Journal of Modern Physics, 2013, 4, 38-48
http://dx.doi.org/10.4236/jmp.2013.44A007 Published Online April 2013 (http://www.scirp.org/journal/jmp)
Light-Front Hamiltonian, Path Integral and BRST
Formulations of the Chern-Simons-Higgs Theory in the
Broken Symmetry Phase*
Usha Kulshreshtha1#, Daya Shankar Kulshreshtha2, James P. Vary3
1Department of Physics, Kirori Mal College, University of Delhi, Delhi, India
2Department of Physics and Astrophysics, University of Delhi, Delhi, India
3Department of Physics and Astronomy, Iowa State University, Ames, USA
Received January 20, 2013; revised February 26, 2013; accepted March 9, 2013
Copyright © 2013 Usha Kulshreshtha et al. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In the present work we study the Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory
in two-space one-time dimensions, in the so-called broken symmetry phase of the Higgs potential (where the phase


x


of the complex matter field

x


carries the charge degree of freedom of the complex matter field and is
akin to the Goldstone boson) on the light-front (i.e., on the hyperplanes defined by the fixed light-cone time). The
theory is seen to possess a set of first-class constraints and the local vector gauge symmetry. The theory being
gauge-invariant is quantized under appropriate gauge-fixing conditions. The explicit Hamiltonian and path integral
quantization is achieved under the above light-cone gauges. The Heisenberg equations of motion of the system are
derived for the physical degrees of freedom of the system. Finally the BRST quantization of the system is achieved
under appropriate BRST gauge-fixing, where the BRST symmetry is maintained even under the BRST light-cone
gauge-fixing.
Keywords: Light-Front Quantization; Hamiltonian Quantization; Path Integral Quantization; BRST Quantization;
Constrained Dynamics; Gauge Symmetry; Chern-Simons-Higgs Theory; Broken Symmetry Phase; Higgs
Potential; Spontaneous Symmetry Breaking
1. Introduction
Gauge theories in two-space one-time dimensions in-
volving Chern-Simons (CS) term coupled to matter fields
describe excitations with fractional statistics [1-8]. Such
studies form a broad field of study [1-15].
The Hamiltonian [16], path integral [17-19] and Bec-
chi, Rouet, Stora and Tyutin (BRST) [20-22], formula-
tions of the pure CS theory have been studied in Refs.
[7,8], in the instant-form (IF) quantization (IFQ) [23,24]
as well as in the light-front (LF) quantization (LFQ)
[23,24].
The CS theory in the presence of a Higgs potential has
been studied in Refs. [9-14], under appropriate gauge-
fixing conditions, in the so-called symmetry phase of the
Higgs potential [9-11] as well as in the so-called broken
(or frozen) symmetry phase [12,13] of the Higgs poten-
tial [12-15], where the phase

x
 of the complex
matter field



x

0constantxt
carries the charge degree of free-
dom of the complex matter field and is, in fact, akin to
the Goldstone boson [12-15].
The IF Hamiltonian, path integral and BRST formula-
tions of this theory have been studied in Ref. [12] on the
hyperplanes defined by the IF time:
[23,24], in the broken (or frozen) symmetry phase of the
Higgs potential [12-15] under appropriate gauge-fixing
conditions.
*Talk presented by UK at the “International Conference on Light-Cone
2011: Applications of Light-Front Coordinates to Highly Relativistic
Systems,” held at the Southern Methodist University, Dallas Campus,
Dallas, USA, May 22-27, 2011 (Published in Conf. Proceedings (Few
Body Syst. 52, 457-461 (2012)).
#Corresponding author.
In this work we study the LF Hamiltonian, path inte-
gral and BRST formulations of this CS-Higgs (CSH)
theory on the hyperplanes defined by the light-cone (LC)
time:
C
opyright © 2013 SciRes. JMP

U. KULSHRESHTHA ET AL. 39

 

01
2
xx
x 

 constant






3
,, dAx
 

[13,23,24], in the BSP of the Higgs potential.
This is in contrast to our earlier work [12] on the
quantization of the same theory, where we have studied
the IFQ of this theory in the BSP of the Higgs potential.
In the present work, on the other hand, we study its LFQ.
We further wish to emphasize that the preliminary re-
sults of this work on the LFQ of this theory are presented
in Ref. [13] and the complete details of this work are
described here in the present work.
It may be important to mention here that the LFQ of
any theory has several advantages over its IFQ for sev-
eral well known reasons [18,19,23,24]. Also because the
LF coordinates are not related to the conventional IF co-
ordinates by a finite Lorentz transformation, the descrip-
tions of the same physical result may be different in the
IF and LF dynamics. In fact, the study of a theory using
the IFQ as well as the LFQ determines the canonical
structure and the constrained dynamics of the theory
rather completely [18,19,23,24]. These are the main mo-
tivations for our present studies.
Also, different aspects of this theory have been studied
by several authors in various contexts [1-8]. For further
details about the motivations for a study of the different
aspects of the CSH theories by various authors including
a comparative description of different studies, we refer to
the work of Refs. [1-8], as well as to our earlier work of
Refs. [7-13].
In this work we present the complete details of the
LFQ of this theory in the broken (or frozen) symmetry
phase of the Higgs potential (cf. the work of Ref. [13] for
our preliminary results). After a brief recap of the theory
in the broken (or frozen) symmetry phase of the Higgs
potential in the next section, its LF Hamiltonian and path
integral formulations are presented in Section 3 and its
LF BRST formulation is described in Section 4. The
summary and discussion is finally given in Section 5.
2. Theory in Broken (or Frozen) Symmetry
Phase: A Recap
The Chern-Simons-Higgs theory in two-space one-time
dimensions is defined by the action [1-15]:
11
S
 (1a)


2
V 


12AA DD
 
 







 (1b)

224
4
  
02
V    (1c)

2
22
2
00
, 0 V    (1d)
2
,,
2π
DieADieA
  


  




(1e)


012
012
:diag1,1,1,
,0,1,2, 1
g


 


(1f)
Here

is the Chern-Simons parameter. We keep the
Higgs potential rather general, i.e., without making any
specific choice for the parameters of the potential except
that they are chosen such that the potential remains a
double well potential with 0. This action thus de-
scribes the theory in the so-called symmetry phase [9-11].
In the following we however, study this theory in the so-
called broken (or frozen) symmetry phase (BSP) [12-15],
0

x





of the complex matter field . For this pur-
pose, for the complex matter field we take [12-15]:
 
00
exp, 0xix





 

(2)


x




 is the phase of the complex matter Here




x
field . The acion of the theory in the BSP [3,4,7]
then becomes:
 
3
d,
1
:22
Sx
AAeA eA
 
 











(3)
It is important to notice here that the vector gauge
boson
A

becmes massive in the BSP. This mass gen-
eration of the vector gauge boson takes place perhaps
through a mechanism similar to the Higgs mechanism
[9-15]. The phase

carries the charge degree of free-
dom of

and is, in fact, akin to the Goldstone boson
and is to be treated as a dynamical field. Also the ground
state in the BSP is not rotational invariant. Such studies
of the theory in the broken-symmetry (superfluid) state
could be relevant for the effective theories in the con-
densed matter, as the action of the theory describes the
low-lying excitations in the BSP [12-15], as well as for
an understanding of the issue of exotic statistics in
gauge-invariant observables [1-8]. In the next section we
study the LF Hamiltonian and path integral formulations
of the above theory in the BSP.
3. Light-Front Hamiltonian and Path
Integral Formulations
The LF action of the theory in the BSP of the Higgs po-
tential reads:
2
dddSxxx

 (4a)

U. KULSHRESHTHA ET AL.
Copyrig JMP
40
ht © 2013 SciRes.




22
2
22
2222
2
:2
1
2
A
AAAAAAAA
eAAeAA eA

 


 





 
AAA





 

(4b)
possesses four primary constraints: Canonical momenta obtained from the above action
are:


12
0, π0eA 

 (6a)




LeA
 π






 (5a)
 
2
2
0,
A
AA









  
 (5b)
2
22
EA
A




 

π,,



2
E
,,
 (5c)
Here and are the momenta ca-
nonically conjugate respectively to
A
A 

and 2
A
.
The above equations however, imply that the theory
324
0, 0
22
AEA





 


(6b)
The symbol

here denotes a weak equality (WE) in
the sense of Dirac [16], and it implies that these above
constraints hold as a strong equality only on the reduced
hypersurface of the constraints and not in the rest of the
phase space of the classical theory (and similarly one can
consider it as a weak operator equality (WOE) for the
corresponding quantum theory) [16,18,19]. The canoni-
cal Hamiltonian density corresponding to is: 

2
2
2
22 2222
:π
1
22
cAAEAL
AAAAAAAAeAeAAeA


 
 
 


 




 


12
,,
(7)
After including the primary constraints 3
 and

in the canonical Hamiltonian density with the
help of the Lagrange multiplier field
c

,,
s
uv w
T

 
and the
total Hamiltonian density could be written as:
4


2
2
2
22 2222
π
22
1
22
TseAwAuEAv
AAAAAAAAeAeAAeA





 


 
 



 
 


2
d
TT
(8)
The physical degrees of freedom of the system are
governed by the reduced Hamiltonian density of the the-
ory (which is obtained by implementing the constraints
of the theory strongly) [8-13]. Also, in the present case,
The Hamilton’s equations of motion of the theory that
preserve the constraints of the theory in the course of
time could be obtained from the total Hamiltonian (and
x.
H
are omitted here for the sake of bravity):

, and
A

play the role of gauge variables and the two
pairs
Demanding that the primary constraint 1

be preserved
in the course of time, one obtains the secondary Gauss-
law constraint of the theory as:


,A





2,
A
E and are the pair of inessential
eliminable variables and a pair describing the physical
degrees of freedom of the system. Accordingly, we choose,
in the present case, the first pair namely,

,A

2
50eeA



 



as
the pair describing the physical degrees of freedom and
the other pair as the pair of inessential eliminable vari-
ables. So for writing the reduced Hamiltonian density of
the theory, we choose
(9)
The preservation of 234
,,  and 5


, for all times
does not give rise to any further constraints. The theory is
thus seen to possess only five constraints i

(with i = 1,
2, 3, 4, 5), where 123 ,,

 and 4

are primary con-
straints and 5

is a secondary constraint. Further, the
matrix of the Poisson brackets among the constraints i

,
with is seen to be a singular matrix im-
plying that the set of constraints i

1, 2, 3, 4, 5i

is first-class and
that the theory under consideration is gauge-invariant.
,π,,
A


 and

 as the
independent variables and the remaining phase space
variables as the dependent variables. The later ones are
then expressed in terms of the independent variables as:

2, πeA 
2
0 , ,
2
EAA


 

  (10)

U. KULSHRESHTHA ET AL. 41
Finally the reduced Hamiltonian density of the theory
describing the physical degrees of freedom of the system
expressed in terms of the independent variables is then
obtained as:
2
12e




  






2
d
RR
2
22
22
R
AA A eAA

 


 (11)
where
H
x is the reduced Hamiltonian of the
theory and it describes the physical degrees of freedom
of the system. Here we reminded ourselves that as an
alternative to the above, we could have equivalently ex-
pressed it in terms of the other pair namely,


,
2
A
E

,A 
instead of the pair. The field equations derived
from the Heisenberg equations of motion are then ob-
tained as:

22
ππ,R iH

  2
2e



 



(12a)

,0
R
iH   

 (12b)
2
2
AeA

 




,0
R
AH
 

2
,R iH 
 


  



(12c)
Ai

 (12d)
2
2
AeA

 




2
,R iH 

  
 (12e)
2
2
42ee
2
,R Ai AHA


 


 




2
,,


 (12f)
The vector gauge current of the theory
J
JJJ
 
 is:

2
2
ddJjxx
A
22
dd 2 xxeeA A


















 (13a)

2
2
ddJjxx
A
22
dd 2 xxeeA A















 (13b)


22
2
ddJjxx
A
22 2
dd 2 xx eeA A












0j


22
,,,0eA A  




 


(13c)
The divergence of the vector gauge current density of
the theory could now be easily seen to vanish satisfying
the continuity equation:

, implying that the
theory possesses at the classical level, a local vec-
tor-gauge symmetry. The action of the theory is indeed
seen to be invariant under the local vector gauge trans-
formations:
(14a)

 
2
,,,π0
22
AE

 


 (14b)

 
,,suew

 
 

  
2,0 suvw
v
(14c)
(14d)

  


 


2
,,
x
where xx 

 is an arbitrary function of its
arguments. In order to quantize the theory using Dirac’s
procedure we now convert the set of first-class con-
straints of the theory i

into a set of second-class con-
straints, by imposing, arbitrarily, some additional con-
straints on the system called gauge-fixing conditions or
the gauge constraints. For this purpose, for the present
theory, we could choose, for example, the following
gauge-fixing condition: 0A


0A. Here the gauge

represents the light-cone coulomb gauge and is
a physically important gauge. Corresponding to this
gauge choice, the theory has the following set of con-
straints under which the quantization of the theory could
e.g. be studied:
11 10



 (15a)


22 2
π0eA  



  (15b)
33 320
2A





 


(15c)
44 40
2
EA





 


(15d)


2
55 50eeA





60A


(15e)
(15f)

 
R The matrix


of the Poisson brackets among the
set of constraints i

with

is seen to
be nonsingular with the determinant given by
1, 2,3,4,5,6i



1
2
23
22
det R
exyxyxy

 
 







 


(16)


R
The other details of the matrix


are omitted here
for the sake of bravity. Finally, following the standard
Dirac quantization procedure, the nonvanishing equal
light-cone-time commutators of the theory, under the
gauge: 0A

are obtained as:






22
22
,, ,π,,
2
x
xx xxx
ixyx y


 





 
(17a)

U. KULSHRESHTHA ET AL.
CJMP
42


,, ,
2





22
22
,,
A
xxx
ixy





xxx
xy





(17b)

opyright © 2013 SciRes.


22
,, ,
2

2
22
,,
A
xxx
ixy




Exxx
xy





(17c)


22
,, ,
2

2
22
,,
x
xx
ie xy





Axxx
xy 
 



(17d)


222
22
,, ,,,
A
xxx Axxx
ixy xy


 







(17e)



22
22
,, ,,,
4
Ex x xx x x
ixy xy 
 





 
0
(17f)
Also, for the later use, for considering the BRST for-
mulation of the theory we convert the total Hamiltonian
density into the first-order Lagrangian density
I
L:





 


02
2
22 222222
1
22 22
suvw T
AAEAsuvw:π
I
A
AA




 






AAAAAeAeAAAAA
 
 


 


  


 
  

(18)


k
Z
In the path integral formulation, the transition to quan-
tum theory is made by writing the vacuum to vacuum
transition amplitude for the theory called the generating
functional J
k
of the theory [17-19] under the
gauge-fixing under consideration, in the presence of the
external sources
J
as:
 
dexpZJ i


3
2
dπ
k
kk suvwT
xJAAEAsuvw 
 
 
 
  
 
 


2
,,,,,,,
k
(19)


π,,,,,,,
ksuvw
E

 
Here, the phase space variables of the theory are: .
A
AAsuvw 

 with the corresponding


The functional measure d

of the generating func-
tional


Z
J respective canonical conjugate momenta: under the above gauge-fixing is obtained as:
k


 

 
   

2
2
ddd
000
22
π000
23
22 2
dd
dd ddddπddddddd
s
uvw
exyxy xyAAA
suvwz E
 
 




 





AE A
eAeeAA

 
 
 






 
 





 


0A
(20)
The Hamiltonian and path integral quantization of the
theory under the gauge: is now complete.
4. Light-Front BRST Formulation
For the BRST formulation of the theory [20-22], we re-
write the theory as a quantum system that possess the
generalized gauge invariance called BRST symmetry.
For this, we first enlarge the Hilbert space of our gauge-
invariant theory and replace the notion of gauge-trans-
formation, which shifts operators by c-number functions,
by a BRST transformation, which mixes operators with
Bose and Fermi statistics, we then introduce new
anti-commuting variables c and c
b
ˆˆˆ
,π0, ,ecA c  


(Grassman numbers
on the classical level, operators in the quantized theory)
and a commuting variable such that [20-22]:

22
ˆˆ
,AcAc 



 (21a)
2
0, , ,
22
ˆˆˆˆ
0
suvw
cE c  
 

ˆˆˆ



 
   
2
ˆˆ
,,
ˆˆ
,
(21b)
s
cuec
vcwc


 


  (21c)

  
ˆˆˆ
0,, 0ccbb 


2
ˆ0

(21d)
with the property

. We now define a BRST-in-

U. KULSHRESHTHA ET AL. 43
variant function f (a function of all the phase space vari-
ables of the BRST-invariant theory) such that ˆ0f 

.
Performing gauge-fixing in the BRST formalism implies
adding to the first-order Lagrangian density 0
I
, a triv-
ial BRST-invariant function. We could thus write e.g.:


2
22 22
22
1
22
11
ˆ
2
BRST AAAAA eAeA
AAc Ab
e

22
22
AAA
AA



 

 
 

 





 






(22)
The last term in the above equation is the extra BRST-invariant gauge-fixing term. After one integration by parts, the
above equation could now be written as:


2
22 2222
1
2
BRST AAAAAAeAeA
AAAAbAbc ccc
e

2
22
22
11
22
AA



 

 
 

 




 
 
 

b



(23)
Proceeeding classically, the Euler Lagrange equation
for reads
1
bA
e




 


0b

(24)
the requirement then implies
ˆ

1
ˆˆˆ
0bA
e



 
c

 (25)
which in turn implies
c (26)
The above equation is also an Euler-Lagrange equation
(ELE) obtained by the variation of
B
RST
ith respect to w
c. I introducing momenta one has to be careful in de-
fining those for the fermionic variables. We thus define
the bosonic momenta in the usual manner so that
n

:BRST b
A






 

(27)
but for the fermionic momenta with directional deriva-
tives we set

:;
:
c BRST
c BRST
c
c
c
c





 









c
(28)
implying that the variable canonically conjugate to is
c


and the variable conjugate to c c is


. For
writing the quantum Hamilotonian density from the La-
grangian density in the usual manner we remember that
the former has to be Hermitian so that:




22
22 22
2
2
22
π
2
111
22
B
RSTsuvwcc BRST
vw
cc
AAEAsuvwcc
suvwAAAAAAAA
eA Acc
e



 
  

 
 

 
  
 

   




 
 
 
su

  

 (29)
We can check the consistency of our definitions of the
the fermionic momenta by looking at the Hamiltons
equations for the Fermionic variables:
;
BRST BRST
cc
cc


 
 

 (30)
Thus we see that
,
B
RST cBRSTc
cc
cc


 
 

 (31)
is in agreement with our definitions of the fermionic
momenta. Also, for the operators ,,ccc and c


,
one needs to satisfy the anticommutation relations of
c


with c or of c


with c, but not of c, with
. In general, c and cc are independent canonical
variables and one assumes that:







,,0,,0,
,1,
cc cc cc
cc cc



 
 (32)


, means an anti-commutator. We thus see that where

U. KULSHRESHTHA ET AL. 44
the anti-commutators in the above equation are non-
trivial and need to be fixed. In order to fix these, we de-
mand that c satisfy the Heisenberg equation:


,BRST cic

 (33)
and using the property 22 0cc one obtains



,ccc

 ,
BRST
c (34)
The last three equations then imply:

,cci


Q
20Q
,1cc

 (35)
Here the minus sign in the above equation is nontrivial
and implies the existence of states with negative norm in
the space of state vectors of the theory. The BRST charge
operator is the generator of the BRST transforma-
tions. It is nilpotent and satisfies . It mixes op-
erators which satisfy Bose and fermi statistics. According
to its conventional definition, its commutators with Bose
operators and its anti-commutators with Fermi operators
for the present theory satisfy:





2
2
,,,
,2
,,
A
QAQAQ
EQ






 
Q c 


(36a)
 
2
π,,,Qcc Qe



  
 2
cec



 


(36b)



22
2
AAA


,
1π
cQ
Ee 
 



  
 



(36c)

,1cQ




2
e eA 



Q
(36d)
All other commutators and anti-commutators involv-
ing vanish. In view of this, the BRST charge opera-
tor of the present theory could be written as:


2
A A




22
d
π
2
QxiceeA
icE eA





 






  





(37)
This equation implies that the set of states satisfying
the conditions:
20
2A


 


0,




 (38a)

0, π
2
EA 



 0
eA


 

(38b)
20eeA



  (38c)
belong to the dynamically stable subspace of states

satisfying 0  Q, i.e., it belongs to the set of BRST-
invariant states. In order to understand the condition
needed for recovering the physical states of the theory we
write the operators c and c
cc
in terms of fermionic
annihilation and creation operators. For this purpose we
consider the equation: 

. The solution of this
equation gives the Heisenberg operator


c

where



x
  is the light-cone time variable, (and corre-
spondingly


c) as:





††
ee;e e
ii ii
cBDcBD
 


 
0
(39)

imply which at the time





††
0;0ccBDccB D (40a)






††
0;
0
cc iBD
cc iBD


 
 (40b)
By imposing the conditions




22 ,,0cc cccc


 (41a)




,,cc icc

 (41b)
one then obtains


22
,BBDD
2
†
B

††
,BD2
†
D0 (42a)







 
††††
,,, ,0BBDDBDB D

 (42b)





 
††††
,,,,0BBDD BD BD

 (42c)






††††
,,, ,1BBDDBD DB

 (42d)





 
††††
,,, ,1BB DD BD DB


22
BD
(42e)
with the solution
2
†
B2
†
D (43a)

0







 
††
††
,,, ,0BD BD BDBD

 

(43b)

††
11
,;,
22
BBDD

 (43c)
We now let 0 denote the fermionic vacuum for
which
000BD (44)


Now by defining to have norm one, we have 0
††
11
00 ,00
22
BB DD
 (45)


so that
††
)0, 00BD (46)


The theory is thus seen to possess negative norm states
in the fermionic sector. The existence of these negative
norm states as free states of the fermionic part of
B
RST


U. KULSHRESHTHA ET AL. 45

is however, irrelevant to the existence of physical states
in the orthogonal subspace of the Hilbert space. In terms
of fermionic annihilation and creation operators the
quantum Hamiltonian density is:


††
2
2
1
2
BRST suvw
suvwBBDD
 



 



 
 



2
22 22 22
11
22
AAAAAAAA eAAe

  



 

(47)
and the BRST charge operator is:


22
2
dπQxiBeeAiEeAAA
A


 




 














2
2
2
π
2
iDee AiEeAA


  


  
(48)
Now because 0Q  
Q, the set of states annihiliated by contains not only the set for which the constraints
of the theory hold but also additional states for which 22
ˆˆ ˆ
,π0, ,
ˆˆ
,
ecA c
AcAc







 
 
(50a)
0  BD  (49a)
2
ˆˆˆ
0,, ,
22
ˆˆˆˆ0
suv w
cEc

 




 

2
0,0
2A




 


(50b)
(49b)

0, π0eA  



2
EA





2
ˆˆ
,,
ˆˆ
,
(49c)

20eeA



 
s
cu ec
vcwc


 


 (50c)

 
(49d)
ˆˆ ˆ
0,, 0ccbb
 (50d)


The Hamiltonian is also invariant under the anti-BRST
transformation given by: with generator or anti-BRST charge
 
22
2
dπ
2
QxiceeAicE eAAA



 







 (51)
which in terms of annihilation and creation operators reads:


2
†2
2
π
2
π
2
BeeAiEeA AA
e AiEeAAA




 

 




 











2?
dQxi
iD e






(52)
We also have The states for which the constraints of the theory hold,
satisfy both of these conditions and are in fact, the only
states satisfying both of these conditions, since although
with (43),


,0;, 0
BRST BRST
QQ


 


d
QQ (53)

2
with
B
RST BRST
x 
Q
(54)

 
†† ††
22BB DDBBDD (56)
and we further impose the dual condition that both
and Q annihilate physical states, implying that:
0and; 0QQ   (55)
there are no states of this operator with †0B  and

†0D


, and hence no free eigenstates of the fer-
mionic part of
B
RST
 that are annihiliated by each of

U. KULSHRESHTHA ET AL. 46
†
,,BB D† D, and . Thus the only states satisfying
0  Q and 0  Q are those that satisfy the con-
straints of the theory. Now because 0Q
 , the set of
states annihilated by Q contains not only the set of
states for which the constraints of the theory hold but
also additional states for which the constraints of the
theory do not hold in particular. This situation is, how-
ever, easily avoided by aditionally imposing on the the-
ory, the dual condition: 0  Q

and 0  Q

. Thus
by imposing both of these conditions on the theory si-
multaneously, one finds that the states for which the con-
straints of the theory hold satisfy both of these conditions
and, in fact, these are the only states satisfying both of
these conditions because in view of the conditions on the
fermionic variables and cc one cannot have simul-
taneously and ,cc

,cc
, applied to

to give
zero. Thus the only states satisfying 0Q 

and
0  Q are those that satisfy the constraints of the
theory and they belong to the set of BRST-invariant as
well as to the set of anti-BRST-invariant states.
Alternatively, one can understand the above point in
terms of fermionic annihiliation and creation operators as
follows. The condition 0
 Q

implies that the set of
states annihiliated by contains not only the states for
which the constraints of the theory hold but also addi-
tional states for which the constraints do not hold. How-
ever,
Q
0  Q guarantees that the set of states anni-
hiliated by Q contains only the states for which the
constraints hold, simply because †0B 

and
†
D0  . Thus in this alternative way also we see that
the states satisfying 0QQ   are only those
states that satisfy the constraints of the theory and also
that these states belong to the set of BRST-invariant and
anti-BRST-invariant states. This completes the BRST
formulation of the theory.
5. Summary and Discussion
In this work we have presented the complete details of
the LF Hamiltonian, path integral and BRST formula-
tions of the CSH theory on the hyperplanes defined by
the light-cone (LC) time:

01
2
xx
xconstant




 



constant

[23,24], in the BSP of the Higgs potential. The prelimi-
nary results of our present investigations are given in Ref.
[13].
Further, our present studies are in contrast with our
earlier work of Ref. [12] on the quantization of the same
theory, where we have studied the IFQ of this theory on
the hyperplanes defined by the IF time:
0
xt
in the BSP of the Higgs potential (instead of its LFQ). In
the present work, on the other hand, we have studied its
LFQ on the hyperplanes defined by the LC time:


01
constant
2
xx
x 




 


.



x
In the BSP of the Higgs potential, the phase


of the complex matter field

x

x
carries the charge
degree of freedom of the complex matter field and is, in
fact, akin to the Goldstone boson [12-15]. The theory in
the so-called symmetry phase of the Higgs potential has
also been studied by us earlier [9-11]. Also, different
aspects of this theory have been studied by several au-
thors in various contexts [1-8]. For a comparative de-
scription of different studies, we refer to the work of Refs.
[1-8], as well as to our earlier work of Refs. [7-13].
What actually necessitates our present studies is an
important fact that the LFQ of any theory has several
advantages over its IFQ for several well known reasons
[23,24]. Also because the LF coordinates are not related
to the conventional IF coordinates by a finite Lorentz
transformation, the descriptions of the same physical
result may be different in the IF and LF dynamics. In fact,
the study of a theory using the IFQ as well as the LFQ
determines the canonical structure and the constrained
dynamics of the theory rather completely [18,19].
The LFQ has several advantages over the conventional
IFQ [23,24]. In particular, for a LF theory seven out of
ten Poincare generators are kinematical while the IF the-
ory has only six kinematical generators. In our treatment,
we have made the convention to regard the light-cone
variable 


as the LF time coordinate [23,24] and
the light-cone variable
x

has been treated as the lon-
gitudinal spatial coordinate [23,24]. The temporal evolu-
tion of the system in
x

is generated by the total Ham-
iltonian of the system. If we consider the invariant dis-
tance between two spacetime points in dimen-
sion [23,24]:

21-



22 2
200 1122
:;xyxyx yxy
IFQ


  







2
222
:2 ;xyx yxyxy
LFQ

(57a)


 




00
(57b)
then we find that in the instant-form, the points on the
x
y

 constant hyperplanes, have space-like separa-
tion except when they are coincident when it becomes
light-like one. On the light-front, however, with
y

x

 constant, the distance becomes independent
of



x
y  and the separation again becomes space-
like. The LF field theory therefore does not necessarily

U. KULSHRESHTHA ET AL. 47
x
need to be local in ,
 even if the corresponding in-
stant-form theory is formulated as a local one. The non-
vanishing equal-time commutators of the IF theory are
nonlocal and nonvanishing for space-like distances and
violate the microcausality principle [23,24]. The nonvan-
ishing equal light-cone-time commutators for the present
theory, on the otherhand would be nonlocal in the
light-cone space variable
x

0A0A
and nonvanishing only on
the light-cone. There would therefore be no conflict with
the microcausality principle for the LF theory unlike the
case of the equal-time commutators in the IF theory. For
further details on the Dirac’s different relativistic forms
of dynamics, we refer to the work of Refs. [23,24].
The constrained dynamics of the present theory in IFQ
as studied by us in Ref. [12], reveals that the theory pos-
sesses a set of four constraints where three constraints are
primary and one secondary Gauss law constraint. The
matrix of the Poission brackets of these two constraints is
singular and therefore they form a set of first-class con-
straints, implying in turn, that the corresponding theory is
gauge-invariant. The theory is indeed seen to possess a
local vector gauge symmetry. For further details of this
work, we refer to the work of Ref. [12].
The LFQ of this theory, on the other hand, reveals that
the LF theory possesses a set of five constraints where
four constraints are primary and one is a secondary
Gauss law constraint. The matrix of the Poission brackets
of these five constraints is singular implying that they
form a set of first-class constraints. This implies in turn,
that the corresponding theory is gauge-invariant. The
theory is indeed seen to possess a local vector gauge
symmetry, and correspondingly there exists a conserved
local vector gauge current.
Now because the set of constraints of the theory is
first-class, one could quantize the theory under some
suitable gauge-fixing as we have done in our present
work for the Hamiltonian and path integral quantization
of our theory. For this we have choosen the gauge-fixing:
. The gauge here represents the light-
cone coulomb gauge. This gauge choice is not only ac-
ceptable and consistent with our quantization procedures
but is also physically more intersting gauge choice rep-
resenting the light-cone coloumb gauge.
However, in the above Hamiltonian and path integral
quantization of the theory under some gauge-fixing con-
ditions the gauge-invariance of the theory gets broken
because the procedure of gauge-fixing converts the set of
first-class constraints of the theory into a set of sec-
ond-class one, by changing the matrix of the Poission
brackets of the constraints of the theory from a singular
one into a non-singular one. In view of this, in order to
achieve the quantization of our gauge-invariant theory,
such that the gauge-invariance of the theory is main-
tained even under gauge-fixing, one of the possible ways
is to go to a more generalized procedure called the BRST
quantization [20-22], where the extended gauge symme-
try called as the BRST symmetry is maintained even un-
der gauge-fixing. It is therefore desirable to achieve this
so-called BRST quantization also if possible. This there-
fore makes a kind of complete quantization of a theory.
The light-cone BRST quantization of the present theory
has been studied by us in the present work, under some
specfic gauge choice (where a particular gauge has been
choosen by us and which is not unique by any means). In
this procedure, when we embed the original gauge-in-
variant theory into a BRST system, the quantum Hamil-
tonian density
B
RST (which includes the gauge-fixing
contribution) commutes with the BRST charge as well as
with the anti-BRST charge. The new (extended) gauge
symmetry which replaces the gauge invariance is main-
tained (even under the BRST gauge-fixing) and hence
projecting any state onto the sector of BRST and
anti-BRST invariant states yields a theory which is iso-
morphic to the original gauge-invariant theory.

6. Acknowledgements
This work was supported in part by the National Science
Foundation under Grant No. PHY0904782 and the De-
partment of Energy under Grant No. DE-FG02-
87ER40371.
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