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. 113
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Proof. Let us reduce eqs.(9) and (10) to the canonical diagonal form, for which the
theorem statements become obvious. Multiplying (9) from the left by
S5? p? ? 1/2
? p5 = p2 ? p2 (11)
V1 = exp i , ? = 0, 3, ,
0 3
p5 2
gives the equation
eH
?
(i?5 p5 ? ?? ?? + k?S12 + m)? = 0, (12)
? = V1 ?, ?= .
2m
This equation may be written in the equivalent form

p5 ? = S5? ?? + i?5 M ? M ?1 ?? ?? ?5 ?? ?? ? M
? ? ? ?,
(13)
? ? |k?| = m.
2
1 + ?5 ?? ?? + M ? = 0, M = m + k?S12 ,
??
With the help of the transformation ? > ? = V2 V3 ? , where

V2 = exp ?M ?1 ?? ?? ?5 = 1 ? M ?1 ?? ?? ?5 ,
? ? ?
2 2

? ?
? ?
?
E + H?
V3 = 1 + ?5 ?1 + ?,
? 2
? ?
?
E 2E + [H, ?]+ (14)
? ?
? ?
?
E + ?H
V3?1 = 1 + ?5 ?1 + ?,
? 2
? ?
?
E 2E + [H, ?]+
On the invariance groups of relativistic equations for the spinning particles 491

H = i?5 ?? ?? M ?1 ?? ?? ? M ,
? M = m + ?S12 ,
? ?
E = |H| = ? = S12 + i?5 1 ? S12 , |?| = m,
2
H 2,

one reduces eq.(13) to the diagonal form
1/2
H c = S12 m2 + ?? ? 2?S12 + ? 2
p5 ? = H c ?, 2
+
1/2
m2 ?? + 2k 2 ? 2
2
(15)
+i?5 S12 (k ? 1)? + i?5 1 ? 2 2
S12 m+ ,
m2 ? k 2 ? 2
2 2 2 2
(1 + ?5 )? = 0, ?? = ?1 + ?2 .

Equations (15) are obviously invariant with respect to transformations ? > QA ?,
where QA are arbitrary matrices, which commute with ?5 and S12 . The complete set
of such matrices may be chosen in the form

Q2 = i?5 1 ? S12 ? S12 ,
2 2
Q1 = i?5 1 + S12 + S12 ,
(16)
Q3 = i?5 1 ? 2S12 ,
2
Q3+a = i?5 Qa , a = 1, 2, 3.

The operators (16) obey the relations

(QA )2 ? = ?,
[QA , QB ]? = 0, Qa Qb = Qc ,
Q3+a Q3+b = Qc , Q3+a Qb = Q3+c , a = b = c = a,

i.e. form the six-dimensional Klein group.
If k = 1, there exist ten linearly independent matrices, which commute with H c
and ?5 . These matrices may be chosen in the form

N12 = (1 ? 2?5 )S12 ?5 ,
2 22 2
N31 = i?5 S12 , N32 = iN31 N12 ,
B1 = i?5 1 ? S12 ,
2
N4a = i?5 S12 Nbc , B1+a = Q3+a .

Operators Bk commute with Bk and with Nk l , and the operators Nkl form the
representation D 1 , 0 ?D 0, 1 ?6D(0, 0) of the Lie algebra of the group SU2 ?SU2 .
2 2
The exact form of the operators QA , Nkl , Nk in the original ?-representation may be
obtained by the formulae

QA = W ?1 QA W , Nkl = W ?1 Nkl W , Bk = W ?1 Bk W ,
? ? ? ? ? ? (17)
? ??? ???
where W = V1 V2 V3 and V1 , V2 , V3 given in (11), (14). The theorem is proved.
Remark 2. The analogous theorem may be proved for the KDP equation, which
describes the motion of a charged particle with anomalous moment in a constant
homogeneous elwctric field E. Such an equation has the form (9), where

?0 = p0 ? Ex3 , Sµ? F µ? = ?2ES03 .
?1 = p1 , ?2 = p2 , ?3 = p3 ,

Let us consider the equation for a particle with an arbitrary spin [5]

?
(18)
Hs ?(t, x) = i ?(t, x),
?t
492 W.I. Fushchych, A.G. Nikitin

where ?(t, x) is a 2(2s + 1)-component wave function,
s
(?1)[?] ?? + (1 + ?1 )?(t, x),
Hs = ? 1 m + ? 3 p
?=?s
(19)
S·p 1/2
? ? (µ ? ?)?1 , p2 p2 p2
?? = p= + + ,
1 2 3
p
µ=?

Sa are generators of the direct sum D(s)?D(s) of the irreducible representation of the
SU2 group, ?1 and ?3 are 2(2s+1)?2(2s+1)-dimensional Pauli matrices, commuting
with Sa , ?(t, x) is an arbitrary potential. If ?(t, x) = 0 eq.(18) coincides with the one
obtained in [5] and describes a free motion of a relativistic spin-s particle.
Theorem 4. Equation (18) is invariant under SU2 algebra. The basis elements of
this algebra have the form

?a = Obc = ?1 Sa + (1 ? ?1 )pa S · pp?2 . (20)

Proof. Using the transformation
Hs > V Hs V ?1 = ?1 m + ?3 p + (1 + ?1 )?(t, x),
s
1
?1 ?1
?a > V ?a V 1 + ?1 + (1 ? ?1 ) (?1)[?] ?? ,
= Sa , V =V =
2 ?=?s

one reduces the Hamiltonian (19) and the operators (20) to such a form, that the
theorem statements become obvious.
For s = 1 eq.(18) coinsides with the Dirac equation with a semirelativistic potential
2
(1+?1 )? ? (1+?0 )?. The SU2 -invariance of such equation has been established in [6].
Theorem 5. The Tamm–Sakata–Taketani equation with a semirelativistic potential
(S · p)2
p2 p2
?
? (21)
i ? = ?1 m + + i?3 + (1 + ?1 )?(t, x) ?,
?t 2m 2m m
is invariant under the Lie algebra of the SU2 group. The basis elements ?A of this
algebra have the form
?7 = i(O23 O31 O12 ? O12 O23 O31 ),
?a = [Oab , Oac ]+ , ?3+a = Obc ,
i
?8 = ? v (O12 O23 O31 + O23 O31 O12 ? 2O31 O12 O23 ),
3
where Oab are given in (20).
We do not give the proof here. The analogous theorem may be formulated for the
KDP equation with the potential ?0 (1 + ?0 )?(t, x).
In conclusion we note that the obtained invariance algebrae may be used for
deriving of new solutions of the equations considered above, if certain partial solution
of these equations is known.
On the invariance groups of relativistic equations for the spinning particles 493

1. Fushchych W.I., Teor. Mat Fiz., 1971, 7, 3 (in Russian); Theor. Math. Phys., 1971, 7, 323 (in
English); Preprint of Institute of Theoretical Physics ITP-70-32, Kiev, 1970.
2. Fushchych W.I., Lett. Nuovo Cimento, 1973, 6, 133; 1974, 11, 508.
3. Nikitin A.G., Segeda Yu.N., Fushchych W.I., Teor. Mat Fiz., 1976, 29, 82 (in Russian); Theor.
Math. Phys., 1976, 29, 943 (in English).
4. Fushchych W.I., Nikitin A.G., Lett. Nuovo Cimento, 1977, 19, 347.
5. Fushchych W.I., Grishchenko A.L., Nikitin AG., Teor. Mat Fiz., 1971, 8, 192 (in Russian); Theor.
Math. Phys., 1971, 8, 766 (in English); Preprint ITP-70-89E, Kiev, 1970 (in English);
Guertin R.F., Ann. Phys., 1974, 88, 504.
6. Smith G.B., Tassie L.I., Ann. Phys., 1971, 65, 352;
Bell I.S., Ruegg H., Nucl. Phys. B, 1975, 98, 151;
Melnikoff M., Zimmerman A.H., Lett. Nuovo Cimento, 1977, 19, 174.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 494–497.

Conformal invariance of relativistic equations
for arbitrary spin particles
W.I. FUSHCHYCH, A.G. NIKITIN
We show that any Poincar?-invariant equation for particles of zero mass and of di-
e
screte spin provide a unitary representation of the conformal group, and find an explicit
expression of the conformal group generators in terms of Poincar? group generators.
e

It is well-known that the relativistic equations for massless particles are invariant
under the conformal transformations. This was first established for the Maxwell
equations [1] and then for the equations describing the massless particles of spin
1/2 [2] and of any spin [3].
L. Gross [4] has demonstrated that the solutions of the Maxwell and of the Rarita–
Schwinger (with mass m = 0) equations provide a unitary representation of the
conformal group C4 . The proof given in [4] is rather tedious and in some sense
non-constructive, since it does not give an algorithm to obtain an explicit form of
Hermitian generators of the group C4 for any conformal invariant equation.
In this note, we shall formulate a theorem, which generalizes the results [1–
4] and give a simple and constructive proof of it. Without restricting ourselves by
any concrete form of equations for massless particles we show that any (generally
speaking, reducible) representation of the Lie algebra of Poincar? group P (1, 3),
e
which corresponds to zero mass and discrete spin, can be extended to provide a
representation of the conformal group Lie algebra, and find the explicit expression of
the generators of the group C4 through the generators of its subgroup P (1, 3).
Theorem 1. Any Poincar?-invariant equation for particles of zero mass and of
e
discrete spin is invariant under the conformal algebra C4 1 , basis elements of which
are given by the operators Pµ , Jµ? and
1
[P0 Pa /P 2 , J0a ]+ ,
D= a, b = 1, 2, 3,
2
1
K0 = [P0 /P 2 , J0a J0a + ?2 ? (1/2)]+ , (1)
2
1
[P0 /P 2 , [J0b , Jab ]+ ]+ ? [Pa /P 2 , J0b J0b + ?2 ? (1/2)]+ ,
Ka =
2
where Pµ and Jµ? are the basis elements of the Poincar? algebra P (1, 3), µ, ? =
e
?1
1 2 2 2 3
0, 1, 2, 3, ? = 2 ?abc Jab Pc P0 ; P = P1 + P2 + P3 ; [A, B]+ = AB + BA 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? satisfy, by definition, the algebra

[Jµ? , P? ]? = i(g?? Pµ ? gµ? P? ),
[Pµ , P? ]? = 0,
(2)
[Jµ? , J?? ]? = i(g?? Jµ? + gµ? J?? ? gµ? J?? ? g?? Jµ? )
Letters in Mathematical Physics, 1978, 2, P. 471–475.
1 We use the same notation for the groups and for the corresponding Lie algebras.
Conformal invariance of relativistic equations for arbitrary spin particles 495

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? ),
(3)
[D, Kµ ]? = ?iKµ ,
[D, Pµ ]? = iPµ , [Kµ , K? ]? = 0, [Jµ? , D]? = 0,

which determine together with (2) the algebra C4 (see e.g. [5]). It is not difficult to
realize such a verification, bearing in mind that, on the set of the 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, Wµ = ?Pµ ,

where Wµ is the Lubanski–Pauli vector

1

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. 113
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