ñòð. 130 |

where ? is any solution of system (3). As an example of symmetry algebra of Eq.(3)

we have the well-known 16-dimensional Lie algebra of the C(1, 3) ? H group. Yet

the Maxwell equations possess certain additional symmetry stated by the following

theorem.

Theorem 1. The Maxwell equations are invariant under the nine-dimensional Lie

algebra A8 , basic elements of which have the form

Q1 = ?3 S · pD, Q3 = ?1 S · pD,

? ?

Q2 = i?2 ,

(6)

Q3+a = i?2 S · pQa , Q8 = i?2 S · p,

? ?

Q7 = 1, a = 1, 2, 3,

where

p2 p2 + p2 p2 ? p2 p2 (1 ? Sa ) + p1 p2 p3 Sa Sb pc ??1 , (7)

D= ab ac bc

a=b=c

1 1/2

? = v p4 p2 ? p2 + p 4 p2 ? p2 + p 4 p2 ? p2 (8)

,

1 2 3 2 1 3 3 1 2

2

?? p2 . Operators

and ?a are the Pauli matrices commuting with Sa , pa = pa /p, p =

(6) satisfy the following relations:

[Qa , Qb ] = ?[Q3+a , Q3+b ] = ??abc Qc , a, b, c = 1, 2, 3,

(9)

[Q3+a , Qb ] = ?abc Q3+c , [Q7 , QA ] = [Q8 , QA ] = 0, A = 1, 2, . . . , 8

forming an algebra isomorphic to Lie algebra of the GL(2) ? GL(2) group.

Proof. One can convince oneself that the statements of Theorem 1 are true by strai-

ghtforward calculation making use of the following relations:

DS · p = ?S · pD,

? ?

D?a = ?a D,

D(S · p)2 = D ? f (p1 + ip2 S3 ? ip3 S2 )S2 , f = p2 p2 + p 2 p2 + p 2 p2 , (10)

? 12 13 32

D2 S · p = S · p, L2 S · p = 0, [D, L2 ] = ?p2 p2 L2 .

? ? ? 23

It is obvious that the additional symmetry algebra of Eq.(3) could not be obtained

within the framework of the classical Lie method, which is based on the infinitesimal

approach.

Since QA in (6) are integro-differential operators, we give the corresponding finite

transformations for the Fourier components of E and H. From the relation

? = (2?)?3/2

? ? ? ?

? > ? = exp(?A QA )?, ?(x) exp(?ip · x)d3 x (11)

574 A.G. Nikitin, W.I. Fushchych, V.A. Vladimirov

we have

? ? ? ?

Ea > Ea = Ea cos ?1 + i?abc pb Dcd Ed sin ?1 ,

?

(12a)

? ? ? ?

Ha > Ha = Ha cos ?1 ? i?abc pb Dcd Hd sin ?1 ,

?

? ? ? ?

Ea > Ea = Ea cos ?2 + Ha sin ?2 ,

(12b)

? ? ? ?

Ha > Ha = Ha cos ?2 ? Ea sin ?2 ,

? ? ? ?

Ea > Ea = Ea cos ?3 ? i?abc pb Dcd Hd sin ?3 ,

?

(12c)

? ? ? ?

Ha > Ha = Ha cos ?3 ? i?abc pb Dcd Ed sin ?3 ,

?

? ? ? ?

Ea > Ea = Ea cosh ?4 ? Dab Hb sinh ?4 ,

(12d)

? ? ? ?

Ha > Ha = Ha cosh ?4 ? Dab Eb sinh ?4 ,

? ? ? ??

Ea > Ea = Ea cosh ?5 + i?abc pb Ec sinh ?5 ,

(12e)

? ? ? ??

Ha > Ha = Ha cosh ?5 + i?abc pb Hc sinh ?5 ,

? ? ? ?

Ea > Ea = Ea cosh ?6 ? Dab Eb sinh ?6 ,

(12f)

? ? ? ?

Ha > Ha = Ha cosh ?6 + Dab Hb sinh ?6 ,

? ? ?

Ea > Ea = Ea exp ?7 ,

(12g)

? ? ?

Ha > Ha = Ha exp ?7 ,

? ? ? ??

Ea > Ea = Ea cos ?8 + i?abc pb Hc sin ?8 ,

(12h)

? ? ? ??

Ha > Ha = Ha cos ?8 ? i?abc pb Ec sin ?8 ,

where ?A (A = 1, 2, . . . , 8) are real parameters,

Dab = ?ab p2 p2 + p2 p2 ? p2 p2 + p1 p2 p3 pc ??1 ,

ad ae de

(13)

c = d = e, c = e, c = a, b.

Using the inverse Fourier transformation one can obtain the finite transformations

generated by (6) in the basic representation:

Ha (t, x) = (2?)?3/2 ?

Ha exp(ipx)d3 p,

(14)

?3/2 ?

Ea (t, x) = (2?) 3

Ea exp(ipx)d p.

Transformations (12a)–(12h) form the representation of the GL(2) ? GL(2) group

which includes the one-parameter HLR group (2).

2. Recently [16] within the framework of the non-Lie approach, group properties

of the equations for vector-potential of the electromagnetic field,

2Aµ = 0,

(15)

?µ Aµ = 0, µ = 0, 1, 2, 3,

New symmetries and conservation laws for electromagnetic fields 575

were investigated. The additional symmetry of Eqs.(15) proved to be even higher than

that of the Maxwell equations.

Theorem 2. Equations (15) are invariant under the Lie algebra of the GL(3) group.

Basic elements of this symmetry algebra on the set of solutions of Eqs.(15) have the

form

1µ

g0 p0 pa ? ga p2 Ab ,

(Fab A)µ = µ

(16)

a, b = 1, 2, 3,

0

2

p

where g? is the metric tensor of the Minowski space and g?? = (1, ?1, ?1, ?1); 1/p2

µ

is the integral operator defined as

f (t, x ) 3

1

f (t, x) = (17)

d x.

|x ? x |

p2

The proof of this theorem is given in Ref. [16]. Obviously, the additional symmetry

algebra of Eqs.(15) generated by nonlocal operators (16) cannot be obtained in the

classical Lie approach.

3. What conservation laws correspond to the symmetries stated by Theorems 1

and 2? Since basic elements of the additional symmetry algebras are nonlocal ope-

rators the traditional method for construction of conserved quantities based on the

Noether theorem is of no use. Another possibility of building up the conserved quan-

tities is to put every element of the invariance algebra of Maxwell equations into

correspondence with a four-vector:

Ja = ?? + M ?2 Sa QA ?

J0 = ? + M QA ?,

A A

(18)

satisfying the continuity equation

?µ (J A )µ = 0, (19)

where ? is vector-function from (3), ?2 , Sa are matrices introduced in (4), M is an

operator which satisfies the following equation

?

+ ?2 S · p, M ? = 0. (20)

i

?t

Employing the Gauss–Ostrogradsky theorem we can conclude from (19) and (20) that

the integrals

d3 xJA =

0

d3 x? + M QA ? (21)

QA =

are independent of time. In this way it is possible to obtain all classical conserved

quantities as well as new conserved quantities which correspond to the non-Lie sym-

metry of Maxwell equations. Operator M must be chosen in accordance with the

demand for integrals (21) to have a clear physical interpretation. The following opera-

tor does satisfy this requirement:

?2 S · p

p0

=? (22)

M= ,

p2

p

where 1/p2 is the integral operator defined in (17). As a matter of fact, substituting

(22) into (21) and choosing QA = {Pµ , Jµ? }, where Pµ and Jµ? are basic elements of

576 A.G. Nikitin, W.I. Fushchych, V.A. Vladimirov

the Poincar? algebra, we obtain classical expressions for energy, momentum, angular

e

momentum and center-of-energy of the electromagnetic field. Inserting (6) into (21)

one obtains

d3 p ? ?

? ? ? ?

f E(t, ?p) · H(t, p) + p2 E a (t, ?p)H a (t, p) , (23a)

Q1 = a

?p a

d3 p ? ? ? ?

p · E(t, ?p) ? E(t, p) + H(t, p) ? H(t, p) (23b)

Q2 = ,

2p2

d3 p ? ? ? ?

f Ha (t, p)Ha (t, ?p) ? Ea (t, p)Ea (t, ?p) +

Q3 =

2?p a

(23c)

? ? ? ?

? ? ? ?

p2 H a (t, p)H a (t, ?p) ? E a (t, p)E a (t, ?p)

+ ,

a

a

d3 p ? ? ? ?

E(t, p) · E(t, ?p) + H(t, p) · H(t, ?p) , (23d)

Q8 =

2p

?A

? (23e)

Q4 = Q5 = Q6 = Q7 = 0, A= .

?t

Thus, the existence of additional symmetry algebras for the electromagnetic field

equations gives rise to the new conserved quantities independent of classical ones.

In a similar way we can show that the additional symmetry (16) of Eqs.(15) leads

us to the following conserved quantities:

i -

? Ab (t, x) p0 Ac (t, x)d3 x, (24a)

Sa = ?abc a, b, c = 1, 2, 3,

2

1 p0 p0

- -

? Aa (t, x) p0 Ab (t, x) + Ab (t, x) p0 Aa (t, x) d3 x. (24b)

?ab =

2 p p

Formulas (24a) express the spin of the vector field [17]. The time independence of

spin components (24a) was originally derived by consideration of properties of energy-

momentum tensor of the vector fields having nothing to do with their symmetry

properties. Now we see that conservation of (24a) as well as the existence of six new

conserved quantities (24b) are the consequences of the non-Lie symmetry of Eqs.(15).

In conclusion we discuss briefly a physical meaning of the new conserved quanti-

ties (23) and (24). It is readily shown that if a monochromatic wave solution of Eq.(1)

is substituted into the following expression

Qa

(25)

Ka = , a = 1, 2, 3,

Q8

the Stokes parameters describing polarization of this wave are obtained. In general

integrals (23a)–(23d) can be regarded as a generalization of these parameters for

arbitrary solutions of Maxwell equations. Equations (24) can be reduced to matrix

ñòð. 130 |