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The author expresses his gratitude to Professor O.S. Parasjuk for the benefits of
many discussions.

1. Lee T.D., Yang C.N., Phys. Rev., 1957, 105, 1671;
Landau L., Nucl. Phys., 1957, 3, 127;
Salam A., Nuovo Cimento, 1957, 5, 1207.
2. Fushchych W.I., Kiev, preprint ITF-70-40, 1970.
3. Shirokov Yu.M., JETP (Sov. Phys.), 1958, 6, 664.
4. Lomont J.S., Moses H.E., J. Math. Phys., 1962, 3, 405.
5. Fronsdal C., Phys. Rev., 1959, 113, 1367;
Voisin J., Nuovo Cimento, 1964, 34, 1257.
6. Tokuoka Z., Prog. Theor. Phys., 1967, 37, 603;
Sen Gupta N.D., Nucl. Phys. B, 1968, 4, 147;
Santhanam T.S., Chandrasekaran P.S., Prog. Theor. Phys., 1969, 41, 264.
7. Fushchych W.I., Kiev, preprint ITF-69-17, 1969.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 207–208.

On the CP -noninvariant equations for
1
the particle with zero mass and spin s = 2
W.I. FUSHCHYCH, A.L. GRISHCHENKO

One of us [1] has shown that for the particle with zero mass and spin s = 1 there
2
are three types of two-component equations (or one four-component equation with
three different subsidiary conditions) which differ from one another by P , T and C
properties. One of these equations is the two-component Weyl equation which, as is
well known, is equivalent to the four-component Dirac equation

?µ pµ ?(t, x) = 0, (1)
µ = 0, 1, 2, 3,

with the subsidiary relativistic invariant condition

(2)
(1 + ?5 )?(t, x) = 0.

Equations (1), (2) may be written in the form of a single equation [2]

{?µ pµ + ?1 (1 + ?5 )} ?(t, x) = 0, (3)

where ?1 is an arbitrary constant (not connected with mass of the particle. Equation
(3) (or eqs. (1) and (2)) is P and C noninvariant, but CP -invariant.
In this note we give two other relativistic invariant equations which differ from (3)
(or from (1) with the subsidiary condition (2)).
These equations have the form
H
?µ pµ + ?2 1 + ?5 (4)
?(t, x) = 0,
E

H
?µ pµ + ?3 1 + (5)
?(t, x) = 0,
E

p2 + p2 + p2 , (6)
H = ?0 ?k pk , k = 1, 2, 3, E= 1 2 3


?2 , ?3 are arbitrary constants.
Equation (4) is equivalent to eq. (1) with the subsidiary condition
H
(7)
1 + ?5 ?(t, x) = 0.
E
Equation (5) is equivalent to eq. (1) with the subsidiary condition
H
(8)
1+ ?(t, x) = 0.
E
Lettere al Nuovo Cimento, 1970, 4, № 20, P. 927–928.
208 W.I. Fushchych, A.L. Grishchenko

The relativistic invariance of eqs. (4) and (5) (or the invariance of the subsidiary
conditions (7) and (8)) follows from the fact that the operators ?5 and H/E are
invariants of the Poincar? group (for the case of zero mass).
e
It easy to verify that eq. (4) (or eq. (1) with condition (7)) is CP and CP T
noninvariant.
Equation (5) (or eq. (1) with condition (8)) is P and T invariant (in the sense of
Wigner time reflection), but C-noninvariant.
Equation (4) coincides with the equation obtained earlier [1] (where the substitu-
tion e3 = p3 /|p3 | > H/E should be made).
Thus, as distinguished from eq. (3) (eqs. (1) and (2)) there are two more eqs. (4)
and (5) which are also relativistic invariant, but CP -noninvariant.
A more detailed analysis of eqs. (4) and (5) will be given in another paper.

1. Fushchych W.I., Nucl. Phys. B, 1970, 21, 321; Preprint, Kiev, ITF-70-29, 1970.
2. Tokuoka Z., Progr. Theor. Phys., 1967, 37, 603;
Sen Gupta N.D., Nucl. Phys. B, 1968, 4, 147.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 209–220.

On two-component equations
for zero mass particles
W.I. FUSHCHYCH, A.L. GRISHCHENKO
The paper presents a detailed theoretical-group analysis of three types of two-component
equations of motion which describe the particle with zero mass and spin 1 . There are
2
studied P -, T - and C-propertias of the equations obtained.
В работе дан детальный теоретико-групповой анализ трех типов двухкомпонентных
уравнений движения, описывающих частиц с нулевой массой и спином 1 . Изучены
2
P -, T -, C-свойства найденных уравнений.

1. Introduction
In the previous paper [1] it was shown by one of the authors that starting from
the four-component Dirac equation with zero mass one can obtain three types of
two-component equations. One of them coincides with the Weyl equation which, as is
known, is P (k) C- and T (1) -invariant but P (k) -, C-noninvariant. Two other equations
are noninvariant with respect to P (k) C-transformations. For one of these two equa-
tions the P T C theorem is not valid, i.e. such an equation is noninvariant with respect
to P (k) T (1) C- and P (k) T (2) C-transformations1 .
This present paper is dedicated to the detailed study of all possible (with an
accuracy of the unitary equivalence) two-component and four-component (with sub-
sidiary conditions) equations describing free notion of a particle with zero mass and
spin s = 1 .2
From the point of view of ideology the previous and the present papers are closely
connected with the papers by Shirokov [2] and Foldy [3] in which for the first
time equations of motion for a particle without antiparticle with non-zero mass and
arbitrary spin were suggested. The Shirokov–Foldy equations are P (k) - and T (1) -
invariant, but T (2) - and C-noninvariant.
2. Three types of two-component equations
1. The helicity and energy sign [2] operators [2]
J12 P3 + J23 P1 + J32 P1 P0
E = p2 + p2 + p2 , (2.1)
?= , ?=
?
1 2 3
E E
are the Casimir operators of the group P (1, 3) for the representations with zero mass
and discrete spin.
Between the operators P , T , C and ?, ? it is easy to establish such relations2 :
?
P (k) ? = ??P (k) , P (k) ? = ?P (k) , (2.2)
?? k = 1, 2, 3,
T (2) ? = ??T (2) ,
T (a) ? = ?T (a) , T (1) ? = ?T (1) , (2.3)
a = 1, 2, ?? ? ?
C ? = ??C. (2.4)
C? = ?C, ? ?

Препринт ИТФ–70-88E, Киев, 1970, № 88, 22 с.
1 Notations and definitions which are gives without explations are the same as in the paper [1].
2 The results of this subsection are valid for the arbitrary spin.
210 W.I. Fushchych, A.L. Grishchenko

Hence it follows such coupling scheme of irreducible representations of the proper
Poincar? group by the operators P , T , C:
e

T (2) P (k) - +
D+ (s)  D (s)
i C,
P P T (2)  
1
P
6 6 (k)
PP
PP
P (k) P

 T (2) PP
?  C,
) q
P?
D+ (?s)  (2) (k) - D? (s)
(k)
T P , CP

It is seen from the scheme (2.5) that there exist three essentially different (with
respect to P -, T and C-transformations) types of two-component equations of motion
on the solutions of which the following representations of the P (1, 3) group are
realized:

D+ (s) ? D? (?s) D? (s) ? D+ (?s),
or (2.6)

D+ (s) ? D? (s) D? (?s) ? D+ (?s),
or (2.7)

D? (s) ? D? (?s).
D+ (s) ? D+ (?s) or (2.8)

Hence it follows such result:
1) the space R1 where the representation (2.6) is realized is invariant with respect
to T (1) - and CP (k) -transformations but noninvariant with respect to T (2) -, P (k) -
and C-transformations;
2) the space R2 where the representation (2.7) is realized is invariant with respect
to T (1) -, T (2) - and C-transformations but noninvariant with respect to P (k) - and
CP (k) -transformations;
3) the space R3 where the representation (2.8) is realized, is invariant with respect
to P (k) - and T (1) -transformations but noninvariant with respect to T (2) - and
C-transformations.
The two-component equations the wave functions of which are transformed accor-
ding to the representations (2.6)–(2.8), have the same P -, T - and C-properties as the
spaces R1 , R2 , R3 have.
2. The Dirac equation

?µ pµ ?(t, x) = 0, (2.9)
µ = 0, 1, 2, 3

is transformed to the form
??(t, x)
(2.10)
i = ?0 ?(t, x),
?t
(2.11)
?(t, x) = U ?(t, x)

with the help of unitary transformation [4]
1 ?k pk
U = v 1+ (2.12)
.
E
2
On two-component equations for zero mass particles 211

In the representation (1.2.6) for the Dirac matrices3 where

?3 0
?0 =
??3
0

eq. (2.10) decomposes into two two-component system

??± (t, x)
= ±?3 E?± (t, x), (2.13)
i
?t
?± (t, x) are two-component wave functions.
If tor the Dirac matrices we choose the representation, where

1 0
?0 =
?1
0

then (2.10) decomposes into the system

? ?± (t, x)
= ±E ?± (t, x), (2.14)
i
?t

?± (t, x) are two-component wave functions.
Eqs. (2.13), (2.14) in themselves (without algebra P (1, 3)) do not unambiguously
determine what particle and antiparticle they describe. Depending on the represen-
tation of the group P (1, 3) with respect to which its wave function is transformed
under transformations from the group P (1, 3), the same (by the form) two-component
equation of motion describes, as is seen below, different particles. In other words, it
means that the equations of motion only together with the algebra P (1, 3) unambi-
guously determine what particle is described by it.
According to the results of the previous subsection for the particle with spin
s = 1 there are three essentially various two-dimension representations for the algebra
2
P (1, 3). They have the following form

?
P0 1 = H?1 = ?3 E, Pk 1 = pk = ?i
? ?
,
?xk
e3 H?1 p3
?
J121 = M12 + , e3 = , p3 = 0,
|p3 |
2E
p2 H ? 1 p1 H ? 1

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