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On the new invariance algebras of relativistic equations 15

Finally, if Pµ and Jµ? are the generators of the irreducible representation of

the Poincar? group in Lomont–Moses [18] form, then the formulae (2.1) give the

e

conformal group generators in the form of Bose and Parker [2].

4. The additional symmetry of the Dirac equation with mass m = 0

Some years ago the new invariance algebra of equation (3.1) was found (Fushchych

[6, 7]); this is different from the algebra of the conformal group generators. The basis

elements of this algebra have the form

1

?ab = Sab ? (?a pb ? ?b pa ) (1 + ?a pa ) ,

? ? ?

2

(4.1)

1 1

?4a = ?4 ?a + ?4 pa (1 + ?b pb ) ,

? ?

2 2

where

1/2

pa = pa p?1 , p = p2 + p2 + p2

? , a, b = 1, 2, 3.

1 2 3

The operators (4.1) realise the representation D 1 , 0 ? D 0, 1 of the Lie algebra

2 2

of the group O(4) ? SU (2) ? SU (2), but do not form the closed algebra together with

(3.3), (3.4) or (3.8). Below we will obtain the 23-dimensional invariance algebra of

equation (3.1), which includes the Lie algebras of the groups C4 and U (2) ? U (2).

Theorem 2. The Dirac equation (3.1) is invariant under the 23-dimensional Lie al-

gebra, which is isomorphic to the algebra of generators of the group C4 ?U (2)?U (2).

The basis elements of this algebra have the form

? ?

Pa = pa = ?i Jab = xa pb ? xb pa + Sab ,

P0 = p 0 = i , ,

?t ?xa

iH ?

J0a = x0 pa ? xa p0 ? (1 ? i?4 )?a ?b pb + ?0a , D = xµ pµ + i,

?

2p

Kµ = ?x? x? + Jab Sab p?2 + p?2 pµ + 2 [xµ + (1 ? ?µ0 )(1 ? ?0 )Sµb pb ] D, (4.2)

?

1 H

? ?

?0c = ?4 (?a + ?0 Sab pb ) ,

p ? ?5 = ,

2 p

1 H?

? ?

?ab = ?abc ?0c , ?6 = 1, a, b, c = 1, 2, 3,

2 p

Proof. Let us transform equation (3.1) and the generators (4.2) to a representation

in which the theorem statements may easily be verified immediately. Using for this

purpose the operator

1

V = V ?1 = [1 + ?0 + (1 ? ?0 )?abc Sab pc ] (4.3)

?

2

one obtains

?

L = V LV ?1 = i ? i?4 p, (4.4)

L ? = 0, ? = V ?,

?t

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

Pµ = V Pµ V ?1 = Pµ , Jab = V Jab V ?1 = Jab ,

1

J0a = V J0a V ?1 = x0 pa ? xa p0 + i?0 ?a ,

2

D = V DV ?1 = D = xµ pµ + i,

(4.5)

Kµ = V Kµ V ?1 = ?x? x? pµ + 2xµ D ,

1

?ab = V ?ab V ?1 = Sab , ?0a = V ?ab V ?1 = i?0 ?a ,

? ? ? ?

2

?5 = V ?5 V ?1 = i?4 , ? = V ?6 V ?1 = ?6 ,

? ? ? ? ?

6

It is not difficult to be convinced that the operators (4.4) and (4.5) satisfy the invari-

ance condition (3.2):

? ?

[L , Pµ ]? = [L , Jab ]? = [L , ?µ? ]? = [L , ?? ]? = 0,

[L , K0 ]? = 2i x0 + (xa pa ? i)i?4 p?1 L , [L , Ka ]? = 2i(xa + i?a x0 ?4 )L ,

p

[L , D]? = iL , [L , J0a ]? = ?4 pa L

?

?

and the commutation relations for QA ? Pµ , Jµ? , Kµ , D , ?µ? , ??

[Pµ , J?? ]? = i(gµ? P? ? g?? Pµ ),

[Pµ , P? ]? = 0,

[Jµ? , J?? ]? = i(gµ? J?? + g?? Jµ? ? gµ? J?? ? g?? Jµ? ),

[Pµ , D ]? = ?iPµ , [Kµ , D ]? = iKµ , [Jµ? , D ]? = 0,

?

[Pµ , K? ]? = 2i(Jµ? ? ?µ? ? gµ? D ),

? ? ? ? ? ? ?

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

? ? ? ?

[?µ? , P? ]? = [?µ? , D ]? = [?µ? , K? ]? = [?? , QA ]? = 0.

The algebra (4.6) is isomorphic to the algebra of generators of the group C4 ?

U (2) ? U (2). The theorem is therefore proved.

We note that the subsidiary condition (3.8) is not invariant under the transforma-

?

tions which are generated by the operators ?µ? . Therefore the Weyl equation (3.7) is

not invariant relative to the whole algebra (4.2), but is invariant with respect to its

subalgebra C4 .

It should be emphasised that the generators (4.2) belong to the class of nonlocal

integro-differential operators, and therefore one cannot obtain them in the classical

Lie approach.

5. The symmetry of Maxwell’s equations

The Maxwell equations for a free electromagnetic field have the form

?H ?E

p?E =i p ? H = ?i

, ,

?t ?t (5.1)

p · E = 0, p · H = 0,

where E and H are the vectors of the electric and magnetic field strengths.

On the new invariance algebras of relativistic equations 17

Equations (5.1) are invariant under the conformal group. It is well known that

these equations are also invariant under the transformations (Heaviside [14], Lar-

mor [15])

Ea > Ha , Ha > ?Ea (5.2)

and under the more general ones (Rainich [25])

Ea > Ea cos ? + Ha sin ?,

(5.3)

Ha > Ha cos ? ? Ea sin ?,

We now demonstrate that the summetry of the Maxwell equations is more exten-

sive, namely that the equations (5.1) are invariant under the set of transformations

which realise the representation of the group U (2) ? U (2) and include (5.3) as a one-

parameter subgroup. The theorem about such an invariance of the Maxwell equations

in the class of transformations of kind (1.1) and (1.2) had been formulated by one of

us (Fushchych [9]) without showing the exact form of the functions g and h. Below

we give the explicit transformation laws for Ea and Ha .

Theorem 3. The Maxwell equations (5.1) are invariant under the transformations

sin ?

Ha > Ha = Ha cos ? + [iDab Eb ?1 ? ?abc pb (Hc ?3 + iDcd Ed ?2 )]

? ,

?

(5.4a)

sin ?

Ea > Ea = Ea cos ? + [iDab Hb ?1 ? ?abc pb (Ec ?3 + iDcd Hd ?2 )]

? ;

?

sin ?

Ha > Ha = Ha cos ? ? [i?abc pb Dcd Hd ?1 + Dad Hd ?2 ? Ea ?3 ]

? ,

?

(5.4b)

sin ?

Ea > Ea = Ea cos ? + [i?abc pb Dcd Ed ?1 + Dad Ed ?2 ? Ha ?3 ]

? ;

?

Ha > Ha = Ha cos ? ? ?abc pb Ec sin ?,

?

(5.4c)

Ea > Ea = Ea cos ? + ?abc pb Hc sin ?;

?

Ha > Ha = exp(i?)Ha ,

(5.4d)

Ea > Ea = exp(i?)Ea ,

where

Dad = p2 p2 + p2 p2 ? p2 p2 ?ad + p1 p2 p3 (pb ?cd + pc ?bd ? pa pd ) L?1 ,

?

ac ab bc

1v 1/2

2 p2 ? p 2 p 4 + p 2 ? p 2 p 4 + p 2 ? p 2 p 4

L= ,

1 2 3 1 3 2 2 3 1

2

and where (a, b, c) is a cyclic permutation of (1, 2, 3);

1/2 1/2

? = ?2 + ? 2 + ? 2 2 2 2

, ? = ?1 + ?2 + ?3 .

1 2 3

?a , ?a , ? and ? are real parameters. The transformations (5.4) realise the representa-

tion of the group U (2) ? U (2).

Proof. One can be convinced by the direct verification that Ea , Ha , Ea , Ha , Ea ,

Ha , Ea , Ha satisfy equation (5.1) as well as the non-transformed vectors E and

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

H but a more elegant and constructive way, which shows the method of obtaining the

group (5.4) is to transform the equations to a form for which the theorem statements

become obvious.

Let us write equations (5.1) in the matrix form (Fushchych and Nikitin [10, 11],

Nikitin and Fushchych [22])

?

(5.5)

i ? = ?a pa ?, ?3 S4a pa ? = 0,

?t

where ? is an eight-component wavefunction

? = column(H1 , H2 , H3 , ?1 , E1 , E2 , E3 , ?2 ) (5.6)

and ?a , S4a are matrices of the form

?a = 2?2 ?a ,

(5.7)

? ?0

? ?I I?

0 ?a

? 0

?2 = i , ?3 = , ?a = ,

?0 ?

? ?I

I? 0 ?a

?

0

? ? ? ?

000 i 0 0i 0

? 0 0 ?i 0? ?0 i?

1 1 00

?1 = ? ?, ?2 = ? ?,

? ?

2? 0 i 0 0? 2 ? ?i 0 0 0?

?i 0 0 0 ?i 0

0 0

? ?

0 ?i 0 0

?i 0 0? ? ?

1 0 S4a 0

?3 = ? ?,

? S4a = ,

?0 0 i? ?

?S4a

?

0

2 0

0 0 ?i 0

? ? ? ?

0 0 0i 00 00

?0 0 0? ?0 0 0 i?

0

=? ?, =? ?,

? ?

S41 S42

?0 0 0? ?0 0 0 0?

0

?i 0 ?i

0 00 00

? ?

0 0 00

?0 0 0?

0

=? ?.

?

S43 ?0 0 i?

0

?i 0

0 0

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