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antiparticles [7].
§ 3. P -, T - and C-properties of two-component equations
In studying P -, T - and C-properties of the equations of motion one does not
indicate as a rule, with what algebra P (1, 3) the given equation is connected. Such an
approach, as it follows from the results of the previous section, is not quite correct
for the studying P -, T , C-properties of Eqs. (2.13), (2.20), (2.28), since the same
equation connected with various algebras P (1, 3) can have various properties with
respect to the space-time reflections.
In order the equation, invariant with respect to the proper group P (1, 3), be P -,
T - and C-invariant it is necessary and sufficient to satisfy such relations:

P (k) , H = 0, k = 1, 2, 3,
?
P (k) , Pl for
=0 k = l,
?
P (k) , Pl for
=0 k = l,
+
P (k) , Jlr for
=0 k = l, k = r, (3.1)
+
P (k) , Jlr for
=0 k = l, k = r,
?
P (k) , J0l for
=0 k = l,
?
P (k) , J0l for
=0 k = l;
+

T (1) , H = T (1) , J0l T (1) , Pk = T (1) , Jkl (3.2)
= 0, = 0,
? ? + +

T (2) , H = T (2) , J0l T (2) , Pk = T (2) , Jkl (3.3)
= 0, = 0,
? ?
+ +

[C, H]+ = [C, Pk ]+ = [C, Jµ? ]+ = 0. (3.4)

Hence it follows that the equation of motion is invariant with respect to P -trans-
formation if all the conditions (3.1) are satisfied. Usually when studying P -properties
of the equations one verifies only the first relation from (3.1) that, evidently, is not
sufficient for the correct conclusion.
How we give the explicit expressions for the operators r(k) , ? (i) (see formulas
(1.3.2)–(1.3.5)) determining the operators of discrete transformations.
On the sets {?1 } and {?1 } the operators P (k) , T (2) and C cannot be determined
since the range of values of these operators does not belong to the sets {?1 } and
{?1 }. The operator T (1) on {?1 }, {?1 } can be defined and it is determined by such
operators

on {?1 },
? (1) = 1 or (3.5)
?3

?3 p2 + ip1
{?1 }.
? (1) = on (3.6)
p2 + p2
1 2
216 W.I. Fushchych, A.L. Grishchenko

T (1) , T (2) and C on the sets {?2 }, {?2 } are given by the operators ? (k) , k = 1, 2, 3
on {?2 },
? (2) = ?1 or ?2 (3.7)

on {?2 }.
? (3) = ?1 or ?2 (3.8)

The operator ? (1) on the sets {?1 } and {?2 } has the form (3.5)
p2 + ip1 ?3 (p2 + ip1 )
on {?2 },
? (1) = or (3.9)
p2 + p2 p2 + p2
1 2 1 2

on {?2 },
? (2) = ?1 or ?2 (3.10)
p2 + ip1
on {?2 }.
? (3) = ?1 (3.11)
p2 + p2
1 2

The operators P (k) are not determined on {?2 } and {?2 }. P (k) and T (1) on the sets
{?3 }, {?3 } are given by:
on {?3 },
? (1) = 1 or (3.12)
?3

on {?3 },
r(k) = ?1 or ?2 (3.13)
p2 + i?3 p1
{?3 },
? (1) = on (3.14)
p2 + p2
1 2

on {?3 },
r(1) = ?1 r(2) = ?2
? a pa (3.15)
on {?3 }.
r(3) =
p2 + p2
1 2

The operators r(k) and ? (k) on the sets {?} and {?w } have the form
p2 ip1 p2 ?3 i?2 p1
=1? ?
1
(1)
or
? +
E(E + |p3 |) E(E + |p3 |) E
(3.16)
p2 ip1 p2 ? 2 p2
1? ? on {?1 }, {?2 },
2
? (1) = ?3 +
E(E + |p3 |) E(E + |p3 |) E

on {?w },
? (1) = ?2 (3.17)
1

p2 ? 2 p1 p2 ? 3 p1
1? ?1 ? ?
1
? (2) = or
E(E + |p3 |) E(E + |p3 |) E
(3.18)
p2 ? 1 p1 p2 ? 3 p2
1? ?2 ? ? on {?2 },
2
? (2) =
E(E + |p3 |) E(E + |p3 |) E
?2 |p3 | p2 ?3 + ip1
? {?2 };
? (3) = ?1 ? (3) =
or on (3.19)
E E
1
(p2 ? ip1 )(p2 ? i?2 p1 E ? i?1 p2 p3 + i?1 p1 p2 )
? (1) = or
2
E
(3.20)
1
= (?ip1 p2 + ?2 p2 E ? ?1 p1 p3 + ?3 p2 )(p2 ? ip1 ) {?w };
(1)
on
? 1 2
E
On two-component equations for zero mass particles 217

? 3 p1 p 1 ? a pa
? (2) = ?1 ? ? {?w },
on or
E(E + |p3 |) 2
E
(3.21)
? 3 p2 p 2 ? a pa
= ?2 ? ? {?w };
(2)
on
?
E(E + |p3 |) 2
E
p2 ? ip1
?1 (p2 + p2 ) ? ip2 p3 + ?3 p1 p3 on {?w };
? (3) = (3.22)
1 2 2
E
1
(p2 ? i?1 p3 + i?3 p1 ) on {?w };
? (1) = (3.23)
3
E

{?w }.
r(k) = ?k , on (3.24)
k = 1, 2, 3 3

The operators (3.16)–(3.24) are obtained from (3.5)–(3.11), (3.14),(3.17) with the
help of transformations (2.19), (2.27). The transformation law of these operators is
given in (D.10)–(D.16).
Summing up all the above said we come to the final conclusion:
1) Eq. (2.13) for the function ?1 (or ?1 ) is T (1) - and P (k) C-invariant, but P (k) -,
T (2) - and C-noninvariant;
2) Eq. (2.13) for the function ?2 (or ?2 ) is T (1) -, T (2) - and C-invariant, but P (k) -
and CP (k) -noninvariant;
3) Eq. (2.14) is P (k) - and T (1) -invariant, but T (2) -, C- and CP (k) -noninvariant.
Evidently Eqs. (2.20), (2.28) have these properties as well.
Note 1. In [1] we established P -, T - and C-properties of Eq. (2.20) starting from the
assumption that r(k) , ? (k) on the set {?} are the 2 ? 2 matrices. As is seen from the
previous such an assumption is limited. On the set {?} r(k) , ? (k) are the operator
functions depending on the momentum components of the particle.
Note 2. Under the four-dimensional rotations in Minkovski space the wave functions
?1 , ?1 , ?2 , ?w , ?3 , ?3 , ?2 are transformed nonlocally.
2

In conclussion of this section we give some corrollaries immediately following from
the previous, which can be useful for the construction of weak interaction models on
the basis of the equations obtained.
Corrollary 1. Any (one-component or two-component) equation of motion for the
particles with zero mass is Invariant with respect to the Wigner reflection of time
T (1) .
Corrollary 2. Eq. (2.20) for the function ?2 (or (2.28) for the function ?w ) is T (1) C-
2
and T (2) C-invariant, but P C-, P T (1) -, P T (2) -, P T (1) C- and P T (2) C-noninvariant.
It means that for such equation neither hypothesis of combined parity conservation,
nor hypothesis of P T C-invariance conservation is valid.
Corrollary 3. Eq. (2.14) is P T (1) -, T (2) C- and P T (2) C-invariant, but P T (2) -, P C-
and P T (2) C-noninvariant.

§ 4. CP -noninvariant subsidiary conditions
The results of previous sections can be rather simply and briefly formulated if one
describes the zero mass particle with the help of four-component wave function. In
218 W.I. Fushchych, A.L. Grishchenko

this case the wave function has the redundant (nonphysical) components which ran
be invariantly separated with the help of relativistic-invariant subsidiary conditions.
From § 2, 3 it follows that there are three types of subsidiary conditions. One of them
is well known and has the form
?
P1 ? = 0 P1 ? = 0,
+
or (4.1)

1
±
P1 = (1 ± ?4 ). (4.2)
2
Eq. (2.9) together with the condition (4.1) is equivalent to Eq. (2.28) for the
function ?w (or (2.20) for the function ?1 ).
1
Now we find two other relativistic-invariant subsidiary conditions. Besides the
matrix ?4 the energy sign operator commutes with the algebra (1.2.1). Hence it is
clear that the operators
H
1
±
P2 = 1 ± ?4 H = ?0 ?k pk , (4.3)
,
2 E

H
1
±
P3 = 1± (4.4)
2 E
± ±
commute with the algebra (1.2.1). The operators P2 , P3 are the projection operators.
We show then that the conditions
?
P2 ? = 0 P2 ? = 0,
+
or (4.5)
?
P3 ? = 0 P3 ? = 0,
+
or (4.6)

can be considered as subsidiary conditions.
± ±
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