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(2.12)

?

?mn = Q1 ?mn ,

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

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

where

Q12 ? s ? 1 + n ?abc

Q+ Q+ ,

s

Pn = , Q1 =

n ?n 2s(s + 1) 0a bc

n =n

k = 0, 1, . . . , 2s ? n

m, n = 1, 2, . . . , 2s + 1,

and akn are the coefficients determined by the recurrent relations

a1n = [n(2s + 1 ? n)]?1/2 ,

a0n = 1,

? = 2, 3, . . . , 2s ? n.

a?n = a??1 n a??1 n+??1 ,

Actually the polynomials of the symmetry operators Q+ given by the relati-

µ?

ons (2.12) manifestly are the symmetry operators of equation (2.1). Operators (2.11)

form the basis of the algebra GL(2s + 1, C) inasmuch as they satisfy the following

commutation relations

?

[?ab , ?cd ] = ?[?ab , ?cd ] = ?bc ?ad ? ?ad ?bc ,

(2.13)

? ? ?

[?ab , ?cd ] = ?bc ?ad ? ?ad ?bc , a, b, c, d = 1, 2, . . . , 2s + 1

which characterise the algebra GL(2s + 1, C). The relations (2.13) are correct on the

set of the equation (2.1) solutions. The validity of these solutions can be verified by

direct calculation using the equivalent matrix representation for the basis elements of

the algebra SL(2, C) (which is evaluated according to (2.11))

Q+ = ?Sa .

Q+ = ?abc Sc , 0a

ab

Here Sa are the matrices which realise the representation D(s) of the SO(3) algebra

in the Gelfand–Zetlin basis [21]. Thus the theorem is proved.

So if equation (2.1) is Poincar? invariant and describes a particle of spin s and

e

mass m > 0, it is invariant also under the algebra GL(2s + 1, C), the basis elements

of which belong to the enveloping algebra of the algebra P (1, 3). The operators (2.12)

together with the Poincar? generators Pµ and Jµ? form the basis of the 10+2(2s+1)-

e

dimensional Lie algebra isomorphic to the algebra P (1, 3) ? GL(2s + 1, C). The last

statement can easily be verified by moving to the new basis Pµ > Pµ , Jµ? >

? ?

Jµ? ? Qµ? , ?mn > ?mn , ?mn > ?mn , where

?

(s ? n + 1)?mn , (s ? n + 1)?mn ,

Q12 = Q03 =

n n

1

Q31 = ?i[Q12 , Q23 ],

Q23 = (?n n+1 + ?n+1 n ),

2a1n

n

Q01 = ?i[Q31 , Q03 ].

Q02 = i[Q23 , Q03 ],

The theorem proved has a constructive character insofar as it gives the explicit

form of the basis elements of additional invariance algebra via the Poincar? generators.

e

Starting, for example, from the Poincar? generators for the Dirac equation

e

? i

Jµ? = xµ p? ? x? pµ + [?µ , ?? ], (2.14)

P µ = pµ = i ,

?xµ 4

On the new invariance algebras and superalgebras 227

where ?µ are the Dirac matrices, one obtains by the formula (2.6) the additional

symmetry operators of this equation found earlier by Fushchych and Nikitin [12]. In

an analogous way to formulae (2.6) and (2.12), the additional invariance algebras of

the Kemmer–Duffin–Petiau and Proca equations may be obtained (see [12, 17, 19,

20]) and even the invariance algebra of infinite-component wave equations [18] may

be found.

Let us note that relativistic wave equations for a particle of spin s > 0 also possess

such additional invariance algebras which belong to the class of integro-differential

operators [5, 8, 9, 16, 17, 29, 27] and, generally speaking, are not numbered among

the enveloping algebras of the algebra P (1, 3).

3. Symmetry operators of the Dirac equation in the class M1

Here we consider in detail the symmetry properties of the Dirac equation

L? ? (? µ pµ ? m)? = 0. (3.1)

It is well known that the symmetry of equation (3.1) which can be found in

the classical Lie approach is exhausted by invariance under the algebra P (1, 3), the

basis elements of which are given in (2.14), and under a corresponding group of

transformations, i.e. the Poincar? group.

e

Theorem 1 gives the possibility of extending the set of symmetry operators of

the Dirac equation. Actually, using formulae (2.6), (2.14) and (3.1) one obtains the

additional symmetry operators [12, 17]

i i

Q± = (?µ p? ? ?? pµ )(1 ± i?4 ). (3.2)

[?µ , ?? ] +

µ?

4 2m

The operators (3.2) are the first-order differential operators with matrix coeffi-

cients (i.e. belong to the class M1 ) and so they cannot be found in the frames of

classical Lie approach. But these operators (with fixed sign ±) form the basis of

16-dimensional Lie algebra together with the Poincar? generators (2.14). It follows

e

from the above that the Dirac equation is invariant under the 16-parameter group

including the Lorentz transformations (generated by Pµ , Jµ? ) and the transformations

which are generated by the operators (3.2). Specifically these transformations have

the form

i ?? ??

? > ? = exp(2i?Q)? = (cos ? ? ?1 ?2 sin ?)? (1 ? i?4 ) sin ? ?1 ??2

m ?x2 ?x1

if Q = Q± etc [12].

12

It may be interesting to know whether the operators (2.14) and (3.3) exhaust all

symmetry operators of the Dirac equation in the class M1 . It turns out that this is

not so.

Here we find the complete set of symmetry operators Q ? M1 for equation (3.1)

which, however, do not form the basis of Lie algebra.

Theorem 2. The Dirac equation has 26 linearly independent symmetry operators

Q ? M1 . These operators include the Poincar? generators (2.14), identity operator

e

and fifteen operators given below

1 1

i?4 (pµ ? m?µ ), ?µ? = mSµ? + i(?µ p? ? ?? pµ ),

?µ =

(3.3)

4 2

Aµ = ?µ? x? + x? ?µ? ? i?µ , B = i?4 (D ? m?µ xµ ),

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

where

3 1

D = xµ pµ + i, (3.3 )

Sµ? = i[?µ , ?? ], µ, ? = 0, 1, 2, 3.

2 4

Proof. To find all linearly independent symmetry operators of the Dirac equation in

the class M1 it is necessary to obtain the general solution of the following operator

equations

(3.4)

[L, Q] = fQ L,

where L = ? µ pµ ? m, Q and fQ are unknown operators belonging to M1 :

? ? ? ?

Q = Aµ pµ + B, fQ = C µ pµ + D,

??? ?

Aµ , Bµ , Cµ and D are 4 ? 4 matrices depending on x = (x0 , x).

Relations (3.4) mean that the operators on the RHS and LHS give the same result

acting on arbitrary solutions of the Dirac equation. On the set of these solutions

operator p0 can be expressed via the operators pa with matrix coefficients: p0 ? =

H? ? (?0 m+?0 ?a pa )?. In other words it is sufficient to restrict ourself by considering

symmetry operators of a form such that

Q = B · p + G, (3.5)

where B and G are 4 ? 4 matrices depending on x. For the operators (3.5) the

invariance condition (3.4) reduces to the following form:

[p0 ? H, Q] = fQ (p0 ? H), (3.6)

where fQ ? 0 insofar as the commutator on the LHS cannot depend on p0 .

An unknown operator (3.5) can be expanded via a complete set of the Dirac

matrices

B = Id0 + i?4 d1 + ?? n? + Sµ? mµ? + ?4 ?? b? ,

(3.7)

G = Ia0 + i?4 a1 + ?? c? + Sµ? f µ? + ?4 ?? g ? ,

where d0 , d1 , n? , mµ? , b? , a0 , a1 , c? , f µ? , g ? are unknown functions on x.

Substituting (3.5) and (3.7) into (3.6) and equating coefficients by the linearly

independent matrices and differential operators one comes to the following system of

partial differential equations:

n0 = b0 = 0, na = i?abc d2 , ba = i?abc d3 ,

b c b c

(3.8)

m0a = i?ab A0 , mab = ?abc A1 , a, b, c = 1, 2, 3,

c

b

?dµ ?dµ

?dµ ?dµ

=? , a = b, m div d0 = 0, m div d1 = 2ima1 ,

a a

b b

, =

?xb ?xa ?xa ?xb

1 1

?2 ?3 ?i

d = ? rot d3 , d = rot d2 , d = ?grad Ai , div di = ?3Ai , i = 0, 1,

2 2

1 1 1

ca = ? (rot d2 )a , c0 = ? div d3 + mA0 , g 0 = div d2 ,

2 3 3 (3.9)

1 1 3 ?0

g = ? (rot d )a ? imda , a = ? i div d , grad a = ? id ,

3 0

a 1 0 0

?

2 2 2

1 1 3 ?1

1 2

? ?

a1 = ? i div d + m div d2 , grad a1 = ?md ? id ,

2 3 2

1? 1 1? 1

f 0a = d0 ? i(rot d1 )a , f ab = ?abc id1 + (rot d0 )c + md2 ,

a c c

2 4 2 4

On the new invariance algebras and superalgebras 229

where the dot denotes the derivative on x0 and there is no sum by the repeated indices.

The symbol dµ denotes a vector with components (dµ , dµ , dµ ) (analogous notation is

1 2 3

used for other vector quantities).

The first line in (3.9) gives the equations in the Killing form. Using this cir-

cumstance it is not difficult to obtain the general solution of the system (3.9) for

m = 0:

d0 = x ? ? + ?x0 + ?, d2 = x ? ? + ?,

d1 = ? + ?x,

g a = ?im(?a + ?xa ),

d3 = ?x0 + µx + ?, g 0 = 0,

1 1

f ab = ?abc (2m?c ? ?c ) + m(xa ?b ? xb ?a ),

f 0a = ?a , (3.10)

2 2

c0 = ?µ ? m(? · x + ?), A0 = ?? · x ? ?,

ca = ?a ,

3

A1 = ??x0 + ?, a1 = ? i?.

a0 = ?,

2

Here the Greek letters denote arbitrary constants.

So the general solution of the system (3.9) depends on 26 arbitrary numerical

parameters. Substituting (3.7), (3.8) and (3.10) into (3.5) and using equation (3.1),

one obtains a general expression for the symmetry operator Q ? M1 for the Dirac

equation as a linear combination of the Poincar? group generators (2.14), identity

e

operator and the operators (3.3). The theorem is proved.

So we have obtained the complete set of the symmetry operators Q ? M1 for

the Dirac equation with m = 0. Besides the Poincar? group generators (2.14) this

e

set includes four operators which coincide on the set of the equation (3.1) solutions

with Lubansky–Pauli vector (2.4), six operators ?µ? = 1 (Q+ + Q? ), trivial identity

µ? µ?

2

operator and five symmetry operators B and Aµ , µ = 0, 1, 2, 3, which belong to the

enveloping algebra generated by the Poincar? generators.

e

The operators (3.3) satisfy the following commutation relations

[B, Pµ ] = ?2i?µ , [B, ?µ ] = ? 1 i(Pµ + mAµ ),

2

1

[Aµ , P? ] = [?µ , ?? ] = ?2i?µ? .

m

However these operators do not form the basis of the Lie algebra inasmuch as the

commutators [?µ? , ??? ] do not belong to the class M1 .

One of the most interesting consequences of the symmetry described in theorem 2

is the existence of new conservation laws for the Dirac equation. Corresponding new

conserved currents have the form

1? ?? ?? ?

??4 ?? µ ?

??µ = ?? ?4 ? + m??4 Sµ? ?,

?xµ

4 ?x

? ?

1 ?? ?? ?? ??

? ? (3.11)

S?µ ? + ?S?? µ ? ?S?µ ? ? µ Sµ? ? +

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