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Path integral for Coulomb system with magnetic charges
Volume 116, number 5
PHYSICS LETTERS A
16 June 1986
P A T H INTEGRAL F O R C O U L O M B S Y S T E M W I T H M A G N E T I C C H A R G E S H. K L E I N E R T
1,2
Umverstty of Cahforma, San D,ego, La Jolla, CA 92093, USA
Received 2 April 1986; accepted for pubhcaUon 18 April 1986
We calculate the path integral for the lagrangtan L = i1m x. 2 + eJc.A - e ~ / r - vZ/2mr 2, where A, ffi g G j x j x 3 / r ( x 2 + x22) is the vector potenttal of a magnetic monopole, ~, ~ts charge, and e the charge of the particle, m orbtt. In addition, we allow for an arbitrary centrifugal bamer 1 / r 2 potential. After the replacement e i ~ e/, + gg,, e~ ~ e ~ - g#, the results apply to dyonium, the bound state between two electrically and magnetically charged particles.
It is a l w a y s f u n to see o l d f r i e n d s i n a n e w dress, especially if this reveals n e w insights. F o r this r e a s o n , the r e c a l c u l a t i o n of the G r e e n f u n c t i o n of the C o u l o m b p r o b l e m b y d i f f e r e n t m e t h o d s [1] has b e e n a p o p u l a r exercise ever since S c h w i n g e r ' s o r i g i n a l s o l u t i o n [2]. R e c e n t l y , p a t h i n t e g r a t i o n has b e c o m e a f a v o r i t e t e c h n i q u e of s o l v i n g o n c e a g a i n w e l l - k n o w n p r o b l e m s . F o r the C o u l o m b case, this was d o n e s o m e y e a r s ago [3,4] #a a n d r e p e a t e d w i t h v a r i o u s m o d i f i c a t i o n s [ 5 - 7 ] ,2.3 I n this n o t e we w a n t to g e n e r a l i z e o u r m e t h o d [3,4] to the case o f a n electric c h a r g e e i n o r b i t a r o u n d a d y o n with c h a r g e ~ a n d m a g n e t i c c h a r g e ~ #4. I n o r d e r to m a k e the D i r a c s t r i n g invisible, ff a n d e fulfil the q u a n t i z a t i o n c o n d i t i o n ~e = q = h a l f - i n t e g e r .
(1)
F o r the sake of b e i n g general, we also a d d a n a r b i t r a r y 1 / r 2 p o t e n t i a l . W e shall c a l c u l a t e the a m p l i t u d e
r
2mr2
,
(2)
where f ~x(t)
- ~--,~lim~
1
~-i__lf~_d3x. =
~ 2~/~/rn
'
via p a t h i n t e g r a t i o n . T h e v e c t o r p o t e n t i a l A a s s o c i a t e d w i t h the m a g n e t i c c h a r g e ~ is g i v e n i n the a b s t r a c t .
i Supported m part by Deutsche Forschungsgememschaft under grant no. K1256 and UCSD/DOE contract DEAT-03-81ER40029. 2 On sabbatical leave from InsUtut far Theone der Elementarteilchen, Frete Umverslt~it Berhn, Armmallee 14, 1000 Berhn 33, Germany. #1 Ref. [4] is the detmled version of ref. [3] including also the two-dimensional case #2 The authors of ref. [5] are more exphclt than refs. [3,4] and write down all expressions in the time shced form but get the correct result only after using a wrong jacoblan 0(x, s)~/O(u)j = 24~z (see the last hne before thetr eq. (13)). The correct jacobian is 24132. In fact, construction of the variable Fj2 must be taken to be equal t o r:_ 1 rather than the average between r~ and r~_ I. #3 In ref. [7] the correct result emerges after using his wrong formula (7), since tins author did not consider the fact that ~,,2is equal to r2_1 by construction. See also ref. [8]. #4 The problem has a long instory, see ref. [9]. For recent &scussions meluchng spin, see ref. [10] 0 3 7 5 - 9 6 0 1 / 8 6 / $ 0 3 . 5 0 © Elsevier Science P u b l i s h e r s B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )
201
Volume 116, number 5
PHYSICS LETTERS A
16 June 1986
Going to the square-root variables
x, = -zto, z,
(3)
r=ztz,
where o, are the Pauli matrices and z I = u 1 - iu 4 = ¢~sin ½0 exp[ - ½i(a + q0)],
z2 = - u 3 +iu2 = - ~ c o s
½0 e x p [ - ½ i ( a -
~)], (4)
each point x, = (r cos 0, r sin 0 cos cp, r sin 0 sin q0) has as m a n y square roots in the four-dimensional u~ space as the angle a has values between 0 and 4~r. The degeneracy of this mapping is removed by introducing • 2 ___k 2 + ~J and using the path (1) an auxiliary fourth variable x 4 extending the kinetic term to x~ integral in four-dimensional space
fd(X4)bf~x(t)exp(iftlbdt½m3c2 =
)
N
f~o_oo(~)4 d(X4)N+
1
j~i__ if (~)4 d4xn
[
N+I
)
explic~nffilE (Xn-- Xn_ 1)2//C 2 / .
(5)
The integral over the final (x 4)~v+ 1 = (x4)b ensures that the extension does not alter the result (see refs. [3,4]). (2) Mapping intervals dx~ into intervals du~ by d x , = 2A,~(a) du~,
(6)
where p .Av. ( U ) =
U3 -- U 2 Ul
U4 -- U1 U2
U1 U4 U3
U2 U3 __U 4
U4
--/23
U2
-- U1
)
has the inverse (1/u2)A T and the determinant u 4 = r 2. Then .~2=4u2/12,
YClX2-~2xl=a[(u 2 + u2)(Ua/~l-Ul//4) + (u 2 +U2)(U3i12--U2~13)],
X2 + x 2 = 4(U 2 + U2 ) ( u 2 + U~). In the new variables, the lagrangian reads
L(u~,, i t n , ) = l [ 4 m u , t i t 2 _ q (
r[ 2
].____~._( U4//l __ Ulil4)+__(U3il2__U2il3))l u~ + u~ u~ + u~ X ( u 2 + u~ - u 2 - u 2 ) + e~
In terms of u - C
(7)
and the angles 0, q~, a, it looks as follows:
L( u, O, ~, a) = 202
_ °___L_ ~ ] 2mu2 .
uai~Z+½mu 6 0 2 + ~ 0 2 + & 2 - 2
a-
q
mu a
~cos0
+eE---
2mu 2
.
(8)
Volume 116, number 5
PHYSICS LETTERS A
16 June 1986
We now perform the change from t to the path dependent pseudo-time s via the idem factor
rbJ° dsJ~-~exp[-iE(tb-ta)]
dsEr(s)
exp i
=1,
(9)
and arrive at the Duru-Kleinert type representation of the Fourier transformed amplitude (2) (see ref. [4] eqs. (100)-(106)):
(XblXa)E= fo4~dabfo°°ds e-'e~s(UbSlU~O) =J0
dOtbJo dse-lees
..
:
[: ~4u(s) expi(jo •ds½M{u'~2+¼u2[8'2+cV2+a'2 10} d/ds.
where M = 4rn and the prime denotes amplitude has a Legendre transform H
We now observe that the lagrangian
L(u, u')
in this
- L
= ( 1 / 2 M ) { pu2 + (4/u2)[
+(4/2Mu2)[q2-q(p~
p2 + (1~sin28)( p2 + (p~ + q)2 + 2(p~
+
q)p~
cos 8)])
+ q) + v2].
(11)
In the canonical version of the path integral (10)
f ~4x f ~;--~--~Pe-P~ exp( ½ifo'ds ( p,u' + poO' +p~r' +p,a' - H ) ), we can therefore easily change the variable of integration p~ into p~ + q, thus picking up a phase factor e x p [ - iq(a e-a,)]. Then, since H does not contain a, the a integration can be performed forcing the new p, to be equal to q. This makes it possible to replace the 1/u 2 potential in (11) by
(4/2MuZ)(,2 _ q2). Keeping this in mind, the path integral in the large brackets of eq. (10), may be rewritten
~lvia~2u~,2 -- a2/2Mu 2)), (UbSlU~O)=exp[iq(ab -- aa)]f~4uexp(ifo~dS(lMu'~ 2-1""
(12)
0:2= ~/- 2E/M = ~/- E/2m
(13)
where
and a2 = 4 ( v 2 - q2). This is the amplitude of a four-dimensional harmonic oscillator with an extra For v = q this is immediately solved in closed form [11] with the result
(UbS l Ua0) =
•
(~/2"ni sin ~0s ) - ' e x p ( , ~
t,)
2
1/u 2 potential.
[(u b + u 2) cos ~os -- 2u b • ua] ).
(14) 203
Volume 116, number 5
PHYSICS LETTERS A
16 June 1986
Going from the variable s to o = e -'e~s, and doing the a b integration this gives immediately the desired amplitude
( X b [ X a > E = _ i m"n p O f o l d O(l__o)2Iq[ 0-" [2po ___Z 1~ 2 ~ ~/Xb " Xa + r b r
a
)
e x p [[- - P o l _ -1- + - ~0( r.
b
+ra)),
(15)
where
Vo = ~
= ~/- ME/2 = ½Mo~, I, = ee/26o.
For v 4: q, a little more work is necessary. Here we first have to do the angular integrals. In a D-dimensional generalization of the method of ref. [12], this gives the partial wave expansion (UbS i Ua0) _
~ (UbSlUaO)t~-'Ytm(~b)Yl,,(~a),
1
(ubua) ° " ,=o
(16)
,.
where Ytm are the D-dimensional spherical harmonics (with m denoting all degenerate quantum numbers). They satisfy the completeness relations 1 21+D-2 EYtm(fib)Ytm(Ua)= St) ~-_--~- Ctn/2-1(fib.fi,) ,
(17)
I
where C[ ~) are the Gegenbauer polynomials ,5
F(a+r)F(l+a-r)
c, °'(cos o)=
n! (--~-'-~ . F - ~
c o s ( 2 r - 1)0
(18)
r~0
and So = 2"~n/2/F(D/2) is the surface of a D sphere. The partial wave amplitudes are given by the radial path integral /=f0 ~ u e x p ~
(½Mu2
½Mo~Eu2) 2Mu 21 ([l+½(D-2)12-¼+a
2)
.
(19)
This can be done [7] giving Mo~Ub~f~ exp[ ½iMo~ ctg o~s( Ub2 + U~)] I~+,n-2)/z(MUbUa/i sin ~0s),
(ubs I u,0>, = ~ sin=0---~
(20)
where 1 is chosen such as to make the natural centrifugal barrier associated with this value of angular-momentum include our extra a2(2Mu 2) potential, i.e. [ i + ( D - 2)/2] 2 = [ l + ( D - 2)/2] = + 4(v 2 - q2).
(21)
We now perform the integral over ]dab exp[iq(ab- a,)]. Since
ftbfa= ~½(Xb.X ~ + rbra) COS[(ab-- a a -- fl)/2], where fl is an angle depending only on 0, q0 (see footnote on p. 420 of ref. [4]), we can integrate (18) directly f04"ndOtbe'qabc}°/2-1)(COS ½ a b )
=
4~ F ( D / 2 - 1 + !+ q/2)F( D/2 - 1 - q/2) (l + q / 2 ) ! ( ! - q / 2 ) ! r 2 ( D / 2 - 1)
* s Notice that m four dimensions, Ct°)(cos O) = ~ t _ o cos(2r - I ) 8 = s i n [ ( / + 1 ) 0 ] / s i n va.
204
(22)
Volume 116, number 5
PHYSICS LETTERS A
16 June 1986
Since D is really equal to 4, the right-hand side is equal to 4~r for 1 = q, q + 2, q + 4. Hence. the integral over a b in (16) gives fo4~ dab
~UbSlUaO)=
1 Mto (UbUa) ~ri sin ws ×exp{ ½iMto ctg[ tos(u 2 + u~)]}
• I=q,q+ 2,
2 ( l ' 1 - 1 ) 1 ] + 1 ( MUbg/a 1
i sin ~0s/"
(23)
As a check, we set v = q such that 1 = ] and we can use the identity 2(1 + 1 ) I , + , ( u ) = h [It(u ) - I,+2(u)] to perform the sum
~_,
2(l+ l)lt+a(u)=hlq(u ).
(24)
l=q,q+ 2, This agrees with a &rect integration of (14) over a b. It is now obvious how the final result (15) changes when allowing for the additional centrifugal barrier a2/2Mu2: We simply have to to use eq. (24) backwards and replace 2
Iq(h) ~ ~
E
(l+
1)I]+l(h ),
(25)
l=q,q+ 2,q+4, such that m 1E = - - ' - E 'IT V~/XbOXa q_rbra I=q,q+2,
(t+l)
£1 . O-~- a/2 . [_ 2V~6- / exp/[ 1 + o, XJO d o ~ _ _ o l]+a[2poT-~__o~/Xb°X a -t- r b r a ) --po~_o(r b_ + r,)) .
(26)
The Fourier transform of this is the desired amplitude for the charged particle moving in the field of a dyon. It goes without saying that the result can trivially be extended to the case that the particle in orbit has itself a magnetic charge ~; in this case the final result merely requires the replacement e~---, e~ + g~, e g ---~ e g - - g e .
The author thanks Dr. Eric D'Hoker for telling him about the supersymmetric aspects of the dyon system. He also thanks Dr. S. Ami for discussions and Professor N. KroU and Professor J. Kuti for their kind hospitality at UCSD.
References [1] M.C. Gutzwdler, J. Math. Phys. 8 (1967) 1979; R.G Storer, J. Math. Phys. 9 (1968) 964, H. Klemert, 1967 Boulder Lectures on Group dynamics of the hydrogen atom, m" Lectures m theoreucal physics, eds. W.E. Bnttln and A.O. Barut (Gordon and Breach, New York, 1968); L. Hostler and R.H. Pratt, Phys. Rev Lett. 10 (1969) 469; L. Hostler, J. Math. Phys. 5 (1964) 551, 11 (1970) 2966; S.M. Blinder, Phys. Rev. Lett. 52 (1984) 20.
205
Volume 116, number 5 [2] [3] [4] [5] [6]
[7] [8] [9]
[10] [11] [12]
206
PHYSICS LETTERS A
J Schwmger, J. Math. Phys 5 (1964) 1607 H. Duru and H. Klemert, Phys. Lett. B 84 (1979) 185 H. Duru and H Klemert, Fortschr. Phys 30 (1982) 401. R. Ho and A Inomata, Phys. Rev. Lett 48 (1982) 231 Ph. Blanchard and M Slrugue, J. Math Phys. 22 (1981) 1372, H Gnnberg, J. Maraflon and H. Vucetl~, Z Phys. C 20 (1983) 147, see also the general d~scusslon, A. Young and C DeW~tt, Texas prepnnt (Austin, 1985) F. Sterner, Phys. Lett A 106 (1984) 356. F. Sterner, Phys. Lett. A 106 (1984) 363. J Schwmger, Science 165 (1969) 757; A.O. Barut, C K E. Schneider and R. Wdson, J Math. Phys. 20 (1979) 2244; R Jaclow, Ann. Phys (NY) 129 (1980) 183. E. D'Hoker and L. Vinet, Nucl. Phys. B 260 (1985) 79. R.P. Feynman and A.R. Hlbbs, Quantum mechamcs and path integrals (McGraw-Hall, New York, 1965). D. Peak and A. Inomata, J. Math. Phys 10 (1969) 1422
16 June 1986
PHYSICS LETTERS A
16 June 1986
P A T H INTEGRAL F O R C O U L O M B S Y S T E M W I T H M A G N E T I C C H A R G E S H. K L E I N E R T
1,2
Umverstty of Cahforma, San D,ego, La Jolla, CA 92093, USA
Received 2 April 1986; accepted for pubhcaUon 18 April 1986
We calculate the path integral for the lagrangtan L = i1m x. 2 + eJc.A - e ~ / r - vZ/2mr 2, where A, ffi g G j x j x 3 / r ( x 2 + x22) is the vector potenttal of a magnetic monopole, ~, ~ts charge, and e the charge of the particle, m orbtt. In addition, we allow for an arbitrary centrifugal bamer 1 / r 2 potential. After the replacement e i ~ e/, + gg,, e~ ~ e ~ - g#, the results apply to dyonium, the bound state between two electrically and magnetically charged particles.
It is a l w a y s f u n to see o l d f r i e n d s i n a n e w dress, especially if this reveals n e w insights. F o r this r e a s o n , the r e c a l c u l a t i o n of the G r e e n f u n c t i o n of the C o u l o m b p r o b l e m b y d i f f e r e n t m e t h o d s [1] has b e e n a p o p u l a r exercise ever since S c h w i n g e r ' s o r i g i n a l s o l u t i o n [2]. R e c e n t l y , p a t h i n t e g r a t i o n has b e c o m e a f a v o r i t e t e c h n i q u e of s o l v i n g o n c e a g a i n w e l l - k n o w n p r o b l e m s . F o r the C o u l o m b case, this was d o n e s o m e y e a r s ago [3,4] #a a n d r e p e a t e d w i t h v a r i o u s m o d i f i c a t i o n s [ 5 - 7 ] ,2.3 I n this n o t e we w a n t to g e n e r a l i z e o u r m e t h o d [3,4] to the case o f a n electric c h a r g e e i n o r b i t a r o u n d a d y o n with c h a r g e ~ a n d m a g n e t i c c h a r g e ~ #4. I n o r d e r to m a k e the D i r a c s t r i n g invisible, ff a n d e fulfil the q u a n t i z a t i o n c o n d i t i o n ~e = q = h a l f - i n t e g e r .
(1)
F o r the sake of b e i n g general, we also a d d a n a r b i t r a r y 1 / r 2 p o t e n t i a l . W e shall c a l c u l a t e the a m p l i t u d e
r
2mr2
,
(2)
where f ~x(t)
- ~--,~lim~
1
~-i__lf~_d3x. =
~ 2~/~/rn
'
via p a t h i n t e g r a t i o n . T h e v e c t o r p o t e n t i a l A a s s o c i a t e d w i t h the m a g n e t i c c h a r g e ~ is g i v e n i n the a b s t r a c t .
i Supported m part by Deutsche Forschungsgememschaft under grant no. K1256 and UCSD/DOE contract DEAT-03-81ER40029. 2 On sabbatical leave from InsUtut far Theone der Elementarteilchen, Frete Umverslt~it Berhn, Armmallee 14, 1000 Berhn 33, Germany. #1 Ref. [4] is the detmled version of ref. [3] including also the two-dimensional case #2 The authors of ref. [5] are more exphclt than refs. [3,4] and write down all expressions in the time shced form but get the correct result only after using a wrong jacoblan 0(x, s)~/O(u)j = 24~z (see the last hne before thetr eq. (13)). The correct jacobian is 24132. In fact, construction of the variable Fj2 must be taken to be equal t o r:_ 1 rather than the average between r~ and r~_ I. #3 In ref. [7] the correct result emerges after using his wrong formula (7), since tins author did not consider the fact that ~,,2is equal to r2_1 by construction. See also ref. [8]. #4 The problem has a long instory, see ref. [9]. For recent &scussions meluchng spin, see ref. [10] 0 3 7 5 - 9 6 0 1 / 8 6 / $ 0 3 . 5 0 © Elsevier Science P u b l i s h e r s B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )
201
Volume 116, number 5
PHYSICS LETTERS A
16 June 1986
Going to the square-root variables
x, = -zto, z,
(3)
r=ztz,
where o, are the Pauli matrices and z I = u 1 - iu 4 = ¢~sin ½0 exp[ - ½i(a + q0)],
z2 = - u 3 +iu2 = - ~ c o s
½0 e x p [ - ½ i ( a -
~)], (4)
each point x, = (r cos 0, r sin 0 cos cp, r sin 0 sin q0) has as m a n y square roots in the four-dimensional u~ space as the angle a has values between 0 and 4~r. The degeneracy of this mapping is removed by introducing • 2 ___k 2 + ~J and using the path (1) an auxiliary fourth variable x 4 extending the kinetic term to x~ integral in four-dimensional space
fd(X4)bf~x(t)exp(iftlbdt½m3c2 =
)
N
f~o_oo(~)4 d(X4)N+
1
j~i__ if (~)4 d4xn
[
N+I
)
explic~nffilE (Xn-- Xn_ 1)2//C 2 / .
(5)
The integral over the final (x 4)~v+ 1 = (x4)b ensures that the extension does not alter the result (see refs. [3,4]). (2) Mapping intervals dx~ into intervals du~ by d x , = 2A,~(a) du~,
(6)
where p .Av. ( U ) =
U3 -- U 2 Ul
U4 -- U1 U2
U1 U4 U3
U2 U3 __U 4
U4
--/23
U2
-- U1
)
has the inverse (1/u2)A T and the determinant u 4 = r 2. Then .~2=4u2/12,
YClX2-~2xl=a[(u 2 + u2)(Ua/~l-Ul//4) + (u 2 +U2)(U3i12--U2~13)],
X2 + x 2 = 4(U 2 + U2 ) ( u 2 + U~). In the new variables, the lagrangian reads
L(u~,, i t n , ) = l [ 4 m u , t i t 2 _ q (
r[ 2
].____~._( U4//l __ Ulil4)+__(U3il2__U2il3))l u~ + u~ u~ + u~ X ( u 2 + u~ - u 2 - u 2 ) + e~
In terms of u - C
(7)
and the angles 0, q~, a, it looks as follows:
L( u, O, ~, a) = 202
_ °___L_ ~ ] 2mu2 .
uai~Z+½mu 6 0 2 + ~ 0 2 + & 2 - 2
a-
q
mu a
~cos0
+eE---
2mu 2
.
(8)
Volume 116, number 5
PHYSICS LETTERS A
16 June 1986
We now perform the change from t to the path dependent pseudo-time s via the idem factor
rbJ° dsJ~-~exp[-iE(tb-ta)]
dsEr(s)
exp i
=1,
(9)
and arrive at the Duru-Kleinert type representation of the Fourier transformed amplitude (2) (see ref. [4] eqs. (100)-(106)):
(XblXa)E= fo4~dabfo°°ds e-'e~s(UbSlU~O) =J0
dOtbJo dse-lees
..
:
[: ~4u(s) expi(jo •ds½M{u'~2+¼u2[8'2+cV2+a'2 10} d/ds.
where M = 4rn and the prime denotes amplitude has a Legendre transform H
We now observe that the lagrangian
L(u, u')
in this
- L
= ( 1 / 2 M ) { pu2 + (4/u2)[
+(4/2Mu2)[q2-q(p~
p2 + (1~sin28)( p2 + (p~ + q)2 + 2(p~
+
q)p~
cos 8)])
+ q) + v2].
(11)
In the canonical version of the path integral (10)
f ~4x f ~;--~--~Pe-P~ exp( ½ifo'ds ( p,u' + poO' +p~r' +p,a' - H ) ), we can therefore easily change the variable of integration p~ into p~ + q, thus picking up a phase factor e x p [ - iq(a e-a,)]. Then, since H does not contain a, the a integration can be performed forcing the new p, to be equal to q. This makes it possible to replace the 1/u 2 potential in (11) by
(4/2MuZ)(,2 _ q2). Keeping this in mind, the path integral in the large brackets of eq. (10), may be rewritten
~lvia~2u~,2 -- a2/2Mu 2)), (UbSlU~O)=exp[iq(ab -- aa)]f~4uexp(ifo~dS(lMu'~ 2-1""
(12)
0:2= ~/- 2E/M = ~/- E/2m
(13)
where
and a2 = 4 ( v 2 - q2). This is the amplitude of a four-dimensional harmonic oscillator with an extra For v = q this is immediately solved in closed form [11] with the result
(UbS l Ua0) =
•
(~/2"ni sin ~0s ) - ' e x p ( , ~
t,)
2
1/u 2 potential.
[(u b + u 2) cos ~os -- 2u b • ua] ).
(14) 203
Volume 116, number 5
PHYSICS LETTERS A
16 June 1986
Going from the variable s to o = e -'e~s, and doing the a b integration this gives immediately the desired amplitude
( X b [ X a > E = _ i m"n p O f o l d O(l__o)2Iq[ 0-" [2po ___Z 1~ 2 ~ ~/Xb " Xa + r b r
a
)
e x p [[- - P o l _ -1- + - ~0( r.
b
+ra)),
(15)
where
Vo = ~
= ~/- ME/2 = ½Mo~, I, = ee/26o.
For v 4: q, a little more work is necessary. Here we first have to do the angular integrals. In a D-dimensional generalization of the method of ref. [12], this gives the partial wave expansion (UbS i Ua0) _
~ (UbSlUaO)t~-'Ytm(~b)Yl,,(~a),
1
(ubua) ° " ,=o
(16)
,.
where Ytm are the D-dimensional spherical harmonics (with m denoting all degenerate quantum numbers). They satisfy the completeness relations 1 21+D-2 EYtm(fib)Ytm(Ua)= St) ~-_--~- Ctn/2-1(fib.fi,) ,
(17)
I
where C[ ~) are the Gegenbauer polynomials ,5
F(a+r)F(l+a-r)
c, °'(cos o)=
n! (--~-'-~ . F - ~
c o s ( 2 r - 1)0
(18)
r~0
and So = 2"~n/2/F(D/2) is the surface of a D sphere. The partial wave amplitudes are given by the radial path integral /=f0 ~ u e x p ~
(½Mu2
½Mo~Eu2) 2Mu 21 ([l+½(D-2)12-¼+a
2)
.
(19)
This can be done [7] giving Mo~Ub~f~ exp[ ½iMo~ ctg o~s( Ub2 + U~)] I~+,n-2)/z(MUbUa/i sin ~0s),
(ubs I u,0>, = ~ sin=0---~
(20)
where 1 is chosen such as to make the natural centrifugal barrier associated with this value of angular-momentum include our extra a2(2Mu 2) potential, i.e. [ i + ( D - 2)/2] 2 = [ l + ( D - 2)/2] = + 4(v 2 - q2).
(21)
We now perform the integral over ]dab exp[iq(ab- a,)]. Since
ftbfa= ~½(Xb.X ~ + rbra) COS[(ab-- a a -- fl)/2], where fl is an angle depending only on 0, q0 (see footnote on p. 420 of ref. [4]), we can integrate (18) directly f04"ndOtbe'qabc}°/2-1)(COS ½ a b )
=
4~ F ( D / 2 - 1 + !+ q/2)F( D/2 - 1 - q/2) (l + q / 2 ) ! ( ! - q / 2 ) ! r 2 ( D / 2 - 1)
* s Notice that m four dimensions, Ct°)(cos O) = ~ t _ o cos(2r - I ) 8 = s i n [ ( / + 1 ) 0 ] / s i n va.
204
(22)
Volume 116, number 5
PHYSICS LETTERS A
16 June 1986
Since D is really equal to 4, the right-hand side is equal to 4~r for 1 = q, q + 2, q + 4. Hence. the integral over a b in (16) gives fo4~ dab
~UbSlUaO)=
1 Mto (UbUa) ~ri sin ws ×exp{ ½iMto ctg[ tos(u 2 + u~)]}
• I=q,q+ 2,
2 ( l ' 1 - 1 ) 1 ] + 1 ( MUbg/a 1
i sin ~0s/"
(23)
As a check, we set v = q such that 1 = ] and we can use the identity 2(1 + 1 ) I , + , ( u ) = h [It(u ) - I,+2(u)] to perform the sum
~_,
2(l+ l)lt+a(u)=hlq(u ).
(24)
l=q,q+ 2, This agrees with a &rect integration of (14) over a b. It is now obvious how the final result (15) changes when allowing for the additional centrifugal barrier a2/2Mu2: We simply have to to use eq. (24) backwards and replace 2
Iq(h) ~ ~
E
(l+
1)I]+l(h ),
(25)
l=q,q+ 2,q+4, such that m 1
(t+l)
£1 . O-~- a/2 . [_ 2V~6- / exp/[ 1 + o, XJO d o ~ _ _ o l]+a[2poT-~__o~/Xb°X a -t- r b r a ) --po~_o(r b_ + r,)) .
(26)
The Fourier transform of this is the desired amplitude for the charged particle moving in the field of a dyon. It goes without saying that the result can trivially be extended to the case that the particle in orbit has itself a magnetic charge ~; in this case the final result merely requires the replacement e~---, e~ + g~, e g ---~ e g - - g e .
The author thanks Dr. Eric D'Hoker for telling him about the supersymmetric aspects of the dyon system. He also thanks Dr. S. Ami for discussions and Professor N. KroU and Professor J. Kuti for their kind hospitality at UCSD.
References [1] M.C. Gutzwdler, J. Math. Phys. 8 (1967) 1979; R.G Storer, J. Math. Phys. 9 (1968) 964, H. Klemert, 1967 Boulder Lectures on Group dynamics of the hydrogen atom, m" Lectures m theoreucal physics, eds. W.E. Bnttln and A.O. Barut (Gordon and Breach, New York, 1968); L. Hostler and R.H. Pratt, Phys. Rev Lett. 10 (1969) 469; L. Hostler, J. Math. Phys. 5 (1964) 551, 11 (1970) 2966; S.M. Blinder, Phys. Rev. Lett. 52 (1984) 20.
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Volume 116, number 5 [2] [3] [4] [5] [6]
[7] [8] [9]
[10] [11] [12]
206
PHYSICS LETTERS A
J Schwmger, J. Math. Phys 5 (1964) 1607 H. Duru and H. Klemert, Phys. Lett. B 84 (1979) 185 H. Duru and H Klemert, Fortschr. Phys 30 (1982) 401. R. Ho and A Inomata, Phys. Rev. Lett 48 (1982) 231 Ph. Blanchard and M Slrugue, J. Math Phys. 22 (1981) 1372, H Gnnberg, J. Maraflon and H. Vucetl~, Z Phys. C 20 (1983) 147, see also the general d~scusslon, A. Young and C DeW~tt, Texas prepnnt (Austin, 1985) F. Sterner, Phys. Lett A 106 (1984) 356. F. Sterner, Phys. Lett. A 106 (1984) 363. J Schwmger, Science 165 (1969) 757; A.O. Barut, C K E. Schneider and R. Wdson, J Math. Phys. 20 (1979) 2244; R Jaclow, Ann. Phys (NY) 129 (1980) 183. E. D'Hoker and L. Vinet, Nucl. Phys. B 260 (1985) 79. R.P. Feynman and A.R. Hlbbs, Quantum mechamcs and path integrals (McGraw-Hall, New York, 1965). D. Peak and A. Inomata, J. Math. Phys 10 (1969) 1422
16 June 1986