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. 47
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= M13 ?
? ?1
J131 , J23 = M23 + ,
2E(E + |p3 |) 2E(E + |p3 |) (2.15)
1 p2 e3
= t 0 p1 ? x1 , H?1 + ?
?
J011 ,
2(E + |p3 |)
2
1 p1 e3
= t 0 p2 ? x2 , H?1 + +
?
J021 ,
2(E + |p3 |)
2
1
= t 0 p3 ? x3 , H?1 + ;
?
J031
2
3 See (2.6) in [1]
212 W.I. Fushchych, A.L. Grishchenko

e3
P0 2 = H?2 = ?3 E,
? ? ?
P k 2 = pk , J122 = M12 + ,
2E
p2 p1
J132 = M13 ?
? ?
J232 = M23 +
, ,
2(E + |p3 |) 2(E + |p3 |)
p2 e3 H?2
1
= t 0 p1 ? x1 , H?2 ?
?
J012 , (2.16)
2E(E + |p3 |)
+
2
p1 e3 H?2
1
= t 0 p2 ? x2 , H?2
?
J022 + ,
2E(E + |p3 |)
+
2
1
= t 0 p3 ? x3 , H?2
?
J032 ;
+
2
e3 ?3
P0 3 = E = H ? 3
? ? ?
P k 3 = pk , J123 = M12 + ,
2
p2 ? 3 p1 ? 3
= M13 ?
? ?
J133 J233 = M23 +
, ,
2(E + |p3 |) 2(E + |p3 |)
1 p2 e3 ?3
= t 0 p1 ? x1 , H?3 + ?
?
J013 , (2.17)
2(E + |p3 |)
2
1 p1 e3 ?3
= t 0 p2 ? x2 , H?3 + +
?
J023 ,
2(E + |p3 |)
2
1
= t 0 p3 ? x3 , H?3 + ,
?
J033
2
where Mkl = xk pl ? xl pk .
By direct verification one can be convinced that the operators (2.15)–(2.17) satisfy
the commutation relations of algebra P (1, 3). These three representations are not
equivalent. Really the operators of energy sign and helicity have the form
Hc 1
for the representation (2.15),
?=
? = ?3 , ? = ??
E 2
1
for the representation (2.16),
? = ?3 ,
? ?=
2
1
for the representation (2.17).
? = 1,
? ? = ?3
2
Hence it is clear that the representations (2.12), (2.16), (2.17) are not equivalent and
are given in the spaces R1 , R2 , R3 respectively.
Besides two-dimensional representations given for the algebra P (1, 3) one can,
evidently, obtain the other ones as well which however, will be unitary-equivalent to
(2.15)–(2.17). If, for example, in (2.15)–(2.17) one performs the substitution
e3 > 1, |p3 | > p3 , (2.18)
then the operators obtained also realize the representations of the algebra P (1, 3). The
explicit form for the generators of the group P (1, 3) obtained from (2.15)–(2.17) with
the help of substitution (2.18) will be denoted in the sequal by (2.15 )–(2.17 ).
If in (2.15)–(2.17) the matrix ?3 is substituted by 1 (or ?1), then such operators
will realize one-dimensional irreducible representations of the algebra P (1, 3) which
are, of course, unitarily equivalent to the corresponding one-dimensional Shirokov
On two-component equations for zero mass particles 213

representations [5]. The representations [5] are obtained without connection with the
equations of motion and are realized on the functions ?(p1 , p2 , p3 ) not depending on
the time.
Summing up all the above presented we come to the conclusion:
1) Eq. (2.13) together with algebra (2.15) (or (2.15 )) describes the particle with
helicity + 1 and the antiparticle with helicity ? 1 4 ;
2 2
2) Eq. (2.13) together with algebra (2.16) (or (2.16 )) describes the particle with
helicity + 1 and the antiparticle with helicity + 1 ;
2 2
3) Eq. (2.14) together with algebra (2.17) (or (2.17 )) describes two particles with
helicity + 1 and ? 1 5 .
2 2
If Eq. (2.13) is connected with the algebra (2.13) (or (2.15 )) the wave function of
such equation is denoted by ?1 (or ?1 ). The wave function in Eq. (2.13) connected
with the algebra (2.16) (or (2.16 )) is denoted by ?2 (or ?2 ). Similarly ?3 (or ?3 )
denotes the wave function in Eq. (2.14) connected with the algebra (2.17) (or (2.17 )).
3. The transition from the canonical equation (2.13) to the non-canonical one of
the type (1.3.1) is realized with the help of unitary transformation [1]
E + |p3 | + i(?1 p2 ? ?2 p1 )
?1
(2.19)
v1 = .
{2E(E + |p3 |)}1/2
Under this transformation Eq. (2.13) takes the form
??(t, x)
= (?1 p1 + ?2 p2 + ?3 |p3 |)?(t, x), (2.20)
i
?t
where
?1
(2.21)
? = ?1 = v1 ?1
or
?1
(2.22)
? = ?2 = v1 ?2 .
The type of Eq. (2.14) under transformation (2.19) is not changed. The operators
(2.15), (2.16) in ?-representation have the form
? ?
P0 a = H = ?1 p1 + ?2 p2 + ?3 |p3 |, Pk a = p k , (2.23)
a = 1, 2,
Jµ? = Jµ? xk > Xk , H?1 > H ,
?1 ?1
(2.24)

Jµ? = Jµ? xk > Xk , H?2 > H ,
?2 ?2
(2.25)
where
p1 (?1 p2 ? ?2 p1 )
?2 ? 3 p2
X1 = x1 ? ?
+ ,
2E(E + |p3 |) 2E 2 (E + |p3 |)
2E
p2 (?1 p2 ? ?2 p1 )
?1 ? 3 p1
? ? (2.26)
X2 = x2 + ,
2E(E + |p3 |) 2E 2 (E + |p3 |)
2E
? 1 p2 ? ? 2 p1
X3 = x3 ? e3 .
2E 2
representation D+ (s) corresponds to the particle and the representation D? (s) to the antiparticle.
4 The
5 Eq. (2.14) can be interpreted as the equation of motion for one particle which can be in two states

differing one from another by the helicity sign.
214 W.I. Fushchych, A.L. Grishchenko

If one connects with Eqs. (2.13) and (2.14) the representations (2.15 )–(2.17 ), but
not the representations (2.l5)–(2.17) in this case the transition from the canonical
equation to the noncanonical one can he conveniently realized with the help of the
unitary transformation [6]
E + p3 + i(?1 p2 ? ?2 p1 )
v ?1 = (2.27)
.
{2E(E + |p3 |)}1/2
Under this transformation Eq. (2.13) takes the form of the Weyl equation
??w (t, x)
= (?1 p1 + ?2 p2 + ?3 p3 )?w (t, x), (2.28)
i
?t
where
?w ? ?w = v ?1 ?1 (2.29)
1

or
?w ? ?w = v ?1 ?2 . (2.30)
2
?w
Eq. (2.14) is unchanged by the transformation (2.27). The generators Jµ? ? Jµ?
w2 2


coincide with (2.25) where the operator H has the form (2.31) and in the operators
(2.26) the substitution (2.18) is performed. This algebra is denoted by (2.25 ). Under
transformation (2.27) the algebra (2.15 ) goes into the algebra
?w
P0 1 = H = ? k pk ,
w
Pk 1 = p k ,
?n
w
Jkl1 = Mkl + k, l, n is the cycle (1,2,3),
, (2.31)
2
1
J0k2 = t0 pk ? [xk , H]+ .
w
2
The algebra (2.17 ) is transformed into the algebra
P0 3 = H = E,
w w
Pk 3 = p k ,
?n
w
Jkl3 = Mkl + k, l, n is the cycle (1,2,3),
, (2.32)
2
? n pl ? ? l pn
1
J0k3 = t0 pk ? [xk , H]+ +
w
.
2 E
The operators Pµ k , Jµ? are defined on the corresponding sets {?w }, k = 1, 2, 3.
w wk
k
From the above given analysis it follows such a result:
1) Eq. (2.20) together with the algebra (2.24) (or Eq. (2.28) together with the
algebra (2.31)) describes the particle with helicity + 1 and the antiparticle with
2
helicity ? 1 ;
2
2) Eq. (2.20) together with the algebra (2.25) (or eq. (2.28)) together with the
algebra (2.25 )) describes the particle with helicity + 1 and the antiparticle with
2
helicity + 1 .
2
Thus, the same (by the form) Eq.(2.20) (or the Weyl equation (2.28)) describes
different types of particles and antiparticles depending on the representation of group
P (1, 3) with respect to which the wave functions ? (or ?w ) are transformed under
transformations from the group P (1, 3).
On two-component equations for zero mass particles 215

Not only the equations invariant with respect to the group P (1, 3) have such
a dual nature but also the equations invariant with respect to the inhomogeneous
de Sitter group P (1, 4). Within the frameworks of the P (1, 4) group the same (by
the form) equation of the Dirac type describes the various types of particles and

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( 122 .)



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