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Induced topological terms, spin and statistics in (2 + 1) dimensions
Volume 159B, number 4,5,6
PHYSICS LETTERS
26 September 1985
INDUCED TOPOLOGICAL TERMS, SPIN AND STATISTICS IN (2 + 1) DIMENSIONS T. JAROSZEWICZ 1 Lyman Laborato~. of Physics, Harvard University, Cambridge, MA 02138, USA Received 15 May 1985
The Hopf invariant responsible for spin and statistics of o-model solitons in (2 + 1) dimensions is written in a local form closely analogous to the Wess-Zumino term in (3 + 1) dimensions. Although a priori not quantized, this term, when generated by radiative corrections, appears with such a coefficient that a soliton which has a unit induced fermion charge, has also a half-integer spin and obeys the Fermi-Dirac statistics.
It is by now well known that topological solitons of purely bosonic classical field theories may have properties of fermions. Their spin and statistics are then not kinematical, but rather dynamical features due to specific terms in the effective action. A classic example is the skyrmion [1] of the nonlinear o-model in (3 + 1) dimensions. Its spin and statistics [2] are due to the Wess-Zumino ( W - Z ) term [3], which for an adiabatic 2rr rotation or interchange of skyrmions takes the value rr and generates the factor exp (i~r) = - 1 in the wave function. In (2 + 1) dimensions the solitons of the nonlinear o-model (we also call them skyrmions) acquire spin and statistics due to the Hopf topological invariant in the action [4]. This mechanism, however, seems to be quite different from that operative in (3 + 1) dimensions: (i) In the o-models (unlike in the case of magnetic monopoles and vortices) there are no physical long range forces. Indeed, no such interactions appear in the W - Z term. The Hopf term, on the other hand, involves long range interactions analogous to magnetic interactions of electric current circuits [5,4]. In fact, it must be so if a topological term responsible for spin and 1 On leave from the Institute of Nuclear Physics, Cracow, Poland.
statistics is written as a space-time integral of a Lorentz invariant density. The reason is this: an adiabatic rotation or interchange of skyrmions is equivalent to some space-dependent Lorentz transformation of the fields; e.g. for a rotation with the angular velocity ~, it is the Lorentz transformation with the velocity ~tr, r being the distance from the rotation axis. If the fields associated with skyrmions were of short range in space, we could make this Lorentz transformation only locally (in a box shown in fig. 1). Then, by Lorentz invariance, the topological term would remain unchanged and could not distinguish between a static and a rotating skyrmion. The Hopf term (eq. (3) below) corresponding to fig. 1 is nonzero only because the antiskyrmion current interacts with the long range gauge potential generated by the rotating skyrmion. (ii) In (3 + 1) dimensions the topological charge and the W - Z term are radiatively induced and related to anomalies [6,2]. In (2 + 1) dimensions the topological charge can also be interpreted as induced [7] by appropriately coupling a dynamical fermion field to the o background field. The Hopf term, on the other hand, can apparently be added to the action with an arbitrary coefficient [4], resulting in a fractional spin and unusual statistics. We briefly discuss these problems in the present letter; more details will be given elsewhere. We find that the o-models in (3 + 1) and (2 + 1)
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
299
Volume 159B, number 4,5,6
PHYSICS LETTERS
26 September 1985
the s p a c e - t i m e integral 1 H = 4~.2
(3)
--f A d A .
Fig. 1. A process of creation and annihilation of a skyrmion-antiskyrmion pair with a 2,r rotation of the skyrmion. The Hopf term of eq. (3) is due to a long range interaction between the rotating skyrmion and the antiskyrmion.
dimensions have m o r e in c o m m o n than it seems: the H o p f invariant can be written in a form analogous to the W - Z term; moreover, it is, in fact, radiatively induced (as suggested in ref. [7]) a n d comes with a calculable coefficient. W e consider an 0(3) nonlinear o-model in (2 + 1) dimensions. The dynamical variable is a unit three-vector no(x), a = 1, 2, 3, obeying some b o u n d a r y condition, say n 3 ~ 1 a s l x [ ---, o0. T h e topological charge Q [8] is then the degree of m a p p i n g S 2---, S 2. In terms of a 2 × 2 matrix r/ = / ' / a q" a ( '7"a are Pauli matrices) we have i
t*
Q = ~
_
i_[_[Tr(n(dn)Z).
(1)
16~r . I
It is convenient to parametrize n in terms of a t w o - c o m p o n e n t complex " s p i n o r " Z~
(zi) Z2
Izxl 2 + Izzl 2 = 1 (i.e. z ~ S 2) and
n = 2zz + - 1.
(2)
T h e last condition determines z uniquely up to a phase. In terms of z i
f
i
f dA,
where A = z + d z can be interpreted as an abelian gauge potential. Similarly, the H o p f invariant is 300
(4)'
with some four-dimensional integration region and s o m e extension of n to an at least 3 × 3 matrix yet to b e specified. Quite generally, in terms of z (now at least t h r e e - c o m p o n e n t and always related to n b y eq. (2)) we have 1
.
+
2
1
.
W = -~-~ jM ( d Z d z ) = - - 4 - ~ J M d ( a d A )
i.e. if the b o u n d a r y of M 4 is the space-time, W = ~rH with H given by eq. (3). But it is not necessarily so, as we shall see now. Consider for definiteness a rotating skyrmion of unit charge, parametrized in polar coordinates p,q~ by z(x)=z(t,p,dp)
cos0(p)/2
=(
exp [i(~ - t)] s i n 0 ( o ) / 2
'
such that z + z =
1 fM T r ( n ( d n ) ' ) W = 128-----~ 4
_
Jd2x'ik Tr(nOinOkn)'
or, in the b y n o w customary differential-form notation,
Q=
All this is well known. But let us now try to write the H o p f invariant in a m o r e unusual way: as a four-dimensional integral of a local function of n, analogous to the W - Z term. A natural c a n d i d a t e (with the coefficient determined a posteriori) is
) '
where 0 ( p ) decreases monotonically from 0(0) = ~r to 0(oo) -- 0; t ranges from 0 to 2~r, so that z ( x ) is periodic in time. Let us now define a field z(~, x ) interpolating between a static skyrmion at = 0 and the rotating one at ~ = 1. A simple choice is
=
cos0(o)/2 ~exp [i(ff - t)] s i n O ( p ) / 2 (1 - ~2) x/2 exp (i~) s i n 0 ( p ) / 2
(6)
Volume 159B, number 4,5,6
The components of
A,, = z +O~,z are
A 0 = 0,
A , = isin20/2,
A t=-i~
2sin 2 0 / 2 ,
PHYSICS LETTERS
26 September 1985
f
then
,-..__.,__,..,-
A~=0.
Note that A vanishes rapidly as p --+ o0. On the other hand, the parametrization (6) is singular .1 at p = 0. Therefore, we have to exclude this point in defining M4: M , = ( $ 2 - {0}) ×D2. Here D 2 stands for a two-dimensional disc on which the polar coordinates are 0 ~<~ ~< 1 and 0 ~< t ~< 2~r. Now the boundary 0M 4 consists of two pieces. The first is just the space-time (S 2 - (0}) × S I, ~ = I, but here the integrand A dA of eq. (5) exactly vanishes (because of the Lorentz invariance mentioned in point (i)). A non-vanishing contribution comes only from the second piece of 0M 4, M 3 = Y'2 × I, where I stands for the unit interval 0 ~<~ ~< 1 and ~2 is a closed surface of an infinitesimally thin tube enclosing the string p = 0,0 ~< t < 2~r (fig. 2). We have then W = -- 4--~-1f o 2 " d * f o 2 " d t f o l d ~ A*
O'At o~ o'
o r ¢2
W= -
4-~1f~, dq~dt A~At~=z"
(7)
Numerically, in our case W = ~r; this can also be checked by directly evaluating eq. (4). Thus, as visualized in fig. 2, W measures the degree of twisting of the surface E 2. The same result can be obtained for other processes, as an interchange of two skyrmions. In all cases W/~r is the linking number of curves n(x) = const, on Y2; W/tr, as given by eqs. (4) or (7) is just another expression for the Hopf invariant H, in terms of local properties of the field n(x). Although our definition of H requires going from three to four dimensions, it is perhaps more natural and We could make a gauge transformation z ~ exp(-iq~)z, A, ---, A , - i. Then the singularity would be at p = oo, resulting in a long-range potential A and the standard expression (3) for W / ~ r = H. ,2 This expression is related to the representation for the Hopf invariant derived in ref. [9].
•1
Fig. 2. The tube ~'2 enclosing the string at O = 0. Because of the periodicity in time, ~2 has the topology of a torus. The two curves shown, corresponding to two different values of n, have the linking number 1.
physical than the definition (3), which involves fictitious long range interactions. In the usual representation of the Hopf invariant, the information about interchange a n d / o r rotation of individual skyrmions is transmitted to the rest of the system by fields of long range in space. In our representation the information about "twists" in the field n(x) is conveyed along the "string", i.e. essentially along the skyrmion world line in space-time. It is easy to see that W is independent of the way in which n(x) is extended to n(¢, x). The values of W corresponding to two different extensions differ by the integral (4) taken over a closed surface. With n given by eq. (2) in terms of a three-component z, this integral describes a mapping S 4 ~ CP 2 and vanishes, because $r4(CP 2) = 0. Thus, unlike the W - Z term [2], W is not topologically quantized. We will see now, however, that when W is generated by radiative corrections, it appears in the effective action with the coefficient 1. This means that skyrmions of unit charge have a half-integer spin and obey the Fermi-Dirac statistics. Let us couple fermions to n(x) via [7]
*gP=~ [iY~'O~- mn( x ) ] ~P.
(81
Assuming that n varies slowly on distances m-1, we may use the method of Goldstone and Wilczek [6] to calculate the induced charge (it is then given by eq. (1)[7]) and induced terms in the effective action. Instead of using (8) directly it is more 301
Volume 159B, number 4,5,6
PHYSICS LETTERS
convenient to make a local gauge transformation ~p(x) ~ g - l ( x ) ~ p ( x ) such that n(x) = g(x)Ng-l(x) where N ( N 2 = 1) is a fixed matrix. T h e lagrangian becomes then ,La= ff [iy~'( O. +
1
128~"
fM Tr(N[v'N]4) 4
= 16qrl f~M4Tr(½(Uv)3_Nv3)"
×
_
-3
f dap f dax T r ( [ ( / ~ -
mN)-l~(x)]3)
1 1 fTr,,Nv,3~ + 16~r 3
g
o
~
(10)
where the dots stand for a term involving Tr ( N v 3). The loop with two insertions contributes
302
T h e a u t h o r is indebted to Professor R o y J. G l a u b e r for making his stay at H a r v a r d University possible. T h a n k s are also due to Dr. Louis A l v a r e z - G a u m e and Dr. Philip Nelson for very useful discussions. This research was supported in part b y the U S Department of Energy under c o n t r a c t n u m b e r DE-AC02-76ER03064.
(9)
Precisely these terms are generated by fermion loops with two and three insertions of the vertex ~ q , . I n the limit m ~ ~ the loop with three insertions can be evaluated at zero external m o m e n t a a n d contributes to the effective action t h e term s (3)o r - -
o n l y Tr(Nv 2d r ) = - Tr (Nv3), so that the normalization of Serf is given by eq. (10) and exactly agrees with eq. (9). This implies that skyrmions of unit topological charge are really fermions.
g-XO,g)-mN]~b.
N o w W can be written in terms of the non-abelian pure gauge potential v = g-ldg: W-
26 September 1985
References {1] T.R. Skyrme, Proc. R. Soc. (London) A260 (1961) 127. [2] E. Witten, Nucl. Phys. B223 (1983) 422, 433. [3] J. Wess and B. Zumino, Phys. Lett. 37B (1971) 95. [4] F. Wilczek and A. Zee, Phys. Rev. Lett. 51 (1983) 2250; Santa Barbara preprint NSF-ITP-84-25 (1984); Y.S. Wu, University of Washington preprint 40048-07P4 (1984). [5] H. Flanders, Differential forms and their applications (Academic Press, New York, 1963). [6] J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47 (1981) 986. [7] T. Jaroszewicz, Phys. Lett. 146B (1984) 337. [8] A.A. Belavin and A.M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. 22 (1975) 503 [JETP Lett. 22 (1975) 245]. [9] A.M. Din and W.J. Zakrzewski, Phys. Lett. 146B (1984) 341; Y.S. Wu and A. Zee, Phys. Lett. 147B (1984) 325.
PHYSICS LETTERS
26 September 1985
INDUCED TOPOLOGICAL TERMS, SPIN AND STATISTICS IN (2 + 1) DIMENSIONS T. JAROSZEWICZ 1 Lyman Laborato~. of Physics, Harvard University, Cambridge, MA 02138, USA Received 15 May 1985
The Hopf invariant responsible for spin and statistics of o-model solitons in (2 + 1) dimensions is written in a local form closely analogous to the Wess-Zumino term in (3 + 1) dimensions. Although a priori not quantized, this term, when generated by radiative corrections, appears with such a coefficient that a soliton which has a unit induced fermion charge, has also a half-integer spin and obeys the Fermi-Dirac statistics.
It is by now well known that topological solitons of purely bosonic classical field theories may have properties of fermions. Their spin and statistics are then not kinematical, but rather dynamical features due to specific terms in the effective action. A classic example is the skyrmion [1] of the nonlinear o-model in (3 + 1) dimensions. Its spin and statistics [2] are due to the Wess-Zumino ( W - Z ) term [3], which for an adiabatic 2rr rotation or interchange of skyrmions takes the value rr and generates the factor exp (i~r) = - 1 in the wave function. In (2 + 1) dimensions the solitons of the nonlinear o-model (we also call them skyrmions) acquire spin and statistics due to the Hopf topological invariant in the action [4]. This mechanism, however, seems to be quite different from that operative in (3 + 1) dimensions: (i) In the o-models (unlike in the case of magnetic monopoles and vortices) there are no physical long range forces. Indeed, no such interactions appear in the W - Z term. The Hopf term, on the other hand, involves long range interactions analogous to magnetic interactions of electric current circuits [5,4]. In fact, it must be so if a topological term responsible for spin and 1 On leave from the Institute of Nuclear Physics, Cracow, Poland.
statistics is written as a space-time integral of a Lorentz invariant density. The reason is this: an adiabatic rotation or interchange of skyrmions is equivalent to some space-dependent Lorentz transformation of the fields; e.g. for a rotation with the angular velocity ~, it is the Lorentz transformation with the velocity ~tr, r being the distance from the rotation axis. If the fields associated with skyrmions were of short range in space, we could make this Lorentz transformation only locally (in a box shown in fig. 1). Then, by Lorentz invariance, the topological term would remain unchanged and could not distinguish between a static and a rotating skyrmion. The Hopf term (eq. (3) below) corresponding to fig. 1 is nonzero only because the antiskyrmion current interacts with the long range gauge potential generated by the rotating skyrmion. (ii) In (3 + 1) dimensions the topological charge and the W - Z term are radiatively induced and related to anomalies [6,2]. In (2 + 1) dimensions the topological charge can also be interpreted as induced [7] by appropriately coupling a dynamical fermion field to the o background field. The Hopf term, on the other hand, can apparently be added to the action with an arbitrary coefficient [4], resulting in a fractional spin and unusual statistics. We briefly discuss these problems in the present letter; more details will be given elsewhere. We find that the o-models in (3 + 1) and (2 + 1)
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
299
Volume 159B, number 4,5,6
PHYSICS LETTERS
26 September 1985
the s p a c e - t i m e integral 1 H = 4~.2
(3)
--f A d A .
Fig. 1. A process of creation and annihilation of a skyrmion-antiskyrmion pair with a 2,r rotation of the skyrmion. The Hopf term of eq. (3) is due to a long range interaction between the rotating skyrmion and the antiskyrmion.
dimensions have m o r e in c o m m o n than it seems: the H o p f invariant can be written in a form analogous to the W - Z term; moreover, it is, in fact, radiatively induced (as suggested in ref. [7]) a n d comes with a calculable coefficient. W e consider an 0(3) nonlinear o-model in (2 + 1) dimensions. The dynamical variable is a unit three-vector no(x), a = 1, 2, 3, obeying some b o u n d a r y condition, say n 3 ~ 1 a s l x [ ---, o0. T h e topological charge Q [8] is then the degree of m a p p i n g S 2---, S 2. In terms of a 2 × 2 matrix r/ = / ' / a q" a ( '7"a are Pauli matrices) we have i
t*
Q = ~
_
i_[_[Tr(n(dn)Z).
(1)
16~r . I
It is convenient to parametrize n in terms of a t w o - c o m p o n e n t complex " s p i n o r " Z~
(zi) Z2
Izxl 2 + Izzl 2 = 1 (i.e. z ~ S 2) and
n = 2zz + - 1.
(2)
T h e last condition determines z uniquely up to a phase. In terms of z i
f
i
f dA,
where A = z + d z can be interpreted as an abelian gauge potential. Similarly, the H o p f invariant is 300
(4)'
with some four-dimensional integration region and s o m e extension of n to an at least 3 × 3 matrix yet to b e specified. Quite generally, in terms of z (now at least t h r e e - c o m p o n e n t and always related to n b y eq. (2)) we have 1
.
+
2
1
.
W = -~-~ jM ( d Z d z ) = - - 4 - ~ J M d ( a d A )
i.e. if the b o u n d a r y of M 4 is the space-time, W = ~rH with H given by eq. (3). But it is not necessarily so, as we shall see now. Consider for definiteness a rotating skyrmion of unit charge, parametrized in polar coordinates p,q~ by z(x)=z(t,p,dp)
cos0(p)/2
=(
exp [i(~ - t)] s i n 0 ( o ) / 2
'
such that z + z =
1 fM T r ( n ( d n ) ' ) W = 128-----~ 4
_
Jd2x'ik Tr(nOinOkn)'
or, in the b y n o w customary differential-form notation,
Q=
All this is well known. But let us now try to write the H o p f invariant in a m o r e unusual way: as a four-dimensional integral of a local function of n, analogous to the W - Z term. A natural c a n d i d a t e (with the coefficient determined a posteriori) is
) '
where 0 ( p ) decreases monotonically from 0(0) = ~r to 0(oo) -- 0; t ranges from 0 to 2~r, so that z ( x ) is periodic in time. Let us now define a field z(~, x ) interpolating between a static skyrmion at = 0 and the rotating one at ~ = 1. A simple choice is
=
cos0(o)/2 ~exp [i(ff - t)] s i n O ( p ) / 2 (1 - ~2) x/2 exp (i~) s i n 0 ( p ) / 2
(6)
Volume 159B, number 4,5,6
The components of
A,, = z +O~,z are
A 0 = 0,
A , = isin20/2,
A t=-i~
2sin 2 0 / 2 ,
PHYSICS LETTERS
26 September 1985
f
then
,-..__.,__,..,-
A~=0.
Note that A vanishes rapidly as p --+ o0. On the other hand, the parametrization (6) is singular .1 at p = 0. Therefore, we have to exclude this point in defining M4: M , = ( $ 2 - {0}) ×D2. Here D 2 stands for a two-dimensional disc on which the polar coordinates are 0 ~<~ ~< 1 and 0 ~< t ~< 2~r. Now the boundary 0M 4 consists of two pieces. The first is just the space-time (S 2 - (0}) × S I, ~ = I, but here the integrand A dA of eq. (5) exactly vanishes (because of the Lorentz invariance mentioned in point (i)). A non-vanishing contribution comes only from the second piece of 0M 4, M 3 = Y'2 × I, where I stands for the unit interval 0 ~<~ ~< 1 and ~2 is a closed surface of an infinitesimally thin tube enclosing the string p = 0,0 ~< t < 2~r (fig. 2). We have then W = -- 4--~-1f o 2 " d * f o 2 " d t f o l d ~ A*
O'At o~ o'
o r ¢2
W= -
4-~1f~, dq~dt A~At~=z"
(7)
Numerically, in our case W = ~r; this can also be checked by directly evaluating eq. (4). Thus, as visualized in fig. 2, W measures the degree of twisting of the surface E 2. The same result can be obtained for other processes, as an interchange of two skyrmions. In all cases W/~r is the linking number of curves n(x) = const, on Y2; W/tr, as given by eqs. (4) or (7) is just another expression for the Hopf invariant H, in terms of local properties of the field n(x). Although our definition of H requires going from three to four dimensions, it is perhaps more natural and We could make a gauge transformation z ~ exp(-iq~)z, A, ---, A , - i. Then the singularity would be at p = oo, resulting in a long-range potential A and the standard expression (3) for W / ~ r = H. ,2 This expression is related to the representation for the Hopf invariant derived in ref. [9].
•1
Fig. 2. The tube ~'2 enclosing the string at O = 0. Because of the periodicity in time, ~2 has the topology of a torus. The two curves shown, corresponding to two different values of n, have the linking number 1.
physical than the definition (3), which involves fictitious long range interactions. In the usual representation of the Hopf invariant, the information about interchange a n d / o r rotation of individual skyrmions is transmitted to the rest of the system by fields of long range in space. In our representation the information about "twists" in the field n(x) is conveyed along the "string", i.e. essentially along the skyrmion world line in space-time. It is easy to see that W is independent of the way in which n(x) is extended to n(¢, x). The values of W corresponding to two different extensions differ by the integral (4) taken over a closed surface. With n given by eq. (2) in terms of a three-component z, this integral describes a mapping S 4 ~ CP 2 and vanishes, because $r4(CP 2) = 0. Thus, unlike the W - Z term [2], W is not topologically quantized. We will see now, however, that when W is generated by radiative corrections, it appears in the effective action with the coefficient 1. This means that skyrmions of unit charge have a half-integer spin and obey the Fermi-Dirac statistics. Let us couple fermions to n(x) via [7]
*gP=~ [iY~'O~- mn( x ) ] ~P.
(81
Assuming that n varies slowly on distances m-1, we may use the method of Goldstone and Wilczek [6] to calculate the induced charge (it is then given by eq. (1)[7]) and induced terms in the effective action. Instead of using (8) directly it is more 301
Volume 159B, number 4,5,6
PHYSICS LETTERS
convenient to make a local gauge transformation ~p(x) ~ g - l ( x ) ~ p ( x ) such that n(x) = g(x)Ng-l(x) where N ( N 2 = 1) is a fixed matrix. T h e lagrangian becomes then ,La= ff [iy~'( O. +
1
128~"
fM Tr(N[v'N]4) 4
= 16qrl f~M4Tr(½(Uv)3_Nv3)"
×
_
-3
f dap f dax T r ( [ ( / ~ -
mN)-l~(x)]3)
1 1 fTr,,Nv,3~ + 16~r 3
g
o
~
(10)
where the dots stand for a term involving Tr ( N v 3). The loop with two insertions contributes
302
T h e a u t h o r is indebted to Professor R o y J. G l a u b e r for making his stay at H a r v a r d University possible. T h a n k s are also due to Dr. Louis A l v a r e z - G a u m e and Dr. Philip Nelson for very useful discussions. This research was supported in part b y the U S Department of Energy under c o n t r a c t n u m b e r DE-AC02-76ER03064.
(9)
Precisely these terms are generated by fermion loops with two and three insertions of the vertex ~ q , . I n the limit m ~ ~ the loop with three insertions can be evaluated at zero external m o m e n t a a n d contributes to the effective action t h e term s (3)o r - -
o n l y Tr(Nv 2d r ) = - Tr (Nv3), so that the normalization of Serf is given by eq. (10) and exactly agrees with eq. (9). This implies that skyrmions of unit topological charge are really fermions.
g-XO,g)-mN]~b.
N o w W can be written in terms of the non-abelian pure gauge potential v = g-ldg: W-
26 September 1985
References {1] T.R. Skyrme, Proc. R. Soc. (London) A260 (1961) 127. [2] E. Witten, Nucl. Phys. B223 (1983) 422, 433. [3] J. Wess and B. Zumino, Phys. Lett. 37B (1971) 95. [4] F. Wilczek and A. Zee, Phys. Rev. Lett. 51 (1983) 2250; Santa Barbara preprint NSF-ITP-84-25 (1984); Y.S. Wu, University of Washington preprint 40048-07P4 (1984). [5] H. Flanders, Differential forms and their applications (Academic Press, New York, 1963). [6] J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47 (1981) 986. [7] T. Jaroszewicz, Phys. Lett. 146B (1984) 337. [8] A.A. Belavin and A.M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. 22 (1975) 503 [JETP Lett. 22 (1975) 245]. [9] A.M. Din and W.J. Zakrzewski, Phys. Lett. 146B (1984) 341; Y.S. Wu and A. Zee, Phys. Lett. 147B (1984) 325.