Journal of Modern Physics, 2013, 4, 57-69
http://dx.doi.org/10.4236/jmp.2013.44A009 Published Online April 2013 (http://www.scirp.org/journal/jmp)
Instant-Form and Light-Front Hamiltonian and Path
Integral Formulations of the Conformally Gauge-Fixed
Polyakov D1-Brane Action in the Presence of a Scalar
Axion Field and an U(1) Gauge Field*
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
Email: #ushakulsh@gmail.com, dskulsh@gmail.com
Received February 9, 2013; revised March 13, 2013; accepted March 25, 2013
Copyright © 2013 Usha Kulshreshtha, Daya Shankar Kulshreshtha. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original
work is properly cited.
ABSTRACT
Recently we have studied the instant-form quantization (IFQ) and the light-front quantization (LFQ) of the conformally
gauge-fixed Polyakov brane action using the Hamiltonian and path integral formulations. The IFQ is studied in the
equal world-sheet time framework on the hyperplanes defined by the world-sheet time and the LFQ
in the equal light-cone world-sheet time framework, on the hyperplanes of the light-front defined by the light-cone
world-sheet time . The light-front theory is seen to be a constrained system in the sense of
Dirac in contrast to the instant-form theory. However, owing to the gauge anomalous nature of these theories, both of
these theories are seen to lack the usual string gauge symmetries defined by the world-sheet reparametrization
invariance (WSRI) and the Weyl invariance (WI). In the present work we show that these theories when considered in
the presence of background gauge fields such as the NSNS 2-form gauge field
D1
constant



constant
0


,B

or in the presence of
1U

,A
gauge field
,C
and the constant scalar axion field
0cost ant


, then they are seen to possess the usual string gauge
symmetries (WSRI and WI). In fact, these background gauge fields are seen to behave as the Wess-Zumino or
Stueckelberg fields and the terms containing these fields are seen to behave as Wess-Zumino or Stueckelberg terms for
these theories.
Keywords: Lagrangi an an d Hamiltonian A pproach; Hamilt oni an Q uantization; Path Integral Quantizat i on; Light - F r o nt
Quantization; Theory of Quantized Fields; Constrained Dynamics; D-Brane Actions; Polyakov Action;
Strings and Branes; String Gau ge Symmetry; Gauge Field Theories
1. Introduction
Study of D-brane actions is a domain of wider interest
[1-20] in string theories. Polyakov D-brane action [1-
8,12-20] does not involve any square root [1-11] and is in
particular, simpler to study. Recently, we have studied
the instant-form (IF) q uantizati on (I FQ) of this action [12]
for the D1 brane in the conformal gauge (CG), using the
Hamiltonian [21] and path integral [22-25] formulations
in the instant-form (IF) of dynamics (on the hyperplanes
defined by the world-sheet (WS) time )
[26,27]. We have also studied its LFQ [13-20] using the
light-front (LF) dynamics (on the hyperplanes of the LF
defined by the light-cone (LC) WS time
constant

 ) [26-33].
The LF theory [13-20] is seen to be a constrained
system in the sense of Dirac [21], which is in contrast to
the corresponding IF theory [12], where the theory
remains unconstrained in the sense of Dirac [21]. The LF
theory is seen to possess a set of twenty six second-class
*Part of this work was presented by DSK as an Invited Talk at “Interna-
tional Conference on Light-Cone Physics LC2011: Applications o
f
Light-Cone Coordinates to Highly Relativistic Systems”, held at the
Southern Methodist University, Dallas, Texas, May 22-27, 2011.
#Corresponding author.
C
opyright © 2013 SciRes. JMP
U. KULSHRESHTHA, D. S. KULSHRESHTHA
58
h
contraints [13-20]. Further, the conformally gauge-fixed
Polyakov D1 brane action (CGFPD1BA) describing a
gauge-noninvariant (GNI) theory (being a gauge-fixed
theory) is seen to describe a gauge-invariant (GI) theory
in the pre sence of an antisymmetric NSNS 2-form gauge
field

,B

B
[13-20].
Recently we have shown [13-20] that this NSNS
2-form gauge field behaves like a Wess-Zumino (WZ)
field and the term involving this field behaves like a WZ
term for the CGFPD1BA [13-20]. We have also studied
the Hamiltonian and path integral formulations of the
CGFPD1BA with and without a scalar dilaton field in the
IF [12] as well as in the LF [13-18] dynamics. In both the
above cases the theory is seen (as expected) to be gauge-
noninvariant (GNI), possessing a set of second-class
constraints in each case, owing to the conformal gauge-
fixing [1-8, 12 - 20] of the theory.
The CGFPD1BA being GNI does not respect the usual
str ing gau ge symmetries defined by the world-sheet (W S)
reparametrization invariance (WSRI) and the Weyl in-
variance (WI) [1-8,12-20]. However, in the presence of a
constant 2-form gauge field
it is seen [13] to de-
scribe a gauge-inavriant (GI) theory [13-20] respecting
the usual string gauge symmetries defined by the WSRI
and the WI.
The IF and the LF Hamiltonian and path integral for-
mulations of the CGFPD1BA have been studied by us in
Refs. [12-18]. The IF and the LF Hamiltonian and path
integral formulations of th is theory in the presence of the
constant 2-form gauge field B
have been studied by
us in Ref. [13].
The question of the string gauge symmetries associ-
ated with the Polyakov D1 brane action in the presence
of some other background fields such as the
1

,A
U
gauge field

, and the constant scalar axion
field C

1U

,A
have been considered by us in Ref. [19].
In the present work, we study the CGFPD1BA in the
presence of some other background fields such as the
gauge field

and the constant scalar
axion field ,C
B
In the next section we recap the basic essentials of the
CGFPD1BA in IFQ as well as in th LF quantization
(LFQ) [12-18]. In Section 3, we recap the basic essentials
of this theory in the presence of the constant NSNS
2-form gauge field
[13]. In Section 4, we study the
IFQ [26,27] as well as the LFQ [26-33] of the
CGFPD1BA in the presence of some other background
fields such as the gauge field
1U

,A
and the
constant scalar axion field
,C
[19,20]. Finally the
summ ar y and discussion is give n in Sectio n 5 .
2. Conformally Gauge-Fixed Polyakov D1
Brane Action (CGFPD1BA): A Recap
The Polyakov D1 brane action in a d-dimensional curved
background
2
dS
is defined by [1-8,12-20]:

(1a)
2
Thh G


 (1b)
det ,hhG XX

 
 (1c)
diag1, 1,, 1

(1d)
 

,0,1,.,2,3,,1,, 01ii dIFQ
 
  (1e)
,,,.,2,3,,1,,,ii dIFQ
 
   (1f)
,
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
G
is the induced metric on the WS
and
,X
h
are the maps of the WS into the d-
dimensional Minkowski space and describe the strings
evolution in space-time [1-8,12-20].
Here
are the auxiliary fields (which turn out to
be proportional to the metric tensor
of the two-
dimensional surface swept out by the string). One can
think of as the action describing d-massless scalar
fields S
X
in two-dimensions moving on a curved
background h
Also because the metric components
.
h
are varied
in the above equation, the 2-dimensional gravitational
field h
is treated not as a given background field, but
rather as an adjustable quantity coupled to the scalar
fields.
The above action has three local gauge symmetries
given by the 2-dimensional (2D) WSRI and WI as fol-
lows:
X
XXX

 
(2a)
XX
 


hhhh
  
(2b)
 
hhhh
(2c)
  
 
 
  (2d)
 
, ,exp2, ,hh
 

  (2e)
where the WSRI is defined for the two parameters
,



; and the WI and is specified by a func-
tion
,

1
. In the following we would, however,
work in the so-called orthonormal gauge where one sets
h
.
Also for the CGFPD1BA one makes use of the fact
that the 2D metric
is also specified by three inde-
pendent functions as it is a symmetric metric. One
can therefore use these gauge symmetries of the theory to
22
Copyright © 2013 SciRes. JMP
U. KULSHRESHTHA, D. S. KULSHRESHTHA 59
choose h
to be of a particular form in the IFQ (on the
hyperplanes defined by ) as follows:
0
:,h
constantxt
:h

 

h
 

10
01
(3)
In the IFQ we take

10
01

(4a)
h
 

(4b)
with

deth

 1h (5)
In LFQ we use the Light-Cone (LC) variables defined
by:

012X
constant

:;:XX


  (6)
In the LFQ (on the hyperplanes defined by
) we take:
x
012
12 0



02
20




:h
 
 (7a)
:h
 
(7b)
with

1
t2
hh


S
2
d
NN
de (8)
The action in the CG (in IFQ as well as in LFQ)
finally becomes: reads [1-8,12-20]:
S
(9a)
2
NT
X
X




0,1,0,1, ,2,3,,25ii IFQ

 

,3, ,25LFQ




(9b)
(9c)
, ,,, ;2ii

 (9d)
This is the CGFPD1BA. In the IFQ it reads:
2
d, T
SLXX XX

11 1
2

 



(10a)


2
2
TXX




12 (10b)
, , ,
X
X
XX




 
 
 
(10c)
The IFQ of this action has been studied by us recently
[12] and we recap it here very briefly. The canonical
momenta conjugate to
X
obtained from the above
action are:

1
:PTX
X


(11)
Here the velocities 1
X
P
T

 are expressible.
Canonical Hamiltonian density for the above theory is:

2
2
11
1;
22
cT
PXP X
T



 



 (12)
The quantization of the system is trivial. The nonvan-
ishing equal WS-time (EWST) commutation relations
(CR’s) for this theory are obtained as:

,, ,XP i






(13)
where

22
ddS
is the one-dimensional Dirac distribu-
tion funct ion.
It is obvious from the above considerations that the
above theory is unconstrained in the sense of Dirac [21].
It may be important to emphasize here that an uncon-
strained system like the above theory is a gauge-nonin-
variant theory and it is some what akin to a gauge-fixed
gauge-invariant theory which makes it a gauge-nonin-
variant system. In the presence of a scalar dilaton field
the theory of co ur se, becomes a constrained system in the
sense of Dirac as shown in our earlier work [12]. For
further details of the IFQ of this theory we refer to our
earleir work of Ref. [12].
The CGFPD1BA in the LFQ reads [12-18]:
(14a)

22
ii
T
X
XXXXX
 
  




,,,,2,3,,25ii
(14b)
 
,PP

i
P
,
(14c)
The canonical momenta and canonically
conjugate respectively to X
 i
and
X
X
, obtained
from the above LF action are:
 
2
:2
T
PX
X



(15a)

2
:2
T
PX
X



(15b)

2
:2
i
ii
T
PX
X


(15c)
Above equations however, imply that the theory pos-
sesses 26 primary constraints:
10
2
T
PX


 


(16a)
Copyright © 2013 SciRes. JMP
U. KULSHRESHTHA, D. S. KULSHRESHTHA
60
2P
 0
2
TX


 


(16b)
0,
2
i
ii
T
PX

 


2,3, ,25.
i

22
ci
PX




2
c
,uv i
w
(16c)
Canonical Hamiltonian density for the above theory is:

0
i
PXPX



(17)
After including the above 26 primary constraints of the
theory in the canonical Hamiltonian density with
the help of Lagrange multiplier fields and , the
total Hamiltonian density could be written as
2
T
2
2
T
i
ii
22
uP XvPX
T
wP X










,uv w
22
d
TT
TT


(18)
We treat and i as dynamical. The Hamiltons
equations obtained from the total Hamiltonian
H
S
, are the equations of motion of the the-
ory that preserve the constraints of the theory in the
course of time. Demanding that these primary constraints
of the theory be preserved in the course of time one does
not get any secondary constraints. The theory is thus seen
to possess only 26 above constraints. Further, the matrix
of the Poisson brackets of the above constraints is seen to
be nonsingular, implying that the set of these constraints
is second-class and consequently the theory is GNI
[12-18]. The theory indeed does not possess the usual
local string gauge symmetries defined by the WSRI and
the WI [12-18].
This action is thus seen to lack the local gauge sym-
metries. This is in contrast to the fact that the original
action had the local gauge symmetries and was
therefore GI. The theory defined by the action
N
S, on
the other hand describes a GNI theory. This is not sur-
prising at all because the theory defined by
N
S is after-
all (conformally) a gauge-fixed theory and consequently
it is not expected to be GI anyway.
Infact, the IF theory defined by
N
S is seen to be un-
constrained [12] whereas the LF theory is seen to possess
a set of 26 second-class constraints [12-18]. In both the
cases theory does not respect the usual local string gauge
symm e t r i e s defined by WSRI and WI [ 12-18].
3. CGFPD1BA in the Presence of a 2-Form
Gauge Field
B
B
: A Recap
We now consider this CGFPD1BA in the presence of a
constant background antisymmetric 2-form NSNS gauge
field
2
d
II
BB
S
studied earlier by Schmidhuber, de Alwis and
Sato, Tseytlin and Abou Zeid and Hull and others de-
fined by [1-8]. This theory has been studied by us earlier
in Ref. [13]. This theory is defined by the action [1-8,13]:
(19a)
I
CB
B

 (19b)
2
CN
T
X
X



 
 

 

 (19c)
2
BTB



 (19d)


201
1,constant,10



 


0
:,
0
B
BXXBB
B

 

 

(19e)
(19f)
01 10
BB BIFQ (19g)
BBB LFQ
 


,0 ,1,0 ,1, ,2 ,3,,25ii IFQ
 
 

,,, ,, ,2,3,,25ii LFQ
 

B
(19h)
(19i)
(19j)
It is important to recollect here that the 2-form gauge
field
is a scalar field in the target-space whereas it
is a constant anti-symmetric tensor field in the world-
sheet space. The above action is seen to be GI in the IFQ
as well as in the LFQ [13]. It is seen to possess only one
constraint in IFQ and a set of 27 constraints in LFQ. The
nature of constraints in both the cases is seen to be
first-class implying that the theory is GI in both the cases
[13]. In Ref. [13], we have studied the Hamiltonian and
path integral formulations of this theory under appropri-
ate gauge-fixing.
3.1. Instant-Form Quantization
In the following, we study the IFQ of this above theory
using the EWST framework of dynamics on the hyper-
planes defined by the WS-time [26,
27]. In the IFQ, the above action reads as:
0constant




2
2
33 3
dd ,2
T
SXXTB







(20)
Canonical momenta are

33
:,:0
B
PTX
B
X





 


P
(21)
and
B
Here
are the canonical momenta con-
jugate respectively to
X
and 0110. The
above theory is thus seen to possess one primary con-
straint:

BB B
10
B
 . Canonical Hamiltonian density of
Copyright © 2013 SciRes. JMP
U. KULSHRESHTHA, D. S. KULSHRESHTHA 61
this theory is:

3
cPX



3BB

(22a)

2
X TB


,u
31
22
cT
PP
T

(22b)
After incorporating the primary constraint of the the-
ory in the canonical Hamilto nian density with th e help of
Lagrange multiplier field
(to be treated as dy-
namical) the total Hamiltonian density of the theory
could be written as:

2
31
22
T
B
T
PP X
T

TB u


p
(23)
Also, the momenta canonically conjugate to μ is de-
noted by
.
The Hamiltons equations obtained from the total Ham-
iltonian: 33
d
TT
H
are the equations of motion that
preserve the constraints of the theory in the course of
time. Demanding that the primary constraint of the the-
ory be preserved in the course of time one does not get
any further constraints. The theory is thus seen to posses
the only one above constraint.
The Poission bracket of the constraint of the theory
with itself is seen to be zero implying that the constraint
is first-class and that the theory is GI. It is indeed seen to
posses three local gauge symmetries given by the 2D
WSRI and the WI defined by:
X
XXX




XX





hh
  


hhhh

(24a)
(24b)
hh


(24c)
 

  
 

 


BBBB
(24d)

 
B


,exp2hh

(24e)
B
 


(24f)




 (24g)
It is important to recollect here that the 2-form gauge
field B
is a scalar field in the target-space whereas it
is a constant anti-symmetric tensor field in the world-
sheet space (and consequently we have 0B

0B 
0
).
Thus the theory is seen to be GI in IFQ as well as in
LFQ. It is therefore gauge-nonanomalous possessing the
three local gauge symmetries defined by the 2D WSRI
and the WI. The theory could therefore be quantized un-
der appropriate gauge-fixing.
Hamiltonian and path integral formulations of this
theory could be studied under appropriate gauge-fixing
e.g., under the gauge: . Corresponding to this
choice of gauge the total set of constraints of the theory
under which the quantization of the theory could e.g., be
studied becomes: 1B
 0B and 2.
The matrix of the Poisson brackets of th ese constraints is
seen to be nonsingular implying that the corresponding
set of constraints is second-class. Following the Dirac
quantization procedure in the Hamiltonian formulation,
the nonvanishing EWST CR’s of the theory under the
above gauge are obtained as:
 
,,, XP i
 




 (25)
3.2. Light-Front Quantization
In LFQ, using the ELCWST framework of dynamics on
the hyperplanes defined by the LC WS-time
constant

 
44
ddS
, the action of the theory reads:
(26a)


42
ii
T
X
XXX
XXTB

 






,, PP

i
P
,, BX X
(26b)
Canonical momenta B, and conjugate
respectively to
i
, and
X
are:

4
:0
BB



(27a)

4
:2
T
PX
X



(27b)

4
:2
T
PX
X



(27c)

4
:,2,3,,25
2
i
ii
T
PXi
X


10
B
(27d)
Above equations however, imply that the theory pos-
sesses the following 27 primary constraints:
(28a)


20
2
T
PX


 


(28b)

30
2
T
PX





(28c)

0,2,3, ,25
2
i
ii
T
PXi

 


4

(28d)
Canonical Hamiltonian density corresponding to is

4
4
c
B
i
i
BP X
PXPX




 
 
(29a)
Copyright © 2013 SciRes. JMP
U. KULSHRESHTHA, D. S. KULSHRESHTHA
62
4
cTB
4
c

,,u


v
(29b)
After including the primary constraints of the theory in
the canonical Hamiltonian density with the help of
Lagrange multiplier fields

,,s


,

and i
,
w

(which we treat as dyna-
mical), the total Hamiltonian density of the theory could
be written as:
421
Tc
su
23ii
v w
 



 (30a)



4
2
2
T
B
i
ii
TB su P
T
vP X
T
wP X

 

 





 




2
TX







44
d
TT
(30b)
Hamiltons equations obtained from the total Hamilto-
nian H
0B
are the equations of motion that
preserve the constraints of the theory in the course of
time. The matrix of the Poisson brackets of these above
constraints is seen to be singular implying that the corre-
sponding set of constraints is first-class and that the cor-
responding theory is GI [13].
The above theory is indeed seen to possess three local
gauge symmetries given by the 2D WSRI and the WI [13]
and the theory could be quantized under appropriate
guage-fixing. To study the Hamiltonian and path integral
formulations of the theory under gauge-fixing, we could
e.g., choose the gauge:

10B
. corresponding to this
gauge choice the total set of 28 constraints of the theory
becomes [13]:


21 0
B
(31a)

 (31b)

0
TX
32 2
P




 

(31c)

0
T
PX
43 2



 

(31d)

0,2,3, ,25
i
ii i
T
PXi
2


 

0B
 (31e)
The matrix of the Poisson brackets of the above 28
constraints is seen to be nonsingular implying that the
corresponding set of constraints is second-class. Now
following the Dirac quantization procedure in the Ham-
iltonian formulation, the nonvanishing ELCWST CR’s of
the theory under the gauge-fixing
 are ob-
tained as [13]:

,,,
2
i
i
XP XP XP
i




 (32a)

,,
2
ii i
XX XXT





(32b)

,,
4
ii
iT
PP PP






1
(32c)
For further details of the Hamiltonian and path integral
formulations of the above theory, we refer to our earlier
work [13].
4. CGFPD1BA in the Presence of a Scalar
Axion Field C and an U(1) Gauge Field Aμ
In this section, we study the IFQ and LFQ of the
CGFPD1BA in the presence of a U gauge field
,AA

and a constant scalar axion field
,CC

1
[19,20]. We find that the CGFPD1BA
describing a GNI theory (being a gauge-fixed theory) is
seen to describe a GI theory when considered in the
presence of above background fields.
,A
We also find that the U gauge field

,
and the constant scalar axion field C
are both
see n to behave like the Wess-Zumino (W Z) fi el ds [1 9, 20]
and the term involving these fields in the action is seen to
behave like a WZ term for the CGFPD1BA [13,19,20].
Here the field
A
is a scalar field in the target space
and a vector field in the WS space and the axion field C
is a constant scalar field in both the target space as well
as in the WS space [19,20].
We find that the resulting theory obtained in the above
manner describes a GI system respecting the usual string
gauge symmetries defined by the 2D WSRI and the WI.
It is seen that the axion field and the
C
1U gauge
field
A
, in the resulting theory behave like the WZ
fields and the term involving these fields behaves like a
WZ term for the CGFPD1BA [19].
The situation in the present case is seen to be exactly
analogous to a theory where one considers the CGFPD-
1BA in the presence of a 2-form gauge field B
as
studied by us in our earlier work [13], where the field
B
behaves like a WZ field and the term involving this
field behaves like a WZ term for the CGFPD1BA [13].
The CGFPD1BA in the presence of a constant back-
ground scalar axion field and an U gauge field C

1
A
is defined by [1-8,19, 2 0] :
2
d
II
AA
S (33a)
I
CA
A

(33b)

2
CN
T
X
X



 


 (33c)

2
ATCF





(33d)
2
1 , constant
 (33e)
Copyright © 2013 SciRes. JMP
U. KULSHRESHTHA, D. S. KULSHRESHTHA 63

01
,
10
F
AA




01



 (33f)
01
f
FFIFQ

 (33g)
f
FFLFQ


,0,1, ,2,3,,25ii IFQ
 

2,3,,25LFQ
0constant




, 0,1

(33h)
(33i)
,,,,,,ii
 
   (33j)
4.1. Instant-Form Quantization
In this section, we study the IF Hamiltonian and path
integral quantization of the above theory using the
EWST framework, on the hyperplanes defined by the
WS-time . The IF action reads:


2
2
,TXXTCf





55
ddS

52 (34)
Overdots and primes denote derivatives with respect to
and
respectively. The canonical momenta ob-
tained are:

5
:PTX




X



(35a)
5
:
cC


0

(35b)
0
 5
0
:0
A



(35c)
15
1
:ETC
A
 

1
,,
 (35d)
where and c are the canonical
momenta conjugate respectively to 01
0
,,PE

X
AA
1


20ETC 
30
c



 
0
50
15
c
PXA
EA C
 


 
and C.
The theory is thus seen to possess three primary con-
straints:
0
0 (36a)
(36b)
(36c)
Canonical Hamiltonian density corresponding to above
Lagrangian density is:
c
(37a)

2
50
22
TXTCA







,, ,uv
1
cPP
T


and
(37b)
After incorporating the primary constraints of the the-
ory in the canonical Hamiltonian density with the help of
Lagrange multiplier fields
,w
55123
Tc
uu w
(treated as dynamical) the total Hamiltonian
density of the theory becomes:
(38a)
 



2
50
0
1
22
T
c
T
PPX TCA
T
uE TCvw


 
,
uv
pp w
p
55
d
TT
H
(38b)
Momenta canonically conjugate to u, v and w are de-
noted respectively by and . Hamiltons equa-
tions obtained from the total Hamiltonian
,
for the closed string with periodic boundary conditions
(BC’s) e.g., are:

5
5
1,
T
T
H
XP
PT
H
PTX
X






 
(39a)

55
0
,
TT
C
C
HH
Cw TAv
C



  (39b)
0
55
000
,0
TT
HH
Au A



 (39c)
55
11
,0
TT
HH
AvE
EA




(39d)
0
55
0,
TT
u
u
HH
up
pu




(39e)

55
0,
TT
v
v
HH
vpETC
pv


 

(39f)
55
0,
TT
wC
w
HH
wp
pw




,
(39g)
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 of the the-
ory be preserved in the course of time one does not get
any further constraints. The theory is thus seen to posses
only three constraint 12
  and .
3
Matrix of the Poission brackets of these constraints is
seen to be singular implying that the constraints form a
set of first-class constraints and that the theory is GI (and
consequently gauge-nonanomalous). It is indeed seen to
posses three local gauge symmetries given by the 2D
WSRI and the WI defined by:
X
XXX

 
(40a)
Copyright © 2013 SciRes. JMP
U. KULSHRESHTHA, D. S. KULSHRESHTHA
64



11 1
14 4123
1
32
MMTM
TM

XX
 




hh
 


hh
(40b)
hh
 

(40c)
hh
  




 
  (40d)
A
AAA



(40e)
A
A



 (40f)
CC CC

C





,exp2hh

(40g)
C (40h)


 

 
0
501
5
c
T
AEA c
 

(40i)
The first order Lagrangian density of the theory is:

IO PX


  
uvw
pu
pvp w

 (41a)

2
51
22
IO T
PPX TC
T

f



00A
(41b)
The theory could be quantized under appropriate
gauge-fixing. To study the Hamiltonian and path integral
formulations of this theory under gauge-fixing, we could
choose e.g., the gauge:

0
11 0
 

22 0ETC
 
33 0
c
(42)
Corresponding to this choice of gauge the total set of
constraints of the theory under which the quantization of
the theory could e.g., be studied becomes:
(43a)
(43b)
  
0
0
(43c)
4
A




:,
(43d)
We now calculate the matrix
P
B
M
 
 
of the Poisson brackets of the constraints i. The non-
vanishing elements of the matrix
M
are obtained as:

14 41
TMTM 23 32
M MT

 (44)
The matrix
M
is seen to be nonsingular implying
that the corresponding set of constraints is a set of sec-
ond-class constraints. The determinant of the matrix
M
is given by:


12 2
det MT



 
 (45)
and the nonvanishing elements of the inverse of the ma-
trix
M
(i.e., the elements of the matrix

1
M
) are
obtained as:

 
 
 (46a)

144
,,dMM

 

1
0A
(46b)
Following the Dirac quantization procedure in the
Hamiltonian formulation, the nonvanishing EWST CR’s
of the theory under the gauge 0
 (with the ar-
guments being suppressed) are obtained as:

,, ,XP i






(47)
In the path integral formulation, the transition to the
quantum theory, is, however, made by writing the vac-
uum to vacuum transition amplitude called the generating
functional
1i
Z
J of the theory under GFC
in the
presence of external sources i
J
as follows:



2
1
2
:dexpd
1
22
i
ii
ZJi J
T
PPX TCf
T



 (48)
where the phase space variables of the theory are
,,,,,,
i01
X
AACuvw
 with the corresponding re-
spective canonical conjugate momenta:
0
,,,,,,
icuvw
PEppp
 .
The functional measure
d
of the generating
functional
1i
J under the GFC
is obtained as:
Z




 


 

201
0
00
dddd
ddddd dd
dddd
000
0
cuvw
c
TXAA
CuvwPE
ppp
A
ETC



 








(49)
 
 
 




00A
The Hamiltonian and path integral quantization of the
above theory under the GFC

66
ddS
is now com-
plete.
4.2. Light-Front Quantization
In LFQ, the action of the theory reads:
(50a)


62
ii
TXX
X
XXXTCf



 





(50b)
In the following we study the LF Hamiltonian and path
integral formulations of the above action. The canonical
Copyright © 2013 SciRes. JMP
U. KULSHRESHTHA, D. S. KULSHRESHTHA 65
momenta and conjugate re-
spectively to , ,,P
 
, ,,
ic

,
i
PP ,
X
XXCA
 
and
A

are obtained
as:
60
A


 (51a)
6TC
A



 (51b)
6
:0
C

c
 (51c)
 
2
T
PX


6
:X


(51d)
 
2
T
PX

6
:X


(51e)
 
:,
2
i
T
PX

10
6
2,3, ,25
ii
X
i


(51f)
Above equations however, imply that the theory pos-
sesses 29 primary constraints:


20TC
 
0
(52a)
(52b)
3c
  (52c)

0
T
PX


 


42 (52d)

0X
 

52
T
P

(52e)

0,
i
ii
T
PX

 




6
ci
i
c
PXPX
C


 

6
c
2
2,3, ,25i

6PX


(52f)
Canonical Hamiltonian density of this theory is:

AA


 

6
cTC A

 

(53a)
(53b)
After including the above 29 primary constraints in the
canonical Hamiltonian density with the help of
Lagrange multiplier fields
1,,v



2,,v



3,,v



5
,, ,v
4
v



,
i
 and v
6611223345
ii
ss suvw
(which we treat as dynamical), the total Hamiltonian
density of the theory could be written as:
Tc
 




 
(54a)
 

6
12 3
2
22
T
i
ii
c
T
TCAu PX
TT
vPXw PX
ss TCs

 






 




 
 
 
 
 
 
 
 
66
d
TT
H
(54b)
The Hamiltons equations of motion of the theory that
preserve the constraints of the theory in the course of
time obtained from the total Hamiltonian:
(55)
e.g., for the closed strings with periodic BC’s are ob-
tained as:

66
,2
TT
HH
T
X
uP v
PX

 






 (56a)

66
,2
TT
HH
T
X
vP u
PX

 



   


 (56b)

66
,2
TT
i
ii i
i
i
HH
T
X
wP w
PX
 


  

 (56c)
66
0, 2
TT
u
u
HH
T
upPX
pu

 


  



(56d)
66
0, 2
TT
v
v
HH
T
vpPX
pv

 


 



(56e)
66
0, 2
i
i
TT
i
iwi
wi
HH
T
wpPX
pw
 


 



(56f)

66
32
,
TT
C
C
HH
Cs TAs
C
 

 
 (56g)
66
2,0
TT
HH
As A





 (56h)
66
1,0
TT
HH
As A





 (56i)
1
1
66
11
0,
TT
s
s
HH
sp
ps




(56j)

2
2
66
22
0,
TT
s
s
HH
s
pTC
ps


  
 (56k)
3
3
66
33
0,
TT
s
C
s
HH
sp
ps




(56l)
Demanding that the primary constraints of the theory
be preserved in the course of time one does not get any
secondary constraints. The theory is thus seen to possess
Copyright © 2013 SciRes. JMP
U. KULSHRESHTHA, D. S. KULSHRESHTHA
66
only 29 constraints: 12345
,,,,

and i
. Further
the matrix of the Poisson brackets of these 29 constraints
among themselves is easily seen to be singular, implying
that the set of these 29 constarints is first-class. This in
turn implies that the theory is GI (and consequently
gauge anomalous). The theory is indeed seen to possess
three local gauge symmetries given by the 2D WSRI and
the WI defined by Equation (40). The theory could now
be quantized under appropriate guage-fixing.
The first-order Lagrangian density of the theory is:



 
 
12
6
123
i
IO
c
i
i
ss
uv
CP X
PX A
psps
pupvp






 
  
 





3
6
s
T
wi
P X
A
ps
w



 
(57a)


62
2
IO TC sA
TuX vX


 


i
i
w X



0A

11 0



0TC
 
33 0
c
(57b)
To study the Hamiltonian and path integral formula-
tions of the theory under gauge-fixing, we could e.g.,
choose the gauge:
(58)
corresponding to this gauge choice, the total set of con-
straints of the theory under which the quantization of the
theory could be studied becomes:
(59a)
22

 (59b)

0A

 
(59c)
4 (59d)

0
T
PX



  

54 2
(59e)

0
T
PX



  


65 2 (59f)

0,2,3, ,25
i
PX
i

 




PB
,
 
 

R
2
ii i
T



 
,:R
 
(59g)
We now calculate the matrix
of the Poisson brackets of these above 30 constraints.
The nonvanishing elements of the matrix
 
56 65,2,3,,25
ii
RRRT i

  
 
1441 2332
TRTR RRT
are
obtained as:

(60a)

 
R
(60b)
The matrix
is seen to be nonsingular implying
that the corresponding set of these 30 constraints is sec-
ond-class. The determinant of the matrix R

is given
by




12
13
2
det R
TT

 



  (61)
R
Nonvanishing elements of the inverse of this matrix

1
R (i.e. the elements of the matrix
 
) are:

111
56 65
1,
2
2,3, ,25
ii
RRR T
i

 
(62a)



11 1
14 4123
1
32
RRTR
TR

 
 
 (62b)
with

130 30
,,dRR
 
 

1
0A
(63)
Finally, following the Dirac quantization procedure in
the Hamiltonian formulation, the nonvanishing ELCWST
commutation relations of this theory under the gauge
(with the arguments being supproted again)
are obtained as:

,,,2
i
i
XP XP XPi

 




(64a)

,,2
ii
XXXXi T



 

 (64b)

,, 4
ii
PP PPiT




 (64c)

,,ATACi

 

 
 (64d)
In the path integral formulation, the transition to the
quantum theory, is, however, made by writing the vac-
uum to vacuum transition amplitude called the generating
functional
2
Z
i
J
i
of the theory in the presence of ex-
ternal sources
J
as follows:



2
2
:dexpdd
2
i
i
i
i
i
ZJ
iJTCsA
TuXvXwX

 

 







 (65)
where the phase space variables of the theory are
,,,,,,,,,,,
ii
CXXXAAvv vvvv
 
 123456
with the
corresponding respective canonical conjugate momenta:
1
,,,,, ,,,,
iv
PPPp ppppp

 23456
i Cvvvvv
. The fun-
ctional measure
d
of the generating functional
Copyright © 2013 SciRes. JMP
U. KULSHRESHTHA, D. S. KULSHRESHTHA 67
Z
2i
J


 
is obtained as:









123
2
12345
d
dd dd
dd dddd
dddd
dddd
0
2
2
i
i
vv vv
TT
CX X X
vvvv
PPP
pppp
T
PX
T
P
 

 

 


 

 

 
 







456
13
6
d d
d
d
dd
c
v v
A A
vv
pp





 
 


 


0
0
2
00
i
ii
c
X
T
PX
TC
A












 







 



 


00
i
2,3, ,25
h
(66)
where denotes the product of similar expressions
for all . The LF Hamiltonian and path in-
tegral quantization of the above theory is now complete.
i
5. Summary and Discussion
The Polyakov D1 brane action in a d-dimensional
courved background
defined by (Equation (1)) is
GI and it possesses the well-known three local gauge
symmetries given by the 2D WSRI and the WI defined
by Equation (2).
However, when we study this action under the con-
formal gauge-fixing defined by Equation (3) to obtain the
CGFPD1BA defined by the action
N
S, we find that the
CGFPD1BA given by
N
S is no longer GI and it de-
scribes a gauge anomalous (and GNI) theory and it also
does not possess the usual local gauge symmetries de-
fined by Equation (2) being a gauge-fixed theory.
Hovever, this GNI theory when considered in the pres-
ence of a contant antisymmetric 2-form gauge field B
is seen to become a GI theory possessing the three local
gauge symmetries defined by the 2D WSRI and the WI
defined by Equation (24).
The 2-form gauge field B
in this case is seen to
behave like a WZ field and the term involving this field
is seen to behave like a WZ term for the CGFPD1BA
which, in the absence of this term, is seen to possess a set
of second-class constraints and consequently describe a
GNI theory which does not respect the local gauge sym-
metries defined by the WSRI and WI given by Equation
(2).
In our earlier work (cf. Section 3) [13], we have stud-
ied the IF and LF Hamiltonian and path integral formula-
tions of this GI theory describing the CGFPD1BA in the
presence of the constant antisymmetric 2-form gauge
field have been studied under appropriate gauge choices
in the abesence of BC’s. The BC’s could however be
taken into account either by imposing them directly in
the usual way for the open and closed strings separately
in an appropriate manner or by considering them as the
Dirac primary constraints [13-20,34] and study them
accordingly.
In the present work [20], we have studied the IF and
LF Hamiltonian and path integral formulations of the
CGFPD1BA in the presence of a constant scalar axion
field and an
1U gauge field C
A
(cf. Section 4).
We find that the scalar axion field and the
C
1U
gauge field
A
are seen to behave like the WZ field and
the term involving these fields is seen to behave like a
WZ term for the CGFPD1BA, which in the absence of
this term is seen to posess a set of second-class con-
straints and consequently describe a GNI theory which
does not respect the local gauge symmetries defined by
the WSRI and WI.
The situation in the present case, as pointed out in the
foregoing, is analogous to the theory where one con-
siderers the CGFPD1BA in the presence of the constant
2-form gauge field B
, where B
which is a scalar
in the target space and an antisymmetric tensor in the WS
space, behaves like a WZ field and the term involving
this field behaves like a WZ term for the CGFPD1BA
[13]. The later theory has been studied by the present
authors in Ref. [13] to which we refer the reader for fur-
ther details [3].
The IF and FF Hamiltonian and path integral formula-
tions of the GI theory describing the CGFPD1BA in the
presence of the constant scalar axion field and the
C
1U gauge field
A
have been studied in this work
under appropriate gauge choices in the abesence of BC’s
[34]. The BC’s, however, could be taken into account in
the usual manner either by imposing them directly in the
usual way for the open and closed strings separately [1-8]
or by considering them as the Dirac primary constraints,
and study them accordingly [13-20,34].
In conclusion, the Polyakov D1 brane action in a
d-dimensional courved background h
defined by
is GI and it possesses the well-known three local string
gauge symmetries. However, under conformal gauge-
fixing, the CGFPD1BA is no longer GI as expected and
it also does not possess the local string gauge symmetries
being a gauge-fixed theory. Hovever, this GNI theory
when considered in the presence of a contant background
scalar axiom field and an U gauge field
S
C

1
A
it
is seen to become a GI theory possessing the three local
string gauge symmetries.
The scalar axion field and the U gauge field C

1
A
are seen to behave like the WZ fields and the term
Copyright © 2013 SciRes. JMP
U. KULSHRESHTHA, D. S. KULSHRESHTHA
68

A
B
involving these fields is seen to behave like a WZ
term for the CGFPD1BA, which in the absence of this
term is seen to posess a set of second-class constraints
and consequently describes a GNI theory which does not
respect the local string gauge symmetries. The situation
in the present case is analogous to a theory where one
considerers the CGFPD1BA in the presence of a constant
2-form gauge field
which behaves like a WZ field
and the term involving this field behaves like a WZ term
for the CGFPD1BA.
6. Acknowledgements
Authors thank Prof. James Vary for several useful dis-
cussions and for his warm hospitality at the Iowa State
University, Ames, Iowa, USA, where a part of this work
was done.
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