ñòð. 55 |

4 ?x0 ?x ?x ?x

1?

B? = 2xµ ?µ? , Aµ? = 2x? ?µ?? .

+ m?[Sµ? , ?? ]+ ?,

2

The tensors ?µ? , ?µ?? , Aµ? and the vector B? correspond to the symmetry operators

?µ , ?µ? , Aµ and B. All quantities (3.11) satisfy the continuity equations

p? ?µ? = 0, p? ?µ?? = 0, p? Aµ? = 0, p? B? = 0

and so generate conservation laws.

230 W.I. Fushchych, A.G. Nikitin

4. Additional symmetry of the Weyl and massless Dirac equations

Here we study the symmetry of the Weyl equation

? µ pµ ? = 0, (4.1)

where ? is the two-component spinor and ? µ the Pauli matrices. Putting

? + ??

(4.2)

?=

i(?? ? ?)

one may rewrite this equation in the Dirac form

? µ pµ ? = 0, (4.3)

where ? µ are the Dirac matrices in the Majorana representation. So we consider the

symmetry properties of equation (4.3) in order to obtain the results which are valued

as for the Weyl equation as for the massless Dirac one.

Theorem 3. The massless Dirac equation has 46 symmetry operators Q ? M1 . These

operators are

(4.4a)

Pµ , Jµ? , Kµ , D, F, F Pµ , F Jµ? , F Kµ , F D, I,

? ?

Aµ = ?µ? xµ + x? ?µ? ? ?µ , ?µ? = ?µ p? ? ?? pµ , (4.4b)

? ? ? F Aµ ,

where Kµ = 2xµ D ? pµ x? x? + 2Sµ? x? , F = i?4 , Pµ , Jµ? and D are given in (2.14)

and (3.5 ).

Proof. This can be carried out in full analogy with the proof of theorem 2. The

general solution of the system (3.8) for the case m = 0 has the form

1

d? = xx · µ? + µ? x2 ? x2 + x ? ? ? + ? ? x + ?? x0 x + ?? + ? ? x0 ,

0

2

d2 = x ? ? + ?x ? ?x0 + ?, d3 = x ? ? + ?x + ?x0 + ?,

? = 0, 1,

1

A? = ? x · µ? x0 + ?? x2 + x2 + ? ? x0 + ? ? x + ?? ,

0

2

(4.5)

3

a = ? i (x · µ + ? x0 + ? ) , x = ??a ,

? ? ? ? 0 a 0

c = ?, g = ?,

2

1

g a = ??a , f 0a = (??a + ?0 xa + µ0 x0 + ?a ),

1 0

a

2

1

f ab = ? ?abc (µ1 x0 + ?1 xc + ?c + ?c

1 0

c 1

2

and includes 46 independent parameters denoted by the Greek letters. Substituting

(3.5) and (4.5) into (3.7) and using equation (4.3) one obtains a general expression

for the symmetry operator of the massless Dirac equation in the form of a linear

combination of the operators (4.4). Thus the theorem is proved.

Among the operators (4.4) there are exactly fourteen symmetry operators, which

do not belong to the enveloping algebra generated by the conformal group generators

Pµ , Jµ? , Kµ , D and by the operator F = i?4 . These essentially new symmetry

operators are given in (4.4b).

Operators (4.4) transform the real wave function ? (4.2) into real wave function

? = Q? and so they are also the symmetry operators for the Weyl equation (4.1).

On the new invariance algebras and superalgebras 231

Incidentally the linear transformations of ? (4.2) generate linear and antilinear trans-

formations of a two-component Weyl spinor.

The operators (4.4) do not form a basis of the Lie algebra. However, one may

consider different subsets of the operators (4.4) which have the structure of the Lie

algebra or superalgebra. Thus the operators (4.4a) form the basis of 32-dimensional

Lie algebra including the Lie algebra of the conformal group. The operators Jµ? , ?µ? ,

?

F and ?µ = F Pµ satisfy the following commutation and anticommutation relations:

[? µ? , ??? ]+ = ?2i[Jµ? , p? p? ] = 2(gµ? p? p? + g?? pµ p? ? gµ? p? p? ? g?? pµ p? ),

? ?

[Jµ? , ??? ] = i(gµ? ??? + g?? ?µ? ? gµ? ??? ? g?? ?µ? ), F 2 = I, (4.6)

? ? ? ? ?

[? µ? , ?? ]+ = [?µ? , F ]+ = 0,

? [?µ , ?? ]+ = 2pµ p? , [?µ , F ]+ = 2Pµ ,

where the symbol [A, B]+ denotes the anticommutator [A, B]+ = AB + BA.

It follows from (2.5) and (4.6) that the set of symmetry operators {Pµ , Jµ? , pµ p? , I,

F, ?µ , ?µ? } form the basis of the Lie superalgebra (which includes ten symmetry

?

operators pµ p? not belonging to the class M1 ).

5. The symmetry and supersymmetry of Maxwell’s equations

We shall write Maxwell’s equations for the electromagnetic field in vacuum in the

following form [17]:

L1 ? ? (i?/?x0 + ?2 S · p)? = 0,

(5.1)

La ? ? (pa ? S · ppa )? = 0.

2

Here

0 ?1 sa 0

(5.2)

?2 = i , Sa = i ,

10 0 sa

where 1 and 0 are unit and zero 3 ? 3 matrices, sa are the generators of irreducible

representation D(1) of the group SO(3) with the matrix elements (sa )bc = i?abc . The

symbol ? denotes the six-component function, ? = column (E1 , E2 , E3 , H1 , H2 , H3 ),

where Ea and Ha are the components of the vectors of electric and magnetic fields

strengths.

It is well known that the Maxwell equations are invariant under the conformal

group C(1, 3) and under the group H of Heaviside–Larmor–Rainich transformations.

Moreover it was found [14, 15, 16, 17] that these equations also have the additional

hidden symmetry in the class of integro-differential operators which is determined by

the algebra GL(2, C). It was demonstrated also that GL(2, C) is the most extensive

invariance algebra of the Maxwell equations if one sopposes the symmetry operators

do not depend on x.

Here we study the symmetry of the Maxwell equations in quite another aspect. The

requirement that the symmetry operators belong to a finite-dimensional Lie algebra is

very important if one is interested in studying the symmetry groups of the equations

considered. However for many applications (e.g. for constructing conservation laws)

this requirement is not essential. So we do not require that the symmetry operators of

Maxwell’s equations should belong to a finite-dimensional Lie algebra but restrict the

class of operators considered by the second-order differential operators with constant

232 W.I. Fushchych, A.G. Nikitin

matrix coefficients. In other words we consider the symmetry operators of a form

such that

(5.3)

Q = dab pa pb + cb pb + g, a, b = 1, 2, 3,

where dab , cb and g are 6 ? 6 numerical matrices. The operators (5.3) do not depend

on po inasmuch as one may always p0 ? via ?2 S · p? according to equation (5.1). Let

us denote the class of the operators (5.3) by the symbol M2 .

We shall see that the Maxwell equations have non-trivial symmetry operators in

the class M2 which do not belong to the enveloping algebra of the conformal group

generators. On the other hand the analysis of more extensive classes of the Maxwell

equation symmetry operators is very complicated and cannot be carried out within

the framework of the present paper.

The invariance condition for equation (5.1) under the operators (5.3) may be wri-

tten in the following general form [17]

2a,d

[L1 , Q] = ?Q L1 + ?Q La ,

1 1a

[La , Q] = ?Q L1 + ?Q Ld ,

2a

(5.4)

2 2 2

where in our case ?Q = ?Q ? 0 since the commutators on the LHS cannot depend

1 2a

on p0 , and

2a,d a,d

?Q = gbc pb pc + fb pb + ha ,

1a a a

?Q = gbc pb pc + fc pc + ha,d ,

a,d

(5.5)

where gbc , fb , hk are numerical matrices, k = a or k = a, d.

k k

Any of the matrices in (5.3) and (5.5) can be represented as a linear combination

of the matrices Dc and G? , where

?

cd

G? = ?? (?cd ? Sc Sd ? Sd Sc ).

?

Dc = ?? Sc , cd

Here ?? are the 6 ? 6 Pauli matrices commuting with Sa of (5.2). Calculating the

commutators in (5.4) and equating the coefficients by the linearly independent matri-

ces and differential operators one may prove the following statement.

Theorem 4. The Maxwell equations (5.1) have ten linearly independent symmetry

operators Q ? M2 which do not belong to the enveloping algebra of the Lie algebra

of the group C(1, 3) ? H. These operators have the form

? (5.6)

Qab = ?1 qab , Qab = ?3 qab ,

where

qab = [(S ? p)a , (S ? p)b ] ? p2 ?ab , p2 = p2 + p2 + p2 .

1 2 3

Proof. The proof can be carried out in full analogy with the proofs of theorems 2 and

3 and so can be omitted. We note only that equations (5.3)–(5.5) are satisfied by the

46 linearly independent operators given below:

?0 S · p, ipa S · p,

?0 , i?0 pa , ? 0 pa pb , i?2 ,

(5.7)

?

?2 S · p, i?2 pa S · p,

? 2 pa , i?2 pa pb , Qab , Qab ,

?

where Qab and Qab are given in (5.6). All operators of (5.7) with the exception of

?

Qab and Qab can be expressed via Pa , S · p = 1 ?abc Jab Pc and ?2 , where Jab and Pa

2

On the new invariance algebras and superalgebras 233

are the Poincar? generators given by the formulae (2.14) with 1 i[?a , ?b ] > ?abc Sc ,

e 4

?2 is the matrix of (5.2) (which is the generator of the Heaviside–Larmor–Rainich

transformations).

Note 1. From twenty operators (5.6) exactly ten are linearly independent in so far as

? ? ?

(Q11 + Q22 + Q33 )? = (Q11 + Q22 + Q33 )? = 0,

where ? is an arbitrary solution of equations (5.1).

1 1a 2a

Note 2. The operators ?Q , ?Q and ?Q from (5.4) which correspond to the symmetry

2c,d 2c,d

operators (5.6) are zero matrices. For ?Qab and ?Qab one obtains by direct calculation

2a,d 2a,d

?Qbc = i?2 ?Qbc = ??1 ?ad [(S ? p)a , (S ? p)b ]+ .

So we have determined the complete set of the Maxwell equation symmetry ope-

rators in the class M2 . Using the notation given in (5.2) and below formula (5.2), it

?

is not difficult to represent the transformations ? > Qab ? and ? > Qab ? generated

by operators (5.6), in the terms of the electromagnetic field strengths

Ec > gab Hd , Hc > gab Ed ,

cd cd

(5.8)

Ec > gab Ed , Hc > ?qab Hd ,

cd cd

(5.9)

where

gab = pa pb ?cd ? pa pc ?bd ? pb pc ?ad + p2 (?ac ?bd + ?bc ?ad ? ?ab ?cd ).

cd

The invariance of Maxwell’s equations under transformations (5.8) and (5.9) can be

easily verified by direct calculation.

The operators (5.6) do not form a basis of a Lie algebra. However, one may

consider subsets of the operators (5.6) which can be extended to the Lie superalgebras.

One of these subsets includes the following operators:

1 1 ?

Q1 = Q2 =

?abc ca Qbc , ?abc ca Qbc ,

2 2 (5.10)

1

Q = S · p,

3 4 5 2

Q = ca cb pa pb , Q =p ,

2

where ca are arbitrary numbers satisfying the condition ca ca = 1. The operators (5.10)

satisfy the relations

2 2 2

2

= Q6 ? Q4 ? Q5

[Qa , Qb ]+ = 2?ab (Qa ) , Q1 = Q2 ,

32

? = Q5 ?, [Qa , Q4 ] = [Qa , Q5 ] = [Q4 , Q5 ] = 0

Q

and so form the basis of the Lie superalgebra together with the operator Q6 . This

superalgebra can be extended by adding the operators Q6+a = i? 2 S · pQa , Q9+a =

i? 2 S · p(Qa )2 and Q12+a = p2 (Qa )2 , a = 1, 2, 3, which satisfy the relations

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