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журн., 1991, 43, № 11, 1456–1470.

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Доп. АН УРСР, Сер. А, 1988, № 1, 28–32.

5. Серов Н.И., Условная инвариантность и точные решения нелинейного уравнения теплопрово-

дности, Укр. мат. журн., 1990, 42, № 10, 1370–1376.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 93–99.

A new conformal-invariant non-linear spinor

equation

W. FUSHCHYCH, W. SHTELEN, P. BASARAB-HORWATH

We propose a new model for a spinor particle, based on a non-linear Dirac equation.

We invoke group invariance and use symmetry reduction in order to obtain a multi-

parameter family of exact solutions of the proposed equation.

1. Introduction

Since the discovery of the electron, many people have proposed and discussed the

hypothesis that the mass of the electron is generated by an electromagnetic field,

which the electron produces itself, so that the electron can be thought of as localized

electromagnetic energy. In other words, this means that the electron is described by

a non-linear dynamical system (see, for instance [1, 2] for these ideas). We propose

a realization of this old and interesting physical idea in the framework of the classical

theory of spinor fields. For the electron, we propose the following Lorentz-invariant

spinor equation

?

(i?? ? m(u, v, ??, jµ j µ ))? = 0, (1.1)

where

?? = ? µ ?µ , µ = 0, 1, 2, 3

and the ? µ are the Dirac matrices

?a

1 0 0

0 a

?= , ?= , a = 1, 2, 3,

?1 ?? a

0 0

where the ? a are the 2 ? 2 Pauli matrices

0 ?i

01 1 0

?1 = , ?2 = ?3 =

, ,

?1

10 i0 0

1 1 ?

u = ? Fµ? F µ? , v = ? Fµ? F µ? ,

2 4

where F µ? is an antisymmetric tensor and

?

?µ F µ? = j ? , ?µ F µ? = 0

with

1

?

Fµ? = ?µ??? F ??

2

and ?µ??? is the antisymmetric Kronecker symbol.

Preprint LiTH-MAT-R-93-05, Department of Mathematics, Link?ping University, Sweden, 8 p.

o

94 W.I. Fushchych, W. Shtelen, P. Basarab-Horwath

The electromagnetic field which the electron itself produces satisfies Maxwell’s

equations:

?

?? F µ? = j? , j? = e?? ? ?,

with (1.2)

where e is the charge of the electron.

We can interpret (1.1) as follows: the mass, m, of an electron is generated by the

electromagnetic field F µ? , and its own spinor field ?. In the usual Dirac equation,

m is a parameter which does not depend on the electromagnetic and spinor fields.

Equation (1.1), in contrast to the standard Dirac equation, is a complicated non-linear

equation, and as a result one has the following problem: how does one find at least

some non-trivial solutions of such an equation?

For the case of m depending only on the spinor field, some classes of exact

solutions of (1.1) have been found [5, 6, 10, 11]. In order to construct solutions of

(1.1), (1.2), we first examine the symmetries of this system, and then we give some

families of exact solutions. The system (1.1), (1.2) is non-linear even for m = const,

and can be thought of as a first modification of the Dirac equation in our approach.

2. Symmetries

?

In the spinor equation (1.1), (1.2), we shall consider the fields F µ? , ?, ? as

independent, and we shall look for symmetry operators of that system in the form

? (1) ? (2) ? ?

(3)

X = ?µ + ?µ + ?µ ? + ?µ? ,

?xµ ?F µ?

??µ ?µ

?

where the coefficients are functions of x, ?, ?, F µ? . In finding these symmetry

operators, we use the method of Lie [4, 8, 9]. Indeed, after a painstaking calculation,

we obtain the following:

Theorem 1. The maximal point symmetry algebra of the system of (1.1), (1.2), with

m = const, has as basis the following vector fields:

? ? ?

Jµ? = xµ ?? ? x? ?µ + (?µ? ?)? ? F ??

?µ = ?/?xµ , + F µ? ,(2.1)

??? ?F ?? ?F µ?

? ?

D = ?µ + F µ? (2.2)

,

??µ ?F µ?

?

P = P µ? (2.3)

,

?F µ?

?

where ?µ P µ? = 0, ?µ P µ? = 0 and

i

?µ? = ? [? µ , ? ? ].

4

Remark 1. The operator D generates scale transformations in the space of the fi-

eld variables ?m , F µ? , not in Minkowski space R(1, 3). The operators ?µ , Jµ? , D ,

generate the extended Poincar? algebra [4].

e

If we assume dependence of the mass on the Lorentz-invariant quantities u, v,

defined in (1.1), (1.2), we retain invariance under the Poincar? group, but not always

e

under the extended Poincar? group. In fact, we have the following result:

e

A new conformal-invariant non-linear spinor equation 95

Theorem 2. The system (1.1), (1.2), where m is a function of the invariants u, v

defined in (2.1), (2.2), is invariant under the algebra generated by (1.3), (1.4) if and

only if

v

m , u = 0,

u

m=

m = const, u=0

Remark 2. Theorem 2 implies that there exists a wide class of non-linear systems

of the form (1.1), (1.2), which are invariant with respect to the extended Poincar?e

algebra. This is so when we assume that the mass depends only on the electromagnetic

field.

3. Conformally invariant equations

In this paragraph, we shall describe equations of the form (1.1), (1.2), which are

invariant under the conformal group, under the assumption that the mass has the

following dependence on the fields:

?

m = ?1 F1 (u, v) + ?2 (??)k . (3.1)

The conformal group, C(1, 3) is well-known (see for instance [4], [5]). It consists

of the Poincar? group together with the following non-linear transformations:

e

xµ ? cµ x2

(3.2)

xµ = ,

?

? (x ) = ?(1 ? (?c)(?x))?(x), (3.3)

Fµ? (x ) = ? 2 Fµ? + 2?{x? [(2(cx) ? 1)(cµ F?? ? c? F?µ ) ?

? c2 (xµ F?? ? x? F?µ )] + c? [x? F?? ? x? F?µ ? (3.4)

? x2 (cµ F?? ? c? F?µ )] + 2(cµ x? ? c? xµ )F?? c? x? },

xµ = e? xµ , (3.5)

3

? (x ) = e? 2 ? ?(x), (3.6)

Fµ? (x ) = e?2? Fµ? , (3.7)

where the primes denote transformed quantities, ? and cµ are arbitrary real constants,

cx = cµ xµ , c2 = cµ cµ , x2 = xµ xµ .

Applying Lie’s method for calculating symmetry operators, one can prove the

following result:

Theorem 3. The system of equations (1.1), (1.2), with mass given by (3.1), is invari-

ant under the conformal group if and only if k = 1 and

3

?

?u 4 F v , u = 0,

1

u (3.8)

F1 (u, v) =

?v 4 ,

1

u = 0,

where F is an arbitrary, smooth function.

96 W.I. Fushchych, W. Shtelen, P. Basarab-Horwath

One can easily verify that (1.1), (1.2), with mass defined by (3.1), is indeed invari-

ant under the scale transformations (2.4)–(2.6). Substituting these into the equations

yields

[i?? ? ?1 F1 (e?4? u, e?4? v) ? ?2 e?(1?3k) (??)k ]? = 0,

?

? ?

?? F µ? = e?? µ ?, ?? F µ? = 0.

The condition of invariance then gives

e? F1 (e?4? u, e?4? v) = F1 (u, v), ?(1 ? 3k) = 0

1

which immediately implies k = 3, and, differentiating with respect to ?, that F1

satisfies the equation

?F1 ?F1

4u + 4v = F1 .

?u ?v

The general solution of this equation is easily shown to be that given by (3.8).

Conformal invariance follows by using the transformations

? ?

?? > ? 3 ??, u > ? 4 u, v > ? 4 v.

Remark 3. Requiring conformal invariance narrows quite considerably the class

of admissible systems (1.1), (1.2). Fixing the function F u , we obtain different

v

conformally-invariant equations for a spinor particle.

4. Exact solutions

We shall construct a class of exact solutions for the simplest conformally-invariant

system (1.1), (1.2), namely for the case F = 1, so that our system becomes

1

?1

i?? ? ?1 u 4 ? ?2 (??) 3 ? = 0,

(4.1)

? ?

?? F µ? = e?? µ ?, ?? F µ? = 0.

We shall look for solutions of this system by the method of reduction [4], that is we

reduce the system of partial differential equations to systems of ordinary differential

equations. For these, we use the following ansatzes [4, 5, 6, 7, 10, 11]:

F µ? (x) = f µ? (?), (4.2)

?(x) = ?(?),

where ?(?) is a four-component vector, f µ? (?) an antisymmetric tensor, ? = ?x,

with ? a constant vector satisfying ? 2 = 1. Substituting (4.2) into (4.1), we obtain

the reduced system of ordinary differential equations

1 1

i(??)? ? ?1 z 4 + ?2 (??) 3 = 0,

? ?

(4.3)

?? f?µ? = e?? µ ?, ?? f µ? = 0

?

with z = ? 1 fµ? f µ? and the dot denotes differentiation with respect to the argu-

2

ment ?. Since f µ? is anisymmetric, it follows that ?µ ?? f?µ? = 0, so that the second

equation in (4.3) yields ?(??)? = 0. Using the relation

?

? µ ? ? + ? ? ? µ = 2g µ?

A new conformal-invariant non-linear spinor equation 97

and the fact that ? is chosen so that ? 2 = 1, it is easy to show that (??)(??) = 1.

Multiplying the first equation of (4.3) on the left by ?(??) we then obtain

?

?? = 0.

??

We therefore find that ? satisfies

(4.4)

?? = const,

? ?(??)? = 0.

?

These equations imply that we should look for solutions ? in the form

(4.5)

? = exp(i(??)g(?))?,

where g(?) is a function we must find and ? is a constant vector which satisfies

?(??)? = 0. Since (??)2 = 1, it follows that

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