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Double-θ vacuum structure and the functional integral in the Schwinger model
Volume 97B, number 1
PHYSICS gETTERS
17 November 1980
DOUBLE- 0 VACUUM STRUCTURE AND THE FUNCTIONAL INTEGRAL IN THE SCHWINGER MODEL
N.V. KRASNIKOV, V.A. MATVEEV, V.A. RUBAKOV, A.N. TAVKHELIDZE and V.F. TOKAREV Institute Jot Nuclear Research of the Academy of Sciences of the USSR, Moscow, USSR
Received 9 September 1980
The double-0 vacuum structure of the Schwinger model is established in the framework of the pseudo-Euclidean functional integal method. The fermion number "spurionization" is found to be connected with the existence of a zero fermion mode in a background gauge field with nontrivial asymptotics. The relation between the pseudo-Euclidean approach and Euclidean one is described.
1. The structure of the ground state is one of the most important problems of the quantum theory o f gauge fields. Recently, strong arguments have appeared in favour of the complicated structure of the Yang-Mills theory vacuum [1 ]. These arguments are based on the Euclidean functional integral formalism, which is believed to be an appropriate one for the discussion of the gauge theory vacuum structure. It is known [2], however, that the similar Euclidean functional integral technique does not describe the complete double-0 vacuum structure [3] of the twodimensional massless electrodynamics (the Schwinger model [4]). In this letter we show that the double-0 vacuum structure of the Schwinger model can be established with the help of the pseudo-Euclidean functional integral and clarify the functional4ntegral mechanism of the fermion number "spurionization". The vacuum structure of the Schwinger model in the transverse gauge is connected with the structure o f the group of residual gauge transformations [5,6]. Using the Lowenstein-Swieca operator solution [3], ore, can show that the c-number gauge transformations
A#(x)-* ~.(x) + el- au~(x),
n+[a] = - ~ f
1
dx l ( 0 0 a ~ o l a )
(1)
(2)
are integer numbers. The operators Un+,n_ of the gauge transformations with nonzero "topological numbers" n± carry the fermion number (n+ + n ) and chirality (n+ - n_). The vacuum 10+, 0 )is fixed by requiring the gauge invariance of the ground state under the residual gauge transformations including those with nonzero n . , i.e. by the condition (cf. [11) Un+,n_[O+,O
)=exp(-in+O+-in
0 )lO+,O ).
(3) From (3) one finds I0+,0 )= ~
exp(in+O++in_O
)ln+,n
7,
H+, n_
n + , n _ = 0 , + 1 .... ,
in+, n_) = Un+ ' n _ [0,0) ,
~(x) -+ eia(x)¢ (x) ,
o.a.,~(x) = o
are implemented by the unitary operators U[a] if and only if the integrals
(4)
where i0,0) is the perturbation theory vacuum (for details see ref. [6]). Recently it has been found [7] that the existence 103
Volume 97B, number 1
PHYSICS LETTERS
of the zero-energy states I - n , n) with chirality ( - 2 n ) and zero fermion number can be understood in terms of the Euclidean functional integral. In particular, the matrix elements of gauge-invariant operators between the states l-1,1 ) and 10,0)have been shown to be described by the instanton4ike Euclidean field configurations, i.e. those with unit winding number ~ d x u =1 ,
2. In this letter we present the functional-integral evaluation of the double-0 vacuum structure (4). We find it convenient to use the pseudo-Euclidean form of the functional integral (the connection between our approach and the Euclidean one will be described at the end of the paper). Our main goal is to show that the unitary operators of the residual gauge transformations (1) with nonzero topological numbers (2) carry nonzero fermion number and chirality; the vacuum structure (4) follows then in a manner discussed above. We note first, that the relevant properties of the operators can be deduced from the matrix elements
U[a]
Z~[l, ~,~] =(O,OlT{exp[i f (Iu~ +~+ ~f)d2x] }U[a]lO,O),
(5)
Iu, ~, ~
Au(x) + 1
x 0 = T(1 - ie), Ouc~(x) -+ 0
e
(x) -+ 0,
X exp[iS +if qu~u +~6 +-~f)d2x] ,
T -+ oo (7a)
x 0 = T(1 - ie),
xO=-T(1-ie),
eia(x)~ (x)-> 0,
r~,
(7b)
x 0=T(1-ie),
x0=-T(l-ie),
e-i~(x)~(x)-+0,
T-~,
(7c)
In eq. (6)
is the action of the Schwinger model, 3`0 = r 1 , 3,1 = _if2. To achieve our goal we consider the quantity
1 6Zal i 5-((x)llf=~:i=O
-
- -
-
(~(x))~
(8)
and choose the gauge function ogical numbers n+[c~]=-l,
n
a(x) to have the topol-
[a]=0.
(9)
According to eq. (5) the quantity (8) is nonzero (the fermion number "spurionizes") if and only if the operator U[c~] carries the fermion number ( - 1 ) . Now we turn to the calculation of Ze using the representation (6), (7). The functional integral (6) for (8) is
× exp
5 ( 5 . M :~)
x 0 = - T ( 1 - ie), '
= faA.60/
where are the sources. Arguments similar to those of ref. [8] show that Z~ is given by the (pseudo-Euclidean) functional integral [9]
-~
)
ff.,FU~d2z
(~(x)~A, ~,
(lO)
where the integration is performed over the fields A u with the boundary conditions (7a) and (6)
with the following boundary conditions * 1
,1 For a = 0 these boundary conditions coincide with the standard Feynman ones [e.g., 8,101. 104
Au(x)~0,
~(x)-+0,
where the superscript E denotes the Euclidean quantities, the integration contour being an infinite circle. However, the existence of the zero-energy states with nonzero fermion numberhas not yet been established in the framework of the functional integral method.
Ze~=f dAu d ~ d ~
i7 November 1980
~,(x)h,~
=f d~d~exp[i f ~iyu(bz-ieAu)$d2zl $(x) (11) xs an integral over fermions with the boundary condi-
Volume 97B, number 1
PHYSICS LETTERS
tions (7b,c) in an external field with the asymptotics (7a). We find, that if the gauge function has the topological numbers (9), the operator _->
ieAu)
D - i'),u(3, -
~(0)(x) = exp [ - i 4 ) ( x ) - i c ~ - ( x ) - i ~ ( x ) ] ( ! 1
(12)
\0 /
with the boundary conditions (7b). Here ~(x) is the potential for the field Au, e.va~(x),
e01 = - e l 0 = 1 ,
a is defined by the equation
a , a are the negative-frequency parts of the functions a, ~. The conjugate operator ~= +- i3,u (O~ +
ieAu)
has no zero modes fulfilling the boundary conditions (7c). The above statements follow from the 4~-field boundary conditions ~b(x) --> 0 ,
qS(x) + Y(x) ~ 0 ,
x 0 = - T ( 1 - ie),
T ~ oo
and the asymptotic behaviour of the functions
(~ + ~)+ 1
( a - Y ) +- - + 0 ,
3. Now we describe the connection between the pseudo-Euclidean and Euclidean functional integrals. In the Euclidean approach one can calculate the matrix elements between the states 10,0) and I - n , n); for the sake of simplicity we choose n = 1. In the pseudo-Euclidean approach these matrix elements are given by the integral (6), (7) with the gauge function with the topological numbers n+ = - 1 , n = +1. To clarify the further discussion we choose the particular form of a(x),
a(x)
7r
a(X) = ~ [C(X0 --X1) - e(x 0 + x l ) ] . Our next goal is to derive the Euclidean prescription of refs. [7] from the integral (6), (7). The boundary conditions (7) make it impossible to perform a Wick rotation directly in the integral (5), so we change the variables in (6) according to
x 0 = T(1 - ie),
(a +y)± ~ ¥
the zero mode. The integral in eq. (13) can be evaluated exactly [9], the result coincides with (0,01 if(x) U[a] 10,0) =- (0,0l ~b( x ) l - 1,0) calculated with the help of the Lowenstein-Swieca operator solution [3] and the explicit operator form [6] of Moreover, using the functional integral (6) with the boundary conditions (7) one can calculate ~2 the matrix elements of all the T-products of the fields Au, ~, ~, between the states 10,0) and In+, n ) and verify that the resulting functions are identical to those obtained in the operator formalism.
U[a].
has the zero mode
A.(x)=l
17 November 1980
A~-+Au + ( - l ~ua ) , log [ - (x_ T- ie) 2 ] , x 0 =¥T(1 -ie),
-+ e-i~-ff,
T ~°°
According to refs. [11 ] the very fact of the existence of the zero mode (i2) of the operator D and the abe_ sence of the zero modes of the conjugate operator D leads to the nonzero value of eq. (11) {and consequently to the nonzero value of (10) and (8)):
~ --> e i ~
.
(14)
Since this change of variables has the form of the gauge transformation, it affects only the source terms in eq. (6). After this change the boundary conditions become Av--1
Ovac_~O,
C
<¢ (x)>A,~ X
i
=
¢(°)(x)
d~d~'exp
(13)
Isi
fiTta(O u -
leAu) d2 . ~,z],
where the new integration variable $' does not contain
e-iac~, elC~c~ ~ 0 ,
x ° = ¢- T(1 - ie),
T --'-oo.
#2 This calculation requires a careful treatment of the zero fernfion modes, the number of which depends on the topological numbers n+_. 105
Volume 97B, number 1
PHYSICS LETTERS
where
% (x) = 0 (x °) ~ - (x) - 0 (_x0)~+ (x). The casual nature o f the function % allows one to perform the rotation into the Euclidean s p a c e - t i m e in the integral (6), the resulting boundary conditions are
~u
E _ - 1 3uo~E --* 0 , e
e-iaE~E ,
IxEI -+ ~ ,
eiaE~E -+ 0 ,
IxE[ -+ oo,
17 November 1980
cannot be found by the standard Euclidean functional-integral methods. Whether the four-dimensional Yang-Mills theory provides the second example, or whether the Yang-Mills vacuum is characterized by more than one O-angle remain open questions. We hope, however, that the pseudo-Euclidean functionalintegral technique developed here will be useful in the .study of the above problems.
(is) (16)
We are deeply indebted to N.N. Bogolubov, A.A. Logunov and D,Ts. Stoyanov for useful discussions.
where
aE(xE,xE1)=O~c(X0 = - - i x E , x 1
1 =XlE)
xE + i x E
=2-( log x0E _ i x E =~0, ¢ is the polar angle in the (x~, x E) plane. Since c~E (x E) is a real function o f x E, the phase factors exp(+ ic~E) in eq. (16) are inessential. From eq. (15) it follows that the Euclidean fields A E have unit winding number, so we come to the Euclidean functional integral o f refs. [7]. The change o f variables (14) does not affect the gauge-invariant quantities, so the matrix elements o f the gauge-invariant operators between the states 10,0} and 1-1,1 ) calculated in the Euclidean and pseudo-Euclidean approaches coincide. It is worth noting that the generalization of the previous considerations to the case of the gauge functions with the nontrivial topological numbers (9) leads to complex-valued functions a E (xE), and hence to the gauge fields s~ uE which are necessarily complex. Moreover, in this case the "phase factors" exp (+ ia E) in eq. (16) cannot be omitted. We believe that these are the main reasons that make the Euclidean functional integral inappropriate for establishing the doubie-O vacuum structure of the Schwinger model. 4. The Schwinger model serves as an example of the theory, in which the complete vacuum structure
106
References [ 1] C.G. Callan, R.F. Dashen and D.L Gross, Phys. Lett. 63B (1976) 334; R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172. [2] B. Schroer, Acta Phys. Austriaca Suppl. XIX (1978) 155. [3] J.H. Lowenstein and J.A. Swieca, Ann. of Phys. 68 (1971) i72. [4] J. Schwinger, Phys. Rev. 128 (1962) 2425. [5] K.D. Rothe and J.A. Swieca, Phys. Rev. D15 (1977) 541 [6] N.V. Krasnikov, V.A. Matveev, V.A. Rubakov, A.N. Tavkhelidze and V.F. Tokarev, Proc. Seminar Group-theoretic methods in physics, Zvenigorod, 1979. [7] N.K. Nielsen and B. Schroer, Nucl. Phys. B120 (1977) 62; Phys. Lett. 63B (1977) 373; G. Maiella and F. Schaposnik, Nucl. Phys. B132 (1978) 357; B. Schroer, ref. [2]; K.D. Rothe and J.A. Swieca, Ann. of Phys. 117 (1979) 382. [8] E.S. Abers and B.W. Lee, Phys. Reports 9C (1973) 1. [9] N.V. Krasnikov, V.A. Matveev, V.A. Rubakov, A.N. Tavkhelidze and V.F. Tokarev, Proc. Seminar Quarks - 80, Sukhumi, 1980. [10] A.N. Vassiliev, Functional methods in quantum field theory and statistics (Leningrad State Univ. Press, 1976); A.A. Slavnov and L.D. Faddeev, Introduction to quantum theory of gauge fields (Nauka, Moscow, 1978). [11] R.D. Peccei and H. Quinn, Nuovo Cimento 41A (1977) 303; N.V. Krasnikov, V.A. Rubakov and V.F. Tokarev, Phys. Lett. 79B (1978)423;Yad. Fiz. 29 (1979) 1127.
PHYSICS gETTERS
17 November 1980
DOUBLE- 0 VACUUM STRUCTURE AND THE FUNCTIONAL INTEGRAL IN THE SCHWINGER MODEL
N.V. KRASNIKOV, V.A. MATVEEV, V.A. RUBAKOV, A.N. TAVKHELIDZE and V.F. TOKAREV Institute Jot Nuclear Research of the Academy of Sciences of the USSR, Moscow, USSR
Received 9 September 1980
The double-0 vacuum structure of the Schwinger model is established in the framework of the pseudo-Euclidean functional integal method. The fermion number "spurionization" is found to be connected with the existence of a zero fermion mode in a background gauge field with nontrivial asymptotics. The relation between the pseudo-Euclidean approach and Euclidean one is described.
1. The structure of the ground state is one of the most important problems of the quantum theory o f gauge fields. Recently, strong arguments have appeared in favour of the complicated structure of the Yang-Mills theory vacuum [1 ]. These arguments are based on the Euclidean functional integral formalism, which is believed to be an appropriate one for the discussion of the gauge theory vacuum structure. It is known [2], however, that the similar Euclidean functional integral technique does not describe the complete double-0 vacuum structure [3] of the twodimensional massless electrodynamics (the Schwinger model [4]). In this letter we show that the double-0 vacuum structure of the Schwinger model can be established with the help of the pseudo-Euclidean functional integral and clarify the functional4ntegral mechanism of the fermion number "spurionization". The vacuum structure of the Schwinger model in the transverse gauge is connected with the structure o f the group of residual gauge transformations [5,6]. Using the Lowenstein-Swieca operator solution [3], ore, can show that the c-number gauge transformations
A#(x)-* ~.(x) + el- au~(x),
n+[a] = - ~ f
1
dx l ( 0 0 a ~ o l a )
(1)
(2)
are integer numbers. The operators Un+,n_ of the gauge transformations with nonzero "topological numbers" n± carry the fermion number (n+ + n ) and chirality (n+ - n_). The vacuum 10+, 0 )is fixed by requiring the gauge invariance of the ground state under the residual gauge transformations including those with nonzero n . , i.e. by the condition (cf. [11) Un+,n_[O+,O
)=exp(-in+O+-in
0 )lO+,O ).
(3) From (3) one finds I0+,0 )= ~
exp(in+O++in_O
)ln+,n
7,
H+, n_
n + , n _ = 0 , + 1 .... ,
in+, n_) = Un+ ' n _ [0,0) ,
~(x) -+ eia(x)¢ (x) ,
o.a.,~(x) = o
are implemented by the unitary operators U[a] if and only if the integrals
(4)
where i0,0) is the perturbation theory vacuum (for details see ref. [6]). Recently it has been found [7] that the existence 103
Volume 97B, number 1
PHYSICS LETTERS
of the zero-energy states I - n , n) with chirality ( - 2 n ) and zero fermion number can be understood in terms of the Euclidean functional integral. In particular, the matrix elements of gauge-invariant operators between the states l-1,1 ) and 10,0)have been shown to be described by the instanton4ike Euclidean field configurations, i.e. those with unit winding number ~ d x u =1 ,
2. In this letter we present the functional-integral evaluation of the double-0 vacuum structure (4). We find it convenient to use the pseudo-Euclidean form of the functional integral (the connection between our approach and the Euclidean one will be described at the end of the paper). Our main goal is to show that the unitary operators of the residual gauge transformations (1) with nonzero topological numbers (2) carry nonzero fermion number and chirality; the vacuum structure (4) follows then in a manner discussed above. We note first, that the relevant properties of the operators can be deduced from the matrix elements
U[a]
Z~[l, ~,~] =(O,OlT{exp[i f (Iu~ +~+ ~f)d2x] }U[a]lO,O),
(5)
Iu, ~, ~
Au(x) + 1
x 0 = T(1 - ie), Ouc~(x) -+ 0
e
(x) -+ 0,
X exp[iS +if qu~u +~6 +-~f)d2x] ,
T -+ oo (7a)
x 0 = T(1 - ie),
xO=-T(1-ie),
eia(x)~ (x)-> 0,
r~,
(7b)
x 0=T(1-ie),
x0=-T(l-ie),
e-i~(x)~(x)-+0,
T-~,
(7c)
In eq. (6)
is the action of the Schwinger model, 3`0 = r 1 , 3,1 = _if2. To achieve our goal we consider the quantity
1 6Zal i 5-((x)llf=~:i=O
-
- -
-
(~(x))~
(8)
and choose the gauge function ogical numbers n+[c~]=-l,
n
a(x) to have the topol-
[a]=0.
(9)
According to eq. (5) the quantity (8) is nonzero (the fermion number "spurionizes") if and only if the operator U[c~] carries the fermion number ( - 1 ) . Now we turn to the calculation of Ze using the representation (6), (7). The functional integral (6) for (8) is
× exp
5 ( 5 . M :~)
x 0 = - T ( 1 - ie), '
= faA.60/
where are the sources. Arguments similar to those of ref. [8] show that Z~ is given by the (pseudo-Euclidean) functional integral [9]
-~
)
ff.,FU~d2z
(~(x)~A, ~,
(lO)
where the integration is performed over the fields A u with the boundary conditions (7a) and (6)
with the following boundary conditions * 1
,1 For a = 0 these boundary conditions coincide with the standard Feynman ones [e.g., 8,101. 104
Au(x)~0,
~(x)-+0,
where the superscript E denotes the Euclidean quantities, the integration contour being an infinite circle. However, the existence of the zero-energy states with nonzero fermion numberhas not yet been established in the framework of the functional integral method.
Ze~=f dAu d ~ d ~
i7 November 1980
~,(x)h,~
=f d~d~exp[i f ~iyu(bz-ieAu)$d2zl $(x) (11) xs an integral over fermions with the boundary condi-
Volume 97B, number 1
PHYSICS LETTERS
tions (7b,c) in an external field with the asymptotics (7a). We find, that if the gauge function has the topological numbers (9), the operator _->
ieAu)
D - i'),u(3, -
~(0)(x) = exp [ - i 4 ) ( x ) - i c ~ - ( x ) - i ~ ( x ) ] ( ! 1
(12)
\0 /
with the boundary conditions (7b). Here ~(x) is the potential for the field Au, e.va~(x),
e01 = - e l 0 = 1 ,
a is defined by the equation
a , a are the negative-frequency parts of the functions a, ~. The conjugate operator ~= +- i3,u (O~ +
ieAu)
has no zero modes fulfilling the boundary conditions (7c). The above statements follow from the 4~-field boundary conditions ~b(x) --> 0 ,
qS(x) + Y(x) ~ 0 ,
x 0 = - T ( 1 - ie),
T ~ oo
and the asymptotic behaviour of the functions
(~ + ~)+ 1
( a - Y ) +- - + 0 ,
3. Now we describe the connection between the pseudo-Euclidean and Euclidean functional integrals. In the Euclidean approach one can calculate the matrix elements between the states 10,0) and I - n , n); for the sake of simplicity we choose n = 1. In the pseudo-Euclidean approach these matrix elements are given by the integral (6), (7) with the gauge function with the topological numbers n+ = - 1 , n = +1. To clarify the further discussion we choose the particular form of a(x),
a(x)
7r
a(X) = ~ [C(X0 --X1) - e(x 0 + x l ) ] . Our next goal is to derive the Euclidean prescription of refs. [7] from the integral (6), (7). The boundary conditions (7) make it impossible to perform a Wick rotation directly in the integral (5), so we change the variables in (6) according to
x 0 = T(1 - ie),
(a +y)± ~ ¥
the zero mode. The integral in eq. (13) can be evaluated exactly [9], the result coincides with (0,01 if(x) U[a] 10,0) =- (0,0l ~b( x ) l - 1,0) calculated with the help of the Lowenstein-Swieca operator solution [3] and the explicit operator form [6] of Moreover, using the functional integral (6) with the boundary conditions (7) one can calculate ~2 the matrix elements of all the T-products of the fields Au, ~, ~, between the states 10,0) and In+, n ) and verify that the resulting functions are identical to those obtained in the operator formalism.
U[a].
has the zero mode
A.(x)=l
17 November 1980
A~-+Au + ( - l ~ua ) , log [ - (x_ T- ie) 2 ] , x 0 =¥T(1 -ie),
-+ e-i~-ff,
T ~°°
According to refs. [11 ] the very fact of the existence of the zero mode (i2) of the operator D and the abe_ sence of the zero modes of the conjugate operator D leads to the nonzero value of eq. (11) {and consequently to the nonzero value of (10) and (8)):
~ --> e i ~
.
(14)
Since this change of variables has the form of the gauge transformation, it affects only the source terms in eq. (6). After this change the boundary conditions become Av--1
Ovac_~O,
C
<¢ (x)>A,~ X
i
=
¢(°)(x)
d~d~'exp
(13)
Isi
fiTta(O u -
leAu) d2 . ~,z],
where the new integration variable $' does not contain
e-iac~, elC~c~ ~ 0 ,
x ° = ¢- T(1 - ie),
T --'-oo.
#2 This calculation requires a careful treatment of the zero fernfion modes, the number of which depends on the topological numbers n+_. 105
Volume 97B, number 1
PHYSICS LETTERS
where
% (x) = 0 (x °) ~ - (x) - 0 (_x0)~+ (x). The casual nature o f the function % allows one to perform the rotation into the Euclidean s p a c e - t i m e in the integral (6), the resulting boundary conditions are
~u
E _ - 1 3uo~E --* 0 , e
e-iaE~E ,
IxEI -+ ~ ,
eiaE~E -+ 0 ,
IxE[ -+ oo,
17 November 1980
cannot be found by the standard Euclidean functional-integral methods. Whether the four-dimensional Yang-Mills theory provides the second example, or whether the Yang-Mills vacuum is characterized by more than one O-angle remain open questions. We hope, however, that the pseudo-Euclidean functionalintegral technique developed here will be useful in the .study of the above problems.
(is) (16)
We are deeply indebted to N.N. Bogolubov, A.A. Logunov and D,Ts. Stoyanov for useful discussions.
where
aE(xE,xE1)=O~c(X0 = - - i x E , x 1
1 =XlE)
xE + i x E
=2-( log x0E _ i x E =~0, ¢ is the polar angle in the (x~, x E) plane. Since c~E (x E) is a real function o f x E, the phase factors exp(+ ic~E) in eq. (16) are inessential. From eq. (15) it follows that the Euclidean fields A E have unit winding number, so we come to the Euclidean functional integral o f refs. [7]. The change o f variables (14) does not affect the gauge-invariant quantities, so the matrix elements o f the gauge-invariant operators between the states 10,0} and 1-1,1 ) calculated in the Euclidean and pseudo-Euclidean approaches coincide. It is worth noting that the generalization of the previous considerations to the case of the gauge functions with the nontrivial topological numbers (9) leads to complex-valued functions a E (xE), and hence to the gauge fields s~ uE which are necessarily complex. Moreover, in this case the "phase factors" exp (+ ia E) in eq. (16) cannot be omitted. We believe that these are the main reasons that make the Euclidean functional integral inappropriate for establishing the doubie-O vacuum structure of the Schwinger model. 4. The Schwinger model serves as an example of the theory, in which the complete vacuum structure
106
References [ 1] C.G. Callan, R.F. Dashen and D.L Gross, Phys. Lett. 63B (1976) 334; R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172. [2] B. Schroer, Acta Phys. Austriaca Suppl. XIX (1978) 155. [3] J.H. Lowenstein and J.A. Swieca, Ann. of Phys. 68 (1971) i72. [4] J. Schwinger, Phys. Rev. 128 (1962) 2425. [5] K.D. Rothe and J.A. Swieca, Phys. Rev. D15 (1977) 541 [6] N.V. Krasnikov, V.A. Matveev, V.A. Rubakov, A.N. Tavkhelidze and V.F. Tokarev, Proc. Seminar Group-theoretic methods in physics, Zvenigorod, 1979. [7] N.K. Nielsen and B. Schroer, Nucl. Phys. B120 (1977) 62; Phys. Lett. 63B (1977) 373; G. Maiella and F. Schaposnik, Nucl. Phys. B132 (1978) 357; B. Schroer, ref. [2]; K.D. Rothe and J.A. Swieca, Ann. of Phys. 117 (1979) 382. [8] E.S. Abers and B.W. Lee, Phys. Reports 9C (1973) 1. [9] N.V. Krasnikov, V.A. Matveev, V.A. Rubakov, A.N. Tavkhelidze and V.F. Tokarev, Proc. Seminar Quarks - 80, Sukhumi, 1980. [10] A.N. Vassiliev, Functional methods in quantum field theory and statistics (Leningrad State Univ. Press, 1976); A.A. Slavnov and L.D. Faddeev, Introduction to quantum theory of gauge fields (Nauka, Moscow, 1978). [11] R.D. Peccei and H. Quinn, Nuovo Cimento 41A (1977) 303; N.V. Krasnikov, V.A. Rubakov and V.F. Tokarev, Phys. Lett. 79B (1978)423;Yad. Fiz. 29 (1979) 1127.