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Substituting (10), (10 ) into (12) one obtains a nonlinear ODE for F (?)
...
? ? ?
?3 F + 4?2 F F + 3?2 F 2 + 6?F F 2 + F 4 = 0, (13)
?
where F = dF/d?.
The general solution of eq. (13) is given by formulae (11). The theorem is proved.
Note 1. Compatibility of the three-dimensional d’Alembert–Hamilton system has been
investigated in detail by Collins [5]. Collins essentially used geometrical methods
which could not be generalized to higher dimensions.
Using Lie’s method (see e.g. ref. [6]) one can prove the following statement.
Theorem 2. System (7), (8) is invariant under the 15-parameter conformal group
C(1, 3) iff
F (?) = 3?(? + C)?1 , (14)
? > 0, C = const.
Note 2. Formula (14) can be obtained from (11) by putting C2 = C3 = 0. So Theorem
2 demonstrates the close connection between compatibility of a system of PDEs and
its symmetry.
Note 3. It is common knowledge that the PDE (7) is invariant under the group C(1, 3)
iff F (?) = ?? 3 [3]. Consequently, an additional constraint (8) changes essentially the
symmetry properties of the d’Alembert equation (choosing F3 (?) in a proper way one
can obtain a conformally-invariant system of the form (4), (5) under arbitrary F2 (?)).
In Table 1 we list the explicit form of some exact solutions of the d’Alembert–
Hamilton system (7), (8) and the reduced ODEs for the function ?(?). h1 , g1 are
arbitrary smooth functions on aµ xµ + dµ xµ ; h2 , g2 on ? + dµ xµ ; and aµ , bµ , cµ , dµ
are arbitrary real parameters satisfying conditions of the form
?aµ aµ = bµ bµ = cµ cµ = dµ dµ = ?1,
aµ bµ = aµ cµ = aµ dµ = bµ cµ = bµ dµ = cµ dµ = 0.
Table 1

F (?) ? = ?(x)
No. QDE for ?(?)
?
1 0 ? = F1 (?)
?
aµ xµ
1
? ?1 [(aµ xµ )2 ? (bµ xµ )2 ]1/2 ? + ? ?1 ? = F1 (?)
1 ? ?
2
2? ?1 [(aµ xµ )2 ? (bµ xµ )2 ? (cµ xµ )2 ]1/2 ? + 2? ?1 ? = F1 (?)
1 ? ?
3
3? ?1 (xµ xµ )1/2 ? + 3? ?1 ? = F1 (?)
1 ? ?
4
?1 0 bµ xµ cos h1 + cµ xµ sin h1 + g1 ? = ?F1 (?)
?
5
aµ x ? bµ x cos h2 ? cµ x sin h2 ? g2 = 0
µ µ µ

?1
[(bµ xµ + h1 )2 + (cµ xµ + h2 )2 ]1/2 ? + ? ?1 ? = F1 (?)
?1 ?? ? ?
6
?2? ?1 [(bµ xµ )2 + (cµ xµ )2 + (dµ xµ )2 ]1/2 ? + 2? ?1 ? = F1 (?)
?1 ? ?
7
0 0 0 = F1 (?)
8 h1
576 W.I. Fushchych, R.Z. Zhdanov

Choosing in a proper way constants aµ , bµ , cµ , dµ and functions fi , gi one can
obtain from Table 1 symmetry ansatze constructed by Winternitz et al. [4]. Such an
approach based on the d’Alembert–Hamilton system makes it possible to obtain a
wider family of ansatze for the nonlinear d’Alembert equation (1) (see also ref. [7]).

1. Fushchych W.I., in Algebraic-theoretical studies in mathematical physics, Kyiv, Institute of Mathe-
matics, 1981, 6–28.
2. Fushchych W.I., in Theoretical-algebraic methods in mathematical physics problems, Kyiv, Institute
of Mathematics, 1983, 4–23.
3. Fushchych W.I., Serov N.I., J. Phys. A, 1983, 16, 3645.
4. Grundland A.M., Harnad J., Winternitz P., J. Math. Phys., 1984, 25, 791.
5. Collins C.B., Math. Proc. Cambridge Philos. Soc., 1976, 80, 165.
6. Olver P.J., Applications of Lie groups to differential equations, Berlin, Springer, 1986.
7. Fushchych W.I., Zhdanov R.Z., On some new exact solutions of nonlinear d’Alembert and Hamilton
equations, Preprint N 468, Minneapolis, Institute for Mathematics and its Applications, 1988.
W.I. Fushchych, Scientific Works 2001, Vol. 3, 577–622.

Symmetry and exact solutions
of nonlinear spinor equations
W.I. FUSHCHYCH, R.Z. ZHDANOV
This review is devoted to the application of algebraic-theoretical methods to the problem
of constructing exact solutions of the many-dimensional nonlinear systems of partial di-
fferential equations for spinor, vector and scalar fields widely used in quantum field
theory. Large classes of nonlinear spinor equations invariant under the Poincar? group
e
P (1, 3), Weyl group (i.e. Poincar? group supplemented by a group of scale transformati-
e
ons), and the conformal group C(1, 3) are described. Ans?tze invariant under the
a
Poincar? and the Weyl groups are constructed. Using these we reduce the Poincar?-
e e
invariant nonlinear Dirac equations to systems of ordinary differential equations and
construct large families of exact solutions of the nonlinear Dirac–Heisenberg equation
depending on arbitrary parameters and functions. In a similar way we have obtained
new families of exact solutions of the nonlinear Maxwell–Dirac and Klein–Gordon–Dirac
equations. The obtained solutions can be used for quantization of nonlinear equations.

1. Introduction
The Maxwell equations for the electromagnetic field and the Dirac equation for
the spinor field,

(?µ pµ ? m)? = 0, (1.1)

discovered 60 years ago, are the fundament of modern physics. In eq. (1.1) ? = ?(x)
is a four-component complex-valued function, x = (x0 ? t, x1 , x2 , x3 ) ? R(1, 3), four-
dimensional Minkowski space, ?µ are 4 ? 4 matrices satisfying the Clifford–Dirac
algebra

(1.2)
?µ ?? + ?? ?µ = 2gµ? ,

where gµ? = diag (1, ?1, ?1, ?1), m is the particle mass. We use two equivalent
representations of the ?-matrices,

0 I 0 ?a
(1.2a)
?0 = , ?a = ,
??a
I 0 0
or
I 0 0 ?a
(1.2b)
?0 = , ?a = , a = 1, 2, 3,
?I ??a
0 0

?a are the 2 ? 2 Pauli matrices.
Fifteen years ago D. lvanenko [1] made an attempt to obtain a nonlinear generali-
zation of the Dirac equation, and suggested the following equation:
? ?
[?µ pµ ? m + ?(??)]?(x) = 0, ? = ? + ?0 . (1.3)
Physics Reports, 1989, 172, 4, 123–174.
578 W.I. Fushchych, R.Z. Zhdanov

In the early fifties W. Heisenberg [2–5] put forward a vast program to construct
a unified field theory based on the nonlinear spinor equation
?
[?µ pµ + ?(??µ ?4 ?)? µ ?4 ]?(x) = 0. (1.4)

Heisenberg and his collaborators [2–5] did their best to construct the quantum field
theory, to establish the quantization rules, and to calculate the mass spectrum of the
elementary particles.
In two papers by R. Finkelstein and collaborators [6, 7] published in the early
fifties, nonlinear spinor fields were investigated from the classical point of view,
i.e., approximate and exact solutions of partial differential equations (PDE) were
studied. From the classical point of view scalar field was studied by L. Schiff [8] and
B. Malenka [9].
Like the general theory of relativity nonlinear spinor field theory is a mathematical
model of physical reality based on a complicated multi-dimensional nonlinear system
of PDE.
Up to now there exists a vast literature on exact solutions of the equations for
the gravitational field. It is well-known which important role has been played in
gravitation theory by Schwarzschild’s, Friedman’s and Kerr’s exact solutions. So far
many of the obtained solutions have no adequate physical interpretation. Nevertheless
the number of exact solutions of the Einstein equations grows rapidly.
Nothing of the kind takes place in nonlinear field theory. There are few enough
classical solutions of nonlinear spinor equations [10–18] although these equations are
essentially simpler than those of gravitation theory. This surprising situation seems
to be explained by the fact that many investigators underestimate the importance of
exact solutions in the theory of quantized fields and expect the great successes in
other domains of quantum field theory.
We think that a thorough investigation of nonlinear spinor equations and a con-
struction of exact solutions for them sooner or later will lead to important physical
results and to new physical ideas and methods. Let us recall that in this way the
theory of solitons was created.
We will not adduce a concrete physical interpretation to the solutions of nonlinear
spinor equations because we think that they speak for themselves. Nevertheless we
will show how to construct nonlinear scalar fields (equations) using exact solutions
of nonlinear spinor equations. In other words, we have a dynamical realization of de
Broglie’s idea to construct an arbitrary field by using a field with spin s = 1 [19]. The
2
kinematical realization of this idea is well known. It is reduced to a decomposition
of a direct product of linear irreducible representations of the Lorentz and Poincare
groups (with spin s = 1 ).
2
It will be shown that the interaction of spinor and scalar fields gives rise to some
mass spectrum (section 4). It is of interest that discrete relations connecting the
masses of spinor and scalar fields are determined by the geometry of the solutions.
Exact solutions obtained by us can be used as a pattern to check the already
known approximate methods and to create new ones. For example, solutions which
depend on the coupling constant A in a singular way cannot be obtained by standard
methods of perturbation theory.
Solutions (classes of solutions) with the same symmetry as the initial equation
of motion seem to be of particular importance. These solutions (not the equation)
Symmetry and exact solutions of nonlinear spinor equations 579

can be used in the quantization procedure. From the set of solutions one can pick
out ones that do not lead to an indefinite metric. This review is based on our papers
[20–30], and the symmetry properties of PDE are used extensively. That is, we apply
the classical ideas and methods of S. Lie to nonlinear spinor equations. The symmetry
properties of nonlinear field equations make it possible to reduce multi-dimensional
spinor equations to systems of ordinary differential equations (ODE). Integration of
these ODE gives rise to exact solutions of the initial equation. Let us note that all
the exact solutions of the nonlinear Dirac equation known to us are included in the
set of solutions obtained in such a way.
The structure and content of the review are as follows. In section 2 we investigate
the symmetry of the nonlinear Dirac equation
?
[?µ pµ + F (?, ?)]?(x) = 0, (1.5)
?
where F (?, ?) is an arbitrary four-component matrix depending on eight field va-
? ?
riables ?, ?. All the matrices F (?, ?) ensuring invariance of eq. (1.5) under the
?
Poincar? group P (1, 3), extended Poincar? group P (1, 3) and conformal group C(1, 3)
e e
are described.
In section 3 we take the ansatz

(1.6)
?(x) = A(x)?(?),

suggested in ref. [30] and described systematically in refs. [23, 24, 29], which reduces
the system of equations (1.5) to systems of equations for the four functions ?0 , ?1 ,
?2 , ?3 depending on three new invariant variables ? = {?1 (x), ?2 (x), ?3 (x)}. In
(1.6) A(x) is a variable nonsingular 4 ? 4 matrix, whose explicit form is given in
section 3. If ? depends on one independent variable then ansatz (1.6) reduces eq.
(1.5) to a system of ODE. Most of them prove to be integrable. Integrating these and
substituting the obtained results into the ansatz (1.6) one obtains particular solutions
of eq. (1.5). Using this approach we have constructed large classes of exact solutions
of the nonlinear Dirac–Heisenberg equation (DHE) for a spinor field.
In sections 4 and 5 multi-parameter families of exact solutions of the Dirac–Klein–
Gordon and the Maxwell–Dirac systems, describing the interaction of a spinor field
with scalar and electromagnetic fields are constructed.
2. Nonlinear spinor equations invariant under the Poincar? group P (1, 3)
e
? (1, 3) and C(1, 3)
and its extensions, the groups P
It is clear that arbitrary equations of the form (1.5) can not be taken as a physically
acceptable generalization of the linear Dirac equation. A natural restriction of the
?
form of the nonlinearity F (?, ?) is imposed by demanding relativistic invariance.
This condition ensures independence of the physical processes described by eq. (1.5)
of the choice of inertial reference system (i.e., the nonlinear equation in question has
to satisfy the Poincar?–Einstein relativity principle). It is common knowledge that
e
the Dirac equation with zero mass admits the conformal group C(1, 3) (see e.g. ref.
[31] and the literature cited there). Therefore it is of interest to choose from the set
of Poincar?-invariant equations of the form (1.5) equations that are invariant under
e
the conformal group.
In this section we describe all equations of the form (1.5) that are invariant under
?
the Poincar? group P (1, 3) and its extensions, the group P (1, 3) and the conformal
e
580 W.I. Fushchych, R.Z. Zhdanov

?
group C(1, 3). Let us recall that the extended Poincar? group P (1, 3) (or Weyl group)
e
is an 11-parameter group of transformations {P (1, 3), D(1)}, where D(1) is a one-
parameter group of scale transformations,

? (x ) = e?k? ?(x),
xµ = e? xµ , (2.1)
k, ? = const.
?
The 15-parameter conformal group C(1, 3) consists of the group P (1, 3) and the four-
parameter group of special conformal transformations

xµ = (xµ ? ?µ x · x)? ?1 (x), ? (x ) = ?(x)[1 ? (? · ?)(? · x)]?(x), (2.2)

where ?(x) = 1 ? 2? · x + (? · ?)(x · x), ?µ are parameters of the group, µ = 0, 1, 2, 3.
Hereafter we use the following notation for the scalar product in Minkowski space
R(1, 3):

a · b ? aµ bµ ? g µ? aµ b? , µ, ? = 0, 1, 2, 3,

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( 145 .)



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