ñòð. 78 |

? = i?? ? 1 ? , (2.12)

? ?2 ?

?3

where ?0 , . . . , ?3 are arbitrary solutions of the wave equation, that is 2?µ = 0. Since

(i??)2 = 2, then the ?-function (2.12) automatically satisfies the Dirac equation (2.1)

for any set of ?µ , 2?µ = 0. So, (2.12) and (2.4), (2.5) give the following chain

of solutions: scalar wave equation > massless Dirac equation > vacuum Maxwell

equations.

It will be noted that Shtelen [14] and Fushchych et al [6] described a simple pres-

cription for obtaining solutions of linear partial differential equations with nontrivial

symmetry. It consists of the following. Let there be given a solution u of the wave

equation (2u = 0). Then the functions

u1 = Ku, u2 = Ku1 , (2.13)

...

where K = 2cxx? ? x2 c? + 2cx (generator of conformal transformations) and cµ are

arbitrary constant, will be also solutions u = 1 we get from (2.13)

1 1

u2 = (cx)2 ? c2 x2 , u3 = (cx)3 ? (cx)c2 x2 , (2.14)

u1 = cx, ...

4 2

For further analysis it is convenient to consider the Dirac equation (2.1) together

with its conjugation and write it uniformly as

i?µ ?µ ? = 0, (2.15)

where ? = ?(x) = column (??), ? = ?0 ?? , ?µ are 8 ? 8 matrices,

??

?µ 04

µ

(2.16)

?= ,

?(? µ )T

04

? µ are Dirac matrices (2.2), 04 is a 4 ? 4 zero matrix.

Symmetry properties of (2.15) were studied first by Dirac who showed that the

equation is conformally invariant. Later, Pauli and Touschhek found that this equation

also admits an eight-parameter group, G8 , of component transformations. And, finally,

Ibragimov [8] proved that a 23-parameter group, G23 = C(1, 3) ? G8 , is the maximal

in the sence of the Lie invariance group of the equation. Relativistic invariance of

(2.15) is usually understood as invariance with respect to the spinor representation

1 1 1 1

, 0 ? D 0, ?D , 0 ? D 0, (2.17)

D

2 2 2 2

On the connection between solutions of Dirac and Maxwell equations 325

of the Poincar? group P (1, 3) (it means that ? is transformed under the Lorentz

e

boost as a spinor). However, the invariance of (2.15) under the Pauli–Touschek eight-

parameter group allows two additional representations of AP (1, 3), which are realized

on the set of solutions of (2.15), namely

D(1, 0) ? D(0, 1) ? D(0, 0) ? D(0, 0) (2.18)

and

11 11

?D (2.19)

D , , .

22 22

The explicit form of basis elements of AP (1, 3) for representations (2.17)–(2.19) is

?

, Jµ? = xµ P? ? x? Pµ + Sµ? ,

AP (k) (1, 3) = (k) (k)

(2.20)

Pµ = µ

?x

where k = 1, 2, 3 corresponds to (2.17)–(2.19), respectively;

gµ? = {1, ?1, ?1, ?1}?µ?

xµ = gµ? x? ,

(k)

and matrices Sµ? are

1 (3) (2) (3) (2)

Sµ? = ? [?µ , ?? ],

(1) (2) (1)

Sµ? = Sµ? + Qµ? , S01 = S01 , S02 = S02 ,

4 (2.21)

(3) (2) (3) (2) (3) (2) (3) (2)

? 2Q03 , ? 2Q13 , ? 2Q23 .

S03 = S03 S12 = S12 , S13 = S13 S23 = S23

Here ?µ are the same as in (2.16); Qµ? are six basis elements of the Pauli–Touschek

algebra, they are 8 ? 8 matrices of the form

?i? 0 ? 2 ?? 0 ? 2

1 1

04 04

Q01 = , Q02 = ,

?i? 0 ? 2 ?? 0 ? 2

04 04

2 2

??5

1 i

0 I4 04

(2.22)

Q03 = , Q12 = ,

?I4

04 ?5 04

2 2

?? 1 ? 3 ?1?3

1 i

04 04

Q13 = , Q23 = ,

?? 1 ? 3 ?? 1 ? 3

04 04

2 2

where

02 I2

?5 = i? 0 ? 1 ? 2 ? 3 =

I2 02

I2 , I4 are 2 ? 2 and 4 ? 4 unit matrices. It will be noted that the action of operators

(2.20) is defined in the space of the eight-component function introduced in (2.15).

Invariance of (2.15) under AP (2) (1, 3) results in the possibility of representing this

equation in the form (2.7), and invariance of (2.15) under AP (3) (1, 3) allows us to

rewrite it as [9]

1

?µ A? ? ?? Aµ ? ?µ??? (? ? B ? ? ? ? B ? ) = 0,

(2.23)

2

?? A? = ?? B ? = 0,

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

where

? ? ? ?

?A2 ?A1

? ?B 0 ? ? ?

3

? + i ? A2

=? ?. (2.24)

? = ?real + i?imag ? ?B 1 ? ?B ?

?A0

B3

Now consider the following three sets of symmetry operators of (2.15):

SA(k) = Pµ , Jµ? , ?4 , I; Qµ? ,

(k)

(2.25)

(k)

where Pµ , Jµ? and Qµ? are defined in (2.20) and (2.22), ?µ are given in (2.16),

?4 = ?0 ?1 ?2 ?3 . There sets of operators form Lie algebra as well as superalgebras.

(k)

Operators Pµ , Jµ? , ?µ , I are even and Qµ? are odd in corresponding superalgebras.

To prove this statement we write down commutation and anticommutation relations

for these operators.

(k)

Operators Pµ and Jµ? satisfy standard commutation relations of the Poincar? e

algebra AP (1, 3)

[Pµ , P? ] = 0, [P? , Jµ? ] = g?µ P? ? g?? Pµ ,

(2.26)

[Jµ? , J?? ] = g?? Jµ? + gµ? J?? ? gµ? J?? ? g?? Jµ? ,

?4 and I commute with all elements of SA(k) . Further, it is convenient to introduce

the notation

1 1

(k) (k)

(k) (k)

(2.27)

Ra = Q0a , Ta = ?abc Qbc , Na = J0a , Ma = ?abc Jbc .

2 2

It is easy to check that

1

{Ra , Rb } ? Ra Rb + Rb Ra = ?ab ,

2 (2.28)

1

{Ta , Tb } = ? ?ab I, {Ra Tb } = ?ab ?4 .

2

Operators Ra , Ta from SA(1) commute with all even operators of SA(1) . For SA(2)

we have

(2)

[Pµ , Ra ] = [Pµ , Ta ] = 0, [Na , Rb ] = [Ra , Rb ] = ?abc Tc ,

(2) (2)

[Na , Tb ] = [Ra , Tb ] = ??abc Rc , [Ma , Rb ] = [Ta , Rc ] = ??abc Rc , (2.29)

(2)

[Ma , Tb ] = [Ta , Tb ] = ??abc Tc .

Subalgebra SA(3) is isomorphis to SA(2) . The isomorphism is achived by means of

the transformations

R3 > R3 = ?R3 , T1 > T1 = ?T1 , T2 > T2 = ?T2 . (2.30)

So, the structure of superalgebras (2.25) is fully described. The superalgebras (2.25)

do not belong to the semi-simple family, but the quotient by their radical is simply

SO(1, 3).

On the connection between solutions of Dirac and Maxwell equations 327

3. Dirac equations with non-zero mass

possessing dual Poincar? invariance

e

The Dirac equation for a massive particle (field)

(i? µ ?µ ? m)? = 0, (3.1)

where ? µ are given in (2.2) and m is an arbitrary real constant (mass of the particle),

is invariant under a 14-parameter group only [8], which includes the Poincar? group,

e

and identical, phase and two charge-type transformations. As always, we are factoring

out an infinite-demensional ideal, present for any linear equation, and corresponding

to the linear superposition principle. It is to be emphasized that we are considering

group action on the field of real numbers, and therefore identical ? = e? ? (? is an

arbitrary real constant) and phase transformations ? = ei? ? should be distinguished.

The above-mentioned four-parameter group of component transformations is not

sufficient to construct a non-spinor representation of AP (1, 3), as was done in the

case of the massless field. The situation can be improved by considering the system

of two Dirac equations

(i?? ? m)?? = 0, (3.2)

(i?? + m)?+ = 0.

The full information on Lie symmetry of this system gives the following statement.

Theorem 3. The maximal in the sense of the Lie invariance algebra of system (3.2)

is a 26-dimensional Lie algebra A26 = AP (1) (1, 3) ? A16 , with basis elements having

the form

? ?(k) ?(1)

, Jµ? = xµ P? ? x? Pµ + Sµ? ,

AP (1) (1, 3) = Pµ = µ

?x

(3.3)

??

matrices 16 ? 16 of the form ? ?

A16 = ,

??

where

1? ? ?µ 08

?(1) ? ?

Sµ? = ? [?µ , ?? ], ?µ = , ?, ? = I, Q01 , Q02 , Q03 ,

08 ??µ

4 (3.4)

?

?, ? = Q12 , Q13 , Q23 , ?4 , ?4 = ?0 ?1 ?2 ?3

(matrices 8 ? 8 ?µ and Qµ? are defined in (2.16)), and acting in the space of

16-component functions

? = column (?? ?+ ) ? column (?? , ?? = ?0 ?? , ?+ , ?+ = ?0 ?? ).

? ? ? (3.5)

? +

Proof. First of all we write system (3.2) together with its conjugation as

? ?

(i?µ ?µ ? m)? = 0, (3.6)

? ?

where ? = ?(x) is defined in (3.5). To prove the theorem is to find the general form

of infinitesimal operator of invariance

Q = ? µ (x)?µ + ?(x), (3.7)

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

where ? µ (x) are scalar functions and ?(x) is a 16 ? 16 matrix. It can be done by

means of the standard Lie algorithm (see [10]), but the simplest way is to use the

invariance condition in the form

(3.8)

[L, Q] = ?(x)L,

?

where L is the operator of (3.6), L ? i?µ ?µ ? m and ?(x) is some scalar smooth

function. Starting from (3.8) one gets, after some simple but tedious calculations, the

proof of the theorem.

Invariance of the system (3.6) with respect to the matrix algebra A16 (3.3) allows

a vector representation of AP (1, 3), which can be realized on the set of solutions of

this system. This representation is

L(D(1, 0) ? D(0, 1)) ? 4D(0, 0). (3.9)

It is defined by the basis elements

?(2) ?(1) ?

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