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?
X3 .
E2
202 W.I. Fushchych

(iii) If one performs a transformation on eq. (1.1)
1
U1 = v (1 + ?3 ) (2.14)
2
and then a transformation
E + p3 + ?a pa
(2.15)
U2 = ,
{2E(E + |p3 |)}1/2
it will transform into the equation
? ?(t, x)
(2.16)
i = ?0 ?(t, x), ? = U2 U1 ?.
?t
The generators of the group P (1, 3) on {?} coincide with (2.11) where the substi-
tution was made e3 > 1, |p3 | > p3 .
3. P -, T - and C-properties of two-component equation
Here we shall study the properties of one of the two-component eqs. (2.7)1
??(t, x)
= (i?3 ?a pa + ?3 |p3 |)?(t, x), (3.1)
i
?t
under the discrete transformations.
We shall denote through P (k) (k = 1, 2, 3) the space inversion operator of one axis
which is determined as
P (1) ?(t, x1 , x2 , x3 ) = r(1) ?(t, ?x1 , x2 , x3 ). (3.2)
Analogously P (2) and P (3) are determined.
As is well known, two non-equivalent definitions of the time-reflection operator
exist. According to Wigner the time-inversion operator is
T (1) ?(t, x) = ? (1) ?? (?t, x). (3.3)
According to Pauli it is:
T (2) ?(t, x) = ? (2) ?(?t, x). (3.4)
The operator of the charge conjugation can be defined as the product of the
operators T (1) , T (2) or as
C?(t, x) = ? (3) ?? (t, x), (3.5)
where r(k) , ? (k) are the 2 ? 2 matrices.
The operators P , T , C with the group P (1, 3) generators satisfy the usual com-
mutation relations.
The generators of the group P (1, 3) on the solutions {?} of eq. (3.1) have the form
of eq. (2.5) where
H? > i?3 ?a pa + ?3 |p3 | = ??2 p1 + ?2 p2 + ?3 |p3 |,
(3.6)
1 1
Sab > i(?b ?a ? ?a ?b ), Sa3 ?3 > ? ?a ,
4 2
and the matrix ?0 is substituted for the matrix ?3 .
1 In what follows, under ? we shall understand the two-component spinor ?+ .
On the P - and T -non-invariant two-component equation for the neutrino 203

Using the definitions (3.2)–(3.5) it is not difficult to verify that eq. (3.1) is P (3) -,
C-invariant but P (1) -, P (2) -, T (1) -, T (2) -non-invariant.
Thus, eq. (3.1) is P (3) C-, P (1) P (2) P (3) C- and P (a) CT (a) -invariant but P (3) CT (a) -
and P (a) C-non-invariant.
We note the following:
(i) The result obtained is a consequence of the fact that the projection operators
Q± , with the operators of the discrete transformations, satisfy the following relations
P (a) Q± = Q? P (a) , T (a) Q± = Q? T (a) ,
(3.7)
P (3) Q± = Q± P (3) , CQ± = Q± C.
(ii) The two-component equations for the functions ?+ and ?? are equivalent to
the four-component one (2.3) with the subsidiary relativistic-invariant conditions
1 1
? iS43 ? = (1 ? ?3 ?4 )? = 0, (3.8)
Q? ? =
2 2
1 1
(3.9)
Q+ ? = + iS43 ? = (1 + ?3 ?4 )? = 0,
2 2
respectively. For eq. (1.1) these conditions look like
1 1
+ ie3 S45 ? = (1 ? e3 ?4 )? = 0, (3.8 )
2 2
1 1
? ie3 S45 ? = (1 + e3 ?4 )? = 0. (3.9 )
2 2
Eq. (1.1) with the subsidiary conditions (3.8 ) and (3.9 ) can be joined and can
be written in the form of two P (a) - and T (b) -non-invariant but P (3) - and C-invariant
equations
{?µ pµ + ?(1 + e3 ?4 )} ?1 (t, x) = 0, {?µ pµ + ?(1 ? e3 ?4 )} ?2 (t, x) = 0,
where ? is some constant value. The four-component equations for the neutrino,
which are the union of eq. (1.1) and the usual subsidiary condition, were recently
considered in ref. [6]. These equations, as well as the Weyl equations (1.2), are P -
and C-non-invariant but T (1) -invariant.
The unitary operator of type U2 for the two-component eq. (3.1) has the form

p2
S a pa 1
a
arctg (3.10)
V1 = exp i , Sk = ?kln Sln ,
|p3 | 2
2
pa
or
E + |p3 | + i?a pa
(3.11)
V1 = .
{2E(E + |p3 |)}1/2
The position operator on the set of solutions {?} of eqs. (3.1) looks as follows
(?a ?c ? ?c ?a )pc
?a ? c pc pa
Xa + = V1?1 xa V1 = xa ? ?i
?
+ ,
2E 2 (E + |p3 |) 4E(E + |p3 |)
2E
(3.12)
? b pb
= V1?1 x3 V1 = x3 + e3
?
X3 + .
2E 2
204 W.I. Fushchych

To complete our treatment, we find the position operator for the neutrino which is
described by the Weyl equation (1.2), for example for the function ?+ . This equation
under a transformation
E + |p3 | + i?k ?k
v (3.13)
V= ,
2 ?k pk
where the vector ? has the following components
?k ? {p1 ? p2 e3 , p2 + e3 p1 , e3 (E + |p3 |)} ,
takes a canonical form
??+ (t, x)
(3.14)
i = ?3 E?+ (t, x), ?3 = ?3 e3 , ?+ (t, x) = V ?+ (t, x).
?t
The position operator for a neutrino which is described by the Weyl equation (1.2)
(for ?+ ) looks like
(?a ?c ? ?c ?a )pc
?3 ?a e3 ?3 ?c pc pa
Xa = V ?1 xa V = xa + ie3 ?i 2 ?i
W
,
2E (E + |p3 |) 4E(E + |p3 |)
2E
? 3 ? b pb
= V ?1 x3 V = x3 ? i
W
X3 .
2E 2
The other definitions of the operators Xk and V for the neutrino are given in
ref. [5].
(iii) From Dirac eq. (1.1) one can, generally speaking, obtain three types of non-
equivalent two-component equations. On the set of solutions of eq. (1.1) a direct sum
of four irreducible representations D? (s) of the group P (1, 3)
1 1 1 1
? D?=?1 s = ? ? D?=1 s = ? ? D?=?1 s =
D?=1 s = (3.15)
2 2 2 2
is realized, where ? is an energy sign, s is a helicity. Hence it follows that there exist
three types of two-component equations on the set of which the following representa-
tion of the group P (1, 3)
1 1
? D?=?1 s = ?
D?=1 s = ,
2 2
or
1 1 1 1
D?=1 s = ? ? D?=?1 s = ? D?=?1 s =
D?=1 s = , (3.16)
,
2 2 2 2
or
1 1 1 1
D?=1 s = ? ?D?=?1 s = ? ?D?=?1 s = ?
D?=1 s =
, ,(3.17)
2 2 2 2
or
1 1
? D?=?1 s = ?
D?=1 s = (3.18)
2 2
are realized. If on the solutions of two-component equation there realizes the represen-
tation (3.16) then this equation will be T (1) -invariant but C-, P -, T (2) -non-invariant,
On the P - and T -non-invariant two-component equation for the neutrino 205

if the representation (3.17) does then it will be T (1) -, T (2) -, C-invariant but P -
non-invariant, and if the representation (3.18) it will be T (1) -, P -invariant but C-,
T (2) -non-invariant. This problem will be considered in more detail in another paper.
4. Equation for a flat neutrino
The motion group in the Minkovski three-space is the P (1, 2) group of rotations
and translations conserving the form

x2 = x2 ? x2 ? x2 .
0 1 2

In this case the simplest spinor equation is
??± (t, x1 , x2 )
= (i?3 ?a pa ± ?3 m)?± (t, x1 , x2 ), (4.1)
i
?t
2
?± is the two-component spinor and m is the eigenvalue of the operator Pµ .
Eq. (4.1) for ?+ (or ?? ) like eq. (3.1) is invariant under the P (1) P (2) - and C-
operations but non-invariant under the P (a) and T (b) -operations.
Thus, eq. (4.1) for the wave function ?+ (or ?? ) is P (1) P (2) C-, T (a) P (b) - and
P (a) CT (b) -invariant but P (a) C- and CT (a) -non-invariant.
It should be noted that the equation being the “direct sum” of the equation for
?+ (t, x1 , x2 ) and ?? (t, x1 , x2 ) is invariant under the P -, T - and C-transformations [7].
Finally, we quote one more example of the P - and C-non-invariant equation which
is invariant with respect to the inhomogeneous De Sitter group. Such is the Dirac
equation:
??(t, x, x4 )
= (?0 ?k pk + ?0 ?)?(t, x, x4 ), (4.2)
i k = 1, 2, 3, 4.
?t
This equation as is shown in refs. [2, 7] is T (1) -, T (2) C-invariant but P (k) -, T (2) - and
C-non-invariant.
All the results obtained in this paper can be generalized for the arbitrary spin s
case, if one uses for this the purpose the equation (ref. [2]):
??(t, x)
(4.3)
i = ?S0l pl ?(t, x), l = 1, 2, 3,
?t
where ? is some fixed parameter (for the Dirac equation ? = ?2i), and Sµ? , Sµ4 , S45
are the matrices (not 4 ? 4 ones) realizing the algebra O(1, 5) representation.
(i) If we transform the usual Dirac equation describing the motion of the non-zero
mass particle m with a spin 1 as
2

?3 p3 + q3 + m
q3 ? p 2 + m2 , (4.4)
V2 = ,
{2q3 (q3 + m)}1/2 3

it has the form
?? (t, x)
(4.5)
i = H ? (t, x),
?t

(4.6)
H = ?0 ?a pa + ?0 q3 , ? = V2 ?, a = 1, 2.
206 W.I. Fushchych

Choosing the representation (2.6) for the Dirac matrices eq. (4.5) is decomposed into
the set of two independent equations
??+ (t, x)
(4.7)
i = (??2 p1 + ?1 p2 + ?3 q3 )?+ (t, x),
?t
??? (t, x)
= (??2 p1 + ?1 p2 ? ?3 q3 )?? (t, x), (4.8)
i
?t
where ?+ and ?? are two-component wave functions.
Eq. (4.7) or (4.8) describes a free motion of spinless particle and antiparticle
with the mass m. Thus besides of the Klein–Gordon equation there exist the other
equations of the type (4.7) and (4.8) which are also relativistically invariant and
describe the spinless particle motion with non-zero mass. The two-component eq. (4.7)
is equivalent to the four-component Dirac equation
??(t, x)
(4.9)
i = (?0 ?k pk + ?0 m)?(t, x), k = 1, 2, 3
?t
with such subsidiary condition
?3 ?4 m + ?4 p3
1? (4.10)
?(t, x) = 0.
q3
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