First order generalized symmetries of Maxwell's equations

Volume 129, number 3

FIRST ORDER GENERALIZED

PHYSICS LETTERS A

SYMMETRIES

16 May 1988

OF MAXWELL’S

EQUATIONS

Juha POHJANPELTO School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Received 28 August 1987; revised manuscript received 2 March 1988; accepted for publication 14 March 1988 Communicated by D.D. Holm

The results of the calculation of first order generalized symmetries of Maxwell’s equations in vacuum are summarized. The Lie algebra structure of the symmetries is also discussed.

1. Introduction It has been long known [ 1,2] that the maximal group of geometrical symmetries of Maxwell’s equations in vacuum d,E=curl I~,B=

-curl

B,

2. First order generalized symmetries

(1)

H,

(2)

div E=O ,

(3)

div B=O

(4)

is the 17-parameter group C( 1,3)@G, where C( 1,3) is the conformal group of Minkowski space and G is the abelian group of dilatations and rotations of the field vectors. Recently Fushchich and Nikitin have found additional non-geometrical symmetries of Maxwell’s equations. In ref. [ 3 ] under the condition that the symbol of an infinitesimal operator does not depend on the space variables they construct an invariance algebra which is isomorphic to u ( 2 ) @u ( 2 ) . In ref. [ 41, dropping the requirement that symmetries form a Lie algebra, they construct new symmetries in the class of second order differential operators which do not belong to the enveloping algebra of the Lie algebra of C( 1,3)@G. In this paper we summarize the results of the calculation of first order generalized symmetries of Maxwell’s equations in vacuum. It is found that in suitably chosen basis for the generalized symmetries the non-geometrical symmetries can be obtained from the geometrical ones with the help of a single discrete 148

transformation. Using this correspondence we show that the first order generalized symmetries form a closed Lie algebra isomorphic to so( 6,C),@g, where g is a 2-dimensional abelian algebra.

We consider an infinitesimal olutionary form

symmetry

where the characteristic (Q, R)T= depends on the space and x1, x2, x3 and t, the field variables der derivatives. The infinitesimal criterion for the the form R2, R3)=

in its ev-

(Q’, Q2, Q3, R ‘, time coordinates and their first orsymmetry

u takes

R ,

D,Q=Curl D,R=-CurlQ, DivQ=O,

DivR=O,

which must be satisfied whenever E and B satisfy ( 1 )-( 4). Here Curl and Div stand for the total curl and divergence. We define the matrices

0375-9601/88/$ ( North-Holland

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B.V.

Volume

129, number

3

PHYSICS

LETTERS A

16 May 1988

1

O3

s=[ $

$

o3

’

i,j=l,2,3. We also write

( 4 ) Translations:

ay

P~=dx,>

ri,= i

/=O

”

Pi’az,

ay

i=O T 123 9 r .

( 5 ) Scaling:

w,’ > i=O, 1,2, 3,

3 ,&“’

i=O

’ dxi ’

ay

k~ox’f7-.

dxi

( 6 ) Dilatation: T, =4x& +do, ; 2x,+7 J=l

d=Y. (7) Rotations of the field vectors:

i J,k=

2f*jkxkSj+2tSia

,

I

h=aY.

>

For proof. cf. ref. [ 7 1.

i=O, 1,2, 3,

1

where etik is the permutation symbol and 1 cr;,,:,=o=

1

1

[ ;

_;

-;

1

-1

-1

3. Lie algebra structure of the symmetries

1 1;

.

1

Theorem 1. The first order generalized symmetries of Maxwell’s equations in vacuum are generated by vector fields with the following characteristics: ( 1) Conformal transformations:

Cl= i

&,Gg+Tig!E’,

k=O

for any symmetries vi and v2 in the basis of theorem 1. (For the definition of Lie brackets of evolutionary vector fields cf. ref. [ 6 1.) Hence we have Theorem 2. The first order generalized symmetries form a closed Lie algebra isomorphic to so ( 6,C) R@g, where g is a 2-dimensional abelian algebra. The basis elements in theorem 1 satisfy the following commutation relations:

i=O, 1,2, 3.

k

(2 ) Lorentz transformations:

1,=xE+x E+s.oY 1ax, Oax, 1 ’

t=x,g+x0.$ 0

(3) Rotations:

I

The Lie algebra of the geometrical symmetries of Maxwell’s equations is isomorphic to so( 4,2)@g, where g is a 2-dimensional abelian algebra [ 3 1. We note that the non-geometrical elements in the basis given in theorem 1 can be obtained from the geometrical ones by applying the transformation Y+aY. It is easy to see that

[[T”ci,Cj] =O 3 [C”Ci, fj] S,Y,

i= 1, 2, 3

.

[onCi,

r,k]

[(T”cI,

PJ]

=a”(6ijck_dikikCj) =20”[d~S+&j(

=(T”(dijCO

+dO,Cj)

,

,

1 -SO,)/;

+SO~(l-60j)1,-(1-60~)(1-60j)rijl

3

149

Volume 129, number 3

[dk,,

s] =u”c,

PHYSICS LETTERS A

[a%41=a”r,,

)

,

References

[a%r,kl=~“(6,,1~-6,~1,) , [b”li,P,l=~“(6,jPO+60jPI) [ unrlj>

r,k

1 =

gnrik

[Yr,,,s]=O,

>

T

[~nrlJ,

Pkl

[ I] H. Bateman, Proc. London Math. Sot. 8 ( 1910) 228. [2] N. Kh. Ibragimov, Sov. Phys. Dokl. 13 (1968) 18. [ 31 V.I. Fushchich and A.G. Nikitin, Sov. J. Part. Nucl. 14 ( 1983)

[O”~~T~]=O 3 =@(6jkPi--6ikPj)

7

[U"pi,S]=-f7"p,,

[ CT’V, anything] =O ,

[ o”h, anything] =O ,

where n is an integer and a0 = 16. Acknowledgement

I would like to thank Professor Peter Olver for his help in preparation of this article.

150

16 May 1988

[4] :.I. Fushchich and A.G. Nikitin, J. Phys. A 20 ( 1987) 537. [ 51 V.I. Fushchich and A.G. Nikitin, Symmetries of Maxwell’s equations (Reidel, Dordrecht, 1987). [ 61 P.J. Olver, Applications of Lie groups to differential equations (Springer, Berlin, 1986). [ 71 J. Pohjanpelto, Ph.D. Thesis, University ofMinnesota, Minneapolis, Minnesota.