ñòð. 3 |

mation (3h) and the transformation (3f), which coincides with Heaviside–Larmor–

Rainich one). It is to be emphasized, that tin transformations (3) have nothing

to do with Lorentz transformations, since they realize unitary finite-dimensional

representation of compact group U2 ? U2 .

In analogy with theorem 1 it can be demonstrated, that the first couple pair of

Maxwell equations

?H ?E

p?E =i p ? H = ?i

,

?t ?t

is invariant under the group U2 ? U2 ? U2 .

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

Let us note in conclusion that transformations (3) do not form a closed group

jointly with ordinary local Lorentz and conformal transformations. But the represen-

tation of the comformal group C4 on the set of solutions of eqs. (1) may be realized

also in the class of nonlocal integro-differential operators [12]. In such a way the

following statement may be proved [10]:

Theorem 2. Maxwell equations (1) are invariant under 23-dimensional Lie algebra

of group C4 ? U2 ? U2 .

We do not give here the explicit form of this algebra basis elements.

1. Heaviside O., Phil. Trans. Roy. Soc. A, 1893, 183, 423.

2. Larmor J., Collected Papers, London, 1928.

3. Rainich G.Y., Trans. Amer. Math. Soc., 1925, 27, 106.

4. Bateman H., Proc. Math. Soc., 1909, 8, 77;

Cunningham E., Proc. Math. Soc., 1909, 8, 77.

5. Ovsjannikov L.V., The Group Analysis of Differential Equations, Moscow, 1978 (in Russian).

6. Fushchych W.I., Teor. Mat. Fiz., 1971, 7, 3 (in Russian).

7. Fushchych W.I., Lett. Nuovo Cimento, 1974, 11, ¹ 10, 508.

8. Fushchych W.I., Nikitin A.G., Lett. Nuovo Cimento, 1977, 19, 347; 1978, 21, 541.

9. Fushchych W.I., On the new method of incestigations of group properties of systems of parti-

al differential equations, in Group-Theoretical Methods in Mathematical Physics, (Kiev, 1978 (in

Russian).

10. Fushchych W.I., Nikitin A.G., Group properties of Maxwell equations, in Group-Theoretical Methods

in Mathematical Physics, Kiev, 1978 (in Russian).

11. Good R.H., Phys. Rev., 1957, 105, 1914;

Beckers J., Nuovo Cimento, 1965, 33, 1362.

12. Fushchych W.I., Nikitin A.G., Lett. Math. Phys., 1978, 2, 471.

W.I. Fushchych, Scientific Works 2000, Vol. 2, 11–22.

On the new invariance algebras of relativistic

equations for massless particles

W.I. FUSHCHYCH, A.G. NIKITIN

We show that the massless Dirac equation and Maxwell equations are invariant under

a 23-dimensional Lie algebra, which is isomorphic to the Lie algebra of the group

C4 ? U (2) ? U (2). It is also demonstrated that any Poincar?-invariant equation for

e

a particle of zero mass and of discrete spin provide a unitary representation of the

conformal group and that the conformal group generators may be expressed via the

generators of the Poincar? group.

e

1. Introduction

Bateman [1] and Cunningham [3] discovered that Maxwell’s equations for a free

electromagnetic field were invariant under conformal transformations. Nearly fifty

years ago the conformal invariance of an arbitrary relativistic equation, for a massless

1

particle with discrete spin was established by Dirac [4] for a spin- 2 particle and by

McLennan [20] for a particle of any spin.

Until now the question of whether the conformal group is the maximally extensive

symmetry group for the equations of motion for massless particles remained unsettled.

A positive answer to this question has been obtained only in the frame of the classical

Sofus Lie approach (Ovsjannicov [24]), but as has been found recently, Lie methods

do not permit the possibility to obtain all possible symmetry groups of differential

equations.

The restriction of the Lie method is that it applies only to those symmetry groups

whose generators belong to the class of differential operators of first order. Using the

non-Lie approach, in which the group generators may be differential operators of any

order and even integro-differential operators, the new invariance groups of relativistic

wave equations have been found (Fushchych [6–9]). It was demonstrated that any

Poincar?-invariant equation for a free particle of spin s ? 12 possessed additional

e

invariance under the group SU (2) ? SU (2) (Fushchych [6, 7]); that the Kemmer–

Duffin–Petiau equation was invariant under the group SU (3) ? SU (3), and that the

Rarita–Schwinger equation was invariant under the group O(6) ? O(6) was demi-

nstrated by Nikitin et al [23] and by Fushchych and Nikitin [10]. The non-Lie approach

was also used successfully to obtain the symmetry groups of the Dirac and Kemmer–

Duffin–Petiau equations describing the particles in an external electromagnetic field

(Fushchych and Nikitin [12]). Other examples of symmetries which cannot be obtai-

ned in the classical Lie approach are the symmetry groups of the non-relativistic

oscillator (Levi–Leblond [16]) and of the hydrogen atom (Fock [5]).

In the present paper, we have found the new symmetry groups of the massless

Dirac equation and of Maxwell’s equations using a non-Lie approach. These groups

are generated not by the transformations of coordinates, but by the transformations

J. Phys. A: Math. Gen, 1979, 12, ¹ 6, P. 747–757.

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

of the Dirac wave function ? and the vectors of the electric field E and the magnetic

field H of the type

?2?

??

?>? =f (1.1)

?, , ,... ,

?xa ?xa ?xb

?E ?H ? 2 E ?2H

E > E = g E, H, , , , ,... ,

?xa ?xa ?xa ?xb ?xa ?xb

(1.2)

?E ?H ? 2 E ?2H

H > H = h E, H, , , , ,... ,

?xa ?xa ?xa ?xb ?xa ?xb

where the functions f and g, h may depend on any order derivatives of ? and E, H

respectively.

It is demonstrated that Maxwell’s equations are invariant under the group U (2) ?

U (2); the explicit forms of the functions g and h in (1.2), which generate the transfor-

mations of such a group, are found. It is also shown that the Dirac equation (with

m = 0) and Maxwell’s equations are invariant under a 23-parametrical Lie group,

which is isomorphic to the group C4 ? U (2) ? U (2). The results obtained admit

immediate generalisation to the relativistic wave equations for massless particles of

any spin. The conformal group generators which leave the Weyl equation and the

massless Dirac equation invariant are expressed in a form which is transparently

Hermitian. It is demonstrated that any (generally speaking, reducible) representation

of a Poincar? group, which corresponds to zero mass and discrete spin, may be

e

extended to the conformal group representation. The explicit expression for the ge-

nerators of the conformal group C4 via the generators of the Poincar? group P (1, 3)

e

has been found. We therefore give a constructive proof of the statement that any

relativistic equation for a discrete spin and zero-mass particle provides the unitary

representation of the conformal group (for Maxwell and Bargman–Wigner equations

this has been demonstrated by Gross [13]).

2. The Hermitian representation

of the conformal group generators for any spin

The conformal invariance properties of any relativistic equation of motion for

a particle of zero mass and of discrete spin may be formulated by the following

statement.

Theorem 1. Any Poincar?-invariant equation for a zero-mass and discrete spin

e

particle is invariant under the conformal algebra C4 ? , basis elements of which are

given by the operators Pµ , Jµ? and

1

D = ? [P0 Pa /P 2 , J0a ]+ ,

2

(2.1)

1 1

[P0 /P 2 , [J0b , Jµb ]+ ]+ ? [Pµ /P 2 , J0b J0b ]+ + gµ? P? /P 2 ?2 ?

Kµ = ,

2 2

where Pµ and Jµ? are the basis elements of algebra P (1, 3),

1 ?1

P 2 = P1 + P 2 + P 3 ,

2 2 2

[A, B]+ = AB + BA, ?= ?abc Jab Pc P0

2

? We use the same notation for the groups and for the corresponding Lie algebras.

On the new invariance algebras of relativistic equations 13

and D, Kµ are the operators which extend the algebra P (1, 3) to the algebra C4 .

Proof. Inasmuch as the operators Pµ and Jµ? by definition satisfy the algebra

[Jµ? , P? ]? = i(g?? Pµ ? gµ? P? ),

[Pµ , P? ]? = 0,

(2.2)

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

the theorem proof is reduced to the verification of the correctness of the following

commutation relations:

[Jµ? , K? ]? = i(g?? Kµ ? gµ? K? ),

[Kµ , P? ]? = 2i(gµ? D ? Jµ? ),

(2.3)

[D, Kµ ]? = ?iKµ ,

[D, Pµ ]? = iPµ ,

[Kµ , K? ]? = 0, [Jµ? , D]? = 0,

which determine together with (2.2) the algebra C4 (see, e.g., Mack and Salam [19]).

It is not difficult to carry out such a verification, bearing in mind that for the set of

solutions of any relativistic equation for a particle of zero mass and of discrete spin

the following relations are satisfied:

Pµ P µ = 0, Wµ W µ = 0, (2.4)

Wµ = ?Pµ ,

where Wµ is the Lubansky–Pauli vector

1

Wµ = ?µ??? J?? P? .

2

So the formulae (2.1) have determined the explicit form of the conformal group

generators via the given generators Pµ , Jµ? of the group P (1, 3). The theorem is

proved.

We note that the generators Kµ and D are written in a transparently Hermitian

form, and hence they generate the unitary representation of the conformal group. The

constructive character of theorem 1 will be demonstrated in the next section.

3. Manifestly Hermitian representation of the conformal group

generators for Dirac and Weyl equations

The results given above may be used to find the explicit form of the generators of

the conformal group representation, which is realised on the set of solutions of any

relativistic equation for a massless particle. In this section we shall demonstrate it by

the examples of the massless Dirac equation and of the Weyl equation.

The Dirac equation for a massless particle of spin 1 may be written in the form

2

? ?

? ?0 ?a pa , pa = ?i (3.1)

L? = 0, L=i ,

?t ?xa

where ?µ are the four-row Dirac matrices.

{QA } denotes the set of the generators of some Lie group G. Equation (3.1) is by

definition invariant under G if the operators QA satisfy the relations

(3.2)

[L, QA ]? = FA L,

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

where FA are some operators which are defined on the set of the solutions of equa-

tion (3.1).

A well known example of such operators is the set of Poincar? group generators

e

P0 = H = ?0 ?a pa , Pa = p a ,

(3.3)

1

Jab = xa pb ? xb pa + Sab , J0a = x0 pa ? [xa , H]+ ,

2

where

1

i(?a ?b ? ?b ?a ).

x0 = t, Sab =

4

According to theorem 1, the representation (3.3) may be extended to the represen-

tation of Lie algebra of the conformal group. Substituting (3.3) into (2.4), one obtains

the operators

1

D= [xµ , Pµ ],

2

(3.4)

1

? ?

Kµ = [Jµ? , x ]+ + [Pµ , x? x ]+

2

which satisfy the invariance condition (3.2) (where FA ? 0) and the commutation

relations (2.5). The operators (3.3) and (3.4) are transparently Hermitian under the

usual scalar product

d3 x ?† ?2 (3.5)

(?1 , ?2 ) = 1

and therefore generate the unitary representation of the conformal group.

Let us note that on the set of solutions of equation (3.1) the generators (3.3) and

(3.4) may also be written in the usual form (see e.g. Mack and Salam [19])

? 3

D = xµ pµ + i,

Pµ = pµ = igµ? ,

?x? 2

1

Jµ? = xµ p? ? x? pµ + i[?µ , ?? ]? , (3.6)

4

1

K? = 2x? D ? xµ xµ p? ? xµ [?? , ?µ ]? ,

2

which is not, however, manifestly Hermitian.

The Weyl equation for the neutrino,

??

(3.7)

i = ?a pa ?,

?t

where ?a are Pauli matrices, is equivalent to the equation (3.1) with the Poincar?-

e

invariant subsidiary condition

(3.8)

(1 + i?4 )? = 0, ?4 = ?0 ?1 ?2 ?3 .

The exact form of the Hermitian generators of the conformal group which are provided

by equation (3.7) may be obtained from (3.3) and (3.4) by the substitution

1

p0 > ? a pa , Sab > i(?a ?b ? ?b ?a ). (3.9)

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