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?µ?? = i
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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