Journal of Modern Physics, 2011, 2, 335-340
doi:10.4236/jmp.2011.25041 Published Online May 2011 (http://www.SciRP.org/journal/jmp)
Light-Front Hamiltonian and Path Integral Formulations
of the Conformally Gauge-Fixed
Polyakov D1 Brane Action
Usha Kulshreshtha1, Daya Shankar Kulshreshtha2
1Department of Physics, Kirori Mal College University of Delhi, Delhi, India
2Department of Physics and Astrophysics University of Delhi, Delhi, India
Received January 6, 2011; revised March 25, 2011; accepted April 5, 2011
Abstract
In a recent paper we have studied the Hamiltonian and path integral quantizations of the conformally
gauge-fixed Polyakov D1 brane action in the instant-form of dynamics using the equal world-sheet time
framework on the hyperplanes defined by the world-sheet time . In the present work we
quantize the same theory in the equal light-cone world-sheet time framework, on the hyperplanes of the
light-front defined by the light-cone world-sheet time
0==constant



= constant  = 
, using the standard constraint
quantization techniques in the Hamiltonian and path integral formulations. The light-front theory is seen to
be a constrained system in the sense of Dirac, which is in contrast to the corresponding case of the
instant-form theory, where the theory remains unconstrained in the sense of Dirac. The light-front theory is
seen to possess a set of twenty six primary second-class contraints. In the present work Hamiltonian and path
integral quantizations of this theory are studied on the light-front.
Keywords: Light-Front Quantization, Hamiltonian Quantization, Path Integral Quantization, Constrained
Dynamics, Constraint Quantization, Gauge Symmetry, String Gauge Symmetry, String Theory,
D-Brane Actions, Polyakov Action, Light-Cone Quantization.
1. Introduction
Polyakov D1 brane action [1-12] is one of the most
widely studied topics in string theories [1-15]. The action
possesses three well-known local gauge symmetries
given by the two-dimensional world-sheet (WS) repara-
metrization invariance (WSRI) and the Weyl invariance
(WI) [1-10]. When the action is considered under the
conformal gauge-fixing it looses the above said string
gauge symmetries as expected (owing to the conformal
gauge-fixing). In a very recent paper [11,12], we have
studied the conformally gauge-fixed Polyakov D1 brane
action (CGFPD1BA) with and without a scalar dilaton
field in the usual instant-form (IF) of dynamics, using the
equal world-sheet (WS)-time (EWST) framework, on the
hyperplanes defined by the WS-time
[9-15]. The theory without a scalar dilaton field is seen to
be an unconstrained system in the sense of Dirac [16],
whereas in the presence of a scalar dilaton field it is seen
to be a constrained system, possessing one primary and
one secondary Gauss law constraint [9-15]. In the present
work the same theory is studied on the light-front (LF)
(in the front-form (FF) of dynamics) using the equal
light-cone world-sheet time (ELCWST) framework on
the hyperplanes of the LF defined by the light-cone
world-sheet time
0==constant



== constant 
. The LF
theory becomes a constrained system in the sense of
Dirac (in contrast to the corresponding case of the IF
theory [3]), and it is seen to possess a set of 26 primary
second-class constraints. The LF theory is quantized using
the standard constraint quantization techniques in the
Hamiltonian and path integral formulations [9-21]. It is
needless to say that the LF quantization (LFQ) has several
distinct advantages over the usual IF quantization (IFQ)
[22-27]. For a recent review on LFQ of field theories we
refer to the work of Brodsky, Pauli and Pinsky [22-27].
In the next section we briefly recapitulate the IF theory
[11,12]. In Section 3, we study the LFQ of the theory

U. KULSHRESHTHA ET AL. 336
using the Hamiltonian and path integral formulations and
finally the summary and discussion is given in Section 4.
2. Recapitulation of Instant-Form Theory
In this section, we recapitulate very briefly the IF theory.
The Polyakov D1 brane action describes the propagation
of the D1 brane in a -dimensional curved background
d

h (with for the fermionic and for
bosonic D1 brane) defined by [1-15]:
=10d=26d
2
=dS


 (1a)

=; =
2
Thh


det






hG h


,1

(1b)

=; =diag1,1,XX

 
  G  (1c)

, =0,1,,1; ,=0,1d   

(1d)
Here

,

  are the two parameters describing
the worldsheet (WS). The overdots and primes would
denote the derivatives with respect to

and

. is
the string tension.
T


is the induced metric on the WS
and
G

,

X

 are the maps of the WS into the -
dimensional Minkowski space and describe the strings
evolution in space-time [1-10].
d


are the auxiliary
fields (which turn out to be proportional to the metric
tensor
h



of the two-dimensional surface swept out by
the string). One can think of as the action describing
massless scalar fields
S

d
X

in two dimensions mov-
ing on a background


. Also because the metric com-
ponents
h


h are varied in the above equation, the 2-
dimensional gravitational field


is treated not as a
given background field, but rather as an adjustable quan-
tity coupled to the scalar fields [1-8]. The action pos-
sesses the well-known three local gauge symmetries given
by the two-dimensional WS reparametrization invariance
(WSRI) and the Weyl invariance (WI) [1-12]:
h
S

=
X
XXX









X


(2a)

δ=X
 




 (2b)
=







hhhh


(2c)
δ=

  
 


 

hhhh

(2d)
 

;,=exp2,
 



 

hh



(2e)
Here

,

 is a gauge parameter corresponding
to the WSRI and
  
,=exp2 ,


   is a gauge
parameter corresponding to the Weyl symmetry. The
WSRI is defined by the first four equations involving the
two gauge parameters


and the WI is defined by the
last equation and is specified by the gauge parameter

(or equivalently by

). Also the above theory being a
gauge-invariant theory (possessing the local gauge sym-
metries including two WSRI and one WI symmetries),
could be studied under approriate gauge-fixing the way
one likes. However, one could also use the above three
local gauge symmeties of the theory to choose


h to
be of a particular form [1-12], e.g., as follows:
1 0
==
01
 





h

 (3)
This is the so-called conformal gauge (CG). In this CG
we have

=det =1

hh

(4)
and the action in this CG now becomes: S

2
11
=dS

 (5a)
1=2
T



h


hG (5b)
=2
TXX

 




 (5c)
=2
T

X
X

(5d)



=2
T
X
XXX




 

 

(5e)

 
2
2
=2
TXX








 (5f)
and =
X
X
XX










 (5g)
This is the CGFPD1BA. The IFQ of this action has
been studied by us recently in Ref. 3 and we recap it here
very briefly. The canonical momenta conjugate to
X

obtained from are:
1
L

1
:= =PT
XX








 (6)
Here the velocities 1
=
X
P
T





are expressible. The
canonical Hamiltonian density corresponding to is:
1


2
2
11
1
==
22
cT
PXLP X
T

;

 

 




1
S
(7)
The quantization of the system is trivial. The non-
vanishing equal WS-time commutation relations for the
theory described by are obtained as [1-4]:




, =δ δi



,,XP

 
 




 (8)
where


δ



 is the one-dimensional Dirac distri-
bution function.

U. KULSHRESHTHA ET AL.337
It is obvious from the above considerations that the
above theory is unconstrained in the sense of Dirac. It
may be important to emphasize here that an unconstrained
system like the above theory is a gauge-noninvariant
theory and is some what akin to a gauge-fixed gauge-
invariant theory which makes it a gauge-noninvariant
system. In the presence of a scalar dilaton field the theory
of course, becomes a constrained system in the sense of
Dirac as shown in our earlier work [11,12]. For further
details of the IF theory we refer to our earleir work of
Ref. 11 and 12. In the next section, we study the LFQ of
this theory [22-27].
3. Light-Front Quantization
In LFQ of the theory we use the three local gauge sym-
metries of the theory to choose


h to be of a particular
form as follows:
012
:== 12 0
 





h 




(9a)
and
02
:== 2 0
 





h 
(9b)
with

=det =12h

 h (9c)
This is the so-called conformal gauge (CG) in the LFQ
of the theory. Also, in the LFQ we use the LC variables
defined by [1-8]:


01
:=and :=2XXX 

 (10)
The action in the above CG in the LFQ reads S

22
=ddSL



 (11a)
2=2
=2
TXX
TXX
 








 
(11b)


=2
ii
T
X
XXX
XX
 
 








 


(11c)
 
,=,,2,,1; =2,3,,1.di d    (11d)
We now study the LFQ of the above Polyakov D1
brane action. The canonical momenta and
obtained from the above equation are:
, PP


=2,3,,25 ,
i
Pi 
 
2
:== 2
T
PX
X







 
2
:== 2
T
PX
X









 (12b)
 
2
:== 2
i
ii
T
PX
X





 (12c)
The above equations, however, imply that the theory
possesses twenty six primary constraints
1=
2
T
PX



0 





(13a)
2=
2
T
PX 


0 





(13b)
=0,=2,3
2
i
ii
T
PX i



 

 ,,25.
0
(13c)
The canonical Hamiltonian density corresponding to
is
2

22
=() ()()
ci
i
PXPXPX






 (14)
After including the primary constraints i

in the ca-
nonical Hamiltonian density 2 with the help of La-
grange multiplier fields and i, the total Hamil-
tonian density could be written as
c

,uv w
2
T

2=22
2
T
i
ii
TT
uPX vPX
T
wP X





 






 






(15)
We now treat and i as dynamical. The Hamil-
tons equations obtained from the total Hamiltonian
,uv w
22
=d
TT
HH


 (16)
e.g., for the closed bosonic strings with periodic boundary
conditions are now obtained as:
22
22
22
22
2
== ==
2
== ==
2
== ==
2
==0 ==2
==0 =
TT
TT
TT
i
ii i
i
i
TT
u
u
T
v
v
HH T
XuP v
PX
HH T
XvP u
PX
HHT
XwP w
PX
HHT
upP
pu
HH
vp
p










 

 



 







  





 




 

，
，
，
，
，
X



2
22
=2
==0 ==2
T
TT
i
iwi
i
wi
i
T
PX
v
HH T
wpP
pw









 



，X

(17)

(12a)

U. KULSHRESHTHA ET AL. 338
These are the equations of motion of the theory that
preserve the constraints of the theory in the course of
time. Demanding that the primary constraints 12 ,


and be preserved in the course of
time one does not get any secondary constraints. The
theory is thus seen to possess only twenty six constraints
12

, =2,3,,25,
ii
 

,


i
and

. The first-order Lagrangian density of
the theory is

  

I
2
2
=
=2
Oi
i
T
uvwi
i
i
i
PXPXPX
pupvp w
TuXvXwX


 

 



 











(18)
The matrix of the Poisson brackets of the constrains
j

namely,
 

P
,:=,
B
 


M is then
calculated and the nonvanishing elements of this matrix
are obtained as:

12 21
===δ
ii T




MMM (19)
The matrix


is seen to be nonsingular with the
determinant given by
M


12 13
det= δT



 




M

(20)
and the nonvanishing elements of the inverse of the
matrix


(i.e. the elements of the matrix M


1



M)
are obtained as:


111
12 21
1
===
2
ii
MMMT





 (21)
with
 
1
26 26
,,d=1δMM


 


  



(22)
Here the step function



  is defined as
 

1for >0
:= 1for <0

 







 (23)
Now, following the standard Dirac quantization pro-
cedure in the Hamiltonian formulation [5], the nonva-
nishing equal light-cone world-sheet time (ELCWST)
() commutators of the theory
describing the Polyakov D1 brane action
=== constant  


N
S are finally
obtained as [9-21]:
 
,,, =δ
2
i
XP  
 






(24a)
 
,,, =δ
2
i
XP  







(24b)
 
,,, =δ
2
i
i
i
XP  
 





,,,=
2
i
XX T










(24d)
 
,,,=
2
ii i
XX T
 






 

(24e)

,,, =δ
4
iT
PP  






 

(24f)

,,,=δ
4
ii
iT
PP
 


 








(24g)
In the path integral formulation, the transition to the
quantum theory, is, however, made by writing the va-
cuum to vacuum transition amplitude called the genera-
ting functional


2k
Z
J
k
of the theory in the presence of
external sources
J
as follows [9-21]:
 







I
22
2
:= d expdd
=dexp dd
=dexp dd2
Ok
kk
i
i
Tk
uvwi k
i
ik
ik
ZJiL J
iPXPXP
pupvp wH J
T
iuX
vXw XJ




 

 
 






 










 





X

(25)
where the phase space variables of the theory are


,,,,,
ki
i
X
XXuvw

 with the corresponding res-
pective canonical conjugate momenta:


,,,,,
kiu
PPPppp


[d ]
vw
i. The functional measure

of the generating functional


2k
Z
J is obtained
as [9-21]:



 

13
d= δdddd
dddddddd
δ0δ0
22
δ0.
2
i
uvi iw
i
i
i
TXX
PPpp XPwp
TT
PX PX
T
PX




 




 


 




  









 





(24c)
uv



under the conformal gauge-fixing it looses the above said
(26)
The LF Hamiltonian and path integral quantizations of
our theory described by the Polyakov D1 brane action
are now complete.
2
S
4. Summary and Discussion
Polyakov D1 brane action [1-12] possesses three well-
known local gauge symmetries given by the two-di-
mensional world-sheet reparametrization invariance and
the Weyl invariance [1-10]. When the action is considered

U. KULSHRESHTHA ET AL.339
udied on the
LF
portant points
he
e standard con-
st
few comments about further
so
the canonical quantization of the theory
w
ur work
w
5. References
[1] D. Luest and S. Theisen, “Lectures on String Theory,”
d M. Henneaux, “Principles of String Theory,”
Plenum Press, New York, 1988.
string gauge symmetries as expected. In a very recent
paper [11,12], we have studied the conformally gauge-
fixed Polyakov D1 brane action with and without a scalar
dilaton field in the usual instant-form (IF) of dynamics,
using the equal world-sheet-time framework, on the hy-
perplanes defined by the WS-time 0==constant

[1-21]. The theory without a scalar dilat
be an unconstrained system in the sense of Dirac [16],
whereas in the presence of a scalar dilaton field it is seen
to be a constrained system, possessing one primary and
one secondary Gauss law constraint [9-21].
In the present work the same theory is st
on field is seen to
(using the front-form of dynamics) in the equal light-
cone world-sheet time (ELCWST) framework on the
hyperplanes of the LF defined by the light-cone world-
sheet time

==constant 
 [9,10,22-27]. The
LF theory is ained system in the
sense of Dirac (which is in contrast to the corresponding
case of the IF theory [11,12]), and it is seen to possess a
set of 26 primary second-class constraints.
It may be worthwhile to record a few im
seen to become a constr
re. The unconstrained IF theory as mentioned above is
in fact analogues to the usual Klein-Gordon field theory
in three-space one-time dimension which is an uncons-
trained theory in the sense of Dirac. This in turn is equi-
valent to a gauge-noninvariant or equivalently a gauge-
fixed theory which does not have a gauge symmetry
(owing to the gauge-fixing). The corresponding LF theory
is on the other hand is a constrained system in the sense
of Dirac as seen in Section 3, where it has been explicitly
shown to be a constrained system possessing a set of 26
second-class constraints. This theory could then we used
to construct an equivalent gauge-invariant theory using
the techniques of constrained dynamics (albeit constraint
quantization) which is however, outside the scope of the
present work. The same may however, not be true of the
corresponding unconstrained IF theory.
The LF theory is quantized using th
raint quantization techniques in the Hamiltonian and
path integral formulations. It is needless to say that the
LF quantization (LFQ) has undisputedly several distinct
advantages [6] over the usual IF quantization (IFQ)
[22-27]. One of the most important advantages of the LF
framework e.g., is that the LF theory provides the largest
number of kinematical generators of the Poincare trans-
formations in Hamiltonian dynamics. For a recent review
on LFQ of field theories we refer to the work of Brodsky,
Pauli and Pinsky [22-27].
Also, we like to make a
lving the LF theory. It is possible to write down the
solutions of the LF theory on the reduced hypersurface of
the constraints of the theory where one implements the
constraints of the theory strongly and this could be
achieved in the Hamiltonian as well as in the path in-
tegral formulation of the theory. This is however, outside
the scope of the present work. If one tryies to do a simi-
lar thing with the IF theory then that would simply not be
possible because even though the IF theory is gauge-
noninvariant and is equivalent to a gauge-fixed gauge-
invariant theory but one simply can not do any of the
manipulations which could be done within the acceptable
framework of the techniques of constraint quantization
simply because (the IF theory) does not have any con-
straint structure. One well known examples of this con-
cerns the Batalin-Fradkin-Vilkovisky quantization of a
gauge-noninvariant theory where one enlarges the phase
space of a classical theory or the Hilbert space of the
corresponding quantum theory by introducing some ad-
ditional fields in to the theory by modifying the second-
class constraints of the theory in such a manner that each
of the second-class constraint of the theory becomes a
first-class constraint. This in principle, could be done
with the LF theory because its constraint structure is
known, but it would simply be impossible to do such a
thing with the IF theory for which the constraint structure
is not known.
Further, in
hile going from ELCWST Dirac brackets of the theory
to the corresponding ELCWST commutation relations
one could encounter the problem of operator ordering
[28] because the product of canoncial variable of the
theory are involved in the classical description of the
theory e.g., in the calculation of the Dirac brackets. These
variables are envisaged as noncommuting operators in the
quantized theory leading to the problem of so-called
operator ordering [28]. This problem could, however, be
resolved [28] by demanding that all the string fields and
momenta of the theory are Hermitian operators and that
all the canoncial commutation relations be con- sistent
with the Hermiticity of the operators [28].
Also it is important to mention here that in o
e have not imposed any boundary conditions for the
open and closed strings separately. There are two ways
to take these boundary conditions into account: (a) one
way is to impose them directly in the usual way for the
open and closed strings separately in an appropriate man-
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boundary conditions as the Dirac primary constraints [29]
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