Classical Monopole

Abstract We derive a particular monopole solution of the modified toda molecule equation with a simple analytical form valid for any Lie Algebra. Using the Laplace transform, we proceed to examine monople solutions of physical interest for the Lie Algebras An, Bn and Cn , deriving expressions for the fundamental component of a monopole solution from which the remaining components of the solution may be obtained. We arive at expressions for the fundamental components of monopole solutions using three different techniques: an integral method, application of a differential “raising operator”, and by application of superposition rules. Next, we investigate the solutions of physical interest for the Lie Algebra Dn, deriving solutions for the fundamental components of non-spinor solutions, and proposing ansatz that allows us a to obtain “candidates” for the spinor Dn solutions. Finally, we explore relationships between these fundamental components of the solutions of the modified toda molecule equation for the Lie algebras Bn, Cn, and Dn and exhibit expressions for the fundamental components in terms of modified spherical bessel functions.

Magnetic monopole solutions for classical Lie algebras Sean Fitzsimmons 7215 Jefferson Ave, Des Moines, Iowa 50322 We use the Laplace transform to investigate properties of a class of spherically symmetric self-dual magnetic monopole solutions of the grand unified field equations. Simple expressions for the monopole solutions for the Lie Algebras An, Bn,Cn, and Dn are obtained. Superposition rules for constructing high rank solutions from lower rank solutions are developed. Bn, Cn, and Dn solutions are expressed in terms of Modified Spherical Bessel functions.

I.

INTRODUCTION 1

Ganoulis, Goddard, and Olive (GGO) have shown that under a certain set of simplifying 2, 3, 4, 5 the problem of exhibiting finite magnetic monopole solutions of a grand unified assumptions model containing gauge fields and single Higgs field Φ in the adjoint representation reduces to the solution of the modified Toda molecule equation (MTME): l d 2θi = exp( K ijθ j ), ∑ dr 2 j =1

i = 1, 2,..., l.

(1.1) 6

with a particular set of boundary conditions. In this expression, K is the Cartan matrix of the Lie algebra G of the gauge group and l denotes the rank of G. One of the GGO assumptions is that the solutions are spherically symmetric; the value of Φ and the radial component of B along the z-axis characterize the radial behavior of the solution. The relationship between the physical fields along the z-axis and the components θi is given by

R  2α ⋅ H R  2α ⋅ H 1 l  1 l  eΦ(kˆ , r ) = − ∑ θi′ + i  i 2 i , eB r (kˆ , r ) = ∑ θ i′′− 2i  i 2 i r  αi r  αi 2 i =1  2 i =1  where

(1.2)

H i is an element of the Cartan subalgebra of G, normalized so that Tr (H i H j ) = δ ij ,

α i are the simple roots of G, and Ri are the components of the level vector in the dual basis (with 2α i basis vectors ) in the l -dimensional weight space of G. 2 αi 7

The MTME is a Hamiltonian system ; the total energy associated with this solution is given by E = T + V , where:

T=

l l 1 l 2 2 ′ ′ = − K θ θ V K ijθ j ). , exp( ∑ ∑ ∑ ij i j 2 2 i , j =1 α i 2 i =1 α i i =1

(1.3)

GGO have argued that if q is any vector in the weight space of G that lies in the positive Weyl chamber there exists a finite energy solution of the MTME of the form

e−θi (q, r ) =

∑

asi e

r q ⋅µis

(1.4)

µ is ∈∏ ( λi )

where the sum is over all the weights in the representation of G with highest weight

λi .

Page 1 of 12

GGO determined that a sufficient condition to obtain a finite energy monopole solution is that the magnetic field (1.2) vanishes at the origin;

θi′′ !

Ri for r near zero. Inserting this expression into r2

the left had side of the MTME they obtained the finite energy conditions:

dj dr j where the constants

0, j = 0,1,..., Ri − 1 e −θi =  , ci (G ), j = Ri r =0

(1.5)

ci (G) are given by ci (G) = Ri !∏ ( R j ) l

− K −1ij

.

j =1

II.

BOSE AND MCGLINN’S SOLUTION TECHNIQUE 8

Bose and McGlinn observed that if

d i represents the number of different values in {q ⋅ µ s } , then

the condition

d i = Ri + 1

(1.6)

means that the number of constants in (1.4) is equal to the number of constraints in (1.5), and solved for the constants

a ij , obtaining a ij = ci (G )

∏ q ⋅ (µ k≠ j

i j

− µ ik )

(1.7)

They noted that the fundamental representations of the Lie Algebras An, Bn, and Cn, G2 (i.e. the representations with highest weight λ1 for An, Bn, and Cn, and with highest weight λ 2 for G2) each satisfy (1.6). For these Lie algebras, they had obtained the fundamental component of an MTME solution for any q in the positive Weyl chamber. Monopole solutions of physical interest correspond to fundamental co-weights of G

9,10

q vectors that are parallel to the

. The fundamental co-weights

λ *i of G satisfy the relations

λ *i ⋅ α j = δ ij where α j are the simple roots of G, and are located on the boundary of the positive Weyl chamber, the region in weight space of G consisting of all all

q vectors such that q ⋅ α j > 0 for

α j . Bose and McGlinn argued that they could construct a solution for any λ *i on the boundary

of the positive Weyl chamber by constructing a parametrized path q(t) from the level vector (at t=1) to the co-weight

λ *i (at t=0), solving for the constants (1.7) along this path, and then

obtaining the fundamental component of the solution with q

= λ *i by taking the limit as t goes to

zero of the expression (1.4). Having described a technique to obtain the fundamental component of the MTME for several Lie algebras, they proceeded to develop a technique for obtaining the remaining components of the solution from the fundamental component. Taking two derivatives of the identity

θi = − ln(e −θ ) , i

Page 2 of 12

substituting the RHS of the MTME for

θi′′ , and making use of the fact that K ii = 2

for any Lie

Algebra, they obtained the relation

( )

W (e −θi ) = ∏ e j ≠i

−θ j

− Kij

i = 1,..., n

(1.8)

where

W (e

−θi

2

  ) ≡  e −θi ′  − e−θi e −θi ′′  

( )

( )

Beginning with the fundamental component proceeding to the opposite end of the Dynkin diagram, there will never be never more than one undetermined component on the right hand side of the expression (1.8), provided the Dynkin diagram of G does not branch. For An, Bn, and Cn, G2 all solution components may be determined from the fundamental component. Finally, Bose and McGlinn noted that for a q vector on the boundary of the positive Weyl chamber, the behavior of solutions for large r had the form:

e−θi (q, r ) ∼ r mi er q⋅µ s Where mi is a non-negative integer. For a component i with mi > 0 they noted: i

θi = − ln(e −θ ) = − ( q ⋅ λ i ) r − mi ln r ,

θ i′ = − ( q ⋅ λ i ) ,

i

θi′′ =

mi r2

Substituting these expressions into the MTME they obtained: l

θi = −mi exp(−2 ln r ) = exp(∑ K ij (− ( q ⋅ λ i ) r − mi ln r )) j =1

Equating powers of r on both sides they found:

∑ K ( q ⋅ λ ) = 0 , and ∑ K l

j =1

l

ij

j

j =1

ij

mj = 2 ,

or

2α i αi

2

⋅ q = 0 , and

In the dual basis of G, the non-vanishing components

2α i m ⋅ = 1. 2 2 αi

(1.9)

mi correspond to the dual components of

the level vector(s) of the simple Lie algebra(s) obtained by removing the all dots corresponding to each to α i that is not orthogonal to q from the Dynkin diagram of G. For q on in positive Weyl chamber or on the boundary of the positive Weyl chamber, the asymptotic physical fields (1.2) have the form:

eΦ =

q , 2

eB r =

Q , r2

(1.10)

with

Q≡

R (K) R (G) − 2 2

Page 3 of 12

(G )

(K)

where R is the level vector of G, and R is the level vector of the semi-simple Lie algebra K obtained by removing all dots corresponding to simple roots of G that are not orthogonal to q to from the Dynkin diagram of G.

q lies in the positive Weyl chamber, then there are no simple roots normal to q, R ( ) = 0 , and the exact symmetry group H is a product of l U(1) factors. This is the case for the MTME solution K

If

(1.11) obtained below. For the physically important case where q is parallel to a fundamental co-weight

λ *i and the exact

symmetry group has the form H = U(1) x K, where K is a semisimple Lie algebra, Bose and McGlinn argued that because both Q and Φ are both normal to all l -1 simple roots, α j , j ≠ i they must lie in the same one-dimensional subspace. The magnetic and Higgs fields are parallel. III.

A SIMPLE, GENERAL SOLUTION

For the special case where q = R / 2 ,

q ⋅ α i = 1 for all simple roots α i of G, and H is a product of

U(1) factors. The inner product q ⋅ u represents the height of the weight i s

usi , and the elements of

{q ⋅ u } are the R + 1 levels of i basic representation of G. The height of the highest weight is R / 2 ; the elements of the set {q ⋅ u } are half integers that run from R / 2 to - R / 2 in integer th

i s

i

i s

i

i

i

steps. Equations (1.4) and (1.7) yield:

e

−θ i

Ri 2

∑

= ci (G )

j =−

with

k=

Ri

Ri 2

−j

(−1) 2 e jr  Ri   Ri   2 − j  ! 2 + j  !   

R + j , this is 2 −

e

−θ i

c (G )e = i Ri !

Ri r 2

−

(−1) − k Ri ! kr ci (G )e e = ∑ Ri ! k =0 ( Ri − k ) !k ! Ri

Ri r 2

(e

r

− 1)

Ri

− c (G )  2 2  = i e −e  Ri !   r

r

Ri

so that

e So for

2 Ri = ci (G )sinh Ri (r / 2) Ri !

−θ i

q = κ R , a solution to the initial value problem is: c (G ) e−θi = iR sinh Ri (κ r ) i κ Ri !

 ci (G )   − Ri ln(sinh(κ r )) Ri  κ Ri ! 

θi = − ln(e−θ ) = − ln  i

(1.11)

θi′ = −κ Ri coth(κ r )), θi′′ = −κ 2 Ri csch(κ r ) Substituting the expressions (1.11) for

θi and its derivatives into (1.3) yields

Page 4 of 12

T=

κ2

l

2

coth 2 (κ r ) ∑

2

i , j =1

αi

l

K ij Ri R j = κ 2 coth 2 (κ r )∑ 2 i

2 Ri

αi

2

,

and l

V = −∑ i =1

2

αi

l

θ ′′ = −κ 2 csch 2 (κ r )∑ 2 i i =1

2 Ri

αi

2

,

so that l

2 Ri

E = κ 2 (coth 2 κ r − csch 2 κ r )∑

αi

i =1

2

l

2 Ri

= κ2∑

αi

i =1

2

.

The total energy E is constant for any radius. At the origin, singularities in the kinetic and potential energies cancel to produce a finite result.

IV.

GENERATION OF PHYSICAL SOLTIONS WITH THE LAPLACE TRANSFORM METHOD

The function

e−θi (q, r ) in (1.4) is the unique solution of the boundary value problem

 di  d   −θ ∏  − q ⋅ µ t   e i (q, r ) = 0 ,   t =1  dr

(1.12)

with the constraints (1.5). It is convenient to solve this initial value problem using the Laplace transform method. The Laplace transform g ( s ) of a function g (r ) is given by ∞

g ( s ) ≡ L { g (r )} = ∫ e− sr g (r ) dr ,

(1.13)

0

and has the property m −1

L{g ( m ) (r )} = s m g ( s ) − ∑ s m − k −1 g ( k ) (r = 0). The Laplace transform

e

−θ f

k =0 −θ f

(q, s ) of the solution e e

−θ f

(1.14)

(q, r ) is given by

di

∏(s − q⋅µ )

(q, s ) = ci (G )

i s

j =1

(1.15)

but according to (1.4)

e

−θ f

di

aki i k =1 s − q ⋅ µ k

(q, s ) = ∑

clearing the denominators gives di

∑ a ∏ ( s − q ⋅ µ ) = c (G ) . j =1

i j

k≠ j

i k

i

The substitution s= q ⋅ µ j reduces all but one of the terms in the sum to zero, yielding (1.7). i

For the physically important case where H=U(1) x K ,

q # λ *i and (1.15) develops higher order

poles. Obtaining fundamental components of MTME solutions with is a simpler operation in sspace that in coordinate space, so we proceed along these lines.

Page 5 of 12

We choose a normalization for q to make the relationships between fundamental components for different Lie algebras apparent. Let

ε i ≡ ki λ *i (no sum over i)

(1.16)

ki = 1, i = 1,..., n ,

(1.17)

For An and Bn

for Cn

1, ki =   2,

i = 1,..., n − 1

1, ki =   2,

i = 1,..., n − 2

i=n

and for Dn

i = n − 1, n

,

(1.18)

.

(1.19)

We look for the fundamental component of the solution to the MTME with The constant

q = εi .

c f (G) and the inner products λ ⋅ µ may be evaluated from Lie algebra tables11, f s

* i

and (1.15) becomes

e−θ1 (εi , r ) = exp{

12

−ir −1 i }L [ An (s)], n +1

e−θ1 (εi , r ) = 2L−1[ Bni (s)],

e−θ1 (εi , r ) = L−1[Cni (s)] .

(1.20)

where

Ani (s) =

s

n+1−i

1 , (s − 1)i

Bni (s) =

s

2 n− 2i +1

1 , (s 2 − 1)i

Cni (s) =

s

2 n − 2i

1 . (1.21) (s 2 − 1)i

We also define

Ani (r ) ≡ L−1 [ Ani ( s )], V.

Bni (r ) ≡ L−1 [ Bni ( s )],

Cni (r ) ≡ L−1 [Cni ( s )] .

(1.22)

INTEGRAL EXPRESSIONS FOR THE FUNDAMENTAL COMPONENTS OF An, Bn and Cn

From a table of Laplace transforms

L−1[

1

( s − 1)

]= i

13

r i −1 r e, (i − 1)!

L−1[

and

(s

1 2

− 1)

where the modified spherical bessel function

in (r ) is given by

n

π

 1 d   sinh r  in (r ) = r n    =  r dr   r 

2r

]= i

⋅I

n+

1 2

ri ii −1 ( r ) . 2i −1 ( i − 1) !

(1.23)

(r ) .

With the definition rn

r

I

n

[ f (r )] ≡ ∫ dr ∫ dr n

0

0

n −1

r3

... ∫ dr 0

r2 2

∫ f (r )dr 1

1

,

0

A property of the Laplace transform is

L−1  I n [ f (r ) ] =

f ( s) sn

Page 6 of 12

If

f ( m ) (r = 0), m = 1, 2,..n − 1 , and (1.14) reduces to:

 dn  L  n [ f ( r ) ] = s n f ( s )  dr  i But from (1.4) and the definitions (1.1.4), An ( r ) and its derivatives are zero up to order n , −1

Bni (r ) is zero to order 2n and Cni (r ) is zero to order 2n − 1 . We have

Cni (r ) = I1  Bni (r )  ,

Bni (r ) =

d Cni (r )  , dr

(1.24)

and

e−θ1 (εi , r ) = exp{

−ir i −ir n−i +1  r i −1er  }An (r ) = exp{ }I  , n +1 n +1 i 1 ! − ( )  

 r i i (r )  e−θ1 (εi , r ) = 2Bni (r ) = I 2n−2i 2Bnn (r )  = I 2n−2i +1  i −2 i −1 ,  2 (i −1)! i −θ1 i 2 n−2i n 2 n −2i  r ii −1 (r )    e (εi , r ) = Cn (r ) = I  i −1 . Cn (r )  = I  2 (i − 1)! VI.

(1.25)

RAISING OPERATOR

We derive a raising operator in coordinate space through some simple manipulations in i n

i n

s space.

i n

Expressions for A ( r ), B ( r ), C ( r ) follow from repeated applications of the raising operator. Defining

f mq ( s ) ≡

1 s ( s p − 1)q m

,

p = 1, 2

(1.26)

we have

qp (s p −1) + 1 d q ms −s fm (s) = m+1 p + m p , ds s (s −1)q s (s −1)q+1 = m f mq + qp f mq + qp f mq +1 .

(1.27)

In coordinate space

d q d f m ( s ) → (r f mq (r )) . ds dr q +1 Converting (1.27) to coordinate space and solving for f m yields −s

f mq +1 (r ) =

1 d 1  d   r − ( pq + m)  f mq = r − ( pq + m − 1)  f mq .   pq  dr pq  dr  

(1.28)

Now

d − ( pq + m −1) q  d  r f m = r − ( pq + m )  r − ( pq + m − 1)  f mq . dr  dr 

{

}

Inserting this expression into equation (1.28) results in the expression for the raising operator:

Page 7 of 12

 r pq + m d − ( pq + m −1)  q f mq +1 (r ) =  r  f m (r ) ,  pq dr  1 applying the raising operator q − 1 times to f m gives: r m + p ( q −1)  d − p +1  f (r ) = q −1  r  p (q − 1)!  dr 

q −2

q m

VII.

 d − ( p + m −1) 1  fm ) .  (r  dr 

(1.29)

GENERATION OF FUNDAMENTAL COMPONENTS OF An SOLUTIONS USING THE RAISING OPERATOR

For An, p = 1,

q = i, m = n − i + 1 , and (1.29) is

Ani =

r n d i −1  − ( n −i +1) r n −i r l  (e − ∑ )  r (i − 1)! dr i −1  l =0 l ! 

Using the product rule in the form n n n n! ( fg )( n ) = ∑ C   f ( q ) g ( n − q ) , where C   ≡ , q =0  q  ( n − q )!q ! q

we have n n (m + q − 1)! − ( m + q ) r d n −m r = r e C   (−1) q r e , ∑ n dr ( m − 1)! q=0 q

(

)

so that n −i  n − l − 1 r l (−1)i −1− k (n − k − 1)! r k + (−1)i ∑ C   . k = 0 ( n − i )!( i − k − 1) ! k ! l =0  i −1  l! i −1

Ani = e r ∑ Given

 −ir  i e−θ1 (ε i , r ) = exp   An ,  n +1 14

we have −θ1

e (εi , r ) = e VIII.

(1 −

 ir 

n −i −   n − l − 1 r l (−1)i −1− k (n − k − 1)! r k i  n +1  + − ( 1) e C ∑ ∑   . k = 0 ( n − i )!( i − k − 1) ! k ! l =0  i −1  l !

i ) r i −1 n +1

(1.30)

GENERATION OF FUNDAMENTAL COMPONENTS OF Bn SOLUTIONS USING THE RAISING OPERATOR

For Bn, p = 2, q = i,

m = 2n − 2i + 1 , and (1.29) is:

r 2n 1 d  B (r ) = i −1   2 (i − 1)!  r dr  i n

i −1

n −i  −2 n + 2 i − 2 r 2l  (cosh r − ∑ ) . r l = 0 (2l )!  

Applying the product rule,

1 d     r dr 

k

k −l

(2q + 2l − 1)!! −(2q+2l +1)  1 d  r −2q k  k   −(2q+1) cosh r  = r C   (−1)l r ∑     (2q − 1)!! r  ( q − 1)! l =0  l    r dr 

 cosh r   ,  r 

so that

Page 8 of 12

e−θ1 (εi , r ) =

(−1)i 2i − 2

 (2(n − k ) − 1)!! (−1) k

i

rk

n −i  n − l − 1 r 2 l i ( r ) 2( 1) C + − .(1.31) ∑   − ( k −1) l =0  i − 1  (2l )!

∑  (2(n − i) − 1)!!  (i − k )! (k − 1)! i k =1





where the modified spherical bessel function of negative order, i− n ( r ) is given by

π  1 d   cosh r  i− n (r ) = r  I 1 (r ) .   = 2r − n − 2  r dr   r  n

n

IX.

GENERATION OF FUNDAMENTAL COMPONENTS OF Cn SOLUTIONS USING THE RAISING OPERATOR

For Cn: p = 2, q = i,

m = 2n − 2i , and (1.29) becomes

Cni (r ) =

r 2 n−1  1 d    2i −1 (i − 1)!  r dr 

i −1

n −i  −2 n + 2i −1 r 2l −1  r (sinh r ) −  ∑ l =1 (2l − 1)!  

Application of the product rule yields: k −l

r −2q k  k  l  1 d  −2q  sinh r   1 d   sinh r  C   ( −1) 2l (q + l −1)!r −2l  ∑   r  =   ,  r dr   r  ( q −1)! l =0  l   r dr   r  k

so that n −i  n − k − 1 (−1) k r k  n − l − 1  r 2l −1 i e (εi , r ) = (−1) ∑ C  ik −1 ( r ) + ( −1) ∑ C  .(1.32)  k −1  (k − 1)! k =1 l =1  i−k  2  i − 1  (2l − 1)! i

−θ1

VII.

i

GENERATION OF FUNDAMENTAL COMPONENTS OF SOLUTIONS BY SUPERPOSITION

The expression (1.26) may be rewritten

f mq ( s ) =

 s p −1  q +1 q +1  = f m− p (s ) − f m (s) , m p q  p s ( s − 1)  s − 1  1

so that

f mq ( s ) = f mq− p ( s ) − f mq−−1p ( s ) , which we write as

χ ni = χ ni −1 − χ ni −−11

χ = A, B, C

i ≠ 1, n .

(1.33)

Unless the index i corresponds to an endpoint of the Dynkin diagram, a solution for the Lie algebra of rank n is simply the difference of two solutions of rank n − 1 . The solutions that cannot be expressed as the difference of two solutions of lower rank play the role of basis vectors. Proceeding along these lines, χ n can be expanded in terms of i

χ 1j and χ jj , with j < n

as follows:

Page 9 of 12

i  n −1− k  1  n −1− j  j i ( 1) (−1) j C  χ + − ∑  k χj j =1  i −1   i− j 

n −i

χ ni = ∑ (−1)k C  k =1

i ≠ 1, n

(1.34)

where l j  1  r k −1 rl 1  −1  j j r l r Ak1 (r) = L−1  k e , A ( r ) L ( 1) j ! e ( 1) = − = = − − , ∑ ∑   j j  l! l =0 l ! l =0  s (s −1)   s(s −1)  k   1 r2l = − Bk1 (r) = L−1  2k +1 2 cosh r , ∑  l =0 ( 2l ) !  s (s −1) 

j −1

  1 r2 j  1 d  cosh r j = Bjj (r) = L−1  2   [ 2 ] + (−1) , j  j −1 r dr − − s ( s 1) 2 ( j 1)! r     j −1

k   1 r2l−1 1  r2 j−1  1 d  sinh r j −1  Ck1 (r) = L−1  2k 2 sinh r , C ( r ) L ]. = − = =  ∑  2 j   [ j  j −1 r l =1 ( 2l −1) !  s (s −1)   (s −1)  2 ( j −1)! r dr 

X.

Non-Spinor Dn Solutions

Finally, we consider the properties of physical solutions for the remaining family of classical Lie groups, Dn. There are 2n weights in the defining representation of Dn, but only 2n-1 levels. Using λ1) we find that the two weights Dynkin’s technique for generating Π(λ n−2

n−2

µ s = λ1 − ∑ α i − α n −1 ,

µ s = λ1 − ∑ α i − α n

i =1

i =1

are at the same level; R 2 ⋅ µ s = R 2 ⋅ µ t = 0 . If we select the subspace of the positive Weyl chamber defined by q ≡ q ⋅ α n −1 = q ⋅ α n , then points the points in the subspace lie on the n-1 dimensional plane defined by

q1 λ1* + q2 λ *2 + ... + qn −2 λ *n − 2 + q (λ *n−1 + λ *n −1 ) This plane contains points that are arbitrarily close to each of the non-spinor fundamental coweights, λ i provided i ≠ n − 1, n . *

Confined to this subspace, the number of unique exponentials in (1.4) is reduced by one, and this allows us to solve for the coefficient of the exponentials using the constraints (1.5).

{

}

From Lie algebra tables, the constant c1 (D n ) =2, and evaluating the inner products λ i ⋅ u s for *

u s ∈ Π (λ1 ) , i ≠ n − 1, n , we obtain e−θ1 (ε i , r ) = 2L−1[ Dni ( s )] where,

Dni ( s ) =

s

2 n − 2 i −1

1 , ( s 2 − 1)i

i ≠ n − 1, n .

By inspection of (1.21)

Dni ( s ) = Bni −1 ( s ),

i ≠ n − 1, n ,

The integral form of solutions, raising operator and superposition rules follow from this expression for the non-spinor solutions. In particular, we have the result

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 r i i (r )  e−θ1 (ε i , r ) = I 2 n − 2 i −1  i i −1 ,  2 (i − 1)!

i = 1, 2,...n − 2

(1.35)

The Dynkin diagram for Dn contains a branch at the dot n − 2 , but solving for the spinor components of the MTME equation using (1.8) presents no particular problem when i ≠ n − 1, n ; the spinor components of a Dn solution are both identical to the spinor component of a Bn-1 solution. XI.

Spinor Dn Solutions

For the spinor Dn solutions, we cannot obtain solutions by approaching the spinor co-weights from the subspace defined above. However, if we make the ansatz that the spinor solutions share the property with the non-spinor solutions that they are even under the substitution r → − r , we gain an additional condition at the origin that allows us to solve the initial value problem:

Dni ( s ) =

s , ( s − 1) n 2

i = n − 1, n

It follows that:

e−θ1 (ε i , r ) = 2L−1[

(s

s

2

− 1)

]= n

 d  rn in −1 ( r )  ,  n dr  2 ( n − 1) ! 

i = n − 1, n

(1.36)

The branch in the Dynkin diagram at dot n − 2 means that the product of the spinor components is determined by component n − 2 of (1.8). To factor the individual spinor components is nontrivial, and we have not been able to develop expressions for the spinor component of Dn solutions in the general case. We start by assuming that the spinor components of the MTME solution are degenerate forms of (1.4), normalize the components so that they satisfy the condition (1.5), check that the consistency conditions that arise from the spinor components of (1.8) are satisfied, and use the asymptotic conditions (1.10) to differentiate the two spinor components. XII.

Spinor Solutions for D4

Using the technique described in the previous section, we arrive at the following MTME solution for D4:

 r3 r  r2 e−θ1 (ε 4 , r ) =  +  sinh r − cosh r 8  24 8   r3 r   r4 r2 1  e (ε 4 , r ) = −  +  sinh(2r ) +  + +  cosh(2r )  96 64   384 64 128  r6 r4 1 + + − 576 384 128 3 r r r2 −θ 3 e (ε 4 , r ) =  +  sinh r − cosh r 8  24 8  −θ 2

e−θ4 (ε 4 , r ) =

(1.37)

1 1 1 1 cosh ( 2r ) − r 4 − r 2 − 32 48 16 32

The components of the solution for

e−θi (ε 3 , r ) are identical to the components indicated above,

except for an interchange of the spinor components.

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XIII.

CONCLUSION

Using the Laplace transform to examine the properties the fundamental component of the MTME solution corresponding to finite energy monopole solutions provides insight into the relationships between monopole solutions for the classical Lie algebras. ACKNOWLEDGMENTS I would like to acknowledge discussions and encouragement from C.A. Raffensperger, E.M. Pircher, R. J.Thornburg S.K. Bose, and W.D. McGlinn. 1

N. Ganoulis, P. Goddard, and D. Olive, Nucl. Phys. B 205, 601 (1982). E. B. Bogomolny, Sov. J. Nucl. Phys. 24, 449 (1976). 3 M. K. Prasad and C. M. Sommerfield, Phys. Rev. Let. 35, 760 (1974). 4 A. N. Leznov and M. V. Saveliev, Lett. Math. Phys. 3, 489 (1979). 5 D. Wilkinson and A. S. Goldhaber, Phys. Rev. D 16, 1221 (1977). 6 R. Slansky, Phys. Rep. 79, 1 (1981). 7 A. N. Leznov and M. V. Saveliev, I. A. Fedoseev, Il Nuovo Cimento 76A, 593 (1983). 8 S. K. Bose and W. D. McGlinn, Phys. Rev. D 29, 1819 (1983). 9 P. Goddard and D. Olive, Nucl. Phys. B 191, 511 (1981). 10 E. J. Weinberg, Phys. Lett. B 119, 151 (1982). 11 G. Feldman, T. Fulton, and P. T. Mathews, J. Math. Phys. 25, 1222 (1984). 12 S. Fitzsimmons, J. Math. Phys. 34, 2115 (1993). 13 M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. 14 D. Wilkinson and F. A. Bais, Phys. Rev. D 19, 2410 (1979). 2

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