ñòð. 77 |

11. Lloyd S.P., Acta Mechanica, 1981, 38, 85–98.

12. Ovsyannikov L.V., Group analysis of differential equations, Moscow, Nauka, 1978 (Engl. transl.

New York, Academic, 1982).

13. Patera J., Winternitz P., Zassenhaus H., J. Math. Phys., 1989, 16, 1597–1624.

W.I. Fushchych, Scientific Works 2002, Vol. 4, 320–336.

On the connection between solutions of Dirac

?

and Maxwell equations, dual Poincare

invariance and superalgebras of invariance

and solutions of nonlinear Dirac equations

W.I. FUSHCHYCH, W.M. SHTELEN, S.V. SPICHAK

The connection between solutions of massless Dirac and Maxwell equations is establi-

shed. It is shown that the massless Dirac equation is invariant under three different

representation of the Poincar? algebra corresponding to spins 1 and 1 and 0, and under

e 2

three superalgebras. All generators of these symmetry algebras and superalgebras are

local (differential operators of first order). A system of two Dirac equations with masses

m and ?m has analogous symmetry properties. Invariant nonlinear generalizations of

this system are described. We construct the complete set of P (1, 3)-inequivalent ans?tze

a

of codimension 1 for all representations of Poincar? algebra discused. These ans?tze are

e a

used for reduction and finding exact solutions of some nonlinear Dirac equations.

1. Introduction

It is well known that the Dirac equation describes a particle with spin- 1 , or2

a fermionic field, because it is invariant with respect to the representation D( 1 , 0) ?

2

D(0, 1 ) of the Poincar? algebra AP (1, 3). In this paper we will show that the massless

e

2

Dirac equation as well as the system of two coupled Dirac equations with masses m

and ?m are invariant not only with respect to the spin- 2 representation of AP (1, 3)

1

but also under integer spin representations of AP (1, 3). This means that Dirac equa-

tions describe not only fermionic fields but also bosonic ones.

In section 2 we obtain formulae of connection between solutions of the massless

Dirac equation and Maxwell equations for a vacuum, so that one can construct soluti-

ons of the Dirac equation knowing solutions of the Maxwell equations and vice versa.

Further, we show that the massless Dirac equation is invariant under three different

representations of the Poincar? algebra AP (1, 3) and under three superalgebras. All

e

generators of these symmetries are differential operators of first order and belong to

the maximal in the sense of Lie invariance algebra of the equation. We shall call

invariance of an equation, with respect to different representations of the Poincar? e

algebra, dual Poincar? invariance.

e

In section 3 we study dual Poincar? invariance of the Dirac equation with non-zero

e

mass and prove that the system of two coupled Dirac equations with masses m and

?m possesses this symmetry. It is worthwhile to note that the same Dirac system

was studied by Fushchych [1, 2] and by Petroni et al [12, 13]. Fushchych [1, 2] had

shown that the most symmetric (including discrete symmetries) spinor representation

of the Poincar? algebra is realized only on the system of two coupled Dirac equations

e

and such a realization is impossible on a single Dirac equation with non-zero mass.

We prove that the Dirac system under study is also invariant under two superalgebras.

Nonlinear dual Poincar? invariant generalizations of the equations are considered.

e

J. Phys. A: Math. Gen., 1991, 24, P. 1683–1698.

On the connection between solutions of Dirac and Maxwell equations 321

In section 4 we construct the complete set of the P (1, 3)-inequivalent ans?tze of

a

codimension 1 for all representations of AP (1, 3) discussed in the previous sections.

These ans?tze reduce corresponding Poincar? invariant equation to a system of ordina-

a e

ry differential equations (ODEs). Here we essentially used results on the subalgebraic

classification of AP (1, 3) of Patera et al [11] and Grundland et al [7]. It will be noted

that the P (1, 3)-inequivalent ans?tze of codimenions 1 and 3 for the spin- 1 Dirac

a 2

field are fully described in Fushchych and Zhdanov [5], Fushchych and Shtelen [4]

and Fushchych et al [6]. Using ans?tze constructed, we make reductions and find

a

exact solutions of some nonlinear Dirac equations. An example solution of a linear

Dirac equation is considered. This solution is obtained by making use of the vector

representation of AP (1, 3) of the coupled Dirac equations. It has an unusual structure

and can be obtained as the invariant solution of the non-Lie symmetry operator of

second order. In conclusion we give operators which transform the fermionic ans?tze a

into bosonic ones.

The massless Dirac equation and Maxwell equations

Consider the massless Dirac equation

i??? ? i? µ ?µ ? = 0, (2.1)

where ? = ?(x) is a four-component complex function (column), x = {x0 = t, x} ?

R(1, 3), µ = 0, 3, ?µ = ?/?xµ and ? µ are 4 ? 4 Dirac matrices,

? ? ? ?

1 00 0 0 0 0 1

?0 ? ?0 0?

10 0? 0 1

?0 = ? , ?1 = ? ?,

?0 0? ?0 0?

0 ?1 ?1 0

?1 ?1

0 00 0 0 0

? ? ? ? (2.2)

?i

0 00 0 0 10

?0 ? ?0 0 ?1 ?

0i 0? 0

?2 = ? , ?3 = ? ?.

?0 0? ? ?1 0 0?

i0 0

?i 00 0 0 1 00

There is a connection between solutions of (2.1) and the Maxwell equations for

a vacuum [15]:

?E

?

E? = rot H, div E = 0,

?t

(2.3)

?H

?

H? = ?rot H, div H = 0,

?t

where E = (E1 , E2 , E3 ) and H = (H1 , H2 , H3 ) are vectors of electric and magnetic

field. To establish this connection let us decompose an arbitrary solution of (2.1)

intoreal and imaginary parts using the notation of Ljolje [9]:

? ? ? ?

?D1 D2

? D3 ? ? ?F ?

=? ? ? ?. (2.4)

? = ?real + i?imag ? ?B2 ? + i ? ?B1 ?

?G B3

322 W.I. Fushchych, W.M. Shtelen, S.V. Spichak

Theorem 1. Let ? defined by (2.4) be an arbitrary solution of the massless Dirac

equation (2.1). Then the functions

t

?

E =D+? G(?, x)d? + ?G(t0 , x),

t0

(2.5)

t

?

H =B+? F (?, x)d? + ?F (t0 , x),

t0

? ?

where G(t0 , x) and F (t0 , x) satisfy the Poisson equations

?G(?, x) ?F (?, x)

? ? (2.6)

?G(t0 , x) = , ?F (t0 , x) = ,

?? ??

? =t0 ? =t0

t0 is an arbitrary constant, are solutions of the Maxwell equations (2.3).

Prof. First of all we note that after substitution of (2.4) into (2.1) and separation into

real and imaginary parts we get Maxwell equations with currents

? ?

D ? rot B = ??G, div D = ?G,

(2.7)

? ?

B + rot D = ??F, div B = ?F ,

where D = (D1 , D2 , D3 ), B = (B1 , B2 , B3 ) and the dot means differentiation with

respect to t. So, the Dirac equation (2.1) and the system (2.7) are fully equivalent.

Therefore, taking into account (2.7) and the well known fact that every component of

the ?-function (2.4) obeying (2.1) satisfies the wave equation 2? = 0 (in particular,

?G(?, x) = ? 2 G(?, x)/?? 2 ) we find after substitution of (2.5) into (2.3)

? ?

E ? rot H = D + ?G ? rot B = 0,

t

?

div E = div D + ?G(?, x)d? + ?G(t0 , x) =

t0

t

? 2 G(?, x) ?

= div D + d? + ?G(t0 , x) =

?? 2

t0

?G(?, x)

? ?

= div D + G ? +?G(t0 , x) = 0.

d? ? =t0

In the last equality we have used (2.6). In the same spirit one can prove the validity

of the theorem for the second pair of Maxwell equations (2.3). Thus, the theorem is

proved.

The inverse statement also holds true.

Theorem 2. Let there be given a solution E, H of the Maxwell equations (2.3) and

two solutions F and G of the scalar wave equation

2F = 2G = 0. (2.8)

Then the ?-function (2.4) witn components F , G and

t

?

Da = Ea ? ?a G(?, x)d? + G(t0 , x) ,

t0

(2.9)

t

?

Ba = Ha ? ?a F (?, x)d? + F (t0 , x) ,

t0

On the connection between solutions of Dirac and Maxwell equations 323

? ?

where a = 1, 2, 3, G(t0 , x) and F (t0 , x) are determined from (2.6), is a solution of

the massless Dirac equation (2.1).

Proof. Let us use the equivalence between the Dirac equation (2.1) and the system

(2.7). Having substituted (2.9) into (2.7) and taking into account (2.3), (2.8) and

(2.6), we get

? ?

D ? rot B + ?G = E ? ?G + ?G ? rot H = 0,

t

? ? ?

?G(?, x)d? ? ?G(t0 , x) + G = 0.

div D + G = div E +

t0

Analogously one has to act to prove the theorem for the rest of the equations of

system (2.7).

Theorem 2 has an important corollary: choosing F = G = 0 we get from (2.9)

D = E, B = H, and in this case formula (2.4) takes the particularly simple form

? ?

?E1 + iE2

? ?

E3

?=? ? (2.10)

? ?H2 ? iH1 ? .

iH3

So, if E and H satisfy the Maxwell equations (2.3), then ? given by (2.10) automati-

cally satisfies the Dirac equation (2.1), and one can consider relation (2.10) as a repre-

sentation of the spinor field ? by an electromagnetic field E, H. It is appropriate

to note that if E and H are transformed under Lorentz boost as an electromagnetic

Maxwell field, then the ?-function (2.10) is not transformed like a Dirac spinor (this

point will be discussed in detail below). It will be also noted that, according to

theorem 1, the procedure of obtaining solutions of the vacuum Maxwell equations

(2.3) from those of the massless Dirac equation (2.1) and the associated Poisson

equations (2.6) is unique to within a gauge transformation, whereas the inverse

procedure, Maxwell > Dirac, involves ambiguities due to the arbitrary choice of

additional scalar fields F and G satisfying (2.8). When we construct solutions of

Maxwell equations via solutions of the massless Dirac equation using formulae (2.5),

? ?

then we have arbitrariness in determining F and G. But this arbitrariness can be

considered as gauge transformetions E > E = E + ?f (x), H > H = H + ?g(x)

(f and g are arbitrary scalar functions satisfying the Laplace equation ?f = ?g = 0),

which leave invariant the Maxwell equations (2.3). An analogous situation is when

considering the inverse procedure (formulae (2.9), Dirac equation in the form (2.7)).

Consider an example. Let us take solutions of the Maxwell equations (2.3) and

wave equations (2.8) in the form

E = ? ? x, H = ?2?t, F = G = 3t2 + x2 , (? = const).

Then, by means of (2.9) and (2.4) one easily finds the following solution of the Dirac

equation (2.1):

? ?

?[(? ? x)1 ? 2tx1 ] + i[(? ? x)2 ? 2tx2 ]

? ?

[(? ? x)3 ? 2tx3 ] ? i(3t2 + x2 )

? ?

?=? ?.

? ?

2t(?2 + x2 ) + 2it(?1 + x1 )

?(3t2 + x2 ) ? 2it(?3 + x3 )

324 W.I. Fushchych, W.M. Shtelen, S.V. Spichak

In terms of D, B, F , G from (2.4)

?

?? = D 2 ? B 2 + F 2 ? G2 (2.11)

and in the case of solution ? considered above we have

?

?? = ?2 x2 ? (? · x)2 ? 4t2 (?2 + 2? · x).

Let us make up a four-component ?-function as

? ?

?0

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