- Email: [email protected]
Dynamics of non-relativistic Chern-Simons solitons
Physics Letters B 308 ( 1993) 286-291 North-Holland
P H YSIC S L ETT ER$ B
Dynamics of non-relativistic Chern-Simons solitons Long Hua a,b and Chihong Chou
a, 1
i Centerfor TheoreticalPhysics, LaboratoryforNuclearScienceandDepartmentofPhysics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA b Department of Physics, New York University, New York, NY 10003, USA
Received l0 March 1992; revised manuscript received 14 December 1992 Editor: M. Dine
The dynamics of non-relativistic Chern-Simons solitons is studied. We find that at low energies solitons move freely in the classical limit. However, there is a quantum mechanical statistical interaction among well-separated solitons. The spin-statistics relation is also confirmed.
Recently, C h e r n - S i m o n s solitons [ 1 ] have been studied extensively. Topological and nontopological solitons are found in a relativistic C h e r n - S i m o n s theory [2,3 ], also non-topological solitons exist in a non-relativistic theory [4 ]. Supersymmetric [ 5 ] and non-abelian [ 6,7 ] generalizations are also considered. In this letter, we shall examine soliton dynamics in the non-relativistic model by Manton's prescription [ 8 ]. Manton's prescription of using the collective coordinate method [ 9 ] was originally proposed in connection with the scattering o f BPS monopoles. The idea is as follows: if there are no forces between static solitons, and the kinetic energy is small, then the full field theory can be truncated to a finite dimensional dynamical system, in which the degrees o f freedom are simply the parameters of the general static solution. This method has been applied to many other systems: Kaluza-Klein monopoles [ 10 ], maximally-charged black holes [ 11 ], the "lumps" o f the CP~ sigma model in ( 2 + 1 ) dimensions [ 12], and the critically coupled abelian Higgs model in ( 2 + l ) dimensions [ 13 ]. A recent application [ 14 ] is to obtain the statistical interactions in relativistic Chern-Simons theory [2,3 ]. In our work, we apply the same idea to the non-relativistic Chern-Simons model [4 ], where explicit static multi-soliton solutions are known. We start with the Jackiw-Pi Lagrangian [4] L=
s(
1 (~/.~)2 ) ,
dEr lx~,~A,~F#7+i~/.Dt~/_½1DvI2+ ~
(1)
where D t = 0 t + i A °, D = V - i A . Let ( z = x + i y ) N
f(z)=
~ i=1
Ci z--ai
N
,
V ( z ) = l-I ( z - a i ) ,
(2)
i=l
then the static solution for N solitons is q/=p I/2ei'° with 4xlf'12
(3)
P= (1 + I)q2)2 '
and This work is supported in part by funds provided by the US Department of Energy (DOE) under contract #DE-AC02-76ER03069. i Present address: Department of Physics, Rockefeller University, New York, NY 1002l, USA. 286
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All fights reserved.
Volume 308, number 3,4
PHYSICS LETTERSB
1 July 1993
o 9 = A r g ( f ' V 2) ,
(4)
w h e r e f ' = df/dz and o9 defined this way cancels all the singularities that may appear otherwise. Note, for static solutions, ai, ci are complex constants: a~ is the position of the ith soliton, and c~ represents its size and phase. Thus, the configuration space of N solitons has dimensionality 4N. We now adopt Manton's idea: let a~, q be time-dependent. By treating these parameters as dynamical variables of the system, we truncate the original infinite dimensional system to a finite (4N) one. Rewriting the Lagrangian as L = f d2r [ (½i~-pdg) +
½x,4X~,l-Ao(xB+p)
- ½1 (Dl -iD2)~'l 2 ] ,
(5)
where various boundary contributions have been dropped, we find that the third term in the integrand can be dropped due to Gauss's law, while the last term vanishes as in the static case due to the self-duality condition
[4] (D, - i D 2 ) g t = 0 .
(6)
Eq. (6) also gives
A= - ½Vxlnp+Vo9
(7)
(thus V.A = 0 because o9 is a harmonic function ). Defining ~ = - ½In p - In I f ' V2 l, we have A = V X q~and
; d2rAX.~= f d2rAX(VX~)= ~ d2r~V.A=O.
(8)
Finally, since d ~ d2r/~= ~ ~ d 2 r p = O ,
(9)
we end with L=-
f dz*
dzpd;.
(10)
Let us first study the two soliton case. For N = 2, eq. (2) becomes f=
c------L-I+ c------L-2.
z--al
(11)
z--a2
For the sake of simplicity, we here consider two solitons with the same size, i.e. [q [ = Ic2[. Also, we choose the center-of-mass frame: a 2 = - a ~ - - a . Solutions in other frames may be obtained by a Galileo boost, as was pointed out in refs. [ 15,4 ]. Furthermore, we pull out an over-all-phase to make c2 = cT -c*. As one can see from eqs. (3) and (4), doing this only shifts o9 by this phase, and consequently modifies the Lagrangian by a total time derivative term, which has no physical meaning, as will be explained below. After these steps, we have C
f= z - a
C* + -z+a' -
(12)
which gives from eqs. (3) and (4), p=
4xuu* /t 2
,
(13)
287
Volume 308, number 3,4 l
PHYSICS LETTERS B
1 July 1993
U
O9= ~ lnu-.,
(14)
with
u= (c+c*) ( z 2 + a 2) + 2az(c-c*) ,
(15)
A= ( z 2 - a 2) (z*2-a .2 ) + [ (c+c*)z+ (c-c*)a] [ ( c + c * ) z * - (c-c*)a*] = 4r2r ] sin 2 ( 0 - Oa) + 4r 2rra sin 20c sin ( 0 -- 0a ) + 4r 2 ( r 2 cos20c + ra2 sin20~) + ( r 2 -- r ] ) 2.
(16)
(We define z = re is, a = rae i°a, and c = r~ei°c). The Lagrangian ( 10 ) becomes
L=_~dz.
dz(pOOg. OO9 .. OO9 OOg..'~ --~a a + p ~ a + P --~c C+ P -~c. C )"
(17)
1
First, we show that the coefficient o f ~ vanishes. Since
u*( z + a ) Z - u ( z * - a * ) Z=8c*( r2 + r2)rra cos( O-Oa) ,
(18)
and because 27t
dO
Cos(O--Oa)
~-2
=0,
(19)
0
we have
~ d z . dzp 0o9
Oc -
2X~dz. dzU*(z+a)2-u(z*-a*)2 "
A2
=0.
(20)
Next, keeping eq. ( 19 ) in mind, we calculate the coefficient of d'. f
4x j" u* dz*dzp 0o9 0a - i dz*dz~ [a(c+c*)+z(c-c*)] oo
27t
0
0
-4Xe-i°~' rdr~i
dO~-~
× {ra(C+C*)2[r2 cos 2 ( 0 - 0a) + r ] ] + ir(3raz - r z ) ( c 2 - c *z ) sin ( 0 - 0a) - 2 r a r 2 ( c - c *) 2) oo
21t
16xf
~
- -~a a qdq 0
s2(l+q2cosZO)+2t2q2+stq(q2-3) sin 0 dO[4q2sin20+8stqsinO+4(qZs2+tz)+(q2-1)2]2'
(21)
0
where s - Re(c/ra), t ~ - Im(c/ra) and q-- r/ra. We have not succeeded in evaluating analytically this integral for arbitrary s and t, although for special values an exact expression is available, see below. Therefore we have performed a numerical calculation with the result ~ d z * dz p
0O9
4nX
Oa
ia
(22)
which gives for the Lagrangian the total time derivative L = - 8nK0a.
(23)
Recalling 0a is the relative angle between two solitons, one sees that (23 ) is nothing but the expected statistical interaction with spin [ 16 ] S = - 4rm. 288
Volume 308, number 3,4
PHYSICS LETTERSB
l July 1993
L=2SOa.
(24)
Thus classically solitons move freely in this approximation. We check the numerical result in three special cases, which are in some 3D subspaces of the original 4D configuration space. In the first case c = - c * ( s = 0 ). 1 -az o9= ~ ln(a-.-~z.) '
(25)
Thus,
L=- ~
(26)
dz* dzp(f/a-~t*/a*)=-8mcOa.
In the second case c=c*(t=O).
(z2+a 2) (z*2 + a .2) a.2) +4c2r 2 ] 2,
(27)
P = 16/~C2 [ (Z2 a 2) (z.2 1 . { zE+a 2
(28)
o9= ~ ln~z.2-5-~-~a.2) , Therefore,
L=
16xc 2
dz* dz
~
=--32/cC2ra2
[(z2a2)(z.2a.2)+4cZr2] 2
i2; rdr
0 oo
=--8KflOalt
a~l(z.E + a.2) -- 2a*it*( zE + a 2) r2cos2(O--Oa) +r2a dO [r4+raa+4C2r2_2r2ar2Cos2(O_Oa)]20a 0
1 Wflot + 3ot 2 dot [(Ot2_l_flot+l)2 4ot213/2
0
fl=__ 4C2
r2)
r-T' ot(29)
= - 8rmOa. Last, we check the case with Icl ~ 0
s=2s~, t=2q
(2=rc/ra),
(s2+t 2_..~0
w i t h s~ t fixed). We change varibles as follows:
(30,31)
R = (R, ~0)= (r 2, 20) = ; t k + n ,
where n is a unit vector and satisfies
R.n=R cos tp=r 2 cos 20,
(32)
and rsin O=x/~ sin ( g / 2 ) = + x / ( R - R . n ) / 2 - , O
(33)
(as 2 ~ 0 ) .
Then the integral I in eq. (21 ) becomes
0
0
s 2 ( l + R . n ) + 2 t 2 R + s , t , x / ~ ( R - 3 ) sin(~o/2) = I + + I _ {(R-n)2+422[Rs 2 +t 2 +2s, llX/~ sin (~o/2)1} 2
(34)
with I+
[" d2R
-- = 1 2 2 J
s2( 1 +R.n) +2tER+_s~ t~v / R ( g - 3) sin(tp/2) - ~ 2 +t 2 +2SI tav/R sin(~a/2)]}2
R{(R-n)2+4~.E[gs
l
d 2 ( kk= +- 4 ) = = } f t .
(35) 289
Volume 308, number 3,4
PHYSICS LETTERS B
1 July 1993
The result (22) is once again regained. We now study the statistical interaction between arbitrary n u m b e r o f well-separated solitons. From (4),
2,c,l_i~,,(z_aj)2
yqc.1]j~i(z._a~)2,
(36)
09=
In
where
a(a*), c(c*) are functions o f time. Thus,
&= 1 (~,c, Yk,,iFIj~,,k(z--aj)2"2(z--ak)(--dk) _ 2i \ YiCiI~j#i(z--aj) 2
(C.C.))+~i (~ici~Ijv~i(z--aj, 2 \ ~ - (c.c.)). 1
(37)
Since the density function o f the ith soliton is almost a &function at ai, for well-separated solitons one can approximately take p as a sum o f ~-functions. Then, 1(~ --2ilkk Idz'dzpdo'~4~zK~ f dz'dzdo(z)d(z-at)=4~rx~t~Xk~.gta~_a
C.C.) + 4ZCXd Arg(iTi. c~)
O(at-ak'+4nxdArg(~ ci) ,
~8~ l
ut
~i
(38,
]
where
O(at - ak) = Arg (at -- ak) •
( 39 )
Therefore,
L=2S Z O(a~-aD,
(40)
l
with the spin as before, S = - 4zex. The total time derivative of the total soliton phase is dropped here, just as in the two body case before. Because it involves the total phase and not paired terms, it is without significance. Some comments are in order. First, we note that the approximate solutions only satisfy the equations of motion o f the original Lagrangian through terms linear in v, and fail at order O(v2), where v is the speed o f the solitons. Second, Manton's prescription is usually used in systems with second order time derivatives. It is not clear how good it is in our case, where only first time derivative terms appear. Thus, the numerical simulation [ 17 ] o f this problem will be interesting. Third, the 90 ° scattering, which is seen in other Manton scatterings does not appear here. Finally, it is worth noticing that the two body result (23) is the same as obtained in the well-separated many-body case when the 8 function approximation for p is made in (38). This may have some deeper explanation. We thank R. Jackiw for suggesting this problem and many helpful discussions. One o f us (L.H.) thanks M. Kaku and H. Min for encouragement, and he would also like to thank New York University for the Dean's Dissertation Fellowship and the Center for Theoretical Physics at M I T for the hospitality during this work.
References
[ 1] For a review, see R. Jaekiw and S.-Y. Pi, MIT preprint CTP#2000 (August 1991). [ 2] R. Jackiw and E. Weinberg, Phys. Rev. Lett. 64 (1990) 2234; R. Jackiw, K. Lee and E. Weinberg, Phys. Rev. D 42 (1990) 3488. [3] J. Hong, Y. Kim and P.Y. Pac, Phys. Rev. Lett. 64 (1990) 2230. [4] R. Jackiw and S.-Y. Pi, Phys. Rev. Lett. 64 (1990) 2969; 66 ( 1991 ) 2682 (C); Phys. Rev. D 42 (1990) 3500. [5] C. Lee, K. Lee and E. Weinberg, Phys. Lett. B 243 (1990) 105. [6] B. Grossman, Phys. Rev. Lett. 65 (1990) 3230. 290
Volume 308, number 3,4
PHYSICS LETTERS B
1 July 1993
[ 7 ] G. Dunne, R. Jackiw, S.-Y. Pi and C. Trugenberger, Phys. Rev. D 43 ( 1991 ) 1332. [8] N.S. Manton, Phys. Lett. B 110 (1982) 54; B 154 (1985) 397. [ 9 ] For example, see N.H. Christ and T.D. Lee, Phys. Rev. D 12 ( 1975 ) 1606; J.-L. Gervais and B. Sakita, Phys. Rev. D 11 ( 1975 ) 2943. [ 10] P.J. Ruback, Commun. Math. Phys. 107 (1986) 93. [ 11 ] G.W. Gibbons and P.J. Ruback, Phys. Rev. Lett. 57 (1986) 1617; R.C. Ferrell and D.M. Eardley, Phys. Rev. Lett. 59 (1987) 1617. [12 ] R. Ward, Phys. Lett. B 158 (1985 ) 424; R. Leese, Nucl. Phys. B 344 (1990) 33. [ 13 ] P.J. Ruback, Nucl. Phys. B 296 ( 1988 ) 669; T.M. Samols, Phys. Lett. B 244 (1990) 285; preprint DAMTP/91-13. [ 14 ] S.-K. Kim and H. Min. preprint SNUTP-91-41. [ 15 ] R. Jackiw and S.-Y. Pi, Phys. Rev. D 44 ( 1991 ). [ 16] R. MacKenzie and F. Wilczek, Intern. J. Mod. Phys. A 3 (1988) 2827; G. Dunne, R. Jackiw and C. Trugenberger, Ann. Phys. (NY) 194 (1990) 197. [ 17 ] R. Strilka, in preparation.
291
P H YSIC S L ETT ER$ B
Dynamics of non-relativistic Chern-Simons solitons Long Hua a,b and Chihong Chou
a, 1
i Centerfor TheoreticalPhysics, LaboratoryforNuclearScienceandDepartmentofPhysics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA b Department of Physics, New York University, New York, NY 10003, USA
Received l0 March 1992; revised manuscript received 14 December 1992 Editor: M. Dine
The dynamics of non-relativistic Chern-Simons solitons is studied. We find that at low energies solitons move freely in the classical limit. However, there is a quantum mechanical statistical interaction among well-separated solitons. The spin-statistics relation is also confirmed.
Recently, C h e r n - S i m o n s solitons [ 1 ] have been studied extensively. Topological and nontopological solitons are found in a relativistic C h e r n - S i m o n s theory [2,3 ], also non-topological solitons exist in a non-relativistic theory [4 ]. Supersymmetric [ 5 ] and non-abelian [ 6,7 ] generalizations are also considered. In this letter, we shall examine soliton dynamics in the non-relativistic model by Manton's prescription [ 8 ]. Manton's prescription of using the collective coordinate method [ 9 ] was originally proposed in connection with the scattering o f BPS monopoles. The idea is as follows: if there are no forces between static solitons, and the kinetic energy is small, then the full field theory can be truncated to a finite dimensional dynamical system, in which the degrees o f freedom are simply the parameters of the general static solution. This method has been applied to many other systems: Kaluza-Klein monopoles [ 10 ], maximally-charged black holes [ 11 ], the "lumps" o f the CP~ sigma model in ( 2 + 1 ) dimensions [ 12], and the critically coupled abelian Higgs model in ( 2 + l ) dimensions [ 13 ]. A recent application [ 14 ] is to obtain the statistical interactions in relativistic Chern-Simons theory [2,3 ]. In our work, we apply the same idea to the non-relativistic Chern-Simons model [4 ], where explicit static multi-soliton solutions are known. We start with the Jackiw-Pi Lagrangian [4] L=
s(
1 (~/.~)2 ) ,
dEr lx~,~A,~F#7+i~/.Dt~/_½1DvI2+ ~
(1)
where D t = 0 t + i A °, D = V - i A . Let ( z = x + i y ) N
f(z)=
~ i=1
Ci z--ai
N
,
V ( z ) = l-I ( z - a i ) ,
(2)
i=l
then the static solution for N solitons is q/=p I/2ei'° with 4xlf'12
(3)
P= (1 + I)q2)2 '
and This work is supported in part by funds provided by the US Department of Energy (DOE) under contract #DE-AC02-76ER03069. i Present address: Department of Physics, Rockefeller University, New York, NY 1002l, USA. 286
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All fights reserved.
Volume 308, number 3,4
PHYSICS LETTERSB
1 July 1993
o 9 = A r g ( f ' V 2) ,
(4)
w h e r e f ' = df/dz and o9 defined this way cancels all the singularities that may appear otherwise. Note, for static solutions, ai, ci are complex constants: a~ is the position of the ith soliton, and c~ represents its size and phase. Thus, the configuration space of N solitons has dimensionality 4N. We now adopt Manton's idea: let a~, q be time-dependent. By treating these parameters as dynamical variables of the system, we truncate the original infinite dimensional system to a finite (4N) one. Rewriting the Lagrangian as L = f d2r [ (½i~-pdg) +
½x,4X~,l-Ao(xB+p)
- ½1 (Dl -iD2)~'l 2 ] ,
(5)
where various boundary contributions have been dropped, we find that the third term in the integrand can be dropped due to Gauss's law, while the last term vanishes as in the static case due to the self-duality condition
[4] (D, - i D 2 ) g t = 0 .
(6)
Eq. (6) also gives
A= - ½Vxlnp+Vo9
(7)
(thus V.A = 0 because o9 is a harmonic function ). Defining ~ = - ½In p - In I f ' V2 l, we have A = V X q~and
; d2rAX.~= f d2rAX(VX~)= ~ d2r~V.A=O.
(8)
Finally, since d ~ d2r/~= ~ ~ d 2 r p = O ,
(9)
we end with L=-
f dz*
dzpd;.
(10)
Let us first study the two soliton case. For N = 2, eq. (2) becomes f=
c------L-I+ c------L-2.
z--al
(11)
z--a2
For the sake of simplicity, we here consider two solitons with the same size, i.e. [q [ = Ic2[. Also, we choose the center-of-mass frame: a 2 = - a ~ - - a . Solutions in other frames may be obtained by a Galileo boost, as was pointed out in refs. [ 15,4 ]. Furthermore, we pull out an over-all-phase to make c2 = cT -c*. As one can see from eqs. (3) and (4), doing this only shifts o9 by this phase, and consequently modifies the Lagrangian by a total time derivative term, which has no physical meaning, as will be explained below. After these steps, we have C
f= z - a
C* + -z+a' -
(12)
which gives from eqs. (3) and (4), p=
4xuu* /t 2
,
(13)
287
Volume 308, number 3,4 l
PHYSICS LETTERS B
1 July 1993
U
O9= ~ lnu-.,
(14)
with
u= (c+c*) ( z 2 + a 2) + 2az(c-c*) ,
(15)
A= ( z 2 - a 2) (z*2-a .2 ) + [ (c+c*)z+ (c-c*)a] [ ( c + c * ) z * - (c-c*)a*] = 4r2r ] sin 2 ( 0 - Oa) + 4r 2rra sin 20c sin ( 0 -- 0a ) + 4r 2 ( r 2 cos20c + ra2 sin20~) + ( r 2 -- r ] ) 2.
(16)
(We define z = re is, a = rae i°a, and c = r~ei°c). The Lagrangian ( 10 ) becomes
L=_~dz.
dz(pOOg. OO9 .. OO9 OOg..'~ --~a a + p ~ a + P --~c C+ P -~c. C )"
(17)
1
First, we show that the coefficient o f ~ vanishes. Since
u*( z + a ) Z - u ( z * - a * ) Z=8c*( r2 + r2)rra cos( O-Oa) ,
(18)
and because 27t
dO
Cos(O--Oa)
~-2
=0,
(19)
0
we have
~ d z . dzp 0o9
Oc -
2X~dz. dzU*(z+a)2-u(z*-a*)2 "
A2
=0.
(20)
Next, keeping eq. ( 19 ) in mind, we calculate the coefficient of d'. f
4x j" u* dz*dzp 0o9 0a - i dz*dz~ [a(c+c*)+z(c-c*)] oo
27t
0
0
-4Xe-i°~' rdr~i
dO~-~
× {ra(C+C*)2[r2 cos 2 ( 0 - 0a) + r ] ] + ir(3raz - r z ) ( c 2 - c *z ) sin ( 0 - 0a) - 2 r a r 2 ( c - c *) 2) oo
21t
16xf
~
- -~a a qdq 0
s2(l+q2cosZO)+2t2q2+stq(q2-3) sin 0 dO[4q2sin20+8stqsinO+4(qZs2+tz)+(q2-1)2]2'
(21)
0
where s - Re(c/ra), t ~ - Im(c/ra) and q-- r/ra. We have not succeeded in evaluating analytically this integral for arbitrary s and t, although for special values an exact expression is available, see below. Therefore we have performed a numerical calculation with the result ~ d z * dz p
0O9
4nX
Oa
ia
(22)
which gives for the Lagrangian the total time derivative L = - 8nK0a.
(23)
Recalling 0a is the relative angle between two solitons, one sees that (23 ) is nothing but the expected statistical interaction with spin [ 16 ] S = - 4rm. 288
Volume 308, number 3,4
PHYSICS LETTERSB
l July 1993
L=2SOa.
(24)
Thus classically solitons move freely in this approximation. We check the numerical result in three special cases, which are in some 3D subspaces of the original 4D configuration space. In the first case c = - c * ( s = 0 ). 1 -az o9= ~ ln(a-.-~z.) '
(25)
Thus,
L=- ~
(26)
dz* dzp(f/a-~t*/a*)=-8mcOa.
In the second case c=c*(t=O).
(z2+a 2) (z*2 + a .2) a.2) +4c2r 2 ] 2,
(27)
P = 16/~C2 [ (Z2 a 2) (z.2 1 . { zE+a 2
(28)
o9= ~ ln~z.2-5-~-~a.2) , Therefore,
L=
16xc 2
dz* dz
~
=--32/cC2ra2
[(z2a2)(z.2a.2)+4cZr2] 2
i2; rdr
0 oo
=--8KflOalt
a~l(z.E + a.2) -- 2a*it*( zE + a 2) r2cos2(O--Oa) +r2a dO [r4+raa+4C2r2_2r2ar2Cos2(O_Oa)]20a 0
1 Wflot + 3ot 2 dot [(Ot2_l_flot+l)2 4ot213/2
0
fl=__ 4C2
r2)
r-T' ot(29)
= - 8rmOa. Last, we check the case with Icl ~ 0
s=2s~, t=2q
(2=rc/ra),
(s2+t 2_..~0
w i t h s~ t fixed). We change varibles as follows:
(30,31)
R = (R, ~0)= (r 2, 20) = ; t k + n ,
where n is a unit vector and satisfies
R.n=R cos tp=r 2 cos 20,
(32)
and rsin O=x/~ sin ( g / 2 ) = + x / ( R - R . n ) / 2 - , O
(33)
(as 2 ~ 0 ) .
Then the integral I in eq. (21 ) becomes
0
0
s 2 ( l + R . n ) + 2 t 2 R + s , t , x / ~ ( R - 3 ) sin(~o/2) = I + + I _ {(R-n)2+422[Rs 2 +t 2 +2s, llX/~ sin (~o/2)1} 2
(34)
with I+
[" d2R
-- = 1 2 2 J
s2( 1 +R.n) +2tER+_s~ t~v / R ( g - 3) sin(tp/2) - ~ 2 +t 2 +2SI tav/R sin(~a/2)]}2
R{(R-n)2+4~.E[gs
l
d 2 ( kk= +- 4 ) = = } f t .
(35) 289
Volume 308, number 3,4
PHYSICS LETTERS B
1 July 1993
The result (22) is once again regained. We now study the statistical interaction between arbitrary n u m b e r o f well-separated solitons. From (4),
2,c,l_i~,,(z_aj)2
yqc.1]j~i(z._a~)2,
(36)
09=
In
where
a(a*), c(c*) are functions o f time. Thus,
&= 1 (~,c, Yk,,iFIj~,,k(z--aj)2"2(z--ak)(--dk) _ 2i \ YiCiI~j#i(z--aj) 2
(C.C.))+~i (~ici~Ijv~i(z--aj, 2 \ ~ - (c.c.)). 1
(37)
Since the density function o f the ith soliton is almost a &function at ai, for well-separated solitons one can approximately take p as a sum o f ~-functions. Then, 1(~ --2ilkk Idz'dzpdo'~4~zK~ f dz'dzdo(z)d(z-at)=4~rx~t~Xk~.gta~_a
C.C.) + 4ZCXd Arg(iTi. c~)
O(at-ak'+4nxdArg(~ ci) ,
~8~ l
ut
~i
(38,
]
where
O(at - ak) = Arg (at -- ak) •
( 39 )
Therefore,
L=2S Z O(a~-aD,
(40)
l
with the spin as before, S = - 4zex. The total time derivative of the total soliton phase is dropped here, just as in the two body case before. Because it involves the total phase and not paired terms, it is without significance. Some comments are in order. First, we note that the approximate solutions only satisfy the equations of motion o f the original Lagrangian through terms linear in v, and fail at order O(v2), where v is the speed o f the solitons. Second, Manton's prescription is usually used in systems with second order time derivatives. It is not clear how good it is in our case, where only first time derivative terms appear. Thus, the numerical simulation [ 17 ] o f this problem will be interesting. Third, the 90 ° scattering, which is seen in other Manton scatterings does not appear here. Finally, it is worth noticing that the two body result (23) is the same as obtained in the well-separated many-body case when the 8 function approximation for p is made in (38). This may have some deeper explanation. We thank R. Jackiw for suggesting this problem and many helpful discussions. One o f us (L.H.) thanks M. Kaku and H. Min for encouragement, and he would also like to thank New York University for the Dean's Dissertation Fellowship and the Center for Theoretical Physics at M I T for the hospitality during this work.
References
[ 1] For a review, see R. Jaekiw and S.-Y. Pi, MIT preprint CTP#2000 (August 1991). [ 2] R. Jackiw and E. Weinberg, Phys. Rev. Lett. 64 (1990) 2234; R. Jackiw, K. Lee and E. Weinberg, Phys. Rev. D 42 (1990) 3488. [3] J. Hong, Y. Kim and P.Y. Pac, Phys. Rev. Lett. 64 (1990) 2230. [4] R. Jackiw and S.-Y. Pi, Phys. Rev. Lett. 64 (1990) 2969; 66 ( 1991 ) 2682 (C); Phys. Rev. D 42 (1990) 3500. [5] C. Lee, K. Lee and E. Weinberg, Phys. Lett. B 243 (1990) 105. [6] B. Grossman, Phys. Rev. Lett. 65 (1990) 3230. 290
Volume 308, number 3,4
PHYSICS LETTERS B
1 July 1993
[ 7 ] G. Dunne, R. Jackiw, S.-Y. Pi and C. Trugenberger, Phys. Rev. D 43 ( 1991 ) 1332. [8] N.S. Manton, Phys. Lett. B 110 (1982) 54; B 154 (1985) 397. [ 9 ] For example, see N.H. Christ and T.D. Lee, Phys. Rev. D 12 ( 1975 ) 1606; J.-L. Gervais and B. Sakita, Phys. Rev. D 11 ( 1975 ) 2943. [ 10] P.J. Ruback, Commun. Math. Phys. 107 (1986) 93. [ 11 ] G.W. Gibbons and P.J. Ruback, Phys. Rev. Lett. 57 (1986) 1617; R.C. Ferrell and D.M. Eardley, Phys. Rev. Lett. 59 (1987) 1617. [12 ] R. Ward, Phys. Lett. B 158 (1985 ) 424; R. Leese, Nucl. Phys. B 344 (1990) 33. [ 13 ] P.J. Ruback, Nucl. Phys. B 296 ( 1988 ) 669; T.M. Samols, Phys. Lett. B 244 (1990) 285; preprint DAMTP/91-13. [ 14 ] S.-K. Kim and H. Min. preprint SNUTP-91-41. [ 15 ] R. Jackiw and S.-Y. Pi, Phys. Rev. D 44 ( 1991 ). [ 16] R. MacKenzie and F. Wilczek, Intern. J. Mod. Phys. A 3 (1988) 2827; G. Dunne, R. Jackiw and C. Trugenberger, Ann. Phys. (NY) 194 (1990) 197. [ 17 ] R. Strilka, in preparation.
291