Open Journal of Technology & Engineering Disciplines (OJTED)
Vol. 2, No. 1, March 2016, pp. 21~27
ISSN: 2455-6971
THE MAXWELL AND QUATERNIONIC DIRAC
EQUATIONS UNDER CHIRAL APPROACH
H. Torres-Silva
Escuela Universitaria de Ingeniería Eléctrica-Electrónica,
Universidad de Tarapacá, Avda. 18 de Septiembre 2222, Casilla Postal 6-D, Arica, Chile.
J. López- Bonilla & R. López-Vázquez
ESIME- Zacatenco, IPN, Edif.5, Col. Lindavista 07738, México DF;
Email: jlopezb@ipn.mx
Article Info
ABSTRACT
Article history:
Received Dec. 11th, 2015
Revised Feb. 14th, 2016
Accepted March. 13th, 2016
Keyword:
Quaternions,
Dirac
equation, Maxwell system.
In the present article we propose a simple model involving the
Dirac quaternion and the Maxwell operators under chiral
approach. This equality establishes a direct connection between
solutions of the two systems and moreover, we show that it is
valid when the electric field E is parallel to the magnetic field
H . Our analysis is based on the quaternionic forms of the
Dirac and Maxwell equations. In both cases the quaternionic
reformulations are completely equivalent to the traditional
form of the Dirac and Maxwell systems for self dual
configuration. This theory can give a new quantum mechanics
(QM) interpretation of the electron. The below research proves
that the QM represents the electrodynamics of the curvilinear
closed chiral waves. It is entirely according to the modern
interpretation and explains the particularities and the results of
the quantum field theory.
Copyright © 2016
Open Journal of Technology & Engineering Disciplines (OJTED)
All rights reserved.
Corresponding Author: J. López- Bonilla
ESIME- Zacatenco, IPN, Edif.5, Col. Lindavista 07738, México DF;
Email: jlopezb@ipn.mx
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Open Journal of Technology & Engineering Disciplines (OJTED)
Vol. 2, No. 1, March 2016, pp. 21~27
ISSN: 2455-6971
1. INTRODUCTION
The relation between the two most important in mathematical physics first order systems
of partial differential equations is among those topics which attract attention because of
their general, even philosophical significance but at the same time do not offer much for
the solution of particular problems concerning physical models. Here we explore the
connection between the Maxwell and Dirac equations under the quaternionic formalism in
different ways [1-9]. Solutions of the Maxwell’s system can be related to solutions of the
Dirac equation through some connection [6-11]. Nevertheless, in spite of these significant
efforts there remain some important conceptual questions. For example, what is the
meaning of this close relation between the Maxwell system and the Dirac equation and how
this relation is connected with the wave-particle dualism? In the present article we propose
a simple equality involving the Dirac and Maxwell operators under chiral approach. This
equality establishes a direct connection between solutions of the two systems and
moreover, we show that it is valid when a quite natural relation between the frequency of
the electromagnetic wave and the energy of the Dirac particle is fulfilled when the wave
electric field is parallel to the magnetic field. Our analysis is based on the quaternionic form
of the Dirac equation [7] and the quaternionic form of the Maxwell equations proposed in
[8] (see also [9]). In both cases the quaternionic reformulation are completely equivalent to
the traditional form of the Dirac equation.
Quaternionic quantum mechanics has been investigated by Adler and others [1-11]; we will
follow here a rather different approach than that of these references. It is interesting to
note that the quaternionic wave function consists of scalar and vector components, hence
the emerging equations can then describe scalar and vector particles, and the resulting
equations would then mix the scalar and vector components. We anticipate that these
particles could be the bosons and fermions, then one can in some respect unify KleinGordon and Dirac equations in a single equation. The spin of the particle shows up when the
particle interacts with the photon field.
2. THE QUATERNIONIC DIRAC EQUATION
The algebra of complex quaternions is denoted by H (C ) ; each complex quaternion q is of
the form q   k  0 qk i k , where qk   C , i0 is the unit and ik
3
k  1, 2,3 are the quaternionic
imaginary units:
i02  i0  ik2 ;
i0 ik  ik i0  ik , k  1, 2,3 ;
i1i2  i2 i1  i3 ; i2 i3  i3i2  i1 ;
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22
i3i1  i1i3  i2 .
Open Journal of Technology & Engineering Disciplines (OJTED)
Vol. 2, No. 1, March 2016, pp. 21~27
ISSN: 2455-6971
The complex imaginary unit i commutes with ik , k  0,3 . We will use the vector
representation
of
Vec(q )  q   k 1 qk ik
3
complex
q  Sc(q)  Vec(q) ,
quaternions:
where
Sc(q)  q0
and
, that is, each complex quaternion is a sum of its scalar part and its
vector part. Complex vectors we identify with complex quaternions whose scalar part is
equal to zero. In vector terms, the multiplication of two arbitrary complex quaternions q
and b can be written as follows:
3
q  b :  qk bk  C ,
qb  q0b0  q  b  q  b  q0b  b0 q ,
k 1
i1
q  b : q1
b1
i2
q2
b2
i3
q3  C 3 .
b3
If q  b is not equal to zero the quaternion product is non-commutative and this property we
use it in our paper. The presence of dot and vector products in the quaternion product is no
coincidence since historically W. Gibbs introduced these two products by splitting the
quaternion product and by taking q0  b0  0 [10].
Following [7], we consider the Dirac equation in its covariant form:
3
 1


   t    k  k   i  mc    t , x   0 ;
k 1

 c


for a wave function with a given energy we have   t , x   a  x  e , where a satisfies the
i t
 i
3
equation     k  k 
k 1
c
i  mc 
 a  x  0 ,

0 1

where we introduce the matrices  and  in chiral
0 
2
representation   
 with    2  1 , and  are the Pauli matrices.
 ,   

0


1 0


i
c
Now we consider a particle described by the quaternionic wave function   ( 0 ,  )  b .
The Dirac’s momentum is defined as:
i
P  ( E , p) ,
c
(1)
i
c
where P  ( E, p)  q ; here we propose a chiral quaternionic expression such that the
quaternionic momentum eigenvalue problem can be written as:
P  i  m0 c ,
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which is in fact similar to Dirac equation [12]. The evolution of this equation is expressed by
two equations:
m c2
1 
    (  i  0 )0  0 ,
c t
0  (
m c2

 i  0 )  ic  
t
(3)
,
(4)
which satisfy the Dirac equation.
Using
 0 
the
identity
     0
and
operating
eq.
(4)
with   ,
we
have
m c2
1 
(  i  0 )    0 . We deduce from Eq. (3):
2
c t
 20 
m c2 
m c2
1 
(  i  0 )(  i  0 )0  0 .
2
t
c t
(5)
Now if we define c    0 , from (3) and (4) and using the identity ic     c( )(      we obtain  0  (
m c2
m c2

1 
 i  0 )    c( )(   )  (  i  0 )0  0
t
c t
So this expression can be reduced to:
 0 
mc
1
0  i  0 0  0 ,
c t
(6)
which is equivalent to spinor representation of ordinary Dirac’s equation in quantum
mechanics [13].
We put    
mc
1
 i  0 and squaring this equation, then:
c t
m 
mc
1 2
   20  2(i  0 ) 0  ( 0 ) 2 0  0 ,
2
2 0
t
c t
(7)
m
mc
1 2
   20  2(ic 0 )  0  ( 0 ) 2 0  0 ,
2
2 0
c t
(8)
these forms of Dirac equation exhibit clearly the wave nature of the electron which can be
compared with the Klein-Gordon equation:
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Vol. 2, No. 1, March 2016, pp. 21~27
ISSN: 2455-6971
mc
1 2
   2  ( 0 ) 2  0 .
c 2 t 2
(9)
Now we consider the mapping similar to gauge transformation  '    A and 0 '  0 
1 
c t
; under these transformations the eqs. (3) and (4) give    ' (  i 
0 ' (
m c2

 i  0 ) '  ic   ' ,
t
m0 c 2

A
t
)0 '  0
and
thus they are invariant under these transformations. This
i
c
transformation is similar to Lorenz gauge of electromagnetism if we put b  A  ( V , A) and
i
P  ( E, p)  q and eqs. (3) and (4) are given by:
c
m c2
1 
  A  (  i  0 )V  0 ,
c t
V  (
(10)
m c2

 i  0 ) A  ic  A ,
t
(11)
where A and  are the complex vector magnetic potential and the scalar electric potential,
respectively. Defining
m c2


 (  i 0 ) ,
t * t
then eq. (10) corresponds to a gauge like Lorenz
gauge:
 A
1 
V  0,
c t *
(12)
and eq. (11) is the instanton solution for the electromagnetism with E parallel to B (selfdual configuration):
V 

A  E  ic A  icB .
t *
(13)
Now we analyze the connection between eq. (7) and the wave equation for the electric
field E . Let us write c    0  c  A  V so:
m 
mc
1 2
V   2V  2(i  0 ) V  ( 0 ) 2 V  0 ,
t
c 2 t 2
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this expression is equivalent to
simplified to c  (
m 
mc
c   2
A  2 c  A  2(i  0 ) c  A  ( 0 )2 c  A  0 , which is
2
2
t
c t
m 
mc
1 2
A   2 A  2i  0
A  ( 0 ) 2 A)  0 ,
t
c 2 t 2
and due to    and c  0 ,
we deduce:
m 
mc
1 2
A   2 A  2i  0 A  ( 0 ) 2 A  0 .
t
c 2 t 2
Taking  of (14) and 
(15)

of (15) and summing we have:
t
m 
mc
1 2
E   2 E  2i  0 E  ( 0 ) 2 E  0 ,
2
2
t
c t
(16)
which is the wave equation of the electric field if we consider 2i 
   (
m0 c
) 2 E  0 ; here the term    (
m0 c
)2 E  0
m0
 
and
is transformed taking the divergence of
both sides, so using the Gauss law we obtain  2   (
m0 c
)2   0
as a wave equation for the
charge density and for the electric field, therefore:
1 2


E   2 E   E 
E 0,
2
2
t

c t
(17)
where the conductivity depends on the mass. Hence the magnetic field B    A inside a
conducting media behaves like a particle with mass defined as 2i 
m0
 
[14]. Our results
can be compared with expressions founded in [13], which use P  m0 c . Here, we are
taking into account the true non-commutative nature of the quaternion equation
P  i  m0 c .
3. CONCLUSION
By adopting the quaternionic quantum mechanics, namely, the quaternionic Dirac
equation, we arrived at a dissipative or generalized Klein-Gordon equation with a particle
spin. The main result of this paper is that the Dirac equation has a deep connection with the
Maxwell’s equation under the chiral approach. The suggested theory is a new quantum
mechanics interpretation, if we consider the instanton solution for the electromagnetic field
E  icB , that is, self-dual solutions possess trivial energy-momentum properties, and the
desired free field configurations are obtainable as superposition of self-dual and anti-selfdual constituents.
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