Nonrelativistic dyon-dyon scattering

ANNALS

OF

PHYSICS

101,

451-495 (1976)

Nonrelativistic

Dyon-Dyon

Scattering*

JULIAN SCHWINGER, KIMBALL A. MILTON, WU-YANG TSAI, LESTER L. DERAAD, JR., AND DONALD C. CLARK Department

of Physics,

University

of California,

Los Angeles,

California

90024

Received April 22, 1976

The nonrelativistic problem of the scattering of two dyons (including the case of electron scattering by magnetic monopoles) is systematically studied, both classically and quantum mechanically, with a view toward the discrimination between various combinations of electric and magnetic charges. We analyze the classical cross section with particular attention to the interesting phenomena which occur for large angle scattering, the “rainbows” and “glory,” where the cross section becomes infinite. Quantum mechanically, we find that these infinities do not occur and that, when the partial wave scattering amplitude is summed, a very elaborate structure emerges for the cross section, which depends sensitively upon the electric and magnetic charges of the particles, as well as on their relative speed. We further discuss a large modification, leading to spin flip and nonflip amplitudes, due to the dipole moments of the particles. Numerical results are presented for a variety of values of these parameters. In principle, these results could be used to distinguish the S-ray distributions produced by the various species of electrically and magnetically charged particles. Quite apart from the experimental implications of our numerical results, we have made a number of theoretical improvements and extensions. Numbered among these are the consideration of dyons and particles having dipole moments, and the explicit demonstration, based on the methods of angular momentum, that the differential cross section is independent of the choice of singularity line.

I. INTRODUCTION Within the past year, there has been a revival of theoretical interest in the subject of magnetic chage [l, 21. Shortly after this resurgence of activity, a cosmic ray event was reported [3] which was thought to have been produced by a magnetically charged particle. Even though this interpretation has subsequently been strenuously criticized [4], it cannot yet be ruled out. Only the observation of further events could affirmatively settle the issue. But, independently of the experimental question, the subject of magnetic charge remains of considerable theoretical interest, for the arguments in favor of its ultimate relevance [l, 5-71 remain as cogent as ever. * Work supported in part by the A. P. Sloan Foundation and the National Science Foundation.

451 Copyright All rights

0 1976 by Academic Press, Inc. of reproduction in any form reserved.

452

SCHWINGER

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AL.

It is our intention to present some results possibly relevant to the discrimination between the various classes of electromagnetically interacting particles: Those possessing pure electric charge (“charged particles”), pure magnetic charge (“monopoles”), and electric plus magnetic charge (“dyons”). Elsewhere, one of us [8] has reconsidered the theory of ionization due to electric and magnetic charges, a central issue in track interpretation. Here, we will review and further develop the nonrelativistic theory of the two body scattering of electrically and magnetically charged particles, with particular emphasis upon the scattering of dyons (equivalently, charged particle-dyon scattering). The results might prove useful in the interpretation of S-rays produced in a nuclear emulsion. But quite apart from any experimental significance, both the classical and quantum mechanical results prove to be of considerable interest, exhibiting such striking features as “rainbows,” “glories,” and intricate interference patterns. The scattering of electric and magnetic charges is a subject of long standing interest, starting with Poincart [9] who first analyzed the classical theory. Tamm [lo] calculated the wavefunctions following Dirac’s [5] realization that magnetic charge could be consistently formulated within a quantum mechanical context. Banderet [ll], whose work followed closely upon that of Fierz [12], was the first to present a partial wave expansion of the scattering amplitude for charged particlemonopole scattering. It remained for Ford and Wheeler [13] to present numerical results and contrast them with the classical theory [14]. In this paper, we propose, first, to incorporate a number of theoretical improvements and extensions, and secondly to obtain numerical results for various circumstances, which are interesting theoretically and possibly relevant experimentally. Accordingly, in our discussion of classical scattering (Section II), we will begin by rederiving the charged particle-monopole results, with particular emphasis on the rainbow and glory phenomena. Then we extend the calculation to the more general dyon-dyon case,l where the number and location of the rainbows depends on the relative speed v of the particles (with the monopole result being recovered in the limit v -+ co), as well as on the sign of the charges. Our efforts in the quantum mechanical direction have been more ambitious. Based upon the vector potential considerations of Appendix A, we discuss, in Section III, the nonrelativistic scattering of two dyons, or of a charged particle and a monopole. We consider the case in which the singularity line of the vector potential is semi-infinite (unsymmetrical case), and that in which it is infinite (symmetrical case). The charge quantization conditions follow immediately from those of angular momentum, and we obtain half-integer quantization for the unsymmetric case (which is Dirac’s condition [5] for charged particle-monopole scattering) and the integer condition for the symmetric case [l, 7, 151. From the 1 Another discussionhas already been presentedelsewhereby one of us [S].

DYON-DYON

SCA'ITERING

453

review of the theory of angular momentum presented in Appendix B, we easily find the wavefunctions, from which the scattering amplitude can be obtained in the form of a modified partial wave expansion. Closed form, analytical expressions are then derived in the small angle and semiclassical limits. The effects of a magnetic dipole moment of the charged particle (or of an electric dipole moment of the magnetically charged particle), which are considered in Section IV, are not small perturbations, being of the same order of magnitude as the magnetic coupling discussed in Section III. Partial wave expansions for both the spin-flip and nonspin-flip amplitudes are obtained, together with their small angle limits. Since we have been unable to proceed analytically with the evaluation of the partial wave expansions for the scattering amplitudes, we have resorted to numerical summation techniques. These methods, together with the results, which differ significantly from the classical ones, are presented in Section V. These representative results might be used as a basis for the development of criteria by which dyons may be distinguished from magnetic monopoles or electrically charged particles, We conclude in Section VI with a few comments on the relevance of our results, together with reasons for believing that the most likely magnetically charged objects to be found are dyons, rather than monopoles, since the former, if they exist, would carry the minimal units of magnetic charge. II. CLASSICAL THEORY We will begin our discussion with the classical nonrelativistic description of the relative motion of two particles, at least one of which carries magnetic charge. This is to be compared with the corresponding quantum mechanical discussion that follows. We will use e to stand for the electric charge and g for the magnetic charge (in Gaussian units); and in the relative coordinate frame, p”, r, and v to be the reduced mass, the relative position vector, and the relative velocity, respectively. A. Charged Particle-Monopole Scattering The equation of motion is /4d2r/dt2) = --KV x (r/F),

(2-l)

where K =

-eg/c,

(2.2)

and c is the speed of light. The scalar product of (2.1) with v yields WO(U

/2) I.@ = 0,

(2.3)

which is the conservation of (kinetic) energy T = (1/2)po2 = constant.

(2.4)

454

SCHWINGER

ET AL.

Therefore the relative speed is a constant of the motion. If we consider the scalar product of r with (2.1) we obtain d2r

E

125

r2

_

02

=

2 dt2

r*p

0

3

which, together with (2.4), implies r2 = v2t2 + d2.

(2.5)

In (2.3, we have chosen the initial condition so that at t = 0, r = d = distance of closest approach. We note that, as t increases, I increases without limit, so that there are no bound states. The conservation of angular momentum can be obtained by considering the vector product of r with (2.1): $

cr

so the total angular momentum

x

pv)

=

-

f

(Kf)

is

J f I+

Kf

= constant,

(2.6)

where Pis the unit vector in the radial direction, and I is the orbital angular momentum I= r x (pv). (2.7) From (2.6), we have J2 = l2 +

K2,

which implies that the magnitude of the orbital angular momentum, of the motion; combining this result with (2.4), we see that d = b,

G.8) 1, is a constant (2.9

where b is the impact parameter. Also note that J-P

=K,

(2.10)

which means that the particle is constrained to move on a cone of half-angle 42 - x/2:

t.Lvb cotx2 = ifi _ IKI’

(2.11)

DYON-DYON

455

SCATTERING

that is, if the interior axis of the cone lies along the +z-axis,2 then x/2 is the angle between the cone and the xy-plane. Thus, in terms of the azimuthal angle 4, we have i = cos 5 cos rp + cos -$ sin ~$j + sin 5 rZ.

(2.12)

By taking the vector product of r with (2.6) and combining the result with (2.5) and (2.9), we find v=+Jxr+

[l + (&)2,l~2

(2.13)

+,

so that the angular velocity of the particle about J is (w = d#dt) WE-

J p-2 *

(2.14)

The azimuthal angle C$is obtained from (2.5), (2.8), and (2.11):

x -tan-l

(0 1t j/b),

if if

t > 0, t < 0,

(2.15)

where we have chosen the initial condition to be 4 = 0 at t = 0. The scattering angle 8 is defined by

cos8 = $ (V)t*+m - wt+--m = -wt+m -VL, 7-Q = 2 cos2 X sin2 ( cos(x/2) 1 2

*,

(2.16)

or (2.17) where we have used (2.12), (2.13), and (2.15). The functional dependence of 1912 on x/2 is plotted in Fig. 1, where we observe that, unlike in the central force scatter2 If K is positive (negative), J lies along the +r(-z)-axis. 595/101/2-S

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SCHWINGER

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AL.

07

-x/z

I.5

a/2

FIG. 1. Scattering angle 0 as a function of impact parameter angle x, for various values of kr = / K 1 u,/q. [The curves should be regarded as reflected about the line O/2 = 42.1

ing problem, x is not a single valued function of 0. The differential cross section is [from (2.11) and (2.17)]

= cx (~~”

sin4&/2)

= (--$ >2g(Q

I !Z3%

I

(2.18) (2.19)

DYON-DYON

SCATTERING

451

where g(e) = ; sin4 &2)

I 2(1 - coy 5) - 5 sin 5 / ’

[=-...?.-

(2.20) (2.21)

cos;x,2) ’

gl8)

and we sum over the various x values that correspond to the same scattering angle 8. The dependence of the cross section on the scattering angle is plotted in Fig. 2.

Monopole h=I--'h;3----

-

1255

FIG. 2. Classical cross section for electron-monopole, k = 1 and 3. [Here, and in Fig. 3, g(0) = (pu/~)~(du/dI2).]

and for dyon-dyon

scattering for

We observe that da/&S becomes infinite under two circumstances (aside from 8 =O): (i) sin 8 = 0 (but sin x # 0), (ii) do/& = 0. These two circumstances are referred to as (i) the glory and (ii) the rainbow [16]. The first situation implies 0 = r (but x # ‘rr), or from (2.17), cos(xJ2) = 1/2n,

II = 1, 2, 3 )...)

(2.22a)

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SCHWINGER

ET AL.

and xs/2 = 1.047, 1.318, 1.403 ,...)

(2.22b)

which approach as

xg/2 = (7r/2) - (1/2n)

n>

1.

(2.22c)

We refer to these as glory angles. For the second condition to hold true, we have [from (2.17) and (2.21)] tan(W)

(2.23)

= L/2,

so that $5, = 4.493, 7.725, 10.904 ,...;

(2.24)

the nth term in the sequence approaches the value ((2n + 1)/2)7r as n becomes large. The corresponding rainbow angles are xT/2 = 1.214, 1.366, 1.426 ,...,

(2.25)

which approach asymptotically cos-‘(1/(2n + I)), for n > 1. The scattering angles at which the rainbows occur are [from (2.17) and (2.291 40, = 1.223, 1.367, 1.427 ,...,

(2.26a)

8, = 140.1”, 156.7”, 163.5” ,..., or, asymptotically, (2.26b)

8, = xr . B. Dyon-Dyon

Scattering

If the charges carried by the dyons are (e, , gl) and (e, , gz), the equation of motion reads KV

x

-!t

r3 ’

(2.27)

(2.28) We may obtain energy and angular momentum manner as in the previous subsection:

conservation

E = (l/2) @’ + (q/r) = constant,

in just the same (2.29)

and J = f +

Kf.

(2.30)

DYON-DYON

459

SCATTERING

The magnitude of the prbital angular momentum as is the radial component of J: J .i

=

is still a constant of the motion, (2.31)

K.

Therefore, as before, the relative motion is confined to the surface of a cone of half angle ~12 - x/2: cot(X/2) = l/i K 1,

(2.32)

I = pbv,, ,

(2.33)

where with b being the impact parameter and v0 the relative speed at infinite separation. The angular velocity w is still given by (2.14), and the equation for the corresponding azimuthal angle 4 is now immediately reduced to quadratures (just as in the central force problem), ~-,;,f

i+=

s:, [(2pd2/P)(E --t/d)

x) - x211/2’

where we have chosen the initial conditions so that at t = 0, r = d = distance of closest approach and 4 = 0. The evaluation of (2.34) yields -P/pqr

= 1 + Ecos(c#J cos(x/2)),

(2.35)

where E =

(1

+

(ho2/q2))“”

=

(1

+

(KV,,/q)2

(2.36)

COt2(X/2))‘/2.

The scattering angle 19can be obtained by forming the scalar product as in (2.16), except that now C$is given by (2.35). The result is (2.37)

where i 6 = / 4 cos 5 17-m = tan-l (y

cot +),

if

q>O, (2.38)

= rr - tan-l

1K / Vo cot x

141

2’ 1

if

q
The dependence of e/2 on x/2 is plotted in Fig. 1, for various values of 1K 1 Vo/q. The differential cross section has the same form as (2.18), the difference lying in the replacement of o by v. , in the definition of K, (2.28), and in the relation between 8 and x, (2.37). The differential cross section, with g(B) defined analogously to (2.19).

460

SCHWINGER ET AL.

132 FIG.

1368

141.6 146.4

151.2

156

IM)fl

,

,

,

165.6

17ofl

175.2

,_ 180 ' (ded

3. Classical dyon-dyon crosssection for k = 10.

is plotted in Figs. 2 and 3 as a function of the scattering angle, for various values of 1K 1 0,/q. The glory angles occur at 8 = 7r (but x # n): UP)

5,

1,2,3 ,*..,

p=

cos(xJ2) = pn9 or, from (2.38), tan

(

pi ~0s 2k2

1

--- ’ K ’ Co cot -26. 2

4

(2.40)

The solutions of (2.40) can be easily read off from Fig. 1. A particularly interesting case (see below) occurs when I K I v,/q = kn, k = + 1, 32, f3,... . For k = 10, the glory angles are &xs = 1.068, 1.340, 1.426, 1.470, 1.496, 1.515, 1.529, 1.541, and 1.552. (2.41) To determine the rainbow angles, we consider the derivative of (2.37), which gives the criterion -tan

5rP (cos(xJ2) >+

L/2 cos(xd2)

-

sin 5, sin xr sin( l/2) xr = 0,

(2.42)

where .$ is given by (2.38). The rainbow angles corresponding to k = 10 are x+/2 = 1.234, 1.388, 1.450, 1.484, 1.506, 1.522, 1.534, 1.546, 1.557. (2.43) The scattering angles at which the rainbows occur (for k = 10) are (see Fig. 3) +& = 1.243, 1.390, 1.450, 1.484, 1.506, 1.522, 1.535, 1.546, 1.557. 0, = 142.5”, 159.3”, 166.2”, 170.1”, 172.6”, 174.4”, 175.9”, 177.1”, 178.4”.

(2.44)

461

DYON-DYON SCATTERING

To see the significance of integer values of k, we expand (2.37) about x = n, and find (2.45) de/& = W K I v,lq), so that the differential cross section becomes (2.46)

Thus a double pole occurs as x -+ 7r whenever k = il,

1K / uoh = kr,

&2, &3 ,... .

(2.47)

Finally, we remark that the differential cross section for dyon-dyon near the forward direction reduces to da 1 -NdG’ - (2pQ2

which is a generalization

w2

- e2gl 2 ele2+glg2 1 + ( c vo

K

of the Rutherford

III.

)I2 (W)”1

scattering

’

(2.48)

scattering formula for small angles.

QUANTUM THEORY

In this section, we will describe the quantum scattering of two dyons, having charges (e, , gl), (ez , g2) respectively. This is more general than the usual description of charged particle-monopole scattering [lO-131, which can be recovered by setting g, = 0 and e2 = 0. In terms of the reduced mass CL,and the two invariant charge combinations [I] ml

=

_

w2

fit

e2gl ’

2P Q = - -phe2 + glg2>,

the Hamiltonian +9=v2+1

of the system is given by” (see Appendix A) 2m’ ‘“-$;+Ps;++ r2 1 - cos 8 i a~$

(3.3)

a For the dyon-dyon case, the problem separates into relative and center of mass coordinates when a common direction for the singularity lines of the vector potentials is chosen.

462

SCHWINGER

in the unsymmetrical or by

ET

AL.

case, with the singularity line lying along the positive z-axis,

2p=va+

2m'~ose12?--!$cot2~+q r2 sm2 0 i a4

(3.4)

in the symmetrical case, with the singularity line lying along the entire z-axis. Since the z-component of the orbital angular momentum I = r x (fi/i)V commutes with &‘, we may choose the wavefunction Y to be an eigenstate of, : (3.5) Then the Schroedinger equation separates into radial and angular parts: W, 0, 4) = R(r) WY @(4>,

(3.6)

CD($) = ein*,

(3.7)

where

i

$

+ f $ + k2

+

+

_

j(j

+

1)2wz!!?)

R = 0,

(3.8) (3.9)

and - [-&-$

(sin 0;)

-

tFi2 - 2m’E cos 0 $ ml2 I 0 = j(j + 1) 0, sin2 19

(3.10)

in which iii = m ==m -m’

symmetric case, unsymmetric

case.

(3.11)

The angular differential operator above is an expression for the square of the total angular momentum of the system (see Appendix B); hence the total angular momentum quantum number j must be integral or half integral. This angular problem may be easily solved in terms of the general rotation matrices [17] (see Appendix B) Us!, = eim’*~ij,(g) eiTrlb, (3.12) uz!,(@

=

(jm'

1ei(B’L)Jy 1jm),

(3.13)

with #, 0, $ being the Eulerian angles characterizing the rotation w. The solution to (3.10) is Uz!% , from which follows the angular momentum restriction j 2 I ffi I, I m’ I.

(3.14)

DYON-DYON

463

SCATTERING

Since i?i and m’ are both either integral or half integral, while m is necessarily integral, we conclude unsymmetric case: symmetric case:

712’= gp,

(3.15a) (3.15b)

ni = p,

where p is an integer. The properties of the U functions are discussed in Appendix B. There, we record the completeness and orthogonality relations satisfied by these functions, the addition theorem, and the relation of these rotation matrices to the Jacobi polynomials. The radial Schroedinger equation is just that of the ordinary Coulomb problem. Hence, the bound state radial function is R&p)

= e-p’2pLL2,L+1(p),

(3.16)

where p = 2Kr,

K = (l/fi)(-2&i”, n = (Q/2K) -L L + 4 = [(j + 3)” - m’2]112.

- 1,

(3.17a) (3.17b)

In order that the solutions be well-behaved, n must be an integer so that L,,” is the generalized Laguerre polynomial, Ln4(x) = (l/n!) e~x-=(d”/dxn)(e-sxn+Q).

(3.18)

The corresponding energy eigenvalues are [ 181

ENj = - @$ {n + $ + [(j + g)” - m’2]1/2)-2.

(3.19)

We are more interested in the scattering wavefunctions, for which E > 0. These may be conveniently expressed in terms of the confluent hypergeometric function

[I91 Rkj = const x e-ikT(kr)L F(L + 1 - iv, 2L + 2, 2ikr),

(3.20)

where 77 = -Q/R

k = (l/h)(2~E)l/~,

(3.21)

and L is defined in (3.17b). The asymptotic form of this is Rkj - (l/r) sin(kr - 7 In 2kr - (7r/2)L + S,),

(3.22)

where 6, = arg T(L + 1 + in) is the Coulomb

phase shift with noninteger L.

(3.23)

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ET

AL.

We now combine angular and radial solutions to express the wavefunction, for kr > 1, as !Pk(r) N C AkiaCV$(O, f$) cP-*)~ jiir

sin(kr - 7 In 2kr - (7r/2) L + S,),

(I/kr)

(3.24)

where the “spherical harmonics,” ?Y$ , are defined in (B.34). The constants Akj, are to be determined so that this represents a distorted plane wave, ei[k.r++(kr-k.r)l, propagating in the Q direction, together with an outgoing spherical wave, representing the scattering. In terms of the ordinary spherical harmonics, the plane wave may be expanded as (Q representing the direction of r) eik.r = 4~ 1 ilj,(kr) 2.m

Y$Ql’)

Y,,(Q).

(3.25)

Using the asymptotic form of the spherical Bessel function, j,(kr)

- (I/kr)

(3.26)

sin(kr - @r/2)),

together with the completeness relation for the spherical harmonics, eik.r

-

g

S(Q

-

!2’)

-

g

S(Q

+

we see that

Q).

(3.27)

Since the scattered wave is purely outgoing, the incoming part of (3.24) must match the incoming part of the modified plane wave. We thus determine, from the completeness relation, (B.36), the expansion coefficient4 ARiE = e- iwm’+?i)~vd(~Q’)*

e-iw2)L-8~~,

If we insert (3.28) into (3.24), and use the addition theorem (B.38), we identify the scattering amplitude, f(o), as the coefficient of the outgoing, distorted spherical wave :

Yout- -l ei&r-sin r

2kdeim’$~(Q,

(3.29)

where (for 0 # 0) 2ikf(f?) =

g

(2j + 1) Ui!m$7r -

e> e-i(rrL-zsL).

j=lm’l The scattering angle, 0, is the angle between k and r, cos 0 = cos 8 cos 0’ + sin 0 sin 8’ cos(+ - c$‘), 4 The replacement of #’ + T by #’ - ?I makes no difference in the linal result, (3.33).

(3.31)

DYON-DYON

465

SCATTERING

while symmetric case,

4’ = 4 + T,

unsymmetric

= (b + (4 - 47,

case,

(3.32)

where 6 is determined by5 1 - sin((8 + 0’)/2) sin((+ - #‘)/2) tan Z + = sin((8 - Q/2) cos((+ - 4’)/2) *

(3.33)

Note that apart from an unobservable phase, our result refers only to the scattering angle, 8. Our calculation has the virtue that, besides the simple generalization to include the Coulomb phase shift, the direction of the incident wave has been chosen arbitrarily relative to the singularity line of the vector potential. By using the general properties of angular momentum, we have been able to demonstrate explicitly that the scattering cross section is independent of the choice of the direction of the singularity line. However, we have been unable to find an analytic means of summing the partial wave expansion. Therefore we have summed the expansion numerically, the results being presented in Section V. Before considering these, let us turn to some approximations and limiting cases. A. Small Angle Scattering6 When 0 < 1, the main contributions to (3.30) come from large j (j> 1). If we further restrict ourselves to values of m’ such that j > j m’ 1,then we may expand L (given by (3.17b)) in a power series in ml/j. If only the leading term is retained, L ej, the partial reduces to

(3.34)

wave expansion (3.30), for charged particle-monopole 2&$((e) N

f (-1)’ j=lrn’l

(2j + 1) U$m$r

scattering,

- ~9).

(3.35)

This can be easily summed by making use of the generating function, (B.26),

2 i=O

,2ix’j’@)

=

1

1 - 2t cos(fi/2) + t2 ’

6The signs of the numerator and denominator imply in which quadrant the angle lies. 6 Subsequently, we will denote the scattering angle by 0.

(3.36)

466

SCHWINGER

ET AL.

where

in which

cos$ = cos; cosw.

(3.38)

Since the two lower indices of the U functions are identical, we can, without loss of generality, set 4 = 0, and take the Fourier transform of (3.36) to obtain a generating function for U$!&vr - 6):

[Here and in the following, we will take m’ to be integral.] Multiplying both sides of (3.39) by t, differentiating the resulting equation with respect to t, and then letting t approach i, we find that the sum, (3.35), is j=g,, (-1)‘(2j

+ 1) U$,(r

- 0) = (-l)“‘(]

m’ I/sin2 (d/2)).

(3.40)

The resulting cross section is ’

do

2

dsz ‘v t-1Fk

1 sin4 (e/2) ’

(3.41)

which coincides with the classical result for small angles [cf. (2.48) in the monopole case]. It is interesting to point out that if we keep the first three terms in the expansion of e--(pL, &nL

N

(-l)i

[l

+

in

&

-

ml4 2 (2j + l)2 ’

(3.42)

(-l)”

(3.43)

Tf

1

and employ the summations f (-l)j j=lm’l

U;!m,(n

-

0) =

U~!,+r

- e) = q-$,

2 sin(8/2) ’

and f (-l)j j=l?n’l

&

eg

1,

(3.44)

DYON-DYON

467

SCATTERING

only the phase of the scattering amplitude

is altered:

2ik.ce) ~ C--l)* I m’ I eim~m~~sink3~2~ , sin2 (d/2) yielding the same cross section as (3.41). [Note the evaluation of (3.40) and (3.43) are exact, while (3.44) is true for 19< 1.1 Our numerical results indicate that this small angle approximation is good up to about 130”, which is near the position of the first classical rainbow. B. Semiclassical Limit7 We may obtain the semiclassical limit by letting all quantum large: j> Im’I>l.

numbers become (3.46)

For convenience, we take m’ to be a positive integer, since the result is independent of the sign of m’ [see (3.30)]. When 8 is not near 0 or V, the asymptotic behavior of U,$$ may be obtained by the stationary phase method. The dominant contribution to

(3.47) with (0 < fl < n)

1

e

*

cos - /3 = sin 5 cos z, 2

(3.48)

is obtained when y$ [(i + $) B - m’#] = 0, that is*

(-=4d* 10

(3.49)

m’ j + (1,2) = sin 5.

]We will see that x corresponds to the classical angle (2.11).] On the other hand, from (3.48), we obtain sin -P2 sin X2 = sin - 0. sm -* 2 2’ ’ More extensive semiclassical approximations,

but in a somewhat different spirit, have been

considered bp Ford and Wheeler [13]. * The subscript 0 stands for evaluation of the quantity at the stationary point.

468

SCHWINGER ET AL.

whence cos -

8 = cos x sin 0B 2 2 2’

(3.52)

The second derivative of /3 at the stationary point is (3.53)

( - d$2 d2P 1,, zzz I2 cot 02 (-OS2 x2 ’

so the asymptotic form of Uz& u$&i-

-

e) ‘v

[7T (j

is9 +

;,

cos

;]y2

[COS2 -g

-

cog

FJ’4

x cos((j + ;, PO- m’~, - +j.

(3.54)

We now insert the above result into the partial wave expansion, (3.30), for the case that Q = 0. In the largej limit, the partial wave summation may be replaced by an integral, which in turn may be evaluated by a stationary phase approximation. The stationary requirement applied to the exponent (for convenience, we henceforth delete the subscript 0) E*(j) = -r[(j

+ $)2 - m’2]1/24 [(j + a)/3 - ml* + 2rN*j],

[where the 2ri periodicity integers)], means

(3.55)

of the exponential function has been recognized (N+ are d&(j)/dj

= 0,

(3.56)

which in turn implies & n/cos(x/2) = /!I + 27~N*.

(3.57)

Notice that because /3 must lie between 0 and rr, only one of the two exponents, E+ or E- , can contribute for any given value of x. The relation between x and 8,

coincides with the classical one, (2.17). The evaluation of thejintegral upon d2E, zzz ~ dj2 i 8 This is a generalization /lo = 57- 8.

2 cos(x/2) sin(8/2)(dO/dx) m’ tan2 4 (co9 (x/2) - co? @/2))1/2’

now depends (3.59)

of the familiar result for Legendre functions when m’ = 0, x = 0,

DYON-DYON

469

SCATTERING

so that we find Zikf(0) - C i(-1)”

m'

x

(3.60)

[

where Y = m’ 7r tan X & 2 sin-l I cos((+~/cos(x/wI [ ( 2 sin( e/2)

)I

+ nN& T $ (1 + sgn Q). The reader is cautioned that this evaluation breaks down when the second derivative, (3.59), vanishes, in which case the third derivative term must be retained. Since x is a multivalued function of 6’ (see Fig. 1) the above scattering amplitude is summed over all values of x corresponding to a given 8. Because of the interference between the branches, the cross section obtained here is not identical with the classical one, (2.18). For the dyon situation, the exponent is replaced by J&‘(j)

= -Q(j) + 2% .

(3.62)

Therefore, we find

’

*@+27rN~)=~&L-2263~

cos(x/2)

or

’

(3.63) (3.64)

where if

rj>O (3.65)

if

77 < 0.

Equations (3.64) and (3.65) correspond to the classical formulas (2.37) and (2.38). The approximate scattering amplitude has the same structure as (3.60) except that now the phase is

y’ = m’

4 [ -i7 ‘Ot 5 + sin(x/2) cos(x/2)

f 2 sin-l

( sin(O/2) l lcosico~~~2))l)

+ TN+ F f

(1 + sgn $),

+ 2s4 (3.66)

470

SCHWINGER ET AL.

where L = m' cot(x/;?).

(3.67)

Finally we remark that at 0 = n, U$‘, ((0) = 1, in which case the stationary phase requirement applied to the exponent E(j) = -7r[(j

+ $))" - m'2]1/2 + 27rNj

(3.68)

means

+7(i)= -

co&)

+ 2rrN=

0,

or 1 cos 5 = 7jp )

N = 1, 2, 3,...,

(3.69)

which coincides with the formula for the classical glory angles, (2.22a). The scattering amplitude is thus reduced to a sum over the different N values: 2ikf(r)

- 2m' Cm (2m’Y’2 2N exp /iv [m'(4N2 - 1)1/z - N + :]I. Nzl (4N2 - 1)5/4

(3.70)

Surprisingly, the resulting cross section agrees with the exact calculation displayed in Fig. 4 to within 2 % for m' 3 2.

FIG. 4. Electron-monopole loo.

backward scattering cross section, g = g(n), for m’ from 1 to

471

DYON-DYON SCATTERING IV. DIPOLE MOMENT MODIFICATION

Realistically, a charged particle will possess a magnetic dipole moment,lO so there will be an additional contribution to the Hamiltonian, which, in this case, is not small, but is of the same order of magnitude as the other magnetic couplings. For concreteness we consider the case of a spin $ particle, where the additional term is -( er2/2pc) yo H

(4.1)

H = g(r/r3)

(4.2)

with and y being the gyromagnetic ratio (near unity for the electron). The solution to this modified problem can be found in a very straightforward manner. For simplicity we consider charged particle-monopole scattering as the generalization to the dyon-dyon case involves only the inclusion of Coulomb phase shifts. In the symmetrical situation the Hamiltonian becomeP [see (3.8) and (3. WI -~ 2psff ii2

a2 I 2 a --(-- 1 = ar2 ’ r 2r r3

$2 A2

ml2 f m’yo . +

(4.3)

1

where

f”- 732

-

&

$

(sin 0 $)

+

m2 + m’2s~2~’

(4.4)

‘OS ’

is the angular operator given in (3.10). This last operator has eigenvaluesj(j + I), j 3 I m I, I m’ 1, and eigenfunctions g$ [see (B.34)]. The total angular momentum is not f but is J = 3 + *cr.

(4.5)

The eigenfunctions of J, belonging to the eigenvalues J = j F $, are z;y1/2

=

[ ‘II

j + U/2) f M v2 v54-l/2 I +> 2jf 1 j + (l/2) T M 112gmfl

1

2j+l

1

i.A4+112

I ->,

where 1 -&) are the spin states, and M is the total magnetic quantum I0 Similarly, one might also imagine the dyon would interact with the electric field of the charged terized by a mass of at least hadronic magnitude, nucleon scattering. I1 The eigenvalue problem for this Hamiltonian

number.

as bearing an electric dipole moment which particle. This latter coupling would be characand therefore might be important for dyonhas been considered by Malkus [18].

472

SCHWINGER

ET AL.

These are not eigenstates of the angular part of the Hamiltonian; diagonalize the matrix of j2/A2 - mf2 + m’y G * r/r: (

J2 - k - ml2 + ym’v m’y(l - v2)lj2

in fact, we must

m’y(1 - v2)lj2 J2 $ 2J + 2 - ml2 - ym’v 1’

(4.7)

where v = m’/(J + (l/2)).

(4.8)

The eigenvalues of this matrix are

L = (J + (1/2>12- m” f (J + (VW G L’f’(L’f’ + I),

(4.9)

where x = [l + v2(y2 - 2r)]‘l”,

(4.10)

or explicitly

L(*) + ; = (J + ;)I[6

41 X - v(1 - y)][&

f x + v(1 - r)]11’2. (4.11)

[The lowest state, for which J = 1m’ 1 - $, can occur in only one way, for which X = 1m’ j (1 - 7) = L,(L, + l),

(4.12)

L, + + = [) + 1m’ 1(1 - r)]‘l”.

(4.1311

where The corresponding eigenvectors are (4.14) where N-2 = (2/(1 - VS))X(1 - yv2 + X).

(4.15)

[For the lowest state, the eigenvector is &il--112.M

=

4kL2,M

*

(4.16)1

The wavefunction thus has the form

(4.17)

DYON-DYON

473

SCATTERING

(where we have displayed the lowest state contribution, corresponding to .I = ] m' / - l/2, explicitly). We will determine the constants AJM for the case that the incoming particle propagates along the +z-axis,12 with spin state I i), 5 = &. It is convenient to introduce the notations (A4 = m' + 95) &z) =

&pt)ei(l/2hL(*)

(4.18)

and = Ab+li2e

wk12

i(112h4.

,

then the boundary condition, that the coefficient incoming spherical wave vanishes, implies

of CY&j2,nlPSCj -5)

1 - yd+1 + [l + d+1(Y2- w2

[J + ; - blj1’2[YY,+& +

(1 -

in the

&G.;

1

$+Jl/Z

_ 5 [J + ; + 5m’]“2 [ 1 - YVJ2+ [l + VJ2(Y2 - WP2 &I _ y”J”q

= (),

(1 - YJ2)l/2 (4.19) where the subscript on v refers to the corresponding J value [cf. (4.8)]. The incoming state, Yin, = eik.r 1 <), (4.20) may be reexpressed, using (3.25), in terms of the c?Y~$by means of the completeness theorem, (B.36), so that the boundary condition for the ] 5) state is

1 - v:+1 + [l + 4+1(Y2- WI”’ (1 - v;+l)1/2

+ 5 (J + Z!j- Imp)1'2 [

l

-

YvJ2

+

r1 (1

+ -

vJ2(?

-

2d11’2 &’

,J2)lj2

1 _ yvJoL$-) 1

= 2(5 + I).

(4.21)

The solutions of (4.19) and (4.21) are (A) = 015

N2

(

J + UPI I” ) [1 + YV+ Xl, lfv

for

5=+,

( J 1+ ~ W2) v )Ip [l ‘F yv + X],

for

5=

(4.22)

and (*) = fN2 a?J

lz It would be preferable, although arbitrary, as in Section III.

-.

more cumbersome, to leave the propagation

(4.23) direction

474

SCHWINGER

ET AL.

[The corresponding solution for the lowest state is OI~,+~/~ = ([I m’ I - 5m’]1/2.

(4.Wl

We now combine all these results and obtain the partial wave expansion for the scattering amplitude for 0 # 0 (j = J + 4):

1 -

i$

+ -y-+ jw

1 +

(1 f

X

5rv

) uTzn+

- @] I 0

- 1 -
[(j

- (j + 1 + ~d)li2

i (1 -

&yz

(1 i

[m’)1’2 U$,l,+Jr

-

U2!,,~+,<~

-

e)

- e)] I - 01

e) 1<)

+ e-“““o{(l m’ j - cm’) UATJ,(77- (I m’ 1 -

l “,““)

U$$+r(7r

e) / -5)).

(4.25)

We note two slight checks of this somewhat elaborate structure: when m’ = 0 the scattering amplitude vanishes (for 0 # 0); and when y = 0 the previous result [(3.30) with sL = 0] is recovered. An interesting special situation is backward scattering, 0 = Z-; then the spin flip amplitude vanishes, while the spin nonflip amplitude is (for m’ > 0) 2ikf(e

z

n)

=

C

f

j@VL’*’

(1

F

‘(’

i j=jm’J+l

X

‘)

).

(4.26)

The case of m’ < 0 may be obtained from (4.26) by the substitutions m’ -+ -m’ and y --+ 2 - y, together with the additional contribution from the lowest J term: 2 1m’ 1eeinLo.

(4.27)

For m’ > 0, when y = 1 [which is approximately the situation pertaining to the electron], L(-) = L(+) - 1, so that it is apparent that, formally, the backward amplitude vanishes. Detailed numerical computations (see Section V) bear this out. In order that L(*) be real for all values of j, there is a restriction on the range of y values: .-I-

3 -T
3

(4.28)

DYON-DYON

475

SCATTERING

which is to be supplemented by 1 YC1

+ 41m’j

(4.29)

for m’ < 0. When these inequalities fail, the energy remains positive but an indefinite metric appears. This situation needs further investigation. To conclude this Section, we present the small angle limit of the scattering cross section, obtained in the same manner as in Section IIIA. The only new feature is the use of the raising and lowering operators to relate the non-diagonal with the diagonal U functions [cf. (B.4)]: d

( T;,+m’

1 - cos e U$m$e) = [(j f m’)(j + 1 7 mr)]1’2 U$,m~T1(B). sin e 1

(4.30)

The results, for the spin-flip and spin nonflip cross sections, are (for 8 < 1) (4.31) (4.32)

Note that the total cross section in this limit remains unchanged from the y = 0 result, (3.41).

V. NUMERICAL

METHODS

AND RESULTS

As mentioned above, since we have been unable to proceed beyond partial wave expansions for the scattering amplitudes describing the interaction of electrically and magnetically charged particles, we have found it necessary to employ numerical summations. Here we will outline the techniques and approximations used, and present the results, for various values of the parameters m’, 7, and y, even though the most realistic case is m’ - 1, 7 < 1, and y N 1 (since these are the parameters corresponding to the scattering of an electron with a particle bearing minimal magnetic charge). Since the differential cross section is a smooth function of m’, we will assume in the following that m’ is an integer (which is the situation in the symmetrical case).

476

SCHWINGER ET AL.

Because the series in question converge at best exceedingly slowly, the Cesaro summation procedure13 was adopted. That is, if 12

sn

=

c k=l

(5.1)

ak

is the nth partial sum formed from the sequence {ak}, the Nth partial Cesaro sum is just the average of the first N partial sums:

Whenever s, converges, I+,! has the same limit. The Cesaro summation tends to average out large fluctuations in the original series. This advantage is gained at a price: Typically where the original series tends to oscillate in such a way as to bound the limit both above and below, the Cesaro sum tends to converge to the limit monotonically (rather slowly). A. Charged Particle-Monopole Scattering When 7 = 0, the partial wave expansion (3.30) becomes 2&f(O) =

g

j=pn’l

(2j + 1) e-iSLU$m,(9r -

@,

(5.3)

where L is given by (3.17b). This series is less divergent if the known sum (3.40), 2ikf’(@ =

2

j=lnz’l

(2j + 1)(-l)’

Ui!&7r

- 0) = (--I)“’

sidem(;l)2),

(5.4)

is subtracted off term by term. The resulting sum is easily evaluated in the backward direction, since there u$m,(o)

= 1.

(5.5)

We have calculated the backward scattering cross section for all integral values of m’ from 1 to 100, including sticient terms so that the error is less than 1%. (For example, for m’ = 100, the first 3000 terms were summed.) Here, as in the following, the differential cross section dcr/dQ = If(O)]”

(5.6)

I8 When the Cesaro summation procedure is applied to the partial wave expansion of the Coulomb scattering amplitude, the exact result is recovered. This is evidence in favor of the validity of this approach.

SCATTERING

477

de) = (k2/m’2)l ml”.

(5.7)

DYON-DYON

is expressed in terms of a function g(8):

(This notation was introduced by Ford and Wheeler [13].) These results are shown in Fig. 4. Even though a general growth of g(r) with increasing m' might have been anticipated, the actual dependence is rather elaborate. To obtain the complete angular dependence, we use the connection of the U functions with the Jacobi polynomials (B.13). The latter are computed using the recusion relation, (B.16-18). The results of computer calculations14 of the cross section in the backward hemisphere are presented in Fig. 5, for m' = 1, 2, 10, 20,

FIG.

5. Electron-monopole

scattering crosssection for m' = 1, 2, 10, 20, and 28.

and 28. [The m' = 50 case is contained in Fig. 9.1 For m' = 1 and 10 our results agree with those of Ford and Wheeler-l5 [13]. The increasingly elaborate structure with increasing m' is characterized by a strong peak at 180” (the incipient “glory”), large peaks at smaller angles (incipient “rainbows”), and an intricate interference pattern that is almost totally destructive at certain angles. As a general rule, the dominant peak occurs at 0 = 180”. However, for m' = 28, for which g(n) has a local minimum [see Fig. 41, the dominant peak occurs at f3 = 165”. I4 The accuracy of the calculation for g(0) is about 5 % or better. 15eOur notation is related to that of Ref. [13] by m' = 2n. l

478

SCHWINGER ET AL.

B. Dyon-Dyon Scattering When 17 # 0, we are confronted with an apparent difficulty: how to calculate efficiently the Coulomb phase shifts for non-integral L [(3.17b) and (3.23)] aL = arg r(L + 1 + i$.

(5.8)

However, because L + 1 is never very small, L + 1 > 1.618...,

(5.9)

the asymptotic evaluation [20] arg r(L + 1 + iT> - 7)(- 1 + In r) + (L + i) 4 sin 3$ -___sin 4 ___-___ 12r + 360r3

sin 54 I 260r5 ’

(5.10)

where r2 = (L + 1)” + y2,

(5.11)

4 = tan-r q/(L + l), is an exceedingly good approximation everywhere. The results of the computer calculation14 for m’ = 1, 10, 20, and 50 are given in Figs. 6-9, for various values of 7. For m’ = 10, the dependence of g(r) on 7 is shown in Fig. 10. Significant

T , =0 and7.3xlO ?)=I ?)=I0

90 FIG.

6. Dyon-dyon

120

-3

--_---__ -

kcalex30)

150

scattering cross section for

I. 180 8

m’

=

1.

DYON-DYON

m’;lO

479

SCATTERING

-3 7 = 0 a7.3x IO

SOL

30-

\ .\

/ ‘L/

/\

\ \

90

FIG.

7.

Dyon-dyon

scattering cross section for

Tj=o ?=I 7=-I 1’10 ll;.,o

FIG.

8. Dyon-dyon

e

150

I20

nz’

=

10.

--------‘-‘-.----

scattering cross section for m’ = 20.

480

SCHWINGER ET AL.

I ’ n=O

___

7=‘0 ‘-‘-‘II ;.,o

6

_ ::..Y..yg (1:s Il.

7=50-------

FIG. 9. Dyon-dyon

7i 1

..” 1:s 2 .El!

me=50 8

scattering cross section for m’ = 50.

scattering cross section on 7 for m' = 10.

FIG. 10. Dependence of backward

quantitative alteration of the differential cross section is produced for 1 r] 1 N 1, which typically corresponds to a relative speed on the order of that of atomic electrons (2 not large), D-

(2/137)c.

There is a significant dependence on the sign of r], with negative values case) having considerably more structure than positive ones. When

9 R Im’l

(5.12) (attractive

(5.13)

DYON-DYON

481

SCATTERING

the oscillatory structure of the scattering amplitude is washed out. However, this does not hold true when 77is large and negative, for then the oscillations seem to be enhanced. C. Efects of the Dipole Moment Generally, except for very small velocities, the effects due to the magnetic moment of the charged particle are more important than the Coulomb phase shifts. For electron-monopole scattering, the expression for the spin-flip (“F”) and nonspinflip (“NP) amplitudes are given in (4.25), for arbitrary gyromagnetic ratio y. The simplified expression for the scattering amplitude in the backward direction (0 = 7r) is given in (4.26). The dependence of g(0 = V) on y for m’ = +l, f2, and 50 are given in Figs. 11-13. [Recall the restriction on y given by (4.28) and (4.29).] Using the recurrence formulas for the Jacobi polynomials (B. 16-18), we can easily compute, for arbitrary angles, the spin-flip and nonspin-flip cross sections, given in terms Of gF and g,, defined analogously to (5.7). These results, for m’ = -1, +l, 10, and 20, and a range of values of y, are given in Figs. 14-17. When m’ > 0, the most conspicuous features are the following: For y = 1, which is the most

FIG.

11. Dependence

of g = g(?r) on gyromagnetic

ratio y for M’ = 1,2.

482

SCHWINGER

ET AL.

12. Dependence of g = g(n) on gyromagnetic

FIG.

ratio y for m’ = -1,

-2.

sl 48-

36 -

a-

:

12If;

01

FIG.

-I

1

I 0

\,i

,

13. Dependence of g = g(w) on gyromagnetic

’ *-Y 3

ratio y for m’ = 50.

DYON-DYON

483

SCATTERING

important case physically, gF = gNF at 8 = n/2, and otherwise gNF is quite insignificant in the backward hemisphere, while g, can have large oscillations. For y = - 1, gNF exhibits the most structure, and tends to be larger than the y = 0 case at 8 = 7~.When m’. < 0, the situation is roughly reversed: the y = 1 case has the most structure, while the y = -1 case is suppressed.

FIG. 14. Spin-tip

(F) and nonspin-flip

(NF) cross sections for electron-monopole

scattering

including gyromagnetic ratio y, for m’ = 1.

FIG. 15. Spin-flip

and nonspin-flip

cross sections for m’ = - 1, 7 = & 1.

484

SCHWINGER ET AL.

FIG. 16.

Spin-flip

and nonspin-flip

cross sections for m’ = 10, various 7.

g(e), 3.0-

/- “\I 5 ‘2 zz. ,= m’=ZO

y=o r=-’ ---Y= ’ -‘-‘-’

2.0-

FIG.

17. Spin-flip

and nonspin-flip

cross section for m’ = 20, y = fl.

DYON-DYON

SCATTERING

485

VI. CONCLUSIONS The existence of magnetic charge is an intriguing theoretical possibility, both from the greater symmetry it imparts to Maxwell’s equations, and, more importantly, from the explanation it provides for the quantization of electric charge. From the notion of magnetic charge, the idea of dyons is a natural extension, and forms the basis for a model of hadronic matter, in which the constituents carry fractional electric and magnetic charges [6, 71. There are reasons for believing that, if the particle seen by Price et al. [3] indeed represents a magnetic charge g = 137e (in units of the electron charge e) it is a dyon and not a monopole. Suppose that there are entities possessing smaller charges than e, specifically, based on simple SU, ideas, e, = (1/3)e.

(6.1)

Independently of whether these objects carry magnetic charge, the minimum value of pure magnetic charge is, supposing that e, is the smallest value of electric charge,

g=3

; x 137e, I!

(6.2)

for the half-integer or integer quantization condition, respectively. Either choice requires a larger minimum value than that reported [3]. If the Price event is a monopole, any hope of building up hadronic matter from fractionally charged objects is impossible. In any case, with the reported value, a pure electric charge of e, is impossible. The only possibility is that the observed event is a dyon and that any object with fractional electric charge must also be a dyon. The charge quantization condition applied to an electron and a dyon carrying the minimum value of magnetic charge, g, , implies

go= I!*1 X 137e

(6.3)

either of which is consistent with the Price event. Unlike simpleminded constituent models of hadronic matter, the dyon model [6, 71 is not excluded by the magnetic charge interpretation of the Price event. So the question is raised: How can a dyon be distinguished from a monopole ? One would hope that our results could be applied to the interpretation of &rays produced by a “magnetic charge” event, and indeed it is clear that the cross sections exhibit a great deal of structure. But it must be admitted that most of the elaborate detail we have discussed is not much in evidence in the most accessible situation, I m’ I -1, I 7 / << 1, y N 1. However, there are significant quantitative alterations in the cross section for small changes in these parameters, as may be seen by con-

486

SCHWINGER ET AL.

suiting Figs. 5, 6, 11, 12, and 15. Thus there is hope that our results might prove useful as the search for magnetic charge proceeds. To conclude, we offer a few remarks concerning the theoretical status of the magnetic charge problem. In the foregoing we have developed and discussed many of the features of nonrelativistic scattering of magnetically charged particles. In most part, the physics is well understood, and it is a matter of the application of known techniques to somewhat novel circumstances. But even here, there are some aspects which are unclear, particularly the behavior for large gyromagnetic ratio. Presumably, a fully relativistic treatment would indicate the proper solution to this problem, as well as provide more accurate scattering amplitudes. However the construction of a relativistic theory of magnetically charged particles remains the outstanding obstacle preventing further progress in this field although a beginning has been made for distant collisions (eikonal approximation) [I 1.

APPENDIX

A: POTENTIALS

The generalized Maxwell equations that include magnetic charges and currents allow two types of solutions: those that do not respect the symmetry of Maxwell’s equations under rotations in the electric-magnetic plane (called unsymmetric solutions) and those that do (called symmetric solutions). Since they yield different charge quantization conditions [l, 5, 151 (for example, the monopole versions of (3.15a) and (3.15b)), we will here demonstrate the general procedure for expressing the solutions in terms of potentials. Maxwell’s equations in vacuum are (in unrationalized CGS units) 1 aE(r, t) V x H(r, t) = ; at V * E(r, t) = 4q&, -V

i $ J,(r, 0,

t),

1 aH(r, t) x E(r, t) = ;at

(A.11 (A.9

+ $ J&, 0,

V * H(r, t) = 4np,(r, t),

(A.3) (A.4)

which imply the equations for charge conservation:

4 p,(r, t) + V - Jdr, 0 = 0, $ pdr, t> + V - J,,kr, 6 = 0.

(A-5)

DYON-DYON

487

SCATTERING

The general solution of (A.4) is H(r, t) = 1 (dr’) f(r - r’) pm(r’, t) + V x A,@, t)

where f(r - r’) is a translationally

67)

invariant vector subject to the constraint

V . f(r - r’) = 47&r - r’),

L4.8)

and, when pm = 0, A, is the conventionally defined vector potential. (A.7) into (A.3) and making use of (A.6), we obtain E(r, 2) = -V4&,

t) - k g A,(r, t) + f 1 (dr’) f(r - r’)

x

Substituting

Jm(r’, t),

where, when J, = 0, $e is the conventionally defined scalar potential. that, in terms of these potentials, we have not only the gauge freedom

(A.9)

We note

A, -+ A, -k VA, (A.lO) 7% -

4e - ; ;

4

but also an extra freedom in choosing f(r - r’), which, apart from the constraint (A.8), is quite arbitrary. To exploit the advantage of a specific choice of gauge, we may invert (A.7) and (A.9), and express the potentials in terms of the fields: 4~ A,@, t) = 4~ Vh(r, t) 47~ 4&r, t) = - $

1 (dr’) f(r - r’) x H(r’, t),

g X(r, t) -

1 (dr’) f(r - r’) . E(r’, t)

(A.1 1) (A.12)

where X(r, t) = 1 (dr’) f(r - r’) . A,(r’,

A convenient choice of gauge is the so called3gauge X(r, t) = 1 (dr’) f(r - r’) * A,(r’,

(A.13)

t).

in which (A.14)

t) = 0.

In this gauge, (A. 11) and (A. 12) reduce to

595/101/Z-10

A,@, t) = -(l/471.)

s (dr’) f(r - r’) x H(r’,

&(r, t) = -(l/47)

1 (dr’) f(r - r’) +E(r’, t),

t),

(A.15) (A.16)

488

SCHWINGER

ET

AL.

which have the particularly simple feature that, knowing H and E, and for a given model off, we may obtain explicit expressions for the potentials. To exploit the symmetry of Maxwell’s equations, we may use (A.l) and (A.2) to obtain an analogous set of equations: E(r, t) = J (dr’) *f(r - r’) p@(r), t) Wr, t) = -V&h,

t) - k f

A,&

V x A&r, t),

t) - f /(dr’)

(A.17)

*f(r - r’) x Je(r’, t); (A.18)

and inversely +&,

t) = -(l/437)

s (dr’) *f(r - r’) * H(r’,

t),

A,@, t) = (1/4n-) / (dr’) *f(r - r’) x E(r’, t),

(A.19) (A.20)

in the *f-gauge:

s

(dr’) *f(r - r’) * A,(r’,

t) = 0.

(A.21)

As before, *f satisfies the constraint V . *f(r - r’) = 4.7~6 (r - r’);

(A.22)

but, a priori, is not necessarily related to f. Further considerations, which are based an the fact that A,, r,& , and A,, #Jo are not independent potentials, show that the consistency of the theory demands *f(r)

= -f(-r).

(A.23)

(The proof of this assertion is most transparent in the relativistic formulation and is given in detail in Refs. [l, 7, 15.1) We now observe that Maxwell’s equations, (A.l)-(A.4), are invariant under the above mentioned rotation in the two-dimensional electric-magnetic plane [for example, E -+ E cos B + H sin 8, H -+ -E sin 0 + H cos 0, and similarly for the charges, currents, and potentials.] The solutions [(A.7), (A.9), (A.17), and (A.18)] may or may not respect this symmetry depending on the choice of f and *f. If, in addition to (A.23), we require *f(r)

= f(r)

(A.24)

DYON-DYON

SCATTERING

489

then the solutions respect this symmetry; otherwise the symmetry is broken. The condition for the symmetric solutions is therefore f(r) = -f(-r),

(A.25)

that is, the S-function is odd. For unsymmetric solutions, there is no additional restriction on f(r). The condition (A.25) is the origin of the integer charge quantization condition [l, 7, 151, as opposed to Dirac’s half-integer one [5]. In order to obtain an explicit expression for the vector potential from (A.15), we need to know the f-function. Since (A.8) describes the flow of a “current” f(r) from a source of strength 47r located at r = 0, its basic solution is the line integral f(r) = 477 jc dx 6(r - x), (A.26) where C is any contour starting from the origin and extending to infinity. Equation (A.26) gives an unsymmetric solution. To obtain a symmetric solution, we use the antisymmetrized form of (A.26) to obtain f(r) = 47~ Ic dx &3(r - x) - S(r + x)].

(A.27)

We can now compute the explicit form of A, [from (A.15)] for given H. In the presence of a static magnetic charge of strength g, located at the origin, the magnetic field strength is H(r) = g(r/r3). (A.28) Furthermore, since we have the freedom of choosing any particular contour C, we make the convenient choice that C is a straight line along a fixed direction, characterized by the unit vector n. We have [from (A.15) and (A.26)], for the unsymmetric case, A,(r) = -g 1 (dr’) Ja dl S(r - r’ - nl) T 0

= -0 x r) jam(r -“.[ (3 zzz-- g

nxr

r r - (n * r) ’

(A.29)

For the symmetric case, we obtain [from (A.15) and (A.27)] (A.30)

490

SCHWINGER

ET AL.

In particular, if we choose II to be along the $2 direction, and use spherical coordinates (r, 6, d), then (A.29) and (A.30) become A,(r) = - 5 cot l 4,

unsymmetric

= - 4 cot e$,

B:

PROPERTIES

(A.31)

symmetric case,

which are the forms used in the Hamiltonians

APPENDIX

case,

(A.32)

(3.3) and (3.4) respectively.

OF

ROTATION

MATRICES

In our investigation of the quantum theory of the scattering of electric and magnetic charges, we encountered an angular differential operator, that of (3.10), for which eigenvalues and eigenfunctions were sought. This same differential operator also appearsl” in the theory of angular momentum. Here, we will simply state a number of useful facts, a detailed exposition being given in Ref. [17]. The fundamental object of interest here is the unitary operator, U, which represents a rotation of the coordinate system. In terms of standard Eulerian angles, c$, 0, 4, this operator is ,y = ei(6/r)J,e”(8/1)Jae’(m/n)r,

(B-1)

The angular momentum operator, J, can be represented by a differential operator, J, , which is defined [17] by its action on U, J,U = UJ.

(B-2)

The components of this differential operator are

~(J,,~iJ,,)=e*“m[*~+~(f~-

cog el

i-&C.

The operator corresponding to the square of the angular momentum

a

)I

(B.4)

is

-&J~=-[&~(sinB$)+-&(~-2cosB+++~)]. (B.5) I6 It also occurs

in the problem

of the symmetric

top (see Ref.

[21 I).

DYON-DYON

491

SCATTERING

The operator ZJ, acting on a state having certain properties relative to a fixed coordinate system, transforms the state into one which has the same properties relative to the rotated coordinate system:
/ jm> = (jm’ / U 1jm> = U$,(w) =e

/ ei(el”)J” I

im't~(jm'

~_.

hn’tiu~!,(e)

=e

eim~

j,n)

03.7)

eimd.

Likewise, taking matrix elements of J,2U = UP,

(B.8)

we have -

&i 1

(sin tii)

m2 - 2mm’ cos

-

ef

sin2 f3

ml2

I u:i!,(e)

=j(j

+ 1) UC' ! (0) . ’ (B.9) mm Equation (B.9) thus displays the eigenvalues and eigenfunctions of the angular differential operator occurring in (3.10). It is convenient to record here some relevant properties of these functions, the definition of whichI is contained in (B.7): u:!,(e)

=

(jm'

/ e’(e’n)Jz 1jm).

(B.lO)

It can be shown that [17] u:!,(e)

= (-I)~-“’

u($,(~

- 0) = (-ly-”

= (-1)“‘~m

,@A,- m(0) = (-l)m’-m

u~!-,(~

- 0)

u(i) mm,(e)

= u!f~-,,(e).

(B.ll)

Accordingly, we only need to calculate these functions for nonnegative values of m and m’. For this case, an explicit expression is

uJf!,(e) = (-l)j-m’

[

x (co, f,--’

(j + ml!

11’2 (sin ;)-“‘“’

(j - m)! (j + m’)! (j - m’)! ($)+-“’

tj+y1

- ty-,’

/t=eos’(e,2) .

(B.12)

I7 The function is commonly denoted by d,$~~(-O). Note also that our present definition of U is the Hermitian conjugate of that defined in Ref. [17].

492

SCHWINGER ET AL.

The relation of the U’s to the Jacobi polynomials

is (in the notation

of Ref. [20])

x py;m.m’+m) (x>, 3m

(B.13)

where x = cos 8,

(B.14)

m' 2 m > 0.

(B. 15)

and The easiest practical way to calculate the U’s is by means of a recurrence relation. This relation can be simply stated in terms of the P's: 4j(j - m' + 1)fj + m' + 1) Pj?>$nz'+m)(x) = [-4mm'(2j

+ 1) + 2j(2j + 1)(2j + 2) x] P~Y~m'nz'+m)(x) (B.16)

- 4(j + l)(j - m)(j + m) PjT>T;nz'+m)(x). Thus for given m and m', starting with the first two Jacobi polynomials, ph’-m,ni+m) (4 = 1, 0 p’“‘-“,“‘+“’ (x) = -m 1

(B.17) (B. 18)

+ (m' + 1) x,

all the rest may be calculated recursively. It is useful to consider integrations over the space of rotations, where the element of volume is dw = .! sin (I de -@ 3 (B.19) 2 4r 4rr ’ # and $ taken to range l8 from 0 to 4rr. The orthonormality then can be written as

relation for the U’s

(B.20) which is complemented

by the completeness statement,

jz, (2 + 1)~-L&J)*u$,(d) = acw_ ol) = 2 w l8 For

m’

- 0 sin e 47T‘(4 - +‘I 477‘(#

an integer, the range of # and 4 is covered twice.

(B.21)

DYON-DYON

493

SCATTERING

On the other hand, an arbitrary rotation can always be expressed as a rotation through an angle /I about a direction n: u(w) = .+‘Jlfi

,

(B.22)

so CI($,,~(W)= (jm’ 1eiRpJlfi 1jm). As an application matrix UzA, :

of this formula,

x@(w) =

(B.23)

we compute the trace (the character) of the

i u,$jn(w) = trtj) eiR".J'* m--j

(B.24)

i 1

= trti) eiaJSifi= eiRn” = SW + U/2)) B sin(l/2) p ’ VI=-; in which we have used the invariance of the trace under unitary transformations. From (B.24), we easily find the relation between /3 and the Eulerian angles *++ cos -yP = cos -e cos ___ 2 2

)

(B-25)

as well as the generating function for the x(j) m

1 I - 2t cos@/2) + t2

= C t2jX(j)(W).

(B.26)

I=0

Equation (B.26) is useful in performing some of the explicit summations in the text. Finally, we are interested in an addition theorem for the Vs. This can be derived from the composition property of successive transformationP u&J - w’) U(d)

= U(w),

(B.27)

or U(w) u-$0’)

= U(w - 0’).

(B.28)

Taking matrix elements of this last relation yields

c u:!,(w) u:!,,(w’)*= u,~!,+J- w’), m

which is the desired addition

(B.29)

theorem. In order to calculate the explicit form of

18The rotation w - w’ carries the coordinate system defining w’ into that defining w.

494

SCHWINGER

ET AL.

cu - o’ (= Y, 0, @) in terms of w (= #, 0, 4) and w’ (= I,L’, 13’, $‘), it suffices to consider the j = 4 case. For 0, we have the well known result: cos 0 = cos e cos 8’ + sin e sin 8’ COS(+- +‘),

(B.30)

and for Y and @ we have @ = -$b’ + $1

y=*+g

(B.31)

where $ and $’ are functions of 0, 8’, and 4 - +‘, that is5 tan$($

+ 4’) =

c0sw + o/2) sin((# - #)/2) - ey2) COS((+ - +‘)/2) 9

(B.32)

- sin((8 + ey2) sin((0 - 0’)/2)

(B.33)

c0g(e

and tan jj (J; - ~5’) =

sin((+ - 55’)/2) - #‘)/2) *

COS(($

For the considerations in the text, # is not a physical variable. We therefore define a set of generalized spherical harmonics appropriate to a given m' value,

&&))

= &P’d

(2j : 1)1/Z c&t

+).

(B.34)

Then, in terms of the CVs, we have (B.35)

I

(B.36) Ma

For the case m' = m", this last result, (B.37), reads

where [see (B.32)] B=$+&

(B.39)

DYON-DYON

495

SCATTERING

ACKNOWLEDGMENTS We would like to thank Dean Harold Ticho and the UCLA for making computer time available to ‘us.

Campus Computing

Network

REFERENCES 1. J. SCHWINGER, Pltys. Rev. D12 (1975), 3105-3111. 2. T. T. Wu AND C. N. YANG, Phys., Rev. D12 (1975), 3845-3857; Y. NAMBU, Phys. Rev. D10 (1974), 4262-4268; G. ‘T HOOFT, Nucl, Phys. B79 (1974), 276-284. 3. P. B. PRICE, E. K. SHIRK, W. Z. OSBORNE, AND L. S. PINSKY, Phys. Rev. Lett. 35 (1975), 487-490. 4. L. ALVAREZ, “Proceedings of the 1975 International Symposium on Lepton and Photon Interactions at High Energies,” Aug. 21-27, 1975, Stanford University, pp. 967-979, 1976. 5. P. A. M. DIRAC, Proc. Roy. Sot. (London) Al33 (1931), 60-72. 6. J. SCHWINGER, Science 165 (1969), 757-761. 7. J. SCHWINGER, “Particles, Sources, and Fields,” Vol. I, Addison-Wesley, Reading, Mass., 1970. 8. J. SCHWINGER, “Gauge Theories and Modern Field Theory” (R. Arnowitt and P. Nath, Eds.), pp. 337-367, MIT Press, Cambridge, Mass., 1976. 9. H. POINCARI?,Compt. Rendus 123 (1896), 530-533. 10. I. TAMM, Z. Phys. 71 (1931), 141-150. 11. P. BANDERET, Helv. Pbys. Acta 19 (1946), 503-522. 12. M. FIERZ, Helv. Phys. Acta 17 (1944), 27-34. 13. K. FORD AND J. A. WHEELER, Ann. Phys. (N.Y.) 7 (1959), 287-322. 14. I. R. LAPILXJS AND J. L. PIETENPOU, Amer. J. Phys. 28 (1960), 17-18; G. NADEAU, Amer. J. Phys. 28 (1960), 566. 15. J. SCHWINGER, Phys. Reu. 144 (1966), 1087-1093; Phys. Rev. 173 (1968). 1536-1544. 16. H. C. VAN DER Hursr, “Scattering of Light by Small Particles,” Wiley, New York, 1957; R. NEWTON, “Scattering Theory of Waves and Particles,” McGraw-Hill, New York, 1966. 17. J. SCHWINGER, “On Angular Momentum,” U.S. Atomic Energy Commission Report No. NYO-3071, 1952 (unpublished); reprinted in “Quantum Theory of Angular Momentum” (L. C. Biedenham and H. Van Dam, Eds.), pp. 229-279. Academic Press, New York, 1965. 18. W. V. R. MALKUS, Phys. Rev. 83 (1951), 899-905. 19. L. D. LANDAU AND E. M. LIFSHITZ, “Quantum Mechanics,” pp. -604. Addison-Wesley, Reading, Mass., 1965. 20. M. ABRAMOWITZ AND I. A. STEGUN, Eds., “Handbook of Mathematical Functions,” National Bureau of Standards, Washington, D.C., 1964. 21. F. REICHE, Z. Phys. 39 (1926), 444-464; S. FL~~GGE, “Practical Quantum Mechanics,” Sect. 46+ Springer-Verlag, New York, 1974.