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Path Integral Solution of the Aharonov–Bohm–Dyon System
Annals of Physics 282, 270284 (2000) doi:10.1006aphy.2000.6031, available online at http:www.idealibrary.com on
Path Integral Solution of the AharonovBohmDyon System De-Hone Lin Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan E-mail: d793314phys.nthu.edu.tw. Received September 29, 1999
An exact path integral solution for a relativistic particle with both electric and magnetic charges (e 1 , q 1 ) moving in the fields created by a flux tube along the z-axis and a particle with charges (e 2 , q 2 ) located at the origin, i.e., the relativistic AharonovBohmDyon system, is given. The solutions of interesting special cases, AharonovBohmCoulomb, Aharonov BohmMonopole, and AharonovBohm, are discussed. 2000 Academic Press
I. INTRODUCTION For the past 17 years, particles associated with questions on fractional angular momenta and quantization of magnetic monopoles moving in AharonovBohm (AB) magnetic fields and long-range potentials have been much debated [15]. In this paper, we present an exact solution of the path integral for a relativistic particle with both electric and magnetic charges (e 1 , q 1 ) moving in the fields created by a charge and a Dirac magnetic monopole of strange (e 2 , q 2 ) at the origin plus an infinitely long and thin solenoid carrying flux g along the z axis, i.e., the Aharonov BohmDyon (ABD) system. The solution provides a concrete example in which to study the above questions.
II. PATH INTEGRAL SOLUTION FOR THE AHARONOVBOHMDYON SYSTEM The starting point is the path integral representation for Green's function of a relativistic particle in external electromagnetic fields [68], G(x b , x a ; E)=
2mc
|
0
|
|
dS D\(*) 8[ \(*)] D 3x(*) exp[iA E [x, x* ]]\(0)
(2.1)
with the action A E [x, x* ]=
|
*b
d* *a
_
m (E&V(x)) 2 mc 2 &\(*) , x* 2(*)+A(x) } x* (*)+\(*) 2\(*) 2mc 2 2 270
0003-491600 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.
&
(2.2)
271
AHARONOVBOHMDYON SYSTEM
where S is defined as S=
|
*b
d*\(*),
(2.3)
*a
in which \(*) is an arbitrary dimensionless fluctuating scale variable, \(0) is the terminal point of the function \(*), and 8[ \(*)] is some convenient gauge-fixing functional [68]. The only condition on 8[ \(*)] is that
| D\(*) 8[ \(*)]=1.
(2.4)
mc is the well-known Compton wavelength of a particle of mass m, A(x) and V(x) stand for the vector and scalar potential of the systems, respectively. E is the system energy, and x is the spatial part of the (3+1) vector x + =(x(*), t(*)). The path integral representation arises from the limiting condition of the *-sliced version G(x b , x a ; E)r
2mc
N+1
_| d\ 8( \ )& d x 1 _ ` | _ (2?i= \ m) & (2?i= \ m) i _exp { A = \(0), |
dS `
0
n
n
n=1
N
32
b
b
n=1
3
&
n
32
n
n
N E
(2.5)
where the sign r becomes an equality for N Ä , and the *-sliced action N+1
AN E= : n=1
_
m(x n &x n&1 ) 2 (E&V(x n )) 2 mc 2 +A(x n ) } (x n &x n&1 )+= n \ n &= n \ n 2 2= n \ n 2mc 2 (2.6)
&
with = n =* n &* n&1 , * b =* N+1 , * a =* 0 , x a =x(* 0 ) and x b =x(* n+1 ). For the ABD system, the interaction potentials are given by V(x)=
&e 2 , r
(2.7)
and the vector potential reads A(x)=q~
& ye^ x +xe^ y & ye^ x +xe^ y , +g^ r(r+z) x2 + y 2
(2.8)
272
DE-HONE LIN
where e^ x, y denotes the basis vectors in the Cartesian coordinate frame. The interacting strength q~ # &(e 1 q 2 &e 2 q 1 )c, g~ #(e 1 + q 1 ) gc, and e 2 # &e 1 e 2 &q 1 q 2 are combinations of the electric and magnetic charges of the two particles, and r#- x 2 + y 2 +z 2 is the radial distance, as usual. The hydrogen atom is a special case of the dyonium system with e 1 =&e 2 =e and q~ =0= g^. An electron around a pure magnetic monopole has e 1 =e, q 2 =q~, e 2 =q 1 =0= g^. In the vector potential we have chosen the monopole to have a Dirac string singularity along the negative z-axis. This is proposed in Ref. [9]. As a consequence of monopole gauge invariance, the parameter q~ has to be an integer or a half-integer number [10], a condition referred to as Dirac's charge quantization. To obtain a tractable path integral for the potential V(x)=&e 2r, we choose the fluctuating scale variable \ n =r n and the gauge-fixing condition N+1
| D\(*) 8[ \(*)]= lim
NÄ
` N=1
dw n
_| (2?i= r m)
12
n n
& { exp
i N+1 m (2w n ) 2 : . n=1 2 = n r n
=
(2.9)
The unity condition is automatically satisfied. With Eq. (2.9), the path integral of Eq. (2.5) turns into G(x b , x a ; E)r
2mc
|
0
|
dS dw b
N rb ` 2 (2?i= b r b m) n=1
d 4x n i N exp A , 2 E & (2?i= n r n m) (2.10)
_|
& { =
where A N E is the action of Eq. (2.6) in which the constant \(0) is chosen as r b and the three-vectors x n of the kinematic term are replaced with the four-vectors x n . This extends the kinetic action to N+1
AN kin = : n=1
m (x n &x n&1 ) . 2 =n rn
(2.11)
The path integral can be simplified by the K-S transformation [11, 12] dx =2A(u ) du
(2.12)
with 4_4 matrix A(u ) chosen as
A(u )=
\
u3 u4 u1 u2
u4 &u 3 u2 &u 1
u1 &u 2 &u 3 u4
u2 u1 &u 4 &u 3
+
.
(2.13)
273
AHARONOVBOHMDYON SYSTEM
The magnetic interaction turns into A(u ) } u $=
q~ (u 1 ) 2 +(u 2 ) 2 &(u 3 ) 2 &(u 4 ) 2 +g^ r
{ _ & = u u* &u u* u u* &u u* _ _ (u ) +(u ) + (u ) +(u ) & . 2 1
1 2
1 2
3 4
2 2
4 3
3 2
(2.14)
4 2
We obtain a path integral in the continuum limit equivalent to Eq. (2.10), G(x b , x a ; E)=
2mc
|
2
2
dSe iSEe mc G(u b , u a ; S),
(2.15)
0
with the four dimensional pseudopropagator G(u b , u a ; S)=
1 dw b D 4u(*) exp[iA E [u , u $]], 16 r b
|
|
(2.16)
where the functional measure has the *-sliced representation N
1
| D u(*)r (2?i= M) 4
b
2
` n=1
_|
d 4u n 2 & (2?i= n M)
&
(2.17)
and the action is given by A E [u , u $]=
|
S
0
d*
_
Mu $ 2 M| 2u 2 4 2: 2 +A(u ) } u $+ + 2 2 2Mu 2
&
(2.18)
in which u 2 =(u 1 ) 2 +(u 2 ) 2 +(u 3 ) 2 +(u 4 ) 2 =r, M=4m, | 2 =
E 2 &m 2c 4 . 4m 2c 2
(2.19)
To go further, let us introduce the Euler angle's representation of the K-S transformation (e.g., 11, 12): u 1 =- r cos(%2) cos[(.+#)2] u 2 =&- r cos(%2) sin[(.+#)2] u 3 =- r sin(%2) cos[(.&#)2] u 4 =- r sin(%2) sin[(.&#)2]
=
\
0 % ?% 0 . 2? 0 # 4?
+
.
(2.20)
274
DE-HONE LIN
Then the action of Eq. (2.18) turns into A E [u , u $]=
|
S
d* 0
M 2 u 2 2 4q~ u* + %4 +.* 2 +#* 2 +2.* #* & cos % 2 4 Mu 2
{ _
\
\
M| 2u 2 4 2: 2 +(q~ &g~ ) .* + 2 2Mu 2
+
+
=
+& (2.21)
with g~ = & g^ for a matter of convention and u= |u |. Here we see that the final term associated to the AB effect just related to a total derivative so that it can factor out of the entire action and be considered in the final step. The path integral in Eq. (2.16) can have a clear expression in terms of canonical representation A E [u , u $]=
|
S
d*[[ p u u* + p % %4 + p . .* + p # #* ]&H E ]
(2.22)
0
with the velocity variables (u* , %4 , .* , #* ) having the representation of canonical momenta as follows, pu u* = , M p% %4 = , ! (2.23) 1 #* = [ p # +'! cos 2%& p . cos %], ! sin 2% .* =
1 [ p &'! cos %& p # cos %], ! sin 2% .
where the variables '# &4q~Mu 2 and !#Mu 24. It is not difficult to find the Hamiltonian H E [u , p ]=
1 4 1 p 2u + 2 p 2% + ( p 2 +( p # &q~ ) 2 &2( p # &q~ ) p . cos %) 2M u sin 2% .
{
+
_
4 M| 2u 2 2 2 2 . [2q ~ p & (: +q ~ )]& # 2Mu 2 2
&=
(2.24)
We obtain the canonical representation of the path integral G(x b , x a ; E)=
2mc
|
1 _ 16
2
2
dSe iSEe mc
0
|
4?
0
|
S
d# b D 4u(*) e i 0 d*(q~ & g~ ) .*
D 4p(*) exp[iA E [u , p ]], 2? & (2.25)
|
275
AHARONOVBOHMDYON SYSTEM
where the integral variables dw b r b in the measure of Eq. (2.16) have been changed by using Eq. (2.12), dw=2(u 2 du 1 &u 1 du 2 +u 4 du 3 &u 3 du 4 ) =r(cos % d.+d#),
(2.26)
and due to the x and thus the angles (%, .) remain fixed during the w integration. In the canonical path integral, the momenta are dummy integration variables so that we can replace ( p # &q~ ) by p # . Then the action becomes A E [u , p ]=
|
S
d*[[ p u u* + p % %4 + p . .* +( p # +q~ ) #* ]&H E [u , p ]]
(2.27)
0
with the Hamiltonian H E [u , p ]=
1 4 1 p 2u + 2 p 2% + 2 ( p 2. + p 2# &2p # p . cos %) 2M u sin %
{
+
_
&=
4 M| 2u 2 [2q~( p # +q~ )& 2(: 2 +q~ 2 )]& . 2 2Mu 2
(2.28)
Action in Eq. (2.27) contains an additional term 2A =q~
|
S
d*#* .
(2.29)
0
This is a pure surface term 2A =q~(# b &# a ). Since the integral over # b forces the momentum p r in the canonical action (2.27) to take the value &q~, this eliminates the term proportional to p # +q~ in Eq. (2.28); therefore the entire path integral becomes G(x b , x a ; E)=
2mc
|
2
2
dSe iSEe mc
0
1 16
|
4?
d# b e iq~(#b &#a )G(u b , u a ; S).
(2.30)
0
Here the pseudopropagator G(u b , u a ; S)=
_|
&
S
D 4u(*) e i 0 d*(q~ & g~ ) .* exp[iA E [u , u $]]
(2.31)
Mu $ 2 M| 2u 2 4(: 2 +q~ 2 ) 2 . + + 2 2 2Mu 2
(2.32)
&
with the action A E [u , u $]=
|
S
0
d*
{
=
276
DE-HONE LIN
The path integral can be performed. After the variable changes d*=&idy, S=&iL, the entire path integral becomes G(x b , x a ; E)=
&i 2mc
|
2
dL e LEe mc
2
0
1 16
|
4?
d# b e iq~(#b &#a )G(u b , u a ; L),
(2.33)
0
with the pseudopropagator (e.g., [11, 12]) G(u b , u a ; S)=
1 : G l (u b , u a ; L) u b u a l=0 l2 L l+1 l2 ik1(.b &.a )+ik2(#b &#a ) i 0 dy(q~ & g~ ) .* : d l2 e _ 2 k1, k2(% b ) d k1, k2(% a ) e 2? k1, k2 =&l2
(2.34) with the radial amplitude G l (u b , u a ; L)=
M|~ 2 2 M |~u b u a , (2.35) e &M|~2(ub +ua ) coth |~LI - (l+1)2 &4(:2 +q~2 ) sinh |~L sinh |~L
\
+
ik1.+ik2# where |~ =- (m 2c 4 &E 2 )2mc and d l2 are the representation funck1, k2(%) e 4? &iq(#b &#a ) tions of the rotation group. The integral 0 d# a e in Eq. (2.33) can be easily done. We arrive at the fixed-energy amplitude of the relativistic ABD system, now labeled by the subscript D,
G(x b , x a ; E D )=
1 - rb ra
lD
:
:
lD =q~
k=&lD
G lD (r b , r a ; E D ) d lk,D q~(% b ) d lk,D*q~(% a )
e ik(.b &.a ) 2?
(2.36)
in which l D is defined as l2 and the fixed-energy Green's function is, G lD (r b , r a ; E D )=
&i 1 2mc 2 _
|
2
L
2
dL e LEDe mc e i 0 dy(q~ & g~ ) .*
0
2 2 M |~u b u a M|~ e &M|~(ub +ua ) coth |~LI - (2lD +1)2 &4(:2 +q~2 ) . sinh |~L sinh |~L
\
+
(2.37) It is now the place to consider the AB influence on the entire system. The term related to the AB effect is given by A A&B =&g~
|
L
dy.* ( y), 0
(2.38)
277
AHARONOVBOHMDYON SYSTEM
where .( y)=.(x( y)), .* =d.dy. Since the particle orbits are present at all times, their worldlines in spacetime can be considered as being closed at infinity, and the integral k=
1 2?
|
L
dy.* ( y)
(2.39)
0
is the topological invariant with integer values of the winding number k. The magnetic interaction is therefore a purely topological one given by A A&B =&g~2k?.
(2.40)
To incorporate the effect, let us first note that [13, p. 58] d lk,D q~(%)=
_
1(1+l D +k) 1(1+l D &k) 1(1+l D +q~ ) 1(1+l D &q~ )
&
12
~ , k+q~ ) _(cos %2) (k+q~ ) (sin %2) (k&q~ ) P (k&q (cos %), lD &k
(2.41)
where P l(x, y)(z) is the Jacobi function. This leads to d lk,D q~(% b ) d lk,D*q~(% a )=
1(1+l D +k) 1(1+l D &k)
_ 1(1+l +q~) 1(1+l &q~) & D
D
~ , k+q~ ) _(cos % b 2 cos % a 2) (k+q~ )(sin % b 2 sin % a 2) (k&q~ ) P (k&q lD &k ~ , k+q~ ) _(cos % b ) P (k&q (cos % a ). lD &k
(2.42)
To go further, let us change the variable l D by defining l D &k= p into p. It is easy to find that Green's function of Eq. (2.36) becomes
G(x b , x a ; E D )=
1 - rb ra
:
:
p=q~
k=&
_G p+k(r b , r a ; E D )
( p+k+12) 1(1+ p+2k) 1(1+ p) 1(1+ p+k+q~ ) 1(1+ p+k&q~ )
e ik(.b &.a ) _ (cos % b 2 cos % a 2) (k+q~ )(sin % b 2 sin % a 2) (k&q~ ) 2? ~ , k+q~ ) _P p(k&q~, k+q~ )(cos % b ) P (k&q (cos % a ). p
(2.43)
278
DE-HONE LIN
The radial Green's function corresponds to G p+k(r b , r a ; E D ) =
&i 1 2mc 2
_
|
2
L
2
dLe LEDe mc e i 0 dy(q~ & g~ ) .*
0
M|~ 2 2 M |~u b u a . e &(M|~2)(ub +ua ) coth |~L I - [2( p+k)+1]2 &4(:2 +q~2 ) sinh |~L sinh |~L
\
+
(2.44) Let us invoke Poisson's summation formula [14, p. 469]
f (k)=
: k=&
|
dy &
:
e 2?nyif ( y).
(2.45)
n=&
By incorporating the AB influence of Eq. (2.40), we arrive at G(x b , x a ; E D )=
1 2? - r b r a
: p=q~
|
dz
:
e iq~(.b &.a )e i(z& g~ )(.b +2k?&.a )
k=&
_G p+z (r b , r a ; E D )
( p+z+12) 1(1+ p+2z) 1(1+ p) 1(1+ p+z+q~ ) 1(1+ p+z&q~ )
_(cos % b 2 cos % a 2) (z+q~ ) (sin % b 2 sin % a 2) (z&q~ ) _P p(z&q~, z+q~ )(cos % b ) P p(z&q~, z+q~ )(cos % a ).
(2.46)
The sum over all k in Eq. (2.46) forces z to be equal to g~ modulo an arbitrary integral number leading to G(x b , x a ; E D )=
1 2? - r b r a
:
:
p=q~
k=&
e iq~(.b &.a )e ik(.b &.a )
_G p+ |k+ g~ | (r b , r a ; E D )
{
( p+ |k+ g~ | +12) 1(1+ p+2 |k+ g~ | ) 1(1+ p) 1(1+ p+ |k+ g~ | +q~ ) 1(1+ p+ |k+ g~ | &q~ )
=
_(cos % b 2 cos % a 2) |k+ g~ | +q~ )(sin % b 2 sin % a 2) ( |k+ g~ | &q~ ) _P p( |k+ g~ | &q~, |k+ g~ | +q~ )(cos % b ) P p( |k+ g~ | &q~, |k+ g~ | +q~)(cos % a ).
(2.47)
279
AHARONOVBOHMDYON SYSTEM
The fixed-energy Green's function is G p+ |k+ g~ | (r b , r a ; E D ) =
&i 1 2mc 2
_
|
2
2
dL e LEDe mc
0
M|~ 2 2 M |~u b u a . e &M|~2(ub +ua ) coth |~L I - [2( p+ |k+ g~ | )+1]2 &4(:2 +q~2 ) sinh |~L sinh |~L (2.48)
\
+
The integration can be done by the formula (e.g., [11, Chap. 9])
|
dy
0
e 2&y t t - `b `a exp & (` a +` b ) coth y I + sinh y 2 sinh y
=
_
& \
1((1++)2&&) t - ` b ` a 1(++1)
+
W &, +2(t` b ) M &, +2(t` a ),
(2.49)
where M +, & and W +, & are the Whittaker functions and the range of validity is given by ` b >` a >0, Re[(1++)2&&]>0, Re(t)>0,
|arg t|
We complete the integration and obtain the exact solution of the path integral for the ABD system G(x b , x a ; E D )=
&i mc 2mc 4?r b r a - m 2c 4 &E 2D
_:
:
p=q~ k=&
{
1(12+- [2( p+ |k+ g~ | )+1] 2 &4(: 2 +q~ 2 )2 &E D :- m 2c 4 &E 2D ) 1(1+- [2( p+ |k+ g~ |)+1] 2 &4(: 2 +q~ 2 ))
_W ED:- m2c4 &E 2 , - [2( p+ |k+ g~ | )+1]2 &4(:2 +q~2 )2 D
_M ED:- m2c4 &E 2 , - [2( p+ |k+ g~ | )+1]2 &4(:2 +q~2 )2 D
2
=
\c - m c &E r + 2 \c - m c &E r + 2 4
2 D b
2 4
2 D a
280
DE-HONE LIN
_
_
(2p+2 |k+ g~ | +1) 1(1+ p+2 |k+ g~ |) 1(1+ p) 1(1+ p+ |k+ g~ | +q~ ) 1(1+ p+ |k+ g~ | &q~ )
&
_e i(q~ +k)(.b &.a )(cos % b 2 cos % a 2) ( |k+ g~| +q~ )(sin % b 2 sin % a 2) (|k+ g~ | &q~ ) g~ | &q~, |k+ g~ | +q~ ) _P p( |k+ g~ | &q~, |k+ g~ | +q~ )(cos % b ) P (|k+ (cos % a ). p
(2.50)
The energy levels can be extracted from the poles. They are determined by the equation 12+- [2( p+ |k+ g~ | )+1] 2 &4(: 2 +q~ 2 )2&E D :- m 2c 4 &E 2D =&n r ,
(2.51)
where the radial quantum number n r =0, 1, 2, ... . With little mathematical manipulation, we obtain
_
E nr, p, k =\mc 2 1+
:2 (12+12- [2( p+ |k+ g~ | )+1] 2 &4(: 2 +q~ 2 )+n r ) 2
&
&12
.( 2.52)
The corresponding wave functions can be obtained according to the methods of e.g., Refs. [11, 12, 19]. There are several interesting special cases as follows: A. The Relativistic AharonovBohmCoulomb (ABC) System if q~ =0 In this case, Green's function reduces to G(x b , x a ; E)=
&i mc 2mc 4?r b r a - m 2c 4 &E 2
_:
:
p=0 k=&
{
1(12+- [2( p+ |k+ g~ | )+1] 2 &4: 22 &E:- m 2c 4 &E 2 ) 1(1+- [2( p+ |k+ g~ | )+1] 2 &4: 2 ) 2
=
\c - m c &E r + 2 _M \c - m c &E r + (2p+2 |k+ g~ | +1) 1(1+ p+2 |k+ g~ | ) 1(1+ p) _ _ & 1 (1+ p+ |k+ g~ | ) _W E:- m2c4 &E2, - [2( p+ |k+ g~ | )+1]2 &4:22
2 4
b
2 4
E:- m2c4 &E2, - [2( p+ |k+ g~ | )+1]2 &4:22
2
2
a
2
_ ik(.b &.a )(cos % b 2 cos % a 2) |k+ g~| (sin % b 2 sin % a 2) |k+ g~| _P (p|k+ g~ |, |k+ g~ | )(cos % b ) P (p|k+ g~ |, |k+ g~ | )(cos % a ).
(2.53)
281
AHARONOVBOHMDYON SYSTEM
a relativistic result was first given in Ref. 8. A further reduction is given by taking g~ =0. In this case, with the help of the relation between the associated Legendre and Jacobi functions [15] P kl (cos %)=(&1) k
1(1+k+l ) k) (cos %2 sin %2) k P (k, l&k (cos %). 1(1+l )
(2.54)
Green's function of the relativistic Coulomb system given in [6, 16] is obtained by G(x b , x a ; E)=
mc &i 1 : 2mc (r b r a ) - m 2c 4 &E 2 l=0
_
l
: Y lm(x^ b ) Y* lm(x^ a ) m=&l
1(12+- (l+12) 2 &: 2 &E:- m 2c 4 &E 2 ) 1(1+2 - (l+ 12) 2 &: 2 )
_W E:- m2c4 &E2, - (l+12)2 &:2 _M E:- m2c4 &E2, - (l+12)2 &:2
2 - m 2c 4 &E 2 r b c
\ + 2 \c - m c &E r + 2 4
2
(2.55)
a
with Y lm(x^ ) the spherical harmonic function. It is worth noting that if the flux is quantized in Eq. (2.53), i.e., 4?g=2?ce_ integer, |k+ g~ | is an integer and the spectrum reduces to that of the relativistic hydrogen atom. In this case, there is also no AB effect. B. The Relativistic AharonovBohmMonopole (ABM) System if :=0 In this case, Green's function of the relativistic ABM system is given by G(x b , x a ; E)=
&i m 2mc 2? - r b r a
_:
:
p=q~ k=&
_
(2p+2 |k+ g~ | +1) 1(1+ p+2 |k+ g~ | ) 1(1+ p) 1(1+ p+ |k+ g~ | +q~ ) 1(1+ p+ |k+ g~ | &q~ )
_I - [2( p+ |k+ g~ | )+1]2 &4q~22
&
1 - m 2c 4 &E 2 [(r b +r a )& |r b &r a | ] 2c
\ + 1 \2c - m c &E [(r +r )+ |r &r | ]+
_K - [2( p+ |k+ g~ | )+1]2 &4q~22
2 4
2
b
a
b
a
_e i(q~ +k)(.b &.a )(cos % b 2 cos % a 2) ( |k+ g~ | +q~ )(sin % b 2 sin % a 2) ( |k+ g~ | &q~ ) _P p( |k+ g~ | &q~, |k+ g~ | +q~)(cos % b ) P (p|k+ g~ | &q~, |k+ g~ | +q~ )(cos % a ),
(2.56)
where I + , K & stand for the modified Bessel functions. The nonrelativistic ABM system is discussed by the Schrodinger equation in Ref. [17]. Nevertheless its
282
DE-HONE LIN
Green function is given in the first time. A further reduction of the ABM system gives us Green's function of a pure relativistic AB system if q~ is taken as vanishing. Green's function is given by [18] G(x b , x a ; E)=
&i m 2mc 2? - r b r a
_:
:
p=0 k=&
_
(2p+2 |k+ g~ | +1) 1(1+ p+2 |k+ g~ |) 1(1+ p) 1(1+ p+ |k+ g~ | ) 1(1+ p+ |k+ g~ | )
_I p+ |k+ g~ | +12
1
&
\2c - m 2c &E [(r +r )& |r &r | ]+ 1 \2c - m 2c &E [(r +r )+ |r &r | ] + 2
4
2
b
2
_K p+ |k+ g~ | +12
4
a
b
a
2
b
a
b
a
_e ik(.b &.a )(cos % b 2 cos % a 2) |k+ g~ | (sin % b 2 sin % a 2) |k+ g~ | _P (p|k+ g~ |, |k+ g~ | )(cos % b ) P (p|k+ g~ |, |k+ g~| )(cos % a ).
(2.57)
C. The Relativistic Dyon System if g~ =0 In the case Green's function of Eq. (2.50) reduces to the pure relativistic dyon system and is given by G(x b , x a ; E D )=
&i mc : 2mc - m 2c 4 &E 2 p=q~ D
lD
:
iq~(.b &.a ) Y p, k, q~(% b , . b ) Y* p, k, q~(% a , . a ) } e
k=&lD
1(12+12 - (2p+1) 2 &4(: 2 +q~ 2 )&E D :- m 2c 4 &E 2D ) _ r b r a 1(- (2p+1) 2 &4(: 2 +q~ 2 )+1) _W ED :- m2c4 &E2 , - (2p+1)2 &4(:2 +q~2 )2 D
_M ED :- m2c4 &E2 , - (2p+1)2 &4(:2 +q~2 )2 D
2
\c - m c &E r + 2 \c - m c &E r + , 2 4
2 D b
2 4
2 D a
(2.58)
where Y p, k, q~(% b , . b ) are the so-called monopole harmonics Y p, k, q~(%, .)=
2p+1 ik. p e d k, q~(%). 4?
(2.59)
The result is given in Ref. [19]. With these, we complete the discussion of the relativistic ABD system.
AHARONOVBOHMDYON SYSTEM
283
III. CONCLUDING REMARKS In this letter, we succeed in calculating Green's function of the relativistic AharonovBohmDyon system, by a path integral approach. The result and its special cases are very interesting in diverse areas of physics such as the angular decomposition of the dyon Green's function being the reduced Wigner matrices, which play a very important part in nuclear physics. There are two other potentials yielding the fields of monopole given by A 1(x)=q~
(& ye^ x +xe^ y ) z , r(x 2 + y 2 )
(3.1)
ye^ x &xe^ y . r(r&z)
(3.2)
and A 2(x)=q~
The field A 1(x) in Eq. (4.1) has two strings of equal strength importing the flux, one along the positive x 3 -axis from plus infinity to the origin and the other along the negative x 3 -axis from minus infinity to the origin. The field A 2(x) in Eq. (3.2) has one string along the positive x 3 -axis from plus infinity to the origin. With the transformation of Eq. (2.20), they have the representations A 1(x) } x* =q~.* cos %
(3.3)
A 2(x) } x* =&q~.* (1+cos %).
(3.4)
and
The new Green's function is just different from the original one of Eq. (2.50) by G 1(x b , x a ; E D )=G(x b , x a ; E D ) e &iq~(.b &.a )
(3.5)
G 2(x b , x a ; E D )=G(x b , x a ; E D ) e &i2q~(.b &.a )
(3.6)
and
where G 1, 2 denote the new Green's function related to the potential A 1, 2(x). We see that the different choices of gauge just relate to the phase changes. This is reasonable for the gauge symmetry. It is our hope that our studies would help to achieve the ultimate goal of obtaining a comprehensive and complete path integral description of quantum mechanics and quantum field theory, including quantum gravity and cosmology.
284
DE-HONE LIN
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
F. Wilczek, Phys. Rev. Lett. 48 (1982), 1144, 1146. A. Goldhaber, Phys. Rev. Lett. 49 (1982), 905. R. Jackiw and A. N. Redlich, Phys. Rev. Lett. 50 (1983), 555. For reviews, see A. Shapere and F. Wilczek (Eds.), ``Geometric Phases in Physics,'' World Scientific, Singapore, 1989, and references therein. For reviews, see A. Khare, ``Fractional Statistics and Quantum Theory,'' World Scientific Press, 1997, and references therein. H. Kleinert, Phys. Lett. A 212 (1996), 15. D. H. Lin, J. Phys. A 31 (1998), 4785. D. H. Lin, J. Math. Phys. 40 (1999), 1246. T. T. Wu and C. N. Yang, Nucl. Phys. B 107 (1976), 365. H. Kleinert, Int. J. Mod. Phys. A 7 (1992), 4693; Phys. Lett. B 246 (1990), 127; Phys. Lett. B 293 (1992), 168. H. Kleinert, ``Path Integrals in Quantum Mechanics, Statistics and Polymer Physics,'' World Scientific, Singapore, 1990. C. Grosche and F. Steiner, ``Handbook of Feynman Path Integrals,'' Springer Tracts in Modern Physics, Vol. 145, Springer-Verlag, Berlin, 1998. A. R. Edmonds, ``Angular Momentum in Quantum Mechanics,'' Princeton Press, Princeton, 1957. W. Magnus, F. Oberhettinger, and R. P. Soni, ``Formulas and Theorems for the Special Functions of Mathematical Physics,'' Springer-Verlag, Berlin, 1966. D. H. Lin, quant-ph9909050. D. H. Lin, J. Phys. A 31 (1998), 7577. S. M. Roy and V. Singh, Phys. Rev. Lett. 51 (1983), 2069. D. S. Chuu and D. H. Lin, chao-dyn9909022. D. S. Chuu and D. H. Lin, J. Phys. A 32 (1999), 4199.
Path Integral Solution of the AharonovBohmDyon System De-Hone Lin Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan E-mail: d793314phys.nthu.edu.tw. Received September 29, 1999
An exact path integral solution for a relativistic particle with both electric and magnetic charges (e 1 , q 1 ) moving in the fields created by a flux tube along the z-axis and a particle with charges (e 2 , q 2 ) located at the origin, i.e., the relativistic AharonovBohmDyon system, is given. The solutions of interesting special cases, AharonovBohmCoulomb, Aharonov BohmMonopole, and AharonovBohm, are discussed. 2000 Academic Press
I. INTRODUCTION For the past 17 years, particles associated with questions on fractional angular momenta and quantization of magnetic monopoles moving in AharonovBohm (AB) magnetic fields and long-range potentials have been much debated [15]. In this paper, we present an exact solution of the path integral for a relativistic particle with both electric and magnetic charges (e 1 , q 1 ) moving in the fields created by a charge and a Dirac magnetic monopole of strange (e 2 , q 2 ) at the origin plus an infinitely long and thin solenoid carrying flux g along the z axis, i.e., the Aharonov BohmDyon (ABD) system. The solution provides a concrete example in which to study the above questions.
II. PATH INTEGRAL SOLUTION FOR THE AHARONOVBOHMDYON SYSTEM The starting point is the path integral representation for Green's function of a relativistic particle in external electromagnetic fields [68], G(x b , x a ; E)=
2mc
|
0
|
|
dS D\(*) 8[ \(*)] D 3x(*) exp[iA E [x, x* ]]\(0)
(2.1)
with the action A E [x, x* ]=
|
*b
d* *a
_
m (E&V(x)) 2 mc 2 &\(*) , x* 2(*)+A(x) } x* (*)+\(*) 2\(*) 2mc 2 2 270
0003-491600 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.
&
(2.2)
271
AHARONOVBOHMDYON SYSTEM
where S is defined as S=
|
*b
d*\(*),
(2.3)
*a
in which \(*) is an arbitrary dimensionless fluctuating scale variable, \(0) is the terminal point of the function \(*), and 8[ \(*)] is some convenient gauge-fixing functional [68]. The only condition on 8[ \(*)] is that
| D\(*) 8[ \(*)]=1.
(2.4)
mc is the well-known Compton wavelength of a particle of mass m, A(x) and V(x) stand for the vector and scalar potential of the systems, respectively. E is the system energy, and x is the spatial part of the (3+1) vector x + =(x(*), t(*)). The path integral representation arises from the limiting condition of the *-sliced version G(x b , x a ; E)r
2mc
N+1
_| d\ 8( \ )& d x 1 _ ` | _ (2?i= \ m) & (2?i= \ m) i _exp { A = \(0), |
dS `
0
n
n
n=1
N
32
b
b
n=1
3
&
n
32
n
n
N E
(2.5)
where the sign r becomes an equality for N Ä , and the *-sliced action N+1
AN E= : n=1
_
m(x n &x n&1 ) 2 (E&V(x n )) 2 mc 2 +A(x n ) } (x n &x n&1 )+= n \ n &= n \ n 2 2= n \ n 2mc 2 (2.6)
&
with = n =* n &* n&1 , * b =* N+1 , * a =* 0 , x a =x(* 0 ) and x b =x(* n+1 ). For the ABD system, the interaction potentials are given by V(x)=
&e 2 , r
(2.7)
and the vector potential reads A(x)=q~
& ye^ x +xe^ y & ye^ x +xe^ y , +g^ r(r+z) x2 + y 2
(2.8)
272
DE-HONE LIN
where e^ x, y denotes the basis vectors in the Cartesian coordinate frame. The interacting strength q~ # &(e 1 q 2 &e 2 q 1 )c, g~ #(e 1 + q 1 ) gc, and e 2 # &e 1 e 2 &q 1 q 2 are combinations of the electric and magnetic charges of the two particles, and r#- x 2 + y 2 +z 2 is the radial distance, as usual. The hydrogen atom is a special case of the dyonium system with e 1 =&e 2 =e and q~ =0= g^. An electron around a pure magnetic monopole has e 1 =e, q 2 =q~, e 2 =q 1 =0= g^. In the vector potential we have chosen the monopole to have a Dirac string singularity along the negative z-axis. This is proposed in Ref. [9]. As a consequence of monopole gauge invariance, the parameter q~ has to be an integer or a half-integer number [10], a condition referred to as Dirac's charge quantization. To obtain a tractable path integral for the potential V(x)=&e 2r, we choose the fluctuating scale variable \ n =r n and the gauge-fixing condition N+1
| D\(*) 8[ \(*)]= lim
NÄ
` N=1
dw n
_| (2?i= r m)
12
n n
& { exp
i N+1 m (2w n ) 2 : . n=1 2 = n r n
=
(2.9)
The unity condition is automatically satisfied. With Eq. (2.9), the path integral of Eq. (2.5) turns into G(x b , x a ; E)r
2mc
|
0
|
dS dw b
N rb ` 2 (2?i= b r b m) n=1
d 4x n i N exp A , 2 E & (2?i= n r n m) (2.10)
_|
& { =
where A N E is the action of Eq. (2.6) in which the constant \(0) is chosen as r b and the three-vectors x n of the kinematic term are replaced with the four-vectors x n . This extends the kinetic action to N+1
AN kin = : n=1
m (x n &x n&1 ) . 2 =n rn
(2.11)
The path integral can be simplified by the K-S transformation [11, 12] dx =2A(u ) du
(2.12)
with 4_4 matrix A(u ) chosen as
A(u )=
\
u3 u4 u1 u2
u4 &u 3 u2 &u 1
u1 &u 2 &u 3 u4
u2 u1 &u 4 &u 3
+
.
(2.13)
273
AHARONOVBOHMDYON SYSTEM
The magnetic interaction turns into A(u ) } u $=
q~ (u 1 ) 2 +(u 2 ) 2 &(u 3 ) 2 &(u 4 ) 2 +g^ r
{ _ & = u u* &u u* u u* &u u* _ _ (u ) +(u ) + (u ) +(u ) & . 2 1
1 2
1 2
3 4
2 2
4 3
3 2
(2.14)
4 2
We obtain a path integral in the continuum limit equivalent to Eq. (2.10), G(x b , x a ; E)=
2mc
|
2
2
dSe iSEe mc G(u b , u a ; S),
(2.15)
0
with the four dimensional pseudopropagator G(u b , u a ; S)=
1 dw b D 4u(*) exp[iA E [u , u $]], 16 r b
|
|
(2.16)
where the functional measure has the *-sliced representation N
1
| D u(*)r (2?i= M) 4
b
2
` n=1
_|
d 4u n 2 & (2?i= n M)
&
(2.17)
and the action is given by A E [u , u $]=
|
S
0
d*
_
Mu $ 2 M| 2u 2 4 2: 2 +A(u ) } u $+ + 2 2 2Mu 2
&
(2.18)
in which u 2 =(u 1 ) 2 +(u 2 ) 2 +(u 3 ) 2 +(u 4 ) 2 =r, M=4m, | 2 =
E 2 &m 2c 4 . 4m 2c 2
(2.19)
To go further, let us introduce the Euler angle's representation of the K-S transformation (e.g., 11, 12): u 1 =- r cos(%2) cos[(.+#)2] u 2 =&- r cos(%2) sin[(.+#)2] u 3 =- r sin(%2) cos[(.&#)2] u 4 =- r sin(%2) sin[(.&#)2]
=
\
0 % ?% 0 . 2? 0 # 4?
+
.
(2.20)
274
DE-HONE LIN
Then the action of Eq. (2.18) turns into A E [u , u $]=
|
S
d* 0
M 2 u 2 2 4q~ u* + %4 +.* 2 +#* 2 +2.* #* & cos % 2 4 Mu 2
{ _
\
\
M| 2u 2 4 2: 2 +(q~ &g~ ) .* + 2 2Mu 2
+
+
=
+& (2.21)
with g~ = & g^ for a matter of convention and u= |u |. Here we see that the final term associated to the AB effect just related to a total derivative so that it can factor out of the entire action and be considered in the final step. The path integral in Eq. (2.16) can have a clear expression in terms of canonical representation A E [u , u $]=
|
S
d*[[ p u u* + p % %4 + p . .* + p # #* ]&H E ]
(2.22)
0
with the velocity variables (u* , %4 , .* , #* ) having the representation of canonical momenta as follows, pu u* = , M p% %4 = , ! (2.23) 1 #* = [ p # +'! cos 2%& p . cos %], ! sin 2% .* =
1 [ p &'! cos %& p # cos %], ! sin 2% .
where the variables '# &4q~Mu 2 and !#Mu 24. It is not difficult to find the Hamiltonian H E [u , p ]=
1 4 1 p 2u + 2 p 2% + ( p 2 +( p # &q~ ) 2 &2( p # &q~ ) p . cos %) 2M u sin 2% .
{
+
_
4 M| 2u 2 2 2 2 . [2q ~ p & (: +q ~ )]& # 2Mu 2 2
&=
(2.24)
We obtain the canonical representation of the path integral G(x b , x a ; E)=
2mc
|
1 _ 16
2
2
dSe iSEe mc
0
|
4?
0
|
S
d# b D 4u(*) e i 0 d*(q~ & g~ ) .*
D 4p(*) exp[iA E [u , p ]], 2? & (2.25)
|
275
AHARONOVBOHMDYON SYSTEM
where the integral variables dw b r b in the measure of Eq. (2.16) have been changed by using Eq. (2.12), dw=2(u 2 du 1 &u 1 du 2 +u 4 du 3 &u 3 du 4 ) =r(cos % d.+d#),
(2.26)
and due to the x and thus the angles (%, .) remain fixed during the w integration. In the canonical path integral, the momenta are dummy integration variables so that we can replace ( p # &q~ ) by p # . Then the action becomes A E [u , p ]=
|
S
d*[[ p u u* + p % %4 + p . .* +( p # +q~ ) #* ]&H E [u , p ]]
(2.27)
0
with the Hamiltonian H E [u , p ]=
1 4 1 p 2u + 2 p 2% + 2 ( p 2. + p 2# &2p # p . cos %) 2M u sin %
{
+
_
&=
4 M| 2u 2 [2q~( p # +q~ )& 2(: 2 +q~ 2 )]& . 2 2Mu 2
(2.28)
Action in Eq. (2.27) contains an additional term 2A =q~
|
S
d*#* .
(2.29)
0
This is a pure surface term 2A =q~(# b &# a ). Since the integral over # b forces the momentum p r in the canonical action (2.27) to take the value &q~, this eliminates the term proportional to p # +q~ in Eq. (2.28); therefore the entire path integral becomes G(x b , x a ; E)=
2mc
|
2
2
dSe iSEe mc
0
1 16
|
4?
d# b e iq~(#b &#a )G(u b , u a ; S).
(2.30)
0
Here the pseudopropagator G(u b , u a ; S)=
_|
&
S
D 4u(*) e i 0 d*(q~ & g~ ) .* exp[iA E [u , u $]]
(2.31)
Mu $ 2 M| 2u 2 4(: 2 +q~ 2 ) 2 . + + 2 2 2Mu 2
(2.32)
&
with the action A E [u , u $]=
|
S
0
d*
{
=
276
DE-HONE LIN
The path integral can be performed. After the variable changes d*=&idy, S=&iL, the entire path integral becomes G(x b , x a ; E)=
&i 2mc
|
2
dL e LEe mc
2
0
1 16
|
4?
d# b e iq~(#b &#a )G(u b , u a ; L),
(2.33)
0
with the pseudopropagator (e.g., [11, 12]) G(u b , u a ; S)=
1 : G l (u b , u a ; L) u b u a l=0 l2 L l+1 l2 ik1(.b &.a )+ik2(#b &#a ) i 0 dy(q~ & g~ ) .* : d l2 e _ 2 k1, k2(% b ) d k1, k2(% a ) e 2? k1, k2 =&l2
(2.34) with the radial amplitude G l (u b , u a ; L)=
M|~ 2 2 M |~u b u a , (2.35) e &M|~2(ub +ua ) coth |~LI - (l+1)2 &4(:2 +q~2 ) sinh |~L sinh |~L
\
+
ik1.+ik2# where |~ =- (m 2c 4 &E 2 )2mc and d l2 are the representation funck1, k2(%) e 4? &iq(#b &#a ) tions of the rotation group. The integral 0 d# a e in Eq. (2.33) can be easily done. We arrive at the fixed-energy amplitude of the relativistic ABD system, now labeled by the subscript D,
G(x b , x a ; E D )=
1 - rb ra
lD
:
:
lD =q~
k=&lD
G lD (r b , r a ; E D ) d lk,D q~(% b ) d lk,D*q~(% a )
e ik(.b &.a ) 2?
(2.36)
in which l D is defined as l2 and the fixed-energy Green's function is, G lD (r b , r a ; E D )=
&i 1 2mc 2 _
|
2
L
2
dL e LEDe mc e i 0 dy(q~ & g~ ) .*
0
2 2 M |~u b u a M|~ e &M|~(ub +ua ) coth |~LI - (2lD +1)2 &4(:2 +q~2 ) . sinh |~L sinh |~L
\
+
(2.37) It is now the place to consider the AB influence on the entire system. The term related to the AB effect is given by A A&B =&g~
|
L
dy.* ( y), 0
(2.38)
277
AHARONOVBOHMDYON SYSTEM
where .( y)=.(x( y)), .* =d.dy. Since the particle orbits are present at all times, their worldlines in spacetime can be considered as being closed at infinity, and the integral k=
1 2?
|
L
dy.* ( y)
(2.39)
0
is the topological invariant with integer values of the winding number k. The magnetic interaction is therefore a purely topological one given by A A&B =&g~2k?.
(2.40)
To incorporate the effect, let us first note that [13, p. 58] d lk,D q~(%)=
_
1(1+l D +k) 1(1+l D &k) 1(1+l D +q~ ) 1(1+l D &q~ )
&
12
~ , k+q~ ) _(cos %2) (k+q~ ) (sin %2) (k&q~ ) P (k&q (cos %), lD &k
(2.41)
where P l(x, y)(z) is the Jacobi function. This leads to d lk,D q~(% b ) d lk,D*q~(% a )=
1(1+l D +k) 1(1+l D &k)
_ 1(1+l +q~) 1(1+l &q~) & D
D
~ , k+q~ ) _(cos % b 2 cos % a 2) (k+q~ )(sin % b 2 sin % a 2) (k&q~ ) P (k&q lD &k ~ , k+q~ ) _(cos % b ) P (k&q (cos % a ). lD &k
(2.42)
To go further, let us change the variable l D by defining l D &k= p into p. It is easy to find that Green's function of Eq. (2.36) becomes
G(x b , x a ; E D )=
1 - rb ra
:
:
p=q~
k=&
_G p+k(r b , r a ; E D )
( p+k+12) 1(1+ p+2k) 1(1+ p) 1(1+ p+k+q~ ) 1(1+ p+k&q~ )
e ik(.b &.a ) _ (cos % b 2 cos % a 2) (k+q~ )(sin % b 2 sin % a 2) (k&q~ ) 2? ~ , k+q~ ) _P p(k&q~, k+q~ )(cos % b ) P (k&q (cos % a ). p
(2.43)
278
DE-HONE LIN
The radial Green's function corresponds to G p+k(r b , r a ; E D ) =
&i 1 2mc 2
_
|
2
L
2
dLe LEDe mc e i 0 dy(q~ & g~ ) .*
0
M|~ 2 2 M |~u b u a . e &(M|~2)(ub +ua ) coth |~L I - [2( p+k)+1]2 &4(:2 +q~2 ) sinh |~L sinh |~L
\
+
(2.44) Let us invoke Poisson's summation formula [14, p. 469]
f (k)=
: k=&
|
dy &
:
e 2?nyif ( y).
(2.45)
n=&
By incorporating the AB influence of Eq. (2.40), we arrive at G(x b , x a ; E D )=
1 2? - r b r a
: p=q~
|
dz
:
e iq~(.b &.a )e i(z& g~ )(.b +2k?&.a )
k=&
_G p+z (r b , r a ; E D )
( p+z+12) 1(1+ p+2z) 1(1+ p) 1(1+ p+z+q~ ) 1(1+ p+z&q~ )
_(cos % b 2 cos % a 2) (z+q~ ) (sin % b 2 sin % a 2) (z&q~ ) _P p(z&q~, z+q~ )(cos % b ) P p(z&q~, z+q~ )(cos % a ).
(2.46)
The sum over all k in Eq. (2.46) forces z to be equal to g~ modulo an arbitrary integral number leading to G(x b , x a ; E D )=
1 2? - r b r a
:
:
p=q~
k=&
e iq~(.b &.a )e ik(.b &.a )
_G p+ |k+ g~ | (r b , r a ; E D )
{
( p+ |k+ g~ | +12) 1(1+ p+2 |k+ g~ | ) 1(1+ p) 1(1+ p+ |k+ g~ | +q~ ) 1(1+ p+ |k+ g~ | &q~ )
=
_(cos % b 2 cos % a 2) |k+ g~ | +q~ )(sin % b 2 sin % a 2) ( |k+ g~ | &q~ ) _P p( |k+ g~ | &q~, |k+ g~ | +q~ )(cos % b ) P p( |k+ g~ | &q~, |k+ g~ | +q~)(cos % a ).
(2.47)
279
AHARONOVBOHMDYON SYSTEM
The fixed-energy Green's function is G p+ |k+ g~ | (r b , r a ; E D ) =
&i 1 2mc 2
_
|
2
2
dL e LEDe mc
0
M|~ 2 2 M |~u b u a . e &M|~2(ub +ua ) coth |~L I - [2( p+ |k+ g~ | )+1]2 &4(:2 +q~2 ) sinh |~L sinh |~L (2.48)
\
+
The integration can be done by the formula (e.g., [11, Chap. 9])
|
dy
0
e 2&y t t - `b `a exp & (` a +` b ) coth y I + sinh y 2 sinh y
=
_
& \
1((1++)2&&) t - ` b ` a 1(++1)
+
W &, +2(t` b ) M &, +2(t` a ),
(2.49)
where M +, & and W +, & are the Whittaker functions and the range of validity is given by ` b >` a >0, Re[(1++)2&&]>0, Re(t)>0,
|arg t|
We complete the integration and obtain the exact solution of the path integral for the ABD system G(x b , x a ; E D )=
&i mc 2mc 4?r b r a - m 2c 4 &E 2D
_:
:
p=q~ k=&
{
1(12+- [2( p+ |k+ g~ | )+1] 2 &4(: 2 +q~ 2 )2 &E D :- m 2c 4 &E 2D ) 1(1+- [2( p+ |k+ g~ |)+1] 2 &4(: 2 +q~ 2 ))
_W ED:- m2c4 &E 2 , - [2( p+ |k+ g~ | )+1]2 &4(:2 +q~2 )2 D
_M ED:- m2c4 &E 2 , - [2( p+ |k+ g~ | )+1]2 &4(:2 +q~2 )2 D
2
=
\c - m c &E r + 2 \c - m c &E r + 2 4
2 D b
2 4
2 D a
280
DE-HONE LIN
_
_
(2p+2 |k+ g~ | +1) 1(1+ p+2 |k+ g~ |) 1(1+ p) 1(1+ p+ |k+ g~ | +q~ ) 1(1+ p+ |k+ g~ | &q~ )
&
_e i(q~ +k)(.b &.a )(cos % b 2 cos % a 2) ( |k+ g~| +q~ )(sin % b 2 sin % a 2) (|k+ g~ | &q~ ) g~ | &q~, |k+ g~ | +q~ ) _P p( |k+ g~ | &q~, |k+ g~ | +q~ )(cos % b ) P (|k+ (cos % a ). p
(2.50)
The energy levels can be extracted from the poles. They are determined by the equation 12+- [2( p+ |k+ g~ | )+1] 2 &4(: 2 +q~ 2 )2&E D :- m 2c 4 &E 2D =&n r ,
(2.51)
where the radial quantum number n r =0, 1, 2, ... . With little mathematical manipulation, we obtain
_
E nr, p, k =\mc 2 1+
:2 (12+12- [2( p+ |k+ g~ | )+1] 2 &4(: 2 +q~ 2 )+n r ) 2
&
&12
.( 2.52)
The corresponding wave functions can be obtained according to the methods of e.g., Refs. [11, 12, 19]. There are several interesting special cases as follows: A. The Relativistic AharonovBohmCoulomb (ABC) System if q~ =0 In this case, Green's function reduces to G(x b , x a ; E)=
&i mc 2mc 4?r b r a - m 2c 4 &E 2
_:
:
p=0 k=&
{
1(12+- [2( p+ |k+ g~ | )+1] 2 &4: 22 &E:- m 2c 4 &E 2 ) 1(1+- [2( p+ |k+ g~ | )+1] 2 &4: 2 ) 2
=
\c - m c &E r + 2 _M \c - m c &E r + (2p+2 |k+ g~ | +1) 1(1+ p+2 |k+ g~ | ) 1(1+ p) _ _ & 1 (1+ p+ |k+ g~ | ) _W E:- m2c4 &E2, - [2( p+ |k+ g~ | )+1]2 &4:22
2 4
b
2 4
E:- m2c4 &E2, - [2( p+ |k+ g~ | )+1]2 &4:22
2
2
a
2
_ ik(.b &.a )(cos % b 2 cos % a 2) |k+ g~| (sin % b 2 sin % a 2) |k+ g~| _P (p|k+ g~ |, |k+ g~ | )(cos % b ) P (p|k+ g~ |, |k+ g~ | )(cos % a ).
(2.53)
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AHARONOVBOHMDYON SYSTEM
a relativistic result was first given in Ref. 8. A further reduction is given by taking g~ =0. In this case, with the help of the relation between the associated Legendre and Jacobi functions [15] P kl (cos %)=(&1) k
1(1+k+l ) k) (cos %2 sin %2) k P (k, l&k (cos %). 1(1+l )
(2.54)
Green's function of the relativistic Coulomb system given in [6, 16] is obtained by G(x b , x a ; E)=
mc &i 1 : 2mc (r b r a ) - m 2c 4 &E 2 l=0
_
l
: Y lm(x^ b ) Y* lm(x^ a ) m=&l
1(12+- (l+12) 2 &: 2 &E:- m 2c 4 &E 2 ) 1(1+2 - (l+ 12) 2 &: 2 )
_W E:- m2c4 &E2, - (l+12)2 &:2 _M E:- m2c4 &E2, - (l+12)2 &:2
2 - m 2c 4 &E 2 r b c
\ + 2 \c - m c &E r + 2 4
2
(2.55)
a
with Y lm(x^ ) the spherical harmonic function. It is worth noting that if the flux is quantized in Eq. (2.53), i.e., 4?g=2?ce_ integer, |k+ g~ | is an integer and the spectrum reduces to that of the relativistic hydrogen atom. In this case, there is also no AB effect. B. The Relativistic AharonovBohmMonopole (ABM) System if :=0 In this case, Green's function of the relativistic ABM system is given by G(x b , x a ; E)=
&i m 2mc 2? - r b r a
_:
:
p=q~ k=&
_
(2p+2 |k+ g~ | +1) 1(1+ p+2 |k+ g~ | ) 1(1+ p) 1(1+ p+ |k+ g~ | +q~ ) 1(1+ p+ |k+ g~ | &q~ )
_I - [2( p+ |k+ g~ | )+1]2 &4q~22
&
1 - m 2c 4 &E 2 [(r b +r a )& |r b &r a | ] 2c
\ + 1 \2c - m c &E [(r +r )+ |r &r | ]+
_K - [2( p+ |k+ g~ | )+1]2 &4q~22
2 4
2
b
a
b
a
_e i(q~ +k)(.b &.a )(cos % b 2 cos % a 2) ( |k+ g~ | +q~ )(sin % b 2 sin % a 2) ( |k+ g~ | &q~ ) _P p( |k+ g~ | &q~, |k+ g~ | +q~)(cos % b ) P (p|k+ g~ | &q~, |k+ g~ | +q~ )(cos % a ),
(2.56)
where I + , K & stand for the modified Bessel functions. The nonrelativistic ABM system is discussed by the Schrodinger equation in Ref. [17]. Nevertheless its
282
DE-HONE LIN
Green function is given in the first time. A further reduction of the ABM system gives us Green's function of a pure relativistic AB system if q~ is taken as vanishing. Green's function is given by [18] G(x b , x a ; E)=
&i m 2mc 2? - r b r a
_:
:
p=0 k=&
_
(2p+2 |k+ g~ | +1) 1(1+ p+2 |k+ g~ |) 1(1+ p) 1(1+ p+ |k+ g~ | ) 1(1+ p+ |k+ g~ | )
_I p+ |k+ g~ | +12
1
&
\2c - m 2c &E [(r +r )& |r &r | ]+ 1 \2c - m 2c &E [(r +r )+ |r &r | ] + 2
4
2
b
2
_K p+ |k+ g~ | +12
4
a
b
a
2
b
a
b
a
_e ik(.b &.a )(cos % b 2 cos % a 2) |k+ g~ | (sin % b 2 sin % a 2) |k+ g~ | _P (p|k+ g~ |, |k+ g~ | )(cos % b ) P (p|k+ g~ |, |k+ g~| )(cos % a ).
(2.57)
C. The Relativistic Dyon System if g~ =0 In the case Green's function of Eq. (2.50) reduces to the pure relativistic dyon system and is given by G(x b , x a ; E D )=
&i mc : 2mc - m 2c 4 &E 2 p=q~ D
lD
:
iq~(.b &.a ) Y p, k, q~(% b , . b ) Y* p, k, q~(% a , . a ) } e
k=&lD
1(12+12 - (2p+1) 2 &4(: 2 +q~ 2 )&E D :- m 2c 4 &E 2D ) _ r b r a 1(- (2p+1) 2 &4(: 2 +q~ 2 )+1) _W ED :- m2c4 &E2 , - (2p+1)2 &4(:2 +q~2 )2 D
_M ED :- m2c4 &E2 , - (2p+1)2 &4(:2 +q~2 )2 D
2
\c - m c &E r + 2 \c - m c &E r + , 2 4
2 D b
2 4
2 D a
(2.58)
where Y p, k, q~(% b , . b ) are the so-called monopole harmonics Y p, k, q~(%, .)=
2p+1 ik. p e d k, q~(%). 4?
(2.59)
The result is given in Ref. [19]. With these, we complete the discussion of the relativistic ABD system.
AHARONOVBOHMDYON SYSTEM
283
III. CONCLUDING REMARKS In this letter, we succeed in calculating Green's function of the relativistic AharonovBohmDyon system, by a path integral approach. The result and its special cases are very interesting in diverse areas of physics such as the angular decomposition of the dyon Green's function being the reduced Wigner matrices, which play a very important part in nuclear physics. There are two other potentials yielding the fields of monopole given by A 1(x)=q~
(& ye^ x +xe^ y ) z , r(x 2 + y 2 )
(3.1)
ye^ x &xe^ y . r(r&z)
(3.2)
and A 2(x)=q~
The field A 1(x) in Eq. (4.1) has two strings of equal strength importing the flux, one along the positive x 3 -axis from plus infinity to the origin and the other along the negative x 3 -axis from minus infinity to the origin. The field A 2(x) in Eq. (3.2) has one string along the positive x 3 -axis from plus infinity to the origin. With the transformation of Eq. (2.20), they have the representations A 1(x) } x* =q~.* cos %
(3.3)
A 2(x) } x* =&q~.* (1+cos %).
(3.4)
and
The new Green's function is just different from the original one of Eq. (2.50) by G 1(x b , x a ; E D )=G(x b , x a ; E D ) e &iq~(.b &.a )
(3.5)
G 2(x b , x a ; E D )=G(x b , x a ; E D ) e &i2q~(.b &.a )
(3.6)
and
where G 1, 2 denote the new Green's function related to the potential A 1, 2(x). We see that the different choices of gauge just relate to the phase changes. This is reasonable for the gauge symmetry. It is our hope that our studies would help to achieve the ultimate goal of obtaining a comprehensive and complete path integral description of quantum mechanics and quantum field theory, including quantum gravity and cosmology.
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DE-HONE LIN
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
F. Wilczek, Phys. Rev. Lett. 48 (1982), 1144, 1146. A. Goldhaber, Phys. Rev. Lett. 49 (1982), 905. R. Jackiw and A. N. Redlich, Phys. Rev. Lett. 50 (1983), 555. For reviews, see A. Shapere and F. Wilczek (Eds.), ``Geometric Phases in Physics,'' World Scientific, Singapore, 1989, and references therein. For reviews, see A. Khare, ``Fractional Statistics and Quantum Theory,'' World Scientific Press, 1997, and references therein. H. Kleinert, Phys. Lett. A 212 (1996), 15. D. H. Lin, J. Phys. A 31 (1998), 4785. D. H. Lin, J. Math. Phys. 40 (1999), 1246. T. T. Wu and C. N. Yang, Nucl. Phys. B 107 (1976), 365. H. Kleinert, Int. J. Mod. Phys. A 7 (1992), 4693; Phys. Lett. B 246 (1990), 127; Phys. Lett. B 293 (1992), 168. H. Kleinert, ``Path Integrals in Quantum Mechanics, Statistics and Polymer Physics,'' World Scientific, Singapore, 1990. C. Grosche and F. Steiner, ``Handbook of Feynman Path Integrals,'' Springer Tracts in Modern Physics, Vol. 145, Springer-Verlag, Berlin, 1998. A. R. Edmonds, ``Angular Momentum in Quantum Mechanics,'' Princeton Press, Princeton, 1957. W. Magnus, F. Oberhettinger, and R. P. Soni, ``Formulas and Theorems for the Special Functions of Mathematical Physics,'' Springer-Verlag, Berlin, 1966. D. H. Lin, quant-ph9909050. D. H. Lin, J. Phys. A 31 (1998), 7577. S. M. Roy and V. Singh, Phys. Rev. Lett. 51 (1983), 2069. D. S. Chuu and D. H. Lin, chao-dyn9909022. D. S. Chuu and D. H. Lin, J. Phys. A 32 (1999), 4199.