P H Y S I C A L R E V I E W D V O L U M E 1 8 , N U M B E R 1 2 1 3 v ~ L E M B E R 1 9 7 8

Note on localized solutions of a nonlinear Klein-Gordon equation related to Riemannian geometry

Eckehard W. Mielke Institut fur Reine und Angewandte Kernphysik der Christian-Albrechts-Universitiit Kiel, Olshausenstrasse 40-60, 2300 Kiel I , Federal

Republic of Germany (Received 17 April 1978)

A nonlinear complex scalar field with a cP2d'(d - ') self-interaction is considered in a flat, ( d + 1)-dimensional space-time. Conformal techniques familiar in differential geometry become operative in the construction of exact, ,'soliton"-type solutions exhibiting finite energies, but exponential decays.

I. INTRODUCTION

The current interest in (meta)stable, localized solutions of nonlinear classical field equations stems f rom the possibility of regarding these "sol- itons" as possible models of extended particles. It i s believed that they play a fundamental role in a nonperturbative treatment o f quantum field theo- ry.' Stimulated by a Euclidean scalar field theory with a (pZN+' self-interaction; the physically more relevant case o f a complex (charged) scalar field q ( x ) will be consider in a flat Minkowskian space- t ime owning d spatial dimensions. The model i s defined by the nonlocal Lagrangian density

- C * ] , (1.1)

where the average of the nonlinear t e rm over a (d - 1)-dimensional topological sphere centered at x produces the self-interaction. For (1.1) to be Lorentz invariant, Sd- '(x) has to be regarded as a standard sphere around x, in a res t frame which has been Lorentz boosted. This dissident choice i s mainly motivated by computational advantages, but it could turn out t o ref lect one's physical in- tuition much more closely. Variation with re- spect to ?* yields the nonlinear field equation

Generalizing a recent analysis3 of a similar model proposed by R o ~ e n , ~ exact spherically sym- metric solutions o f (1.2) will be derived in Sec. I 1 in the case N = 2/(d - 2) by employing conformal techniques5 borrowed f rom Riemannian geometry. Sections I11 and IV, respectively, touch upon the questions of field energy and stability o f these quasisoliton solutions.

It should be noted that the nonlinear interactions considered by Werle6 are rather complementary to those advocated here.

11. LOCALIZED, SPHERICALLY SYMMETRIC SOLUTIONS

Since Eq. (1.2) i s Lorentz invariant, i t ' i s suf- ficient to desire static, spherically symmetric solutions. In order to get "moving solitons," one has to consider boosted solutions,

q ( [x - X , - v(t - t o ) ] / ( l - v 2 / ~ 2 ) 1 / 2 )

which, however, do not contain any new physical information.

By introducing spherical coordinates (r , e l , . . . , Od-,) at the point x =x , in the static case , formula7

can be utilized, where A, i s the Laplacian on the unit sphere Sd-'. The generalized spherical func- tions Yy(B,, . . . , 9,-,) supporting an irreducible representation of the group SO(d) are eigenfunctions of A,:

They are normalized according to

where the particular values

can be inferred f r o m explicit representation^.^ The familiar separation ansatz

) e - i ( t - t O ) ~ g l h c~ = f i ( r )Y ; (@, , . . . , ,-, (2.5)

reduces (1.2) af ter integrating over the angular variables in the nonlinear t e rm to an ordinary

4526 E C K E H A R D W . M I E L K E 18 -

differential equation for the radial function f k(r) The substitution (below abbreviated by f ) :

f a(r) =uI , (r )r - ' / " (2.7) d - 1 l ( l + d - 2 ) fu+ - f l - f + ( ~ + l ) ( c ~ ) ~ ~ f ~ ~ + l = 0 .

Y Y f rees the u and uZN+' terms f rom r-dependent (2.6) multiplicative factors:

-

r 2 u U + ( d - 1 -2 /N)ru1- [ l ( l+d- 2 ) + (d - 2 ) /N- l / ~ ~ ] u + ( ~ + l ) ( c ~ ) ~ ~ u ~ ~ + ' = ~ . (2.8)

Furthermore, the change [it i s not necessary but instructive to choose the same "length" Q as in (1.1)]

d* = y - Z l N + l - d , F z y / Q - dF (2.9)

o f the independent variable eliminates the contribution f rom the derivative term of f i rs t order:

52(2/ N + Z - d ) d2U - [ l ( l+d- 2 ) + ( d - 2 ) / ~ - l / W ] u + ( ~ + l ) ( c Q ) ' ~ u ~ ~ + ~ = 0 . dw2 (2.10)

In the following, the analysis concentrates on the case

~ = 2 / ( d - 2 ) , d > 2 (2.11)

by which (2.10) simpli f ies to

The primes denote the differentiation with respect to

w = lnF, (2.13)

which i s in accordance with (2.9). The resulting equation alludes to a significant

analogy in Riemannian geometry. Consider there the task of relating a geometry of constant scalar curvature R to one o f another constant R by means o f a conformal change

o f the metric. For L =-2, this amounts to solving [see , e.g., Appendix Eq. (A5) of Re f . 51

R=Ry12+ (d - 1)[2ylyl"-d(y11)2] (2.15)

provided that the conformal function yl depends only on the coordinate w. An easily obtained sol- ution o f (2.15) i s

An alteration of the "scale dimension" L = -2 to L = 4/(d - 2) can be achieved by introducing the new function

which consequently [Appendix Eq. (A6) o f Re f . 51 satisfies

Comparing factors

R=(21+d- 2)'(d- l ) ( d - 2)-', (2.19)

R = 4d(d - l ) (d - 2)-'(cQ)*/ ( d - 2 ) (2.20)

in (2.18) with those in (2.12) yields

In view o f the substitutions (2.7) and (2.13), the solutions of the radial equation (2.6) finally read

In the real world, i .e . , d = 3 , the spherically symmetric solutions

represent localized (Fig. 1 ) classical wave pack- e ts . Their maxima

do not exceed the input ("elementary") length Q. The obtained solutions are o f remarkable signif-

icance also in differential geometry. In order to

I8 - N O T E O N L O C A L I Z E D S O L U T I O N S O F A N O N L I N E A R . . . 4527

FIG. 1. Radial solutions for the lowest quantum num- bers 1 of the angular momentum.

see this, apply a change of coordinates similar to that of (2.13) to the solution (2 .16 ) . Then the con- formally related squared line element

RP '[d"' ( d - l ) ( d - 2 ) ::-')dxadxb] (2.25)

generalizes that of a metric with constant Gauss ian curvature ~ = R / d ( d - 1 ) written in Riemann's "iso- thermal" form.g Similarly, the necessary condi- tion for an embedding of a two-dimensional sur- face of constant K into Euclidean three space in-

volves the sine-Gordon equation in two space-time dimension^.^'

111. MASS FORMULA

As an approximation to the energy eigenvalues of the corresponding quantum-theoretical s tates the c lass i ca l field energy

of the "solitons" should be finite and positive-def- inite. I ts calculation will be facilitated by sub- tracting from (3 .1) a total divergence

Because of the field equations (1 .2) this procedure yields

The ansatz (2 .5) allows us to ca r ry out the inte- gration over the angular variables

E - E o = - " Q ~ / ^ { ( F C / E ) Y ~ + ( d - 2 ) - ' [ c ( l , m , 2 / (d - 2))QT' ( d - 2 ) f 'I (~*)")Fd- ldp, P 0

The insertion of expression (2 .22) produces

E - E o = t i2 (21+d- 2)d-2 [ ( p c / 5 ) z ~ * ( 7 4 1 / ( d - 2 )+z + 1 ) ~ - d ~ 2 1 + d - 1 p c ( l , m , 2 / ( d - a ) ) " o dF

f ( ~ 4 1 1 ( d - 2 )+z + 1)-d pzld/ (d-2)+d-ldv ( d - 2 1 4 0

The integrals can be evaluated by means of the formula''

a s these conditions a r e fulfilled in both cases . The result i s

E - E o = E 2 ( d - 2)(21 +d - 2)d-3 2 p c ( l , m , 2 / ( d - 2 ) F

[ ( ( 2 1 + d ) ( d - 2 ) ) r ( d - 2 - ( 2 1 i d ) ( d - 2 ) ) r - l (d - 2 ) + (21 + d - 2)2 (pciE)2r 41 + 2(d - 2) 41+2(d- 2 ) (d - 2 ) Q 2 ~ ( : ) r - ' ( d ) ] . (3 .7 )

4528 E C K E H A R D I. M I E L K E 18 -

In the physical space-t ime (3 .7) simplifies to found in th ree dimensions a s e t of solutions which

a relation which is reminiscent of the Giirsey- Radicati m a s s formulaL2 prominent in hadron physics.

In o r d e r to r e n d e r the energy positive, i.e.,

even in the case I = 0, the Compton wavelength

o f a p a r t i c l e of m a s s M* has to be 'in the range

O<Qs t i / 2 f i P c . (3 .11)

This could be interpreted a s an interesting r e - s t r ic t ion

o < , u G M * / J ~ - * M * (3.12)

on the m a s s parameter y (of a "constituent" quark ?).

IV. STABILITY

With regard to the i s sue of stability the r e a d e r will be r e f e r r e d to the work of Vazquez? who

a r e related to the radial functions (2 .23) by the following identifications :

[1t is well known that fo r 1 = 0 , f :(r) a l so represen ts a solution of Emden's equation,'3 famil iar in the astrophysics of Gaskugeln.] I t tu rns out that the localized solutions (2 .23) a r e not s table , but de- cay exponentially with the character is t ic t ime

7 = l / c ~ , ( l ) . (4 .2)

The eigenvalue y,(l) appears in the f i r s t -o rder perturbation about the exact solutions and has been calculated numerically .3 Assuming (3 . l o ) with M * = 1 GeV,

ge t s slightly s m a l l e r than the decay t imes assoc- iated with hadronic resonances.

ACKNOWLEDGMENT

This work was supported by the Deutsche Forschungsgemeinschaft, Bonn.

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