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The quantization of the Chiral Schwinger model based on the Hamilton-Jacobi formalism
SUPPLEMENTS Nuclear Physics B (Proc. Suppl.) 129&130 (2004) 916-918
ELSEVlER
www.clscvierphysics.com
The quantization of the Chiral Schwinger model based on the Hamilton- Jacobi formalism S. I. Muslih” “Department
of Physics, Al-Azhar
We apply the a = 1. We show the integrability fields. The path
University,
Gaza, Palestine
Hamilton-Jacobi method to the Chiral Schwinger model in the case of regularization ambiguity that one can obtain the integrable set of equation of motion and the action function by using conditions of total differential equations and without any need to introduce unphysical auxiliary integral for this model is obtained by using the canonical path integral method.
1. Introduction
There has been a great progress in the understanding of the physical meaning of the anomalies in quantum field theory through the study of the Chiral Schwinger model (CSM) [l]. Jackiw and Rajaramn [2] showed a consistent and unitary, quantum field theory is even possible in the gauge non-invariant formulation. On the other hand, Faddeev and Shatashvili [3] have proposed that, the gauge non invariant theory can converted into a gauge invariant version [4] by adding a WessZumino action to the original theory . In fact, many authors have a,nalyzed their works as an archetype of anomalous gauge theory [5-71. In an earlier work [8], it was shown that the canonical path integral method [9-141 leads us to obtain the path integral quatnization for the Chiral Schwinger model in the case of regularization ambiguity a > 1, without any need to introduce unphysical auxiliary fields. In the present paper we summarize our results for a similar demonstration for the Chiral Schwinger model in the case of regularization ambiguity a = 1. 2. The
Hamilton-Jacobi
approach
0 2004 Published
dS -a!3 = 0, Hi tp,q,,-%a’ at, Q, p = 0, n - T + 1, . .. . n, 1, . . . . 72 - T,
a =
by Elsevier
B.Y
(1)
where H& = JL(tp,qa,pa)
+~a,
and the Hamiltonian Ho
=
-qt,
Paw,
(2)
Ho is defined as
+ Ppdp~pu=-Hv
qi, Qv, &
=
w,),
p, .v = n-r
+ 1, . ..) n.
(3)
The equations of motion are obtained as total differential equations in many variables as follows [16]:
4, = “Hb,dt 8Pa cY’
dpa = - dllb,& &Lz
a’
(4)
(5) dz=
In this section we shall briefly review the Hamilton-Jacobi method [15-171 for studying the constrained systems. In this formulation, if we start with a singular Lagrangian with Hessian matrix of rank (n - T), 0920-56326 - see front matter doi:10.1016/j.nuclphysbps.2003.12.330
T < n, then the the set of Hamilton-Jacobi partial differential equations [HJPDE] is expressed as follows:
(-&,+pa%)dt,;
where z = S(t,; qa). The set of equations (4-6) is integrable if and only if [B] dH; = 0,
dHI, = 0.
(7)
S.I. Muslih/Nuclear
This is the necessary and sufficient condition that the system (4-6) of total differential equations be completely integrable. If the set of equations (4-6) is integrable, then one can obtain the canonical action function (6) in terms of the canonical coordinates. In this case, the path integral representation may be written as [9-141 da Q’,;
t’,; qa, ta) =
exp i a =
-&+p,d2) I,
.‘.)
n-r,
3. Quantization
dH’ a
a:=o,n-r+l,..., of Chiral
1i &
Schwinger
,
12.
(8)
model
In this section, we perform the canonical path integral procedure for the Chiral Schwinger model which is a second class constrained system. Now we consider the bosonized CSM model in the case of a = 1 [1,4] SCSM = eA,(rfY
J
d2+;F~YF,,,
- P)dti$
+ &+#%+h
+ &~A,A’],
+ (9)
where r]fiV = diag(1, -l), eol = 1 and a is a regularization ambiguity [1,4], which is defined for calculating the fermionic determinant of the fermionic CSM. The canonical momenta are given by x0 =0
d =&-&A,,,
7r4 = i+e(Ao
(10)
-Al),
(11)
where ~0, ~1 and ~4 are the momenta conjugated the the fields Ao, AI and 4 respectively. Following the Hamilton-Jacobi method [16,17], we obtain the set of Hamilton-Jacobi partial differential equations as follows: dS H; = dz(m,) = 0, niTO=m, (12) J H;=Po+Ho;
Po=~,
dS
where the canonical Hamiltonian Ho =
J
dx+r1)2
+ &ro)”
e(r# + &~)(Ao ~e2{(Ao)”
- AI) - Ao%n’ -
- (A1)2} + ie2(Aa - A1)2]. (14)
Now, the set of equations of motion and the action function is obtained as total differential equations in many variables as follows:
bAl = [T’ + d,A,]dt, (15) ST’ = [-e(r$ + 814) - 2e2A1 + e2Ao]dt,(16)
Dqa Dpa x
.I’4.2
917
Physics B (Proc. Suppl.) 129&130 (2004) 916-918
(13)
S$ = [TT~- e(A0 - Al)]dt, STT~= [-&@c#I + e#(Ao - Al)]dt,
(17)
&TO= [din’ + en4 + edI4 + e2Al]dt, 1 1 1 dS = [--(r’)” - -(T$)~ - -(a$)2 + 2 2 2 e(rd + &$)(Ao - AI) + Ao&T’ +
(19)
ie2{(Ao)’
- (AI)~} - ie”(Ao
(18)
- AI)~ +
d(d + &Ao) + x$(r# - e(Ao - Ad)ldt.
(20)
To check whether the set of equations (15-20) is integrable or not, let us consider the total variations of (12). In fact dHi = Hidt = [&T? + e-/r#+ e&cj + e2AJdt.(21) The total variation of Hi leads us to obtain the following condition dHi = [e2n1]dt = HAdt. Taking the total variation condition
(22) of HA, we obtain the
dHA = [-e3(r+, + &c$) - 2e4A1 + e4Ao]dt = H;dt.
(23)
Again, taking the total variation of Hi, gives dHi = -e3[&r,
+ a,“$ + 2er’ + 2e&Al]dt
e4dAo = 0,
+ (24)
HO is given by
which leads us to obtain dAo in terms of dt as follows:
+ ;(L%#J)~-
dAo = [i&m
+ ~@c#I + 2e? + 2e&Al]dt.
(25)
918
S.I. Muslih/Nuclear
The integrabilty for A0 as
condition
Physics B (Proc. Suppl.) 129&130 (2004) 916-918
(23) gives the solution
Ao=~(rrgtald)+2eAl=~~.
(26)
Hence, we obtain the integrable system as follows: ~AI = [T’ + d,Po]dt, (27) 6~’ = [-e(T# + al$) - 2e2A1 + e2/30]dt,(28) 64 = [~4 - e(Po - AI)]& 6-n;@= [-dl#$ + e@(& - Al]dt, dS = [&T~)~ + ;(TQ)~ - $4)”
(29) (30) +
e(t%$ + eAl)Po + -e2(A1)2]dt.
(31)
Since, the action function is integrable, we can make use of (8) to obtain the path integral representation for the CSM as K =
J
n dAi d7+ d4 dx,
J e(&4 + eAl)Po+ -e2(Al)2)1.
exp{i
d2x[i(7r1)2
+ in
- i(@)2
f (32)
The path integral representation (32) is an integration over the canonical phase space coordinates (Ai,#) and (4,~$). 4. Conclusion In this work we have studied the CSM model by using the Hamilton-Jacobi formalism [16, 171. Following the prescriptions of this method, we obtained the set of Hamilton-Jacobi partial differential equations which has two Hamiltonians Hi and HA. The integrability conditions lead us to obtain the canonical reduced phase-space coordinates and the integrable action function in terms of these canonical variables. Then we quantize this model using the canonical path integral method [9-141 to obtain the path integral quantization for our model as an integration over the canonical phase space coordinates (Ai, xi) and ($,rd) without any need to enlarge the initial phase-space by introducing unphysical auxiliary fields (see refs. [I] and the references therein).
Acknowledgments The author would like to thank local organizing committee (LOC) for financial support. REFERENCES 1. J.-H Cha, Y.-W Kim et al, 2. Phys. C69, 175 (1995); W. T. Kim, Y.-W. Kimet al, J. Phys. G23, 325 (1997); M.-I. Park, Y.-J. Park and S. J. Yoon, J. Phys. G24, 2179 (1998). 2. R. Jakiw, R. Rajaraman, Phys. Rev. Lett. 54, 1219 (1985) ; Phys. Rev. Lett. 54, 2060 (1985); K. Harada, Phys. Rev. Lett. 64, 139 ( 1990); Phys.Rev. D42, 4170 (1990). 3. L. D. Faddeev and S. S. Shatashvili, Phys. Lett. B167, 225 ( 1986). 4. F. Schaposnik, C. Viallet, Phys. Lett. B177, 385 (1986) ; K. harada and I. Tsutsui, Phys. Lett. B183, 311 ( 1987). 5. R. Rajaraman, Phys. Lett. B154, 305 (1985); H. 0. Girotti, H. J. Rothe and K. D. Rothe, Phys. Rev. D34, 582 (1986); J.-G. Zhou, Y.-G. Miao and Y.-Y. Liu, Mod. Phys. Lett. A49, 1273 (1994). Y.-W. Kim et al, Phys. Rev.D46, 4574 (1992). R. Banerjee, H. J. Rothe, K. D. Rothe, Phys. Rev. D49, 5438 (1994). S. I. Muslih, Mod.Phys.Lett. A18, 1187 (2003). S. I. Muslih, NUOVO Cimento B115,l (2000); B115, 7 (2000). 10. S. I. Mu&h, H. El-Zalaan and F. El-Sabaa, Int. J. Theor. Phys. 39, 2495 (2000). 11. S. I. Mu&h, Gen. Rel. Grav. 34, 1059 (2002). 12. S. I. Muslih, Czech. J. Phys. 52. 919 (2002). 13. S. I. Muslih, Mod. Phys. Lett. A36, 2382 (2002). 14. S. I. Muslih, Nuovo Cimento B117, 383 (2002). 15. S. I. Muslih, Nucl. Phys. B106, (Proc. Supp.), 879 (2002); 119 C (2003) 968. Nuovo Cimento B107, 1143 16. Y. Giiler, (1992); B107, 1389 (1992). 17. S. I. Muslih and Y. Giiler, Nuovo Cimento BllO, 307 (1995).
ELSEVlER
www.clscvierphysics.com
The quantization of the Chiral Schwinger model based on the Hamilton- Jacobi formalism S. I. Muslih” “Department
of Physics, Al-Azhar
We apply the a = 1. We show the integrability fields. The path
University,
Gaza, Palestine
Hamilton-Jacobi method to the Chiral Schwinger model in the case of regularization ambiguity that one can obtain the integrable set of equation of motion and the action function by using conditions of total differential equations and without any need to introduce unphysical auxiliary integral for this model is obtained by using the canonical path integral method.
1. Introduction
There has been a great progress in the understanding of the physical meaning of the anomalies in quantum field theory through the study of the Chiral Schwinger model (CSM) [l]. Jackiw and Rajaramn [2] showed a consistent and unitary, quantum field theory is even possible in the gauge non-invariant formulation. On the other hand, Faddeev and Shatashvili [3] have proposed that, the gauge non invariant theory can converted into a gauge invariant version [4] by adding a WessZumino action to the original theory . In fact, many authors have a,nalyzed their works as an archetype of anomalous gauge theory [5-71. In an earlier work [8], it was shown that the canonical path integral method [9-141 leads us to obtain the path integral quatnization for the Chiral Schwinger model in the case of regularization ambiguity a > 1, without any need to introduce unphysical auxiliary fields. In the present paper we summarize our results for a similar demonstration for the Chiral Schwinger model in the case of regularization ambiguity a = 1. 2. The
Hamilton-Jacobi
approach
0 2004 Published
dS -a!3 = 0, Hi tp,q,,-%a’ at, Q, p = 0, n - T + 1, . .. . n, 1, . . . . 72 - T,
a =
by Elsevier
B.Y
(1)
where H& = JL(tp,qa,pa)
+~a,
and the Hamiltonian Ho
=
-qt,
Paw,
(2)
Ho is defined as
+ Ppdp~pu=-Hv
qi, Qv, &
=
w,),
p, .v = n-r
+ 1, . ..) n.
(3)
The equations of motion are obtained as total differential equations in many variables as follows [16]:
4, = “Hb,dt 8Pa cY’
dpa = - dllb,& &Lz
a’
(4)
(5) dz=
In this section we shall briefly review the Hamilton-Jacobi method [15-171 for studying the constrained systems. In this formulation, if we start with a singular Lagrangian with Hessian matrix of rank (n - T), 0920-56326 - see front matter doi:10.1016/j.nuclphysbps.2003.12.330
T < n, then the the set of Hamilton-Jacobi partial differential equations [HJPDE] is expressed as follows:
(-&,+pa%)dt,;
where z = S(t,; qa). The set of equations (4-6) is integrable if and only if [B] dH; = 0,
dHI, = 0.
(7)
S.I. Muslih/Nuclear
This is the necessary and sufficient condition that the system (4-6) of total differential equations be completely integrable. If the set of equations (4-6) is integrable, then one can obtain the canonical action function (6) in terms of the canonical coordinates. In this case, the path integral representation may be written as [9-141 da Q’,;
t’,; qa, ta) =
exp i a =
-&+p,d2) I,
.‘.)
n-r,
3. Quantization
dH’ a
a:=o,n-r+l,..., of Chiral
1i &
Schwinger
,
12.
(8)
model
In this section, we perform the canonical path integral procedure for the Chiral Schwinger model which is a second class constrained system. Now we consider the bosonized CSM model in the case of a = 1 [1,4] SCSM = eA,(rfY
J
d2+;F~YF,,,
- P)dti$
+ &+#%+h
+ &~A,A’],
+ (9)
where r]fiV = diag(1, -l), eol = 1 and a is a regularization ambiguity [1,4], which is defined for calculating the fermionic determinant of the fermionic CSM. The canonical momenta are given by x0 =0
d =&-&A,,,
7r4 = i+e(Ao
(10)
-Al),
(11)
where ~0, ~1 and ~4 are the momenta conjugated the the fields Ao, AI and 4 respectively. Following the Hamilton-Jacobi method [16,17], we obtain the set of Hamilton-Jacobi partial differential equations as follows: dS H; = dz(m,) = 0, niTO=m, (12) J H;=Po+Ho;
Po=~,
dS
where the canonical Hamiltonian Ho =
J
dx+r1)2
+ &ro)”
e(r# + &~)(Ao ~e2{(Ao)”
- AI) - Ao%n’ -
- (A1)2} + ie2(Aa - A1)2]. (14)
Now, the set of equations of motion and the action function is obtained as total differential equations in many variables as follows:
bAl = [T’ + d,A,]dt, (15) ST’ = [-e(r$ + 814) - 2e2A1 + e2Ao]dt,(16)
Dqa Dpa x
.I’4.2
917
Physics B (Proc. Suppl.) 129&130 (2004) 916-918
(13)
S$ = [TT~- e(A0 - Al)]dt, STT~= [-&@c#I + e#(Ao - Al)]dt,
(17)
&TO= [din’ + en4 + edI4 + e2Al]dt, 1 1 1 dS = [--(r’)” - -(T$)~ - -(a$)2 + 2 2 2 e(rd + &$)(Ao - AI) + Ao&T’ +
(19)
ie2{(Ao)’
- (AI)~} - ie”(Ao
(18)
- AI)~ +
d(d + &Ao) + x$(r# - e(Ao - Ad)ldt.
(20)
To check whether the set of equations (15-20) is integrable or not, let us consider the total variations of (12). In fact dHi = Hidt = [&T? + e-/r#+ e&cj + e2AJdt.(21) The total variation of Hi leads us to obtain the following condition dHi = [e2n1]dt = HAdt. Taking the total variation condition
(22) of HA, we obtain the
dHA = [-e3(r+, + &c$) - 2e4A1 + e4Ao]dt = H;dt.
(23)
Again, taking the total variation of Hi, gives dHi = -e3[&r,
+ a,“$ + 2er’ + 2e&Al]dt
e4dAo = 0,
+ (24)
HO is given by
which leads us to obtain dAo in terms of dt as follows:
+ ;(L%#J)~-
dAo = [i&m
+ ~@c#I + 2e? + 2e&Al]dt.
(25)
918
S.I. Muslih/Nuclear
The integrabilty for A0 as
condition
Physics B (Proc. Suppl.) 129&130 (2004) 916-918
(23) gives the solution
Ao=~(rrgtald)+2eAl=~~.
(26)
Hence, we obtain the integrable system as follows: ~AI = [T’ + d,Po]dt, (27) 6~’ = [-e(T# + al$) - 2e2A1 + e2/30]dt,(28) 64 = [~4 - e(Po - AI)]& 6-n;@= [-dl#$ + e@(& - Al]dt, dS = [&T~)~ + ;(TQ)~ - $4)”
(29) (30) +
e(t%$ + eAl)Po + -e2(A1)2]dt.
(31)
Since, the action function is integrable, we can make use of (8) to obtain the path integral representation for the CSM as K =
J
n dAi d7+ d4 dx,
J e(&4 + eAl)Po+ -e2(Al)2)1.
exp{i
d2x[i(7r1)2
+ in
- i(@)2
f (32)
The path integral representation (32) is an integration over the canonical phase space coordinates (Ai,#) and (4,~$). 4. Conclusion In this work we have studied the CSM model by using the Hamilton-Jacobi formalism [16, 171. Following the prescriptions of this method, we obtained the set of Hamilton-Jacobi partial differential equations which has two Hamiltonians Hi and HA. The integrability conditions lead us to obtain the canonical reduced phase-space coordinates and the integrable action function in terms of these canonical variables. Then we quantize this model using the canonical path integral method [9-141 to obtain the path integral quantization for our model as an integration over the canonical phase space coordinates (Ai, xi) and ($,rd) without any need to enlarge the initial phase-space by introducing unphysical auxiliary fields (see refs. [I] and the references therein).
Acknowledgments The author would like to thank local organizing committee (LOC) for financial support. REFERENCES 1. J.-H Cha, Y.-W Kim et al, 2. Phys. C69, 175 (1995); W. T. Kim, Y.-W. Kimet al, J. Phys. G23, 325 (1997); M.-I. Park, Y.-J. Park and S. J. Yoon, J. Phys. G24, 2179 (1998). 2. R. Jakiw, R. Rajaraman, Phys. Rev. Lett. 54, 1219 (1985) ; Phys. Rev. Lett. 54, 2060 (1985); K. Harada, Phys. Rev. Lett. 64, 139 ( 1990); Phys.Rev. D42, 4170 (1990). 3. L. D. Faddeev and S. S. Shatashvili, Phys. Lett. B167, 225 ( 1986). 4. F. Schaposnik, C. Viallet, Phys. Lett. B177, 385 (1986) ; K. harada and I. Tsutsui, Phys. Lett. B183, 311 ( 1987). 5. R. Rajaraman, Phys. Lett. B154, 305 (1985); H. 0. Girotti, H. J. Rothe and K. D. Rothe, Phys. Rev. D34, 582 (1986); J.-G. Zhou, Y.-G. Miao and Y.-Y. Liu, Mod. Phys. Lett. A49, 1273 (1994). Y.-W. Kim et al, Phys. Rev.D46, 4574 (1992). R. Banerjee, H. J. Rothe, K. D. Rothe, Phys. Rev. D49, 5438 (1994). S. I. Muslih, Mod.Phys.Lett. A18, 1187 (2003). S. I. Muslih, NUOVO Cimento B115,l (2000); B115, 7 (2000). 10. S. I. Mu&h, H. El-Zalaan and F. El-Sabaa, Int. J. Theor. Phys. 39, 2495 (2000). 11. S. I. Mu&h, Gen. Rel. Grav. 34, 1059 (2002). 12. S. I. Muslih, Czech. J. Phys. 52. 919 (2002). 13. S. I. Muslih, Mod. Phys. Lett. A36, 2382 (2002). 14. S. I. Muslih, Nuovo Cimento B117, 383 (2002). 15. S. I. Muslih, Nucl. Phys. B106, (Proc. Supp.), 879 (2002); 119 C (2003) 968. Nuovo Cimento B107, 1143 16. Y. Giiler, (1992); B107, 1389 (1992). 17. S. I. Muslih and Y. Giiler, Nuovo Cimento BllO, 307 (1995).