equations implies the
continuity equation:
⎛ 1

∂ ⎛ ∈0 2
1 2⎞
(3)
B ⎟ + ∇ • ⎜ E × B ⎟ = 0,
⎜ E +
2 μ0
∂t ⎝ 2

⎝ μ0

for the electromagnetic energy.If relations (2) are rewritten into the
form:

cB '+ iE ' = eiα ( cB + iE ) ,

(4)

the participation of the complex vector [10-13]:

F = cB + iE
follows, and expressions (1) become:
∇ • F = 0,

1 ∂F
− i∇ ×F = 0
c ∂t

Now we show that the Maxwell equations adopt a very compact
structure if we employ quaternions [10,14-23]. In fact, with we
construct the quaternionic vector :
© 2005 C. Roy Keys Inc. — http://redshift.vif.com

(5)

(6)

Apeiron, Vol. 12, No. 4, October 2005

F = IFX + JFY + KFZ ,

373

(7)

and the quaternonic operator [24-26]:
∇=

i ∂



+K ,
+I
+J
c ∂t
∂y
∂z
∂x

(8)

so that the Maxwell equations (1) are carried to the following
quaternionic version:
∇F = 0,

(9)

Conway [27] – Silberstein [28] introduced quaternions as a notation
in the special theory of relativity; Silberstein [24]-Lanczos [25,29]
were the first authors to deduce (9) (this quaternionic expression
reminds us of the Weyl equation of massless ½ spin particles).
Unitary complex quaternions generate [10, 22, 30-33] proper
Lorentz transformations, consequently, we consider as a natural fact
to use quaternions – as in eq.(9) – for the description of the Maxwell
field.

2.

The Fueter conditions as Debye
expressions

If f is an analytic function of the complex variable z=x+iy, then it has
the form f(z)=u(x,y)+iv(x,y) with the fulfillment of the CauchyRiemann relations [34]:
∂u ∂v ∂u
∂v
= ,
=− ,
(10)
∂x
∂x ∂y
∂y
which thereby imply the harmonic character of u and v because:
∇ 2 u = ∇ 2 v = 0, ∇ 2 =

∂2
∂2
+
∂x 2 ∂y 2

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374

Apeiron, Vol. 12, No. 4, October 2005

The conditions (10) allow to obtain two interesting differential
identities for u and v, which have great similarity with the Debye
expressions [2, 35-39] for the electromagnetic potentials, in fact:
r
u = • ∇(ru ) − r × ∇u 3,
(12)
r
where we have employed the known notation from vectorial analysis:

[

r = xiˆ + yˆj ,

r = x2 + y2 ,

[r × ∇g ] ≡ x ∂∂gy − y ∂∂gx ,
3

]



∇ = iˆ + ˆj ,
∂y
∂x

(13)

The function if is also analytic, then if(z)=-v+iu implies that (12) is
correct with the changes u → −v and v → u , that is:


v=

r →
⎡→ → ⎤
• ∇(rv ) + ⎢ r × ∇u ⎥ ,
r

⎦3

(14)

The expressions (12) and (14) are a reformulation of the
Cauchy-Riemann relations, these being a strong motivation for the
existence of Debye generators in electromagnetic theory. The solution
of the source-free Maxwell equations can be written [2,35-39] in
terms of two real scalar generators (Debye potentials) - ψ E and ψ M which satisfy the wave equation:

ψ E = ψ M = 0,

=

∂2
− ∇2
2
2
c ∂t

(15)

in according to:

φ = −c

r
i ∇ ( rψ E ) ,
r

A = − r × ∇ψ M + r

∂ψ E
,
c∂t

© 2005 C. Roy Keys Inc. — http://redshift.vif.com

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Apeiron, Vol. 12, No. 4, October 2005

375

up to gauge transformations. We must note that the existence of and
implicitly follows from results of several authors [40-43].
Now we shall obtain the generalization of (12) and (14) for the
quaternionic case. Fueter[44] founded the theory of functions
G ( q ) = u o + I u 1 + Ju 2 + K u 3 ,
of
a
quaternionic
variable

q = x0 + Iy1 + Jy 2 + Ky3 , and he imposed the following differential
conditions on the , which correspond to the extension of the CauchyRiemann equations (10):
∂u0 ∂u1 ∂u2 ∂u3



=0 ,
∂x0 ∂y1 ∂y2 ∂y3
∂u1 ∂u0 ∂u3 ∂u2
+
+

=0 ,
∂x0 ∂y1 ∂y2 ∂y3
∂u2 ∂u3 ∂u0 ∂u1

+
+
=0 ,
∂x0 ∂y1 ∂y2 ∂y3

(17)

∂u3 ∂u2 ∂u1 ∂u0
+

+
=0 .
∂x0 ∂y1 ∂y2 ∂y3

Imaeda [45] shows that (17) permits to establish a connection
with the Maxwell equations, which leads to a new formulation of
classical electrodynamics. If we introduce the operator (8):




∇=
+I
+J
+K
(18)
∂x0
∂y1
∂y2
∂y3
then (17) are equivalent to:
∇G=0 ,
(19)
It is remarkable the similarity between (9) and (19), of course we
may see to (9) as a particular case of (19).
© 2005 C. Roy Keys Inc. — http://redshift.vif.com

Apeiron, Vol. 12, No. 4, October 2005

376

With the aid of (17) and taking as guide the relation (12), it is not
difficult to deduce the Debye type expression:
u0 =

r

i ∇ ( ru0 ) +
( y1u1 + y2u2 + y3u3 ) + ⎡⎣ r × ∇u1 ⎤⎦ 1 + ⎡⎣ r × ∇u2 ⎤⎦ 2 + ⎡⎣ r × ∇u3 ⎤⎦ 3 , (20)
r
∂x0

where
r = iy1 + jy2 + k y3 ,

∇=i




∂g
∂g
, ⎡ r × ∇g ⎦⎤ ≡ y2
+j
+k
− y3
1
∂y1
∂y2
∂y3 ⎣
∂y3
∂y2

∂g
∂g
⎡ r × ∇g ⎤ ≡ y3
,
− y1

⎦2
∂y1
∂y3

,

∂g
∂g
⎡ r × ∇g ⎤ ≡ y1
.
− y2

⎦3
∂y2
∂y1

(21)

The function - G ( q ) I = u1 − Iu0 − Ju3 + Ku2 - is also analytic,
then in (20) we can make the changes u0 → u1 , u1 → −u0 , u2 → −u3
and u3 → u2 , therefore:
u1 =

r

i ∇ ( ru1 ) +
( − y1u0 − y2u3 + y3u2 ) − ⎡⎣ r × ∇u0 ⎤⎦1 − ⎡⎣ r × ∇u3 ⎤⎦ 2 + ⎡⎣ r × ∇u2 ⎤⎦ 3 . (22)
r
∂x0

Similarly the analytic character of -G ( q ) J and -G ( q ) K leads to:
u2 =


r
i ∇ ( ru2 ) +
( y1u3 − y2u0 − y3u1 ) + ⎡⎣ r × ∇u3 ⎤⎦1 − ⎡⎣ r × ∇u0 ⎤⎦ 2 − ⎡⎣ r × ∇u1 ⎤⎦ 3 ,
∂x0
r

u3 =


r
i ∇ ( ru3 ) +
( − y1u2 + y2u1 − y3u0 ) − ⎡⎣ r × ∇u2 ⎤⎦1 − ⎡⎣ r × ∇u1 ⎤⎦ 2 − ⎡⎣ r × ∇u0 ⎤⎦ 3 .
r
∂x0

(23)

The relations (20), (22) and (23) represent a Debye type
reformulation of the Fueter conditions (17), which are relations not
explicitly found in the literature.

© 2005 C. Roy Keys Inc. — http://redshift.vif.com

Apeiron, Vol. 12, No. 4, October 2005

3.

377

Quaternions, 3-rotations and Lorentz
transformations

In Minkowski space, any real matrix L4×4 = ( L jk ) with the property

LT L = I , that is:

L jk L jl = δ kl ,

(24)

allows to make a Lorentz transformation over an arbitrary event , via
the expression [22,41]:
x ' j = L jk xk
(25)
such that (24) implies the invariance x ' j x ' j = x j x j , this being:
x '2 + y '2 + z '2 − c 2t '2 = x 2 + y 2 + z 2 − c 2t 2 .

(26)

If we define the complex 2x2 matrices:.
⎛ x − ix4
X =⎜ 3
⎝ x1 + ix2
with the properties:

x1 − ix2 ⎞
⎛α
⎟ ,U = ⎜
− x3 − ix4 ⎠
⎝γ

β⎞
,
δ ⎟⎠

(27)

⎛α* γ * ⎞
(28)
U† = ⎜ *
,
*⎟
⎝β δ ⎠
then the construction of a Lorentz transformation L can be
accomplished through the relation [22,41]:

det X = − x j x j ,

X ' = UXU † ,

(29)

with detU = 1 as required by (26). In other words, any four complex
constants α , β , γ , δ subject to the unimodular condition:

αδ − βγ = 1
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Apeiron, Vol. 12, No. 4, October 2005

378

generate also a Lorentz’s matrix. By comparison between (25) and
(29) it results the following expressions [46] of Synge[41] –
Rumer[47]- Aharoni[48]:

1 *
i
α δ + βγ * ) + c.c. ,
L12 = (α *δ + βγ * ) + c.c. ,
(
2
2
1 *
1
L13 = (α γ − β *δ ) + c.c. ,
L14 = (α *γ + β *δ ) + c.c. ,
2
2
1
i
L21 = (αδ * − βγ * ) + c.c. ,
L22 = (α *δ − β *γ ) + c.c. ,
2
2
i
i
L24 = (αγ * − β *δ ) + c.c. ,
L23 = (αγ * + β *δ ) + c.c. ,
2
2
(31)
1
i
L31 = (α * β − γ *δ ) + c.c. ,
L32 = (α * β − γ *δ ) + c.c. ,
2
2
1
1
L33 = (αα * − ββ * − γγ * + δδ * ) , L34 = (αα * + ββ * − γγ * − δδ * ) ,
2
2
1
i
L41 = (α * β + γ *δ ) + c.c. ,
L42 = (α * β + γ *δ ) + c.c. ,
2
2
1
1
L43 = (αα * − ββ * + γγ * − δδ * ) , L44 = (αα * + ββ * + γγ * + δδ * ) ,
2
2
L11 =

where c.c. means the complex conjugate of all the previous terms. It
is evident that the matrices produce the same, thus they are said [32,
33, 49, 50] to constitute a two-valued representation of the Lorentz
transformations.
On the other hand, we may follow Lanczos [10, 22] and introduce
the quaternions [10,14-22]:
R = ct + i (Ix + Jy + Kz ) ,
A = a4 + Ia1 + Ja2 + Ka3 , (32)
together with the definitions:
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379

Apeiron, Vol. 12, No. 4, October 2005

A = a4 - ( Ia1 + Ja2 + Ka3 ) ,

A = a 4 +Ia + Ja 2 + Ka
*

*

*
1

*

*
3

, (33)

so that A generates a Lorentz’s matrix via the quaternionic relation:
R ' = ARA*
with A fulfilling the condition:

(34)

AA=a12 + a22 + a32 + a42 = 1 .

(35)

For example, (35) is verified by:
1
i
a1 = − ( γ + β ) , a2 = − ( γ − β ) ,
2
2
(36)
1
i
a3 = (δ − α ) ,
a4 = − (δ + α ) ,
2
2
and, if the complex numbers satisfy (30) then (34) and (36) imply
(31). Another option is:
a1 = iQ ( λ *e P + η *e − P )
a3 = iQ ( e P − e − P )

,

,

a2 = Q ( λ *e P − η *e − P ) ,
a4 = −Q ( e P + e − P )

,

(37)

−1
1
1
Q = 1 − λ *η * 2 ,
( M + iN ) ,
2
2
where M,N are arbitrary real numbers, and λ ,η are any complex
numbers such that λη ≠ 1 . Eqs. (34) and (37) give us the following
expressions [46] of Greenberg-Knauer [51] for L :

P=

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380

Apeiron, Vol. 12, No. 4, October 2005

(1 + λ η ) + c.c. ,
= Te (η − λ ) + c.c. ,
= iTe (1 + λη ) + c.c. ,
= iTe (η − λ ) + c.c. ,
= T ( λ e − η e ) + c.c. ,
= T ⎡⎣ e (1 − λλ ) + e (1 − ηη ) ⎤⎦ ,
= −T ( λ e + η e ) + c.c. ,
= −T ⎡⎣ e (1 − λλ ) − e (1 − ηη ) ⎤⎦ ,

L11 = Te

iN

L13

iN

L21
L23
L31
L33
L41
L43

*

*

− iN

*

− iN

*

−M

M

M

−M

*

M

M

*

−M

*

−M

*

(1 − λ η ) + c.c. ,
= −Te (η + λ ) + c.c. ,
= −Te ( λη − 1) + c.c. ,
(38)
= −iTe (η + λ ) + c.c. ,
= iT ( λ e + η e ) + c.c. ,
= −T ⎡⎣e (1 + λλ ) − e (1 + ηη ) ⎤⎦ ,
= −iT ( λ e + η e ) + c.c. ,
= T ⎡⎣ e (1 + λλ ) + e (1 + ηη ) ⎤⎦ ,

L12 = iTe
L14
L22
L24
L32
L34
L42
L44

iN

*

iN

*

− iN

*

− iN

*

M

−M

M

−M

*

−M

M

M

*

*

−M

*

1
−1
with T= 1 − λη
well behaved because λη ≠ 1 . Sachs [52]
2
obtained some special cases of (38). The refs. [50, 53] have important
applications of (38) to the Newman-Penrose formalism [54, 55] in
general relativity.
Now we consider that A is a real unitary quaternion with their four
components a j written in terms of two complex numbers α and β :

1
β + β*) ,
(
2
i
1
a3 = (α − α * ) ,
a4 = − (α + α * ) ,
2
2
*
*
αα + ββ = 1
a1 = −

i
β − β*) ,
(
2

a2 = −

(39)

then (34) implies a rotation in the 3-space:
⎛ x '⎞
⎛ x⎞
⎜ ⎟
⎜ ⎟
⎜ y '⎟ ≡ R ⎜ y ⎟ ,
⎜ z'⎟
⎜z⎟
⎝ ⎠
⎝ ⎠

t'=t ,

© 2005 C. Roy Keys Inc. — http://redshift.vif.com

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Apeiron, Vol. 12, No. 4, October 2005

381

where [47,56]:

i 2
⎛1 2
*2
2
*2
*2
2
*2
* * ⎞
⎜ 2 (α + α − β − β ) − 2 (α − α + β − β ) − (αβ + α β ) ⎟


1 2
1 2
*2
2
*2
*2
2
*2
* * ⎟ (41)

R = (α − α − β + β )
(α + α + β + β ) −i (αβ − α β ) ⎟
⎜2
2


i (αβ * − αβ * )
αβ * + α * β
αα * − ββ * ⎟
⎜⎜




is an orthogonal matrix (element of O(3)) because:
RRT = I .

(42)

The representation of an arbitrary rotation of three-space with the
help of a real quaternion of length 1 was known by Euler and it was
employed by him [10, 57].
It is interesting to note that if we make x4 = 0, γ = − β * , δ = α * into
(29), then we obtain (40) and (41), being U an element of SU (2)
because:

β⎞
⎛ α
U =⎜ *
, UU † = I ,
*⎟

β
α


*
detU = αα + ββ * = 1 .

(43)

The unitary matrices ±U generate the same orthogonal matrix R ,
thus SU (2) is a two-valued representation of O(3) [32, 33, 56, 58-62].
On the other hand, (39) is equivalent to:
α = a4 − ia3 , β = − a2 − ia1 .
(44)
so that takes the form:

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Apeiron, Vol. 12, No. 4, October 2005

⎛0 1⎞

⎛1 0⎞

⎛0 − i⎞
⎛1 0 ⎞
⎟⎟ − ia3 ⎜⎜
⎟⎟,
0⎠
⎝ 0 − 1⎠

U = a ⎜⎜⎝ 0 1 ⎟⎟⎠ − ia ⎜⎜⎝ 1 0 ⎟⎟⎠ − ia ⎜⎜⎝ i
= a I − i (a σ + a σ + a σ ),
4

1

2

~

4

1

~

x

2

y

3

382

(45)

z

where σ x , σ y , σ z are the known matrices of Pauli. If now we use the
formal association [21]:
I → 1 , − iσ x → I ,

− iσ y → J ,

− iσ z → K

(46)

it follows that U → A , which motivates the intimate relationship
between complex 2x2 unitary matrices and real 3x3 orthogonal
matrices generated by real quaternions of length 1.
Thus we have seen that (29) or (34) describe completely the
rotations in three and four dimensions.
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