Nuclear Physics 31 (1962) 550---555; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or mlerofilm without written permission from the publisher
A MODEL U N I F I E D FIELD E Q U A T I O N J. K. P E R R I N G and T. H. R. S K Y R M E *
Atomic Energy Research Establishment, Harwell, England Received 29 S e p t e m b e r 1961 The classical solutions of a unified field t h e o r y in a two-dimensional space-time are considered. This system, a model of interacting m e s o n s and b a r y o n s , illustrates h o w the particle can be built f r o m a wave-packet of m e s o n s and h o w reciprocally the m e s o n a p p e a r s as a t i g h t l y b o u n d c o m b i n a t i o n of particle and antiparticle.
Abstract:
1. Introduction One a u t h o r has proposed the consideration of non-linear meson fields with periodic properties for the unified description of mesons and their particle sources 1, 2). In this a n d the following paper 6) some properties of these systems are examined further. A simple model of this t y p e was defined in sect. 2 of ref. 2) b y the field equation ~2oc/~x2--(1/c2)32~/~t2--K 2 sin ~ = 0,
(1)
and the b o u n d a r y condition cos0c---- 1
at
x=
4-00,
(2)
to be satisfied b y the " a n g u l a r " field variable ~(x, t). Units are chosen such t h a t c = K = 1, a n d such t h a t the energy density is
#(x) = (1/8~)[(~oc/~x)2+ (9:c/St) 2] + (1/4n)(1-- cos a).
(3)
The particle number is N----(1/2~)[~(+oo)--~--(--oo)l,
(4)
the integral of the particle number density (1/2~)(3oc/3x), and necessarily an integer on account of the b o u n d a r y condition (2).
2. S i m p l e S o l u t i o n s Simple solutions m a y be found in which ~ is a function only of s ---- (x--vt)/ (l--v2)½ and the field equation reduces to the pendulum-like equation d20c/ds 2 ---~ sin
~.
(5)
t Now at Dept. of lXiathematics, University of Malaya, P a n t a i Valley, Kuala L u m p u r , Malaya. 55O
A MOD~L ~ m r m D rmLD ~UAT~ON If
551
the phase-velocity v > 1 the solution is a t r a v e l l i n g w a v e sin ½ ~ = + k sn
(i(S--So), k),
(6)
reducing to a plane wave as the amplitude k --> 0. This can be made to fit the b o u n d a r y condition (2) in a large box, the condition is evidently equivalent to 0~ = 0 in this case, a n d all these solutions have particle n u m b e r N ----- 0. T h e y are interpreted as meson waves. For a real physical velocity v < 1, there are the solutions sin ½ ~ -~ 4-sech
(S--So),
(7)
which have N = 4-1, and total energy E = (2[~) (1--v 2)-½. T h e y are interpreted as the fields associated with a particle (or antiparticle) of mass (2/~) centred at s ---- s o and moving with velocity v. These are the only simple solutions with finite N. There are also solutions, analogous to those for a pendulum m a k i n g complete revolutions, with infinite N describing a lattice of uniformly spaced particles. Certain other analytical solutions have been found and wilt be described below; their existence depends, however, upon the specially simple nature of sin in the field equation. If sin ~ is replaced b y some other periodic function there are always solutions like (6) a n d (7), but usually numerical analysis will be needed beyond t h a t point. The structure of the wave-equation (1) is sufficiently simple to allow direct numerical integration on a computer, a n d programmes were written to follow the interaction of wave-packets, such as (6) or (7), t h a t are initially far apart. The m e t h o d of integration is outlined in the appendix. 3. Particle-Particle
Interactions
The problem of two colliding particles is equivalent, b y s y m m e t r y , to t h a t of a single particle moving in the half-space x > 0, with the b o u n d a r y conditions (0) -~ 0 and ~ ( ~ ) ---- 2n. To our initial surprise the numerical integration showed t h a t the scattering off the b o u n d a r y at x = 0 was purely elastic, almost like hard-sphere scattering. An analytical solution was then found with these properties; this has tg
~ ~ v sinh(x/(1--v~)½ ) / (cosh(vt/ (1--v2)½ )
(8)
and describes two particles centred approximately at x = ±vt and colliding elastically at x ---- 0 at time t ~ 0 . The positions of the particles are defined more precisely b y the condition t h a t there cos ~ ---- --1; this then gives the 'scattering length' in the collision equal to 2(1--v2)½ log (l/v), in units of 1/K. A similar solution exists describing a particle-antiparticle collision, with tg ¼~ ---- (l/v) sinh
(vt/(1--v2)½)/cosh(x/(1--v~)½).
(9)
552
J.K.
PERRING AND T. H. R. SKYRME
The scattering length is the same, describing here an attractive process in which particle and antiparticle are accelerated through one another. 4.
The
Potential
Although we have found an explicit solution of the two-particle scattering problem in this model, it is interesting also, for the better understanding of the nature of the interactions, to construct a potential. The definition of potential energy is somewhat arbitrary, but that most natural in this problem is the energy of the static field associated with particles held in fixed positions. This is found from the static solution of the field equation (1) which satisfies in addition to (2) the conditions cos ~ - ~ - - 1 at particle positions, (3~¢/~x) :> 0 for particle, (3~/~x) < 0 for antiparticle.
(10)
In general (i.e. except for a free particle) the field gradient will be discontinuous across the particle position; this corresponds with the force that has to be applied at a point to keep the particle fixed. Between two particles separated b y a distance r the field will be given b y an elliptic function whose parameter k is determined b y ~r = 2k K ( k ) .
(11)
The energy of the fields between the particles is then E ---- (2/~)k -1 [ E ( k ) - - { ( 1 - - k 2) K(k)],
(12)
and for two particles the fields outside would be the same as for free particles. For large r, one has k --~ 1 and E, given b y eq. (12), becomes equal to (2/n). For finite r the difference from this value measures the potential, with the approximate formulae V ~-~ (8K/n) e -~r, V ~ (~/2r)
as r - + oo, as r --> 0.
(13)
It is interesting to see the origins of these two limits. At large distances the interaction m a y be treated as a perturbation and will satisfy the linearised wave-equation (V2--K 2) ~¢ = const • 6'(x--x~), (14) giving the exponential dependence in (13); the coefficient (8/~) expresses the effective coupling constant appropriate to the detailed source structure. The sign (repulsion) is that expected for a derivative interaction between similar particles.
A MODEL U N I F I E D F I E L D E Q U A T I O N
553
At small distances there is also repulsion, b u t its origin is distinct, It comes from the n a t u r e of the b o u n d a r y conditions: ~ has to change b y 2z~ over tile small distance r, implying a large field gradient a n d a large addition to the gradient term in the energy density (3). A similar analysis can be made of the potential between a particle a n d an antiparticle; at large distances there is simply a change in sign as would be expected from the a s y m p t o t i c field. At small distances the b o u n d a r y conditions ' m a t c h ' so t h a t '~ assumes a nearly constant value between the sources a n d E -+ 0; there is then a potential well of m a x i m u m depth equal to one particle mass. 5. T h e
Meson
as a Particle
Pair
The most natural aspect of the field equation (1) is t h a t of a self-interacting meson field, and the particles appear as particular types of non-dispersive wavepackets t h a t can be constructed on account of the periodic nature of the boundary conditions. The analytical solution (9) leads to the converse picture of a meson as a t i g h t l y b o u n d state of the particle-antiparticle system 3). F o r real v < 1, this solution (9) describes the scattering of a pair with asymptotic velocities + v , and the total energy is (4/a)(1--v~)-½. E v i d e n t l y it continues to satisfy the field equation when v is continued through zero to imaginary values, giving a state of total energy less t h a n two particle masses, i.e. a b o u n d state. The sinh function is replaced b y a sine, so t h a t the field falls off exponentially at large distances at all times; it describes a localised but oscillating meson wave-packet with N = 0. In the limit as v -+ ioo, one has = ~ exp
(i•t),
an infinitesimal plane meson wave of zero m o m e n t u m . B y a Lorentz transformation states with arbitrary m o m e n t u m appear similarly as the limits of particle-pair systems with this centre-of-mass m o m e n t u m . 6. M e s o n - P a r t i c l e
Interactions
The existence of analytical expressions for meson wave-packets of finite amplitude makes it easier to handle numerically the problem of the interaction of mesons and particles in this model. A n u m b e r (7) of integrations of this t y p e have been made. In these the particle was initially at rest at tile origin, and the meson packet approached it from a distance with chosen values of velocity, amplitude and phase. The results were again unexpectedly simple in character. In all cases the final state consisted of a displaced particle at rest a n d a transm i t t e d meson wave, and appeared to be independent of the choice of phase for the meson wave.
554
J.K.
PERRING
AND
T. H. R. $ K Y R M E
W e have no explanation of these results except that they appear again to be connected with accidental features of the equation with sin ½=. In the case of a small amplitude meson wave we can use perturbation theory, writing =
(x, t),
where %(x) is the solution for a particle at rest at the origin (eq. (7)): sin ½ ~¢e -~ sech x. T h e n the linearised equation for A is ([:]--K 2 cos %) A = ([--]--K2+2K l sech 2 Kx) A =- 0,
(15)
with periodic solutions A = e '~* e ±ik~ ( 1 ± (ilk) t a n h x),
(16)
where o)2 = K2+k 2. This describes meson scattering in which there is no reflected wave and the t r a n s m i t t e d wave has a phase change 8, tg (~ = (l/k). The absence of reflection arises from the p a r t i c u l a r form of the scattering potential and is thus an accidental feature arising from the shape of sin 0c.
Appendix NUMERICAL INTEGRATION PROCEDURE Two m e t h o d s of integration were tried. First, a simple leapfrogging scheme 4) was p r o g r a m m e d for the Harwell Mercury Computer, writing
o~(x, t + z)
-= o~(x, t ) + z&(x, t+½z) &(x, t + l z ) = &(x, t - - ½ z ) + z ( ~ 2 o~l~x2 - sin a)~,,, with ~2 o~/~x2 given as
h -2 [a(x+h)--2:t(x)+o~(x--h)]. In solving such equations stability considerations are always i m p o r t a n t ; the m e t h o d was found to be wildly unstable for z = h, b u t a reduction of z to 0.95 h was sufficient to give stability. The first results, on particle-particle scattering, showed t h a t a n y inelasticity was e x t r e m e l y small, and it became clear t h a t greater a c c u r a c y and a larger lattice were required t h a n was possible with the use of the Mercury core store. A new p r o g r a m m e was therefore written for the IBM 7090, at Aldermaston, which used the slightly more complex m e t h o d of integration along the characteristics 5). The m e t h o d is a d e q u a t e l y described in this reference; the characteristics are known b u t the presence of the source t e r m makes iteration for a at each step of the integration necessary. However, error analysis shows t h a t this m e t h o d is, on the average, four times as fast as the preceding one.
J. K. PERRING AND T. H. R. SKYRM]~
555
W e are i n d e b t e d to Mr. A. R. Curtis for v a l u a b l e discussions.
References 1) 2) 3) 4)
T. H. R. Skyrme, Proc. Roy. Soc. 260 (1961) 127 T. H. R. Skyrme, Proe. Roy. Soc. 262 (1961) 237 E. Fermi and C. N. Yang, Phys. l~ev. 76 (1949) 1739 R. D. Richtmeyer, Difference methods for initial value problems (Interscience, New York, 1957) p. 166 5) National Physical Laboratory, Modern computing methods (1957) p. 66 6) T. H. R. Skyrme, Nuclear Physics 31 (1962) 556