Physica 97A (1979) 139152 @
NorthHolland Publishing Co.
CLASSICAL ELECTRODYNAMICS OF A NONLINEAR DIRAC FIELD FREE SOLUTIONS
Christian WAGNER* and Mario SOLER Junta
de Energia Nuclear, Madrid, Spain
Received 29 September 1978 In previous papers it has been shown that adding a positive scalar selfinteraction (&J)’ to the Dirac field Lagrangian provides a reasonably satisfactory model to describe the barions. In this work, we analyze other solutions of the same nonlinear Dirac equation, making progress in the direction of a systematic analysis. These solutions could provide the ground states for more elaborate interacting schemes of the real particles. Unfortunately the new solutions appear to have energies consistently higher than the ones analyzed in previous papers. Also, the more complicated solutions, whose energy seems to be much higher than little hope for a low minimum energy state. the simplest one, leave us
1. Introduction
There is at present considerable interest in classical nonlinear theories in connexion with models of elementary particles’). On the one hand, gauge theories, even at the classical level, seem to provide a valuable framework for the study of the different particles. On the other hand, the progress made in understanding the soliton solutions of certain equations, renders more attrac tive the possibility of identifying classical solutions with elementary particles. Among the infinite variety of nonlinear theories which might be analyzed, it seems advisable to explore in the first place the simplest models. It has been shown2) that adding a positive scalar selfinteraction (&)’ to the Dirac field Lagrangian provides a reasonably satisfactory model to describe the barions. The nonlinear terms might srcinate from a spinspin selfcoupling, first discovered by Wey13) in connexion with the structure of spacetime, and the particular scalar form of the nonlinearity can be obtained in certain simple models of Universe4). Even if this nonlinear scalar Dirac model has been analyzed fairly exten sively, its simplest solutions have not been systematically studied. The study carried out in ref. 2 (to which we will refer as “I”) showed the existence of a continuum of classical states of bounded energy with one state corresponding
* Present address: Department of Physics, Sciences Faculty of Cadiz, Cadiz, Spain. 139
14 CHRISTIAN WAGNER AND MARIO SOLER
to the minimum energy of the model. It was assumed that in a process of production the state with minimum energy would consistently appear, and might thus be identified with an elementary particle. There exist however other solutions of the same nonlinear Dirac equation which have not been analyzed. The purpose of this work is to make progress in the direction of a systematic analysis. It is conceivable that, in the context of nonlinear differential equations, solutions with very different energies might exist. If solutions widely separate in energy and having a minimum were to exist, they could provide the basis or ground states for more elaborate interacting schemes of the real particles. Unfortunately, as we will see, the above possibility seems not borne out by the mathematical results, since the new solutions appear to have energies consistently higher than the simplest one analyzed in
I
which can only describe the barions. In the more complicated solution, the complexity of the numerical analysis has prevented us from obtaining more than one state out of the continuum, but its energy is much higher than the simplest solution, leaving little hope for a low minimum energy state. The paper is organized as follows: section 2 presents the three possible forms of wave functions, from which a fermion model may be obtained. We outline the techniques employed to obtain numerical solutions and discuss the results for the two simplest forms in section 3 and for the other possible form
in section
4. In the appendix, it is shown that if the radial functions are complex, a more general expression for the wave function will not be obtained. 2.
General form of solutions
The nonlinear Lagrangian considered in paper I in order to have a satis factory classical spinor theory was”) the equation for IC, s ir”a,$  me + 2A ($I&) = 0. (2) Using this equation, we find
L = A( b)*.
ince
L
is real for real A, we need not add to
L
its complex conjugate as is usually done in quantum mechanics. Let us begin by listing four reasonable assumptions concerning the proper ties of the desired wave function in order to be an adequate first step for a fermion model (considered to be at rest at the srcin):
CLASSICAL ELECTRODYNAMICS OF NONLINEAR DIRAC FIELD 141
a) Its total angular momentum and its zcomponent will be well defined and equal to one half. b) Its parity need not be well defined in order to include a possible model for the neutrino. c) I, J will be everywhere finite and quadratically integrable so that the measurable quantities which can be evaluated from it will be finite. d) I, J will be a stationary solution of eq. (2). This we achieve by separating the time in the form * = exp(ior)+ (r), (3) where w is a positive parameter which is supposed to adjust itself to the value corresponding to the minimum (nonzero) energy of the system. We will not be interested in solutions having total angular momentum larger than one half. Therefore, 4 may be expanded in spherical polar coordinates in the form: (4) where g, and fn are real radial functions. For complex g, and fn, the expression (4) will not be more general. This is equivalent to multiplying this expression by an arbitrary phase, as is shown in the appendix. $” are eigenspinor functions which are required to be an orthonormal set of eigen functions of the total angular momentum and of its zcomponent, with eigenvalues equal to one half (j = 4, mj = 4). Consequently, only two different eigenspinor functions, which are related to the two possible values of the orbitalangularmomentum, may be written. *+ = (47r1’2 l 0 0 forI=O(j=O+JJ evenparity, I& = (47~)“*(~:“~,“e) for
I =
1 (j =
1 
k) odd parity. It is easy to show that: where
ur
cos
e
sin 0 emiP =
r
(
in 0 e’* cos 8 >
lgqJ+ = **l/J, *T* = ‘)w+,
where I/J* represents the complex conjugate of I, L
142 CHRISTIAN WAGNER AND MARIO SOLER
The wave function (4) becomes: where 8, y, and E are three parameters to be determined. By writing eq. (2) in the form i&y’+ = [iy’ai  m + 2A(Ij$)]+ (6) it is easy to show that the expression (5) may be only an acceptable solution of this equation if the parameters are taken to be e” = + i, eiy = ? i, and e If = ?i. Therefore, the set of solutions of eq. (2) which verifies the conditions previously required, may be obtained from the following forms: (7) (subscripts will be omitted if unnecessary), where g and
f
are real functions and IJJR will be equivalent to forms I/I~ or $,z when
f.
or
fh
are respectively considered to be zero everywhere. The solution of eq. (6) can be obtained by solving a set of two differential equations in cases r, ~, nd IJQ, or four in case IlrR, which express the coupling of radial equations. By defining r = plm, A =
w/m, g = (m/2h)G,
f = mL’h)F, 8)
eq. (6) gives,
for
I
G’+(l+A4)F=O F’+(2/p)F+(l A4  )G =O’
@a) (9b)
CLASSICAL ELECTRODYNAMICS OF NONLINEAR DIRAC FIELD 143 for
Ccl2 {
G’+(lA++)F=O F’+(2/p)F+(l+A +4)G =O’ where 4 = G2 
F*
(for $J,, C#J 0, and for I&, 4 c 0), for
+R 1
Gb+(l+A$)F, =0 G;,+(lA
 )Fh =0 f F:, +
(2/p)F, +
(1

A  c )G, = 0
Fb+(2/p)F,,+(l+A$)G,=O where+=GiFiGi+Fi,
UW lob)
From the energymomentum tensor (Too = i&y&&l;), by utilizing expressions (7) for +, and making changes (S), we obtain for the energy: for +i and I&
E=&
mh
A(G2+F2)+;42
p’dp,
I
(12)
E=k
mh
A(G;+G;+F;+F;)++
p’dp. I In paper I, it was shown that a minimum for the Adependent energy exists in case 9,. We have now found that this minimum exists in case @I as well.
3. Solutions in cases JI, and I,
We will impose the boundary conditions that the functions
F
and G are finite and regular at every point. Since eqs. (9b) and (lob) have the term 2F/p, the finiteness condition can be met at the srcin only if
F
vanishes there. It follows from eqs. (9a) and (10a) that G’ also vanishes there. The systems of equations (9) and (10) are two differential equations of the first order, so that there are only two boundary conditions and these may be taken to be the values of
F
and G at the srcin. Since
F
necessarily vanishes there, the solutions bounded at the srcin depend only on the positive real G(0) = K. We need not consider negative values of
K
because the solutions for positive or negative values of
K
are symmetrical in respect to the p axis. It turns out, however, that these solutions usually do not vanish at infinity, and that the