J. Electromagnetic Analysis & Applications, 2011, 3, 22-26
doi:10.4236/jemaa.2011.31004 Published Online January 2011 (http://www.SciRP.org/journal/jemaa)
Electron Monopole Duality in Quantum Hall
Effect
Pawan Ku. Joshi, Praveen Singh Bisht, Om Prakash Singh Negi
Department of Physics, Kumaun University, S. S. J. Campus, Almora1(Uttarakhand), India.
Received November 25th, 2010; revised December 20th, 2010; accepted December 29th, 2010.
ABSTRACT
Starting from the duality between electric and magnetic field, we have made an attempt to discuss the quantum hall
effect from the consideration of magnetic monopole in view of electron monopole duality. Starting from the dual dy-
namics of electric and magnetic charges, we have reformulated a consistent theory of quantum hall effect in presence of
monopole. Speculating th e existence of magnetic monop oles in magnetic materials (metals), we have accordingly modi-
fied the parameters; like drift velo city, current density, Hamiltonian and eigen valu es and eigen function for harmoni c
oscillator; resposible to examine the quantum Hall effect in metals.
Keywords: Hall Effect, Monopole, Quantum Mechanics
1. Introduction
The Hall Effect was discovered by Sir Edwin Hall [1] in
1879 while he was under graduate student in Johns Hop-
kins University. But at that time, even the electron was
not experimentally discovered. Clear understanding had
to wait until quantum mechanics came in to appearance.
A hundred years later, the Hall effect was revived as a
source of wonderful physics. In 1980, Klaus von Klitzin g
discovered [2] that two dimensional electron gas, at very
low temperatures and strong magnetic fields, displays a
remarkable quantization of the Hall conductance. Namely,
the graph of the Hall conductance as the function of the
magnetic field, is a staircase function, where the value of
the Hall conductance at the plateaus is, to great accuracy,
an integer multiple of

2=1 25812.807572eh

. This
discovery led to superior standards of resistance and
von-Klitzing was awarded the Nob el Prize in 1985 for his
discovery. In a 1981 Robert Laughlin put forward [3] an
argument for the quantization of the Hall conductance.
This argument played a seminal role in the development
of the theory of the Integer Hall effect. On the other hand,
the lack of symmetry between electric and magnetic
fields is one of the oldest puzzles in physics. One of the
biggest unresolved question s in theoretical physics is that
associated with the quantization of electric charge, i.e.
why the observed electric charges in all the electrically
charged matter is an integer multiple of a “fundamental
charge” ‘e’, the electron charge.Why is it possible to iso-
late positive and negative electric charges, but not north
and south magnetic poles? P. A. M. Dirac [4] introduced
the idea of magnetic charge (monopole) ‘g’ in the uni-
verse to answer this question. As a matter of fact, the
quantum dynamics of a particle with electric charge ‘e’
under the influence of the magnetic field generated by
such particle is well defined if the well known Dirac
Quantization co ndition [4] ‘=12egn c’, (where nZ

is an integer) is satisfied. But unfortunately all experi-
mental searches [5] to discover the magnetic monopoles
have been found to be negative and proved their exis-
tence fruitless. However, a group of physicists has been
claimng [6] now a days the possibility of indirect evi-
dences of magnetic monopoles in the consnsed matter
physics.. So, it is being speculated [7] that that magnetic
materials may provide a new context for observing mag-
netic monopoles which plays an important role in con-
densed matter physics. While a magnetic monopole par-
ticle has never been conclusively observed, there are a
number of phenomena [6,7] in condensed-matter physics
where a material, due to the collective behavior of its
electrons and ions, can show emergent phenomena that
resemble magnetic monopoles in some respect. So,
keeping in view the recent interests on monopole and
their possible role to produce strong magnetic field re-
sposible for quantum Hall effect, in this paper, we have
made an attempt to investigate the consistent theory of
classical and quantum Hall effect for which the magnetic

Electron Monopole Duality in Quantum Hall Effect23
field is produced due to the presence of magnetic mono-
pole like electron produces electric field. Starting from
the duality between electric and magnetic field, we have
tried to study the theory of quantum hall effect from the
consideration of magnetic monopole in view of electron
monopole duality. Taking into account the dual dynamics
of electric and magnetic charges, we have reformulated a
consistent theory of quantum hall effect in presence of
monopole. Speculating the magnetic monopoles in mag-
netic materials, we have accordingly, modified the pa-
rameters; like drift velocity, current density, Hamiltonian,
eigen values and eigen function for harmonic oscillator;
resposible to examine the quantum Hall effect in metals.
2. Dual Electrodynamics
The concept of electromagnetic (EM) duality has been
receiving much attention [8,9] in gauge theories, field
theories, Supersymmetry and super strings. Duality in-
variance is an old idea introduced a century ago in clas-
sical electromagnetism for the following Maxwell’s
equations in vacuum (using natural units ,
==1c
space-time four-vector


=,,,,
x
txyz


=
x
x




and

=1,1,1,1=




  through out the text)
[10],
=0; =0;
=; =
EH
;
H
E
EH
tt
 

 

  

  

(1)
as these were invariant not only under Lorentz and con-
formal transformations but also invariant under the fol-
lowing duality transformations,
cossin ;
sincos ;
EE H
HE H





 
 
  (2)
where and and E

H
 are respectively the the electric
and magnetic fields. For a particular value of =2

 ,
Equations (2) reduces to
;EHH E
 
;

(3)
which can be written as
01 .
10
EE
H
H
 

  

 


 
 

Let us introd
(4)
uce a complex vector =EiH
 

=1i

so that the Maxwell’s Equations (1) be writ-
ten as
=0;= ;it

 

  (5)
which is also invariant under following duality transfor-
mations

exp .i 




(6)
The duality symmetry is lost if electric charge and
current source densities enter to the conventional Max-
well’s equations given by
=; =
=0; =;
eE
EHj
t
H
HE
t


 

 

;



  (7)
where




=,
e
j

 j

is described as four-current source
density and symbol e is used for electric charge. Conse-
quently, Maxwell’s equations may be solved by intro-
ducing the co ncept of vector potential in either two ways.
The conventional choice has been used as
=;=
A
EgradH
t

;A







(8)
where




=,
A
A



 is denoted as the four potential.
Accordingly, the second pa ir of the Maxwell’s Eq uations
(7) becomes kinematical identities. So the dynamics is
contained in the first pair. Equation (5) is now modified
as
=; =
eii
t

 ;j

 




(9)
which is no more invariant under duality transformations
(6). Here if we may consider the another alternative way
to write
=;=
B;
H
grad EB
t










B
(10)
by introducing another potential

, we see

=,B



that source free Maxwell’s Equations (1) are unchanged
but but Maxwell’s Equations (7) are changed as
=0; =;
=; =
m
E
EH
t
;
H
HEk
t


 






 
(11)
where




=,
m
k

 k

. Here, the first pair becomes kin-
ematical whereas the dynamics is contained in the second
pair. Equation (11) may also be obtained if we apply the
transformations (3-4) along with the following duality
transformations for potential and current i.e.
01
;;
10
01
;.
10
AA
ABBA BB
jj
jkkj kk

 




 


 



 
 


 



 
(12)

Electron Monopole Duality in Quantum Hall Effect
24
So, we may identify the potential
 
=,B



B

 as the
dual potential and the current

 
=,
m
kk 


is used as
the dual current. The symbol m is written for magnetic
(dual of electric) case. Correspondingly, the differential
Equations (11) are iden tified as the dual Maxwell’s equa-
tions and accordingly one can develop the dual electro-
dynamics. Thus, the concept of electromagnetic duality in
the Maxwell’s equations establishes the connection be-
tween electric and magnetic charge as,
01
;10
ee
egge
g
g
 

 

 
(13)
where g is described as the dual electric charge (charge of
magnetic monopole). Hence we may recall the dual elec-
trodynamics as the dynamics of pure magnetic monopole
and the corresponding physical variables associated there
are described as the dynamical quantities of magnetic
monopole.
The Lorentz force equation of motion for a dual charge
(i.e magnetic monopole) may now be written from the
duality Equations (3) and (13) as

===
m
dp

F
mxg HvE
d 

 
 (14)
where is the momentum, and ==pmxmv
 

f

 is a
force acting on a prticle of charge g, mass m and moving
with the velocity in electromagnetic fields.
a
v

3. Hall Effect in Presence of Monopoles
The QHE is a manifestation of quantum mechanics ob-
servable at macro- scopic scales. In order to illustrate the
role of magnetic monopole in classical and Quantum Hall
Effect, let us start with the Lorentz force Equation (14). It
is to be noted that for the case of pure monopole, we may
assume the magnetic field as stationary and electric field
is obtained for moving magnetic particle containing
monopole. If the volume charge density of a magnetic
monopole is ,
m

then the magnetic current density k

is described as
=m kv


(15)
and accordingly the continuity equation reduces to
.
m
kt




  (16)
So the Biot-Savart law in presence of magnetic mono-
pole charge is associated with electric field and is thus
defined as

2
0
ˆ
1
=4m
I
r
Er dl
r 


 (17)
where Im is a magnetic current, 0

is permittivity of free
space. So, the Lorentz-Drude theory is modified for mag-
netic conductivity and gives rise to the following expres-
sion for drift velocity as
=
d
vgH
m



 (18)
where

is the relaxation time and m is the mass of
magnetic monopole. So, the magnetic current density is
modified as
0
===
dg
kngvngH H
m






(19)
where 2
0=ng m

 . In the study state case the Lorentz
force Equation (14) is vanishing and thus leads to
=.
vE
Hc




 (20)
Hence, the current (15) is described as
0
=.
vE
kc 



(21)
Thus, the expressions for steady state behavior of
conductivity and resistivity become
=;=
xx xyxx xy
yx yyyx yy
 

 


 .



(22)
Hence, we get
=kH.


 (23)
Accordingly, we may modify the drift velocity for
magnetic monopole. Thus the classical Hall effect for two
dimensional case in presence of monopoles describes
0
0
=;
=;
x
xcy
y
ycx
H
jj
H
jj




 (24)
where c

is the cyclotron frequency given by
=
c.
g
E
mc
 (25)
So, we can easily get
00
1
==, ==
c
xx yyxy yx


 


 (26)
and


2
2
==
1
==
1
o
xx yy
c
oc
xy yx
c

  ;







(27)
Hence, the relation between conductivity and resistiv-
ity is described as

Electron Monopole Duality in Quantum Hall Effect25
2
=, =
22
2
xy
xx
xxxy .
x
xxy xxxy






(28)

for classical Hall effect. In quantum mechanics, the Ham-
iltonian is ( along the X- direction) 
2
1
=
2.
x
gB
pg
mc




H


(29)
Now we choose the Landu gauge in which the vector
potential is indep e ndent of Y— coordinate as

=0, ,0.
x
AE (30)
Let us take a wave function which has a plane-wave
dependence on the Y— coordinate as
 
,=
ik y
y
x
ye x 

 (31)
and substituting the Equation (31) into the Schroedinger
equation, we get

 
22
22
2
1=
22
cmc y
dmxIkgHxxx
mdx


 




(32)
where Ic is the classical cyclotron orbit radius .
=
c
c
I.
g
H





(33)
So, the eigan values and eigan states are described as

2
2
1
=22
iccy
cc
gH
Ei gHIkm




 


 


(34)





20
2
0
,= exp2
ik y
y
ic ci
x
x
xyex xIHI 


 







(35)
where 2
02
=cy
c
g
E
xIkm

.
4. Summary and Conclusions
The precision of the quantization in the Hall effect is re-
markable in that it takes place in systems that are impre-
cisely characterized on the microscopic scale. Different
samples may have dierent distributions of impurities,
different geometry and different concentrations of elec-
trons. Nevertheless, whenever their Hall conductances
are quantized, the quantized values mutually agree with
great precision. The quantum Hall effect may also be
interpreted [11] as a measurement of the fine structure
constant so that the Hall conductance may have topo-
logical signicance. Since magnetic monopoles have the
topological origin [12] and the quantum Hall effect is
described in terms of strong magetic field, we have dis-
cussed here the theory of classical and quantum Hall ef-
fect in presence of magnetic monopole so that the mag-
netic field can be obtained directly instead of rotating an
electric charge. In order to seek the existence of magnetic
monopoles in condensed matter physics [6,7], particu-
larly in case of Hall effect, in the foregoing analysis, we
have discussed the manifestly covariant theory of mag-
netic monopoles and established the connections among
the various parameters of classical and quantum Hall
effect in terms of electron monopole duality invariance.
5. Acknowledgements
One of us OPSN is thankful to Professor H. Dehnen,
Universitt Konstanz, Fachbereich Physik, Postfach-M
677, D-78457 Konstanz, Germany for his hospitality at
Universitt Konstanz. He is also grateful to German Aca-
demic Exchange Service (Deutscher Akademischer
Austausch Dienst), Bonn for financial support under
DAAD re-invitation programme.
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