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23 02 23 02
(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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