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Commutators of Gauss-law operators in chiral gauge theories
Volume 223, number 1
PHYSICS LETTERS B
1 June 1989
COMMUTATORS OF G A U S S - L A W O P E R A T O R S I N C H I R A L G A U G E T H E O R I E S T.D. K I E U Department of Physics, Universityof Edinburgh, EdinburghEH9 3JZ, Scotland, UK and Departmentof TheoreticalPhysics, OxfordOX1 3NP, UK Received 29 October 1988; revised manuscript received 14 February 1989
A field theoretic derivation, by nonperturbative functional methods in even space-time dimensions, of the equal-time commutators of the Gauss-law operators is presented. For the anomalous formulation ofchiral gauge theories, the Schwinger term in the commutators is expressed in a closed form in terms of the consistent current divergence anomaly. However, in the gaugeinvariant quantization, with the addition of the Wess-Zumino lagrangian, the commutators are free of the Schwinger term.
Ever since the suggestion [1 ] o f a connection between anomalies in s p a c e - t i m e , Schwinger terms in equal-time c o m m u t a t o r s and cohomology o f the relevant gauge groups, there have been m a n y efforts to calculate the Schwinger term in the c o m m u t a t o r s o f the Gauss-law operators Ga = - ( D i E i ) a + j o ,
(1)
in a n o m a l o u s chiral gauge theories. Here we a d o p t the metric ( + . . . . ) a n d the convention
[To, Tbl=ifabcTc. Various m e t h o d s ranging from point splitting [ 2,3 ] ~, the BJL limit [ 4 - 7 ] to others [ 8 ] have been used and it seems that the cohomological prediction for the Gauss-law c o m m u t a t o r s is realized up to a c o b o u n d a r y term. However, there also are indications that these resuits could be either inconsistent [ 9 ] in (3 + 1 ) dimensions or erroneous [ 10 ] in ( 1 + 1 ) dimensions. This is not very surprising as the quantization ofchiral gauge theories is not at all well u n d e r s t o o d due to apparent internal inconsistencies [ 2,4,11,12 ]. As it turns out, a closer look at the quantization p r o b l e m has led to the ad hoc inclusion [2,4,13] o f the W e s s - Z u m i n o term [ 14] to the path integral action to keep the theories away from being infected Present address. ~1 See also ref. [4] for a critical review of the method. 72
with gauge anomalies. The appearance o f this term has also been derived formally [ 15 ] by the F a d d e e v P o p o v trick, the insertion o f the resolution o f unity into the naive path integral. However, this derivation begins with the integration over all the gauge orbits and m a y thus u n d e r m i n e unitarity. More recently, in the o p e r a t o r formalism, this extra inclusion has been shown to be compatible with the ambiguity o f the Bogoliubov transformation between the time slices [ 16 ]. In fact, from the o p e r a t o r point o f view, I have shown that the W Z term is a consequence o f the h o l o n o m y phase associated with the first quantized h a m i l t o n i a n s o f chiral gauge theories [ 17 ]. Even though it is suggested that there should be no anomalies associated with either large or small gauge transformations, here only the time i n d e p e n d e n t small gauge transformations, whose generators are the Gauss-law operators, are considered. It is then expected that with such gauge-invariant quantization the Schwinger term,
Gab(x,y) - i [ G a ( x ) , Gb(Y)] +fabcGc(x)~(x--Y) ,
(2)
will d i s a p p e a r with the disappearance o f the gauge anomalies. If it is so, the Gauss-law constraints can be consistently i m p o s e d on the Hilbert space as first class constraints. Some calculation based on the BJL m e t h o d have confirmed this expectation [ 18 ].
0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
Volume 223, number 1
PHYSICSLETTERSB
I present in this letter a field theoretic derivation of the commutators of Gauss-law operators using nonperturbative functional techniques in even spacetime dimensions. The method requires no explicit regulator. For the naive, anomalous path integral quantization the Schwinger term will be expressed in a closed form in terms of the consistent anomaly of the current divergence [ 19 ], Xa - ( D J ' ) a i
=4---~e'~trTuOuAv, 1
- 247r2 Ea"p~ tr
(I+I)D,
TaOa(A, OpA,+ ½AvApAa),
(3+l)D.
(3)
Up to a coboundary term, the result coincides with the cohomology prediction: the inconsistencies of this quantization are thus followed by the usual argument. On the other hand, for the gauge-invariant quantization, the Schwinger term disappears together with the gauge anomalies associated with small gauge transformations. A noble feature of this work is that explicit expressions of the WZ term are not needed. As long as the theories are gauge invariant quantum mechanically, the result should hold. In the temporal gauge Ao = 0 the Gauss-law operators generate time independent gauge transformations,
Aa(x)--,A~(x) --D~b(X)ab(X) , D~b (X) -= O.'~"b+f"'bA~(x) ,
dxaa(x)G~(x)
~ (9(y) - (5- L~ ,
I dxo[Q,~(Xo),~(y)]~(Xo-yo) (6) with o~ (x) is an arbitrary function of spatial coordinates. No assumption has been made about the time dependence of the charges. Nevertheless, if the associated transformations turn out to constitute a symmetry of the theory, the charges will not vary with time and the integral over Xo in the last expression will be trivial to yield the more familiar expression. Exploiting eq. (6), I will consider the VEV of the fermionic currents and the electric fields to deduce the equal-time commutators of Gauss-law operators. I now begin with the anomalous quantization of a chiral gauge theory where only the left-handed current is gauged. That is, in the ghost-free temporal gauge, I take the naive expression for the path integral formulation of the generating function
~ - f ~A~(Ao)exp[i(SyM[A]+W[A])],
(7)
where SyM[A] is the usual Yang-Mills action and W[A ] is defined to be
W[Al--iln[f ~, ×exp(i f ddxVl,~(iOu+A~½(l-iys)g/)].
(8)
(4)
as can be seen from the canonical equal-time commutators of the charges Q•(xo) = j
1 June 1989
(5)
with various elementary field operators ~, ~, and A u. Here we have used the canonical (anti) commutations i[E~(x), A~(y) ] =~abc~o6(x-y) and {~'m*(x), q/~ (y)} =6m"6,~(x--y) at equal time. For a composite operator (9 being gauge transformed into (9' we thus have, up to order 0 (e),
Such quantization, as it turns out, is inconsistent. Strictly speaking, gauge-fixing condition cannot be imposed because of the anomaly; but it is necessary. Only then, the interpretation of the cohomological argument can be either confirmed or denied; and only then, the second-class nature of the Gauss-law constraints can be revealed quantum mechanically; thus, only then, the inconsistencies of the naive path integral quantization of chiral gauge theories are exposed. For the time being, in line with other derivations in the literature, I ignore the question of consistency until later. The consistent fermionic currents [ 19 ] can be obtained as 73
Volume 223, n u m b e r 1
J#(x) =
aW[Al 8A~(x)
PHYSICS LETTERS B
(9)
.o=O"
For the definition of the electric fields, the expression E ~ ( x ) = 5.~a(x )5
(SyM[A]+W[A])
A0=0
(10)
1 June 1989
can be deduced. This expression leads to the same results as in ref. [ 7 ], which were obtained after some lengthy perturbative computation, in the distribution sense since
[A (x), B(y) ] = C(x, y) is really an operator-valued distribution relationship
ensures that E~ are in fact the conjugate momenta of the gauge fields at the semi-classical Poisson bracket level and thus are the right ones to be used in the expression for the Gauss-law operators ( 1 ). This definition automatically takes care of the anomalous contribution of W[ A ]. The change of the time component of the current under a time independent gauge transformation is required; and this task can be simplified by the use of an integrated form of the consistent current divergence anomaly (3)
f ddx oG(x) ( D , J ' ) ~ ( x )
(11)
As a~J°(y)
J" dax aa(x)Di.ac(X) ~aAi(x) Jo(Y) o
{ ddx o~(x)(DiJi)a(x) - 8Abo(y) "J
Ao=0
(12)
ddXaa(X)
+
74
jO(y) ] =
"~
Ao=0 }.
(SvM + W),
,
(15)
where a is now an arbitrary function but dependent on time. The gauge-invariant property of the YangMills action and the defining equation for the consistent anomaly (3) have been employed to arrive at the last expression. I next apply the functional operation
on eq. ( 15 ), then let Ao = 0 and remove the time dependence of a. In comparison with an anomaly free theory, it can be seen that the additional contribution to
f dx o~a(x)D~C(x) ~
8
i Eb(y ), (16)
(13)
__fabcjOr~(x_y)
f . aXo(x)l J OXo~ l A o = O
f d~x c~a(x)D~C(x) ~
5,~E~(y) =
Thus, in comparison with eq. (6), the equal-time commutators
i[Ga(x),
W[A] )
6 Mr(y)
I functionally differentiate expression ( 11 ) with respect to A ~ and let Ao = 0 to get
x(__fabejocS(x_y)+ aXa(x) aAbo(y )
A,(SvM[A ] +
: ~ ddxota(x)xa(x)
+ f ddxaa(x)Xa(x) .
6"J°(Y) = f
B(v) ] = f dx dy u(x)C(x, y)v(y),
where u, v are test functions. The simplicity of this method is hence demonstrated even though it cannot yield any information on the cross term commutators: neither [DE, J] nor [J, J]. To evaluate the gauge variation of the electric fields, I now consider a time-dependent gauge transformation of
=
= - f ddx aa(X) (DoJ°)~(X)
=
[A(u),
(14)
and thus to i [ G a ( x ) ,
Wb(y) ], is
aXe(x) f d~xoLa(x) ~ ~o=O"
(17)
The result is, once again, in agreement with the combination of various commutators in ref. [7], for example.
Volume 223, n u m b e r 1
PHYSICS L E T T E R S B
We are then in a position to deduce the Schwinger term of the Gauss-law operator commutators in a closed form in terms of the consistent anomaly
Gab(x, y)
t'
X'(x)
AO=0"
(18) This result is valid for any even space-time dimensions and is comparable to expressions ofrefs. [4,20]. Note that here A and Jl are treated as independent functional variables when the derivatives are performed as should be clear from the defining equation (10). Furthermore, these functional differentiations in the last expression are to be performed before the integration over Xo, and not in reversed order, as the time dependence of the function c~ has now been removed. The above derivation is nonperturbative in nature as it only hinges on the covariant divergence of the consistent current, which can be derived nonperturbatively [ 21 ]. From a canonical quantization viewpoint [17], however, the path integral (7) has been shown to be the wrong expression for the vacuum-to-vacuum amplitudes for chiral gauge theories. The fight one should have an action incorporated with an extra nonlocal piece, which can be written in a local form with the help of the WZ expression c~ (A, g) f ~A ~(Ao) × exp{i(SvM +
W[A]
+Sno..toca, [A ] )},
W[A ] + az (A, g) ) }.
(19)
The resulting amplitudes are then gauge invariant [2,4],
f dax a~(x) D ac u (x)
Ao=0
(21)
'
~G~(Y)
= f dax Oia(x)Diac(X ), ~ -- ~Abo(y)~
ddX O~a(X) Diac(X),
Gb(Y),~
~S~o~al I
~)Ai(X~--)lAo=oC
"
(22) So all ! need is to functionally differentiate (20) with respect to A ob (y), then let Ao = 0 and remove the time dependence of o~ to obtain
= --fabc f d dx aa(x)Gc~(x-y) •
(23)
Thus, the Gauss-law operators transform covariantly, as can be seen from (6), i[Ga(x), Gb(y) ] =
-f~bcGc~(x--Y) •
(24)
It is worthwhile to note that explicit expressions of the nonlocal addition or the WZ term, up to some gauge-invariant quantity, are not required. If the theory is gauge invariant quantum mechanically, Gausslaw constraints should be able to be consistently imposed as first-class constraints. Stricktly speaking, one should further check that those operators are time independent. This should follow either from the relationship [ 6,11 ] 8oGa= ( D u J U ) a = 0 ,
= f ~ g ~A ~ (Ao) × exp{i (SvM +
~St°tal 6A~(x)
one may derive various commutators in the manner presented above. However, in order just to see that there is no anomalous contribution to their commutators, it is only necessary to consider
= Jdxo
×(~ASUo(Y) D b c ( Y ) ~ )
G,(x) =
1 J u n e 1989
Sto,a~= 0 ,
(20)
where S~o,a~is the argument of the exponential in the path integral. With the corresponding Gauss-law operators,
or, equivalently, from the gauge invariance of the hamiltonian as may be seen in the second quantization [ 17 ] [H, G "1 = 0 = i 3 o G a . I want to thank David McMullan and Dipti Sen for discussions and the authors of ref. [ 20 ] for sending me a copy of their preprint.
75
Volume 223, number 1
PHYSICS LETTERS B
References [ 1 ] L. Faddeev, Phys. Lett. B 145 ( (1984) 81; T. Fujiwara, Phys. Lett. B 152 (1985) 103; J. Mickelsson, Commun. Math. Phys. 97 (1985 ) 361; B.Zumino, Nucl. Phys. B 253 ( 1985 ) 477. [2] L. Faddeev and S.L. Shatashvili, Phys. Lett. B 167 (1986) 225. [3] S. Gosh and R. Banerjee, Z. Phys. 41 (1988) 121; S. Hosono and K. Seo, Phys. Rev. D 38 ( 1988 ) 1296. [4] S.-G. Jo, Phys. Rev. D 35 (1987) 3179. [5] S.-G. Jo, Nucl. Phys. B 259 (1985) 616; M. Kobayashi, K. Seo and A. Sugamoto, Nucl. Phys. B 273 (1986) 607. [6] K. Fujikawa, Phys. Lett. B 171 (1986) 424; B 188 (1987) 115. [ 7 ] S.-G. Jo, Phys. Lett. B 163 ( 1985 ) 353. [8] A.J. Niemi and G.W. Semenoff, Phys. Rev. Lett. 56 (1986) 1019; H. Sonoda, Nucl. Phys. B 266 (1986) 410.
76
1 June 1989
[9] P. Mitra, Phys. Rev. Lett. 60 (1988) 265. [10] G.W. Semenoff, Phys. Rev. Lett. 60 (1988) 680; 1590(E). [ 11 ] D.S. Hwang, Nucl. Phys. B 286 (1987) 231. [ 12 ] F.C. Hansen and Y.C. Chao, Phys. Rev. D 37 ( 1988 ) 1570. [ 13 ] A.J. Niemi and G.W. Semenoff, Phys. Rev. Lett. 56 ( 1986 ) 1019. [ 14 ] J. Wess and B. Zumino, Plays• Lett. B 37 ( 1971 ) 95. [ 15 ] O. Babelon, F.A. Schaposnik and C.M. Viallet, Phys. Lett. B 177 (1986) 385. K. Harada and I. Tsutsui, Phys. Lett. B 183 ( 1987 ) 311. [ 16] K. Odaka and T. Itoh Lett. Math. Phys. 15 (1988) 297• [17]T•D. Kieu, Phys. Lett. B 218 (1989) 221, in: Proc. IMA Conf. on MathemanccsTpartlcle physics interface (Oxford, September 1988), to appear• [ 18 ] K. Harada and I. Tsuts~i, Prog. Theor. Phys. 78 ( 1987 ) 675• [19] W.A. Bardeen and B. Zumino, Nucl. Phys• B 244 (1984) 421. [20] T. Nishikawa and I. Tsutsui, Nucl. Phys. B, to appear. [21 ] A.P. Balachandran, C. Marmo, V.P. Nair and C.G. Tahern, Phys. Rev. D 25 (1982) 2713. •
J
,
,
.
PHYSICS LETTERS B
1 June 1989
COMMUTATORS OF G A U S S - L A W O P E R A T O R S I N C H I R A L G A U G E T H E O R I E S T.D. K I E U Department of Physics, Universityof Edinburgh, EdinburghEH9 3JZ, Scotland, UK and Departmentof TheoreticalPhysics, OxfordOX1 3NP, UK Received 29 October 1988; revised manuscript received 14 February 1989
A field theoretic derivation, by nonperturbative functional methods in even space-time dimensions, of the equal-time commutators of the Gauss-law operators is presented. For the anomalous formulation ofchiral gauge theories, the Schwinger term in the commutators is expressed in a closed form in terms of the consistent current divergence anomaly. However, in the gaugeinvariant quantization, with the addition of the Wess-Zumino lagrangian, the commutators are free of the Schwinger term.
Ever since the suggestion [1 ] o f a connection between anomalies in s p a c e - t i m e , Schwinger terms in equal-time c o m m u t a t o r s and cohomology o f the relevant gauge groups, there have been m a n y efforts to calculate the Schwinger term in the c o m m u t a t o r s o f the Gauss-law operators Ga = - ( D i E i ) a + j o ,
(1)
in a n o m a l o u s chiral gauge theories. Here we a d o p t the metric ( + . . . . ) a n d the convention
[To, Tbl=ifabcTc. Various m e t h o d s ranging from point splitting [ 2,3 ] ~, the BJL limit [ 4 - 7 ] to others [ 8 ] have been used and it seems that the cohomological prediction for the Gauss-law c o m m u t a t o r s is realized up to a c o b o u n d a r y term. However, there also are indications that these resuits could be either inconsistent [ 9 ] in (3 + 1 ) dimensions or erroneous [ 10 ] in ( 1 + 1 ) dimensions. This is not very surprising as the quantization ofchiral gauge theories is not at all well u n d e r s t o o d due to apparent internal inconsistencies [ 2,4,11,12 ]. As it turns out, a closer look at the quantization p r o b l e m has led to the ad hoc inclusion [2,4,13] o f the W e s s - Z u m i n o term [ 14] to the path integral action to keep the theories away from being infected Present address. ~1 See also ref. [4] for a critical review of the method. 72
with gauge anomalies. The appearance o f this term has also been derived formally [ 15 ] by the F a d d e e v P o p o v trick, the insertion o f the resolution o f unity into the naive path integral. However, this derivation begins with the integration over all the gauge orbits and m a y thus u n d e r m i n e unitarity. More recently, in the o p e r a t o r formalism, this extra inclusion has been shown to be compatible with the ambiguity o f the Bogoliubov transformation between the time slices [ 16 ]. In fact, from the o p e r a t o r point o f view, I have shown that the W Z term is a consequence o f the h o l o n o m y phase associated with the first quantized h a m i l t o n i a n s o f chiral gauge theories [ 17 ]. Even though it is suggested that there should be no anomalies associated with either large or small gauge transformations, here only the time i n d e p e n d e n t small gauge transformations, whose generators are the Gauss-law operators, are considered. It is then expected that with such gauge-invariant quantization the Schwinger term,
Gab(x,y) - i [ G a ( x ) , Gb(Y)] +fabcGc(x)~(x--Y) ,
(2)
will d i s a p p e a r with the disappearance o f the gauge anomalies. If it is so, the Gauss-law constraints can be consistently i m p o s e d on the Hilbert space as first class constraints. Some calculation based on the BJL m e t h o d have confirmed this expectation [ 18 ].
0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
Volume 223, number 1
PHYSICSLETTERSB
I present in this letter a field theoretic derivation of the commutators of Gauss-law operators using nonperturbative functional techniques in even spacetime dimensions. The method requires no explicit regulator. For the naive, anomalous path integral quantization the Schwinger term will be expressed in a closed form in terms of the consistent anomaly of the current divergence [ 19 ], Xa - ( D J ' ) a i
=4---~e'~trTuOuAv, 1
- 247r2 Ea"p~ tr
(I+I)D,
TaOa(A, OpA,+ ½AvApAa),
(3+l)D.
(3)
Up to a coboundary term, the result coincides with the cohomology prediction: the inconsistencies of this quantization are thus followed by the usual argument. On the other hand, for the gauge-invariant quantization, the Schwinger term disappears together with the gauge anomalies associated with small gauge transformations. A noble feature of this work is that explicit expressions of the WZ term are not needed. As long as the theories are gauge invariant quantum mechanically, the result should hold. In the temporal gauge Ao = 0 the Gauss-law operators generate time independent gauge transformations,
Aa(x)--,A~(x) --D~b(X)ab(X) , D~b (X) -= O.'~"b+f"'bA~(x) ,
dxaa(x)G~(x)
~ (9(y) - (5- L~ ,
I dxo[Q,~(Xo),~(y)]~(Xo-yo) (6) with o~ (x) is an arbitrary function of spatial coordinates. No assumption has been made about the time dependence of the charges. Nevertheless, if the associated transformations turn out to constitute a symmetry of the theory, the charges will not vary with time and the integral over Xo in the last expression will be trivial to yield the more familiar expression. Exploiting eq. (6), I will consider the VEV of the fermionic currents and the electric fields to deduce the equal-time commutators of Gauss-law operators. I now begin with the anomalous quantization of a chiral gauge theory where only the left-handed current is gauged. That is, in the ghost-free temporal gauge, I take the naive expression for the path integral formulation of the generating function
~ - f ~A~(Ao)exp[i(SyM[A]+W[A])],
(7)
where SyM[A] is the usual Yang-Mills action and W[A ] is defined to be
W[Al--iln[f ~, ×exp(i f ddxVl,~(iOu+A~½(l-iys)g/)].
(8)
(4)
as can be seen from the canonical equal-time commutators of the charges Q•(xo) = j
1 June 1989
(5)
with various elementary field operators ~, ~, and A u. Here we have used the canonical (anti) commutations i[E~(x), A~(y) ] =~abc~o6(x-y) and {~'m*(x), q/~ (y)} =6m"6,~(x--y) at equal time. For a composite operator (9 being gauge transformed into (9' we thus have, up to order 0 (e),
Such quantization, as it turns out, is inconsistent. Strictly speaking, gauge-fixing condition cannot be imposed because of the anomaly; but it is necessary. Only then, the interpretation of the cohomological argument can be either confirmed or denied; and only then, the second-class nature of the Gauss-law constraints can be revealed quantum mechanically; thus, only then, the inconsistencies of the naive path integral quantization of chiral gauge theories are exposed. For the time being, in line with other derivations in the literature, I ignore the question of consistency until later. The consistent fermionic currents [ 19 ] can be obtained as 73
Volume 223, n u m b e r 1
J#(x) =
aW[Al 8A~(x)
PHYSICS LETTERS B
(9)
.o=O"
For the definition of the electric fields, the expression E ~ ( x ) = 5.~a(x )5
(SyM[A]+W[A])
A0=0
(10)
1 June 1989
can be deduced. This expression leads to the same results as in ref. [ 7 ], which were obtained after some lengthy perturbative computation, in the distribution sense since
[A (x), B(y) ] = C(x, y) is really an operator-valued distribution relationship
ensures that E~ are in fact the conjugate momenta of the gauge fields at the semi-classical Poisson bracket level and thus are the right ones to be used in the expression for the Gauss-law operators ( 1 ). This definition automatically takes care of the anomalous contribution of W[ A ]. The change of the time component of the current under a time independent gauge transformation is required; and this task can be simplified by the use of an integrated form of the consistent current divergence anomaly (3)
f ddx oG(x) ( D , J ' ) ~ ( x )
(11)
As a~J°(y)
J" dax aa(x)Di.ac(X) ~aAi(x) Jo(Y) o
{ ddx o~(x)(DiJi)a(x) - 8Abo(y) "J
Ao=0
(12)
ddXaa(X)
+
74
jO(y) ] =
"~
Ao=0 }.
(SvM + W),
,
(15)
where a is now an arbitrary function but dependent on time. The gauge-invariant property of the YangMills action and the defining equation for the consistent anomaly (3) have been employed to arrive at the last expression. I next apply the functional operation
on eq. ( 15 ), then let Ao = 0 and remove the time dependence of a. In comparison with an anomaly free theory, it can be seen that the additional contribution to
f dx o~a(x)D~C(x) ~
8
i Eb(y ), (16)
(13)
__fabcjOr~(x_y)
f . aXo(x)l J OXo~ l A o = O
f d~x c~a(x)D~C(x) ~
5,~E~(y) =
Thus, in comparison with eq. (6), the equal-time commutators
i[Ga(x),
W[A] )
6 Mr(y)
I functionally differentiate expression ( 11 ) with respect to A ~ and let Ao = 0 to get
x(__fabejocS(x_y)+ aXa(x) aAbo(y )
A,(SvM[A ] +
: ~ ddxota(x)xa(x)
+ f ddxaa(x)Xa(x) .
6"J°(Y) = f
B(v) ] = f dx dy u(x)C(x, y)v(y),
where u, v are test functions. The simplicity of this method is hence demonstrated even though it cannot yield any information on the cross term commutators: neither [DE, J] nor [J, J]. To evaluate the gauge variation of the electric fields, I now consider a time-dependent gauge transformation of
=
= - f ddx aa(X) (DoJ°)~(X)
=
[A(u),
(14)
and thus to i [ G a ( x ) ,
Wb(y) ], is
aXe(x) f d~xoLa(x) ~ ~o=O"
(17)
The result is, once again, in agreement with the combination of various commutators in ref. [7], for example.
Volume 223, n u m b e r 1
PHYSICS L E T T E R S B
We are then in a position to deduce the Schwinger term of the Gauss-law operator commutators in a closed form in terms of the consistent anomaly
Gab(x, y)
t'
X'(x)
AO=0"
(18) This result is valid for any even space-time dimensions and is comparable to expressions ofrefs. [4,20]. Note that here A and Jl are treated as independent functional variables when the derivatives are performed as should be clear from the defining equation (10). Furthermore, these functional differentiations in the last expression are to be performed before the integration over Xo, and not in reversed order, as the time dependence of the function c~ has now been removed. The above derivation is nonperturbative in nature as it only hinges on the covariant divergence of the consistent current, which can be derived nonperturbatively [ 21 ]. From a canonical quantization viewpoint [17], however, the path integral (7) has been shown to be the wrong expression for the vacuum-to-vacuum amplitudes for chiral gauge theories. The fight one should have an action incorporated with an extra nonlocal piece, which can be written in a local form with the help of the WZ expression c~ (A, g) f ~A ~(Ao) × exp{i(SvM +
W[A]
+Sno..toca, [A ] )},
W[A ] + az (A, g) ) }.
(19)
The resulting amplitudes are then gauge invariant [2,4],
f dax a~(x) D ac u (x)
Ao=0
(21)
'
~G~(Y)
= f dax Oia(x)Diac(X ), ~ -- ~Abo(y)~
ddX O~a(X) Diac(X),
Gb(Y),~
~S~o~al I
~)Ai(X~--)lAo=oC
"
(22) So all ! need is to functionally differentiate (20) with respect to A ob (y), then let Ao = 0 and remove the time dependence of o~ to obtain
= --fabc f d dx aa(x)Gc~(x-y) •
(23)
Thus, the Gauss-law operators transform covariantly, as can be seen from (6), i[Ga(x), Gb(y) ] =
-f~bcGc~(x--Y) •
(24)
It is worthwhile to note that explicit expressions of the nonlocal addition or the WZ term, up to some gauge-invariant quantity, are not required. If the theory is gauge invariant quantum mechanically, Gausslaw constraints should be able to be consistently imposed as first-class constraints. Stricktly speaking, one should further check that those operators are time independent. This should follow either from the relationship [ 6,11 ] 8oGa= ( D u J U ) a = 0 ,
= f ~ g ~A ~ (Ao) × exp{i (SvM +
~St°tal 6A~(x)
one may derive various commutators in the manner presented above. However, in order just to see that there is no anomalous contribution to their commutators, it is only necessary to consider
= Jdxo
×(~ASUo(Y) D b c ( Y ) ~ )
G,(x) =
1 J u n e 1989
Sto,a~= 0 ,
(20)
where S~o,a~is the argument of the exponential in the path integral. With the corresponding Gauss-law operators,
or, equivalently, from the gauge invariance of the hamiltonian as may be seen in the second quantization [ 17 ] [H, G "1 = 0 = i 3 o G a . I want to thank David McMullan and Dipti Sen for discussions and the authors of ref. [ 20 ] for sending me a copy of their preprint.
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Volume 223, number 1
PHYSICS LETTERS B
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1 June 1989
[9] P. Mitra, Phys. Rev. Lett. 60 (1988) 265. [10] G.W. Semenoff, Phys. Rev. Lett. 60 (1988) 680; 1590(E). [ 11 ] D.S. Hwang, Nucl. Phys. B 286 (1987) 231. [ 12 ] F.C. Hansen and Y.C. Chao, Phys. Rev. D 37 ( 1988 ) 1570. [ 13 ] A.J. Niemi and G.W. Semenoff, Phys. Rev. Lett. 56 ( 1986 ) 1019. [ 14 ] J. Wess and B. Zumino, Plays• Lett. B 37 ( 1971 ) 95. [ 15 ] O. Babelon, F.A. Schaposnik and C.M. Viallet, Phys. Lett. B 177 (1986) 385. K. Harada and I. Tsutsui, Phys. Lett. B 183 ( 1987 ) 311. [ 16] K. Odaka and T. Itoh Lett. Math. Phys. 15 (1988) 297• [17]T•D. Kieu, Phys. Lett. B 218 (1989) 221, in: Proc. IMA Conf. on MathemanccsTpartlcle physics interface (Oxford, September 1988), to appear• [ 18 ] K. Harada and I. Tsuts~i, Prog. Theor. Phys. 78 ( 1987 ) 675• [19] W.A. Bardeen and B. Zumino, Nucl. Phys• B 244 (1984) 421. [20] T. Nishikawa and I. Tsutsui, Nucl. Phys. B, to appear. [21 ] A.P. Balachandran, C. Marmo, V.P. Nair and C.G. Tahern, Phys. Rev. D 25 (1982) 2713. •
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