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Jab
(7.23)
?0 Sab pb + iSa4 p0
J4a = x4 pa ? (xa E3 + E3 xa ) + ?0 ,
2 E3 + ?
?0 iS4a pa
= x4 p0 ? (x0 E3 + E3 x0 ) ? ?0
J40 .
2 E3 + ?
Операторы (7.23), несмотря на то, что их явный вид получен исходя из уравне-
ния Дирака (7.14) и представления
1 3 1 3
, l1 = , ? ? D? l0 = ? , l1 = , ? ,
D + l0 =
2 2 2 2
удовлетворяют коммутационным соотношениям (1.1) независимо от явного вида
матриц Sab и iS4a (например, вида (6.2)). Это означает, что операторы (7.23), где
сделана замена ?0 > ?3 , реализуют неприводимое представление D?3 (?3 l0 , l1 , ?)
алгебры P (1, 4), если матрицы Sab и iSa4 реализуют неприводимое представление
алгебры O(1, 3).
Приведем в заключение выражения для xµ = Vxµ V ?1 :

S4b pb E3 + S5b pb p0 + iS45 p2
S45 0
x0 = x0 + +i ,
2
E3 E3 (E3 + ?)
Sab pb E3 + iSa4 E3 p0 ? iS5b pb pa + p0 pa S45
Sa5 (7.24)
?
xa = xa + i ,
2
E3 E3 (E3 + ?)
x4 = x4 ,
где матрицы Sµ? и S5µ имеют вид (6.2).
Автор выражает свою признательность проф. Ю.М. Широкову за ценные со-
веты, а А.Л. Грищенко, Л.П. Сокуру за плодотворные дискуссии и помощь при
выполнении настоящей работы.
198 В.И. Фущич

1. Hegerfeldt G.C., Henning J., Fort. Phys., 1968, 16, 9.
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3. Соколик Г.А., Групповые методы в теории элементарных частиц, Атомиздат, 1965.
4. Фущич В.И., Укр. физ. ж., 1967, 12, 741; 1968, 13, 363.
5. Fushchych W.I., Krivsky I.Yu., Nucl. Phys. В, 1968, 7, 79; 1969, 14, 573.
6. Фущич В.И., Кривский И.Ю., Препринт ИТФ–72, АН УССР, Киев, 1968.
7. Rоsen S.P., J. Math. Phys., 1968, 9, 1593.
8. Фущич В.И., Препринт ИТФ–17, АН УССР, Киев, 1969.
9. Wignеr E.P., Ann. Math., 1939, 40, 149.
10. Foldу L.. Phys. Rev., 1956, 102, 568.
11. Широков Ю.М., ЖЭТФ, 1957, 33, 1196; 1958, 34, 717.
12. Гельфанд И.М., Минлос Р.А., Шапиро З.Я., Представления группы вращений и группы Лорен-
ца, Физматгиз, 1958.
13. De Vos J.A., Hilgervoord J., Nucl. Phys. B, 1967, 1, 494.
14. Hwa R.C., Nuovo Cim. A, 1968, 56, 107.
15. Newton T.D., Ann. Math., 1950, 51, 730.
16. Комар А.А., Сладь Л.М., ТМФ, 1969, 1, 50.
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18. Фущич В.И., Укр. физ. ж., 1966, 8, 907.
19. Bakri M.M., J. Math. Phys., 1969, 10, 289.
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21. Фущич В.И., Сокур Л.П., Препринт ИТФ–33, АН УССР, Киев, 1969.
22. Широков Ю.М., ЖЭТФ, 1951, 21, 748.
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Rosen J., Roman P., J. Math. Phys., 1966, 7, 2072.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 199–206.

On the P - and T -non-invariant
two-component equation for the neutrino
W.I. FUSHCHYCH
The relativistic two-component equation describing the free motion of particles with
zero mass and spin 1 , which is P - and T -non-invariant but C-invariant, is found. The
2
representation of the Poincar? group for zero mass and discrete spin is constructed. The
e
position operator for such a particle is defined.

1. Introduction
As is known, the Dirac equation for a particle with zero mass:
??(t, x)
(1.1)
i = ?0 ?k pk ?(t, x), k = 1, 2, 3,
?t
is invariant with respect to the space-time reflections. If one chooses for the Dirac
matrices the Weyl representation eq. (1.1) decomposes into a system of two equations
??± (t, x)
= ±?k pk ?± (t, x), (1.2)
i
?t
where ?k are the Pauli matrices and ?± is a two-component spinor. The Weyl equati-
on (1.2) for ?+ (or ?? ) is not invariant under space reflection P and charge conjugati-
on C but is invariant under the CP - and T -operations.
Due to the fact that the space parity in the weak interactions is not conserved it
is usually assumed that neutrino is described, not by the four-component eq. (1.1),
but by a two-component one (1.2). Therefore, in papers [1] an hypothesis was put
forward that the weak interactions are invariant with respect to the CP operation and
consequently to the T operation, if the CP T theorem is valid.
In this paper the two-component equation for a particle with zero mass and spin 1 ,
2
which is non-invariant under the time reflection of T and the CP operation, is found.
2. Equation for a neutrino with “variable mass”
On the solutions of eq. (1.1) the generators of the Poincar? group P (1, 3) have the
e
form
P0 = H? = ?0 ?k pk ,
? ?
Pk = p k ,
(2.1)
1
Jkl = xk pl ? xl pk + Skl , J0k = x0 pk ? xk , H?
? ?
,
+
2
1 1
i(?µ ?? ? ?? ?µ ), Sµ4 = i(?µ ?4 ? ?4 ?µ ),
Sµ? =
4 4
(2.1 )
1 1
Sµ5 = i?µ , S45 = i?4 , µ = 0, 1, 2, 3,
2 2
where ?µ and ?4 are the Dirac matrices.
Nuclear Physics B, 1970, 21, P. 321–330.
200 W.I. Fushchych

If one performs a unitary transformation [2] over eq. (1.1)
1 p3
(2.2)
U1 = exp i?S53 e3 , e3 = , p3 = 0,
|p3 |
2
or
1
U1 = v (1 + ?3 e3 ), (2.2 )
2
eq. (1.1) has the form
??(t, x)
= (?0 ?a pa + ?0 |p3 |)?(t, x), (2.3)
i a = 1, 2,
?t
?+
?? (2.4)
? = U1 ?, ,
??

where ?± is a two-component spinor.
The Poincar? group generators P (1, 3) on {?} being the solution of eq. (2.3) have
e
the form
? ?
P0 = H? = ?0 ?a pa + ?0 |p3 |, Pk = pk ,
? ?
Jab = xa pb ? xb pa + Sab , Ja3 = xa p3 ? x3 pa ? e3 Sa3 ?3 , (2.5)
1
?
J0k = x0 pk ? [xk , H? ]+ .
2
Choosing for the Dirac matrices somewhat unusual representation
?3 0 i?a 0
?0 = , ?a = ,
??3 ?i?a
0 0
(2.6)
0 i 0i
?3 = , ?4 = ,
?i 0
i 0
eq. (2.3) decomposes into a system of two equations
??± (t, x)
= {i?3 ?a pa ± ?3 |p3 |} ?± (t, x),
i
?t (2.7)
1 1
Q± = ± iS43 = (1 ± ?3 ?4 ).
?± = Q± ?,
2 2
Eq. (2.7) for the functions ?+ (t, x) (or ?? (t, x)) has quite the other properties
relative to the discrete transformations than the Weyl equation (1.2).
We note the following:
(i) It is possible to arrive at eq. (2.3) (or (2.7)) in another way. If we “extract the
square root” from the operator equation
p2 ? p2 ? = p2 ?,
0 a 3

we obtain eqs. (2.3) (or eqs. (2.7)).
(ii) The fact that the Dirac equations for zero and non-zero mass are invariant
under the P -, T - and C-transformations is the consequence of the fact that they,
besides being invariant with respect to the group P (1, 3), are invariant under the
On the P - and T -non-invariant two-component equation for the neutrino 201

group SU (2) ? SU (2) ? O(4) (this question will be considered in detail in a following
paper).
Eq. (2.3) coincides in form with a usual Dirac equation for zero mass if |p3 | is
considered as the mass of a particle. Therefore it is possible to say that eq. (2.3)
describes a “flat neutrino” with variable mass |p3 |. Really the operator |p3 | is the
Casimir operator of the group P (1, 2) but not of the group P (1, 3).
Before passing to an investigation of the P -, T - and C-properties of eqs. (2.7) we
shall construct the operator of the position in the space.
For eq. (2.3) the operator of the Foldy–Wouthuysen type has the form

p2
S5a pa a
arctg (2.8)
U2 = exp .
|p3 |
p2a

If the matrix S5a have the form of (2.1 ) then
E + |p3 | + ?a pa
p2 + p2 + p2 . (2.9)
U2 = , E=
{2E(E + |p3 |)}1/2 1 2 3


Eq. (2.3) after the transformation (2.9) transfers into
??(t, x)
= H? (t, x) = ?0 E?(t, x), (2.10)
i ?(t, x) = U2 ?(t, x).
?t
The generators of the group P (1, 3) on {?} have the form

P0 = H? = ?0 E,
? ?
P k = pk ,
Sab pb
Jab = xa pb ? xb pa + Sab , Ja3 = xa p3 ? x3 pa ? e3
? ?
,
E + |p3 | (2.11)
1 Sab pb 1
J0a = x0 pa ? xa , H? ? ?0 J03 = x0 p3 ? x3 , H?
? ?
, .
E + |p3 |
+ +
2 2
It must be noted that the operators (2.11), as it can be immediately verified,
satisfy the algebra P (1, 3) commutation relations not depending on the matrices Sab
explicit form, i.e. the operators (2.11), if ?0 is substituted for 1 (or ?1) and realize
irreducibly the algebra P (1, 3) representation which is characterized by zero mass and
discrete spin. The representation (2.11) differs from the corresponding Shirokov [3],
Lomont–Moses [4] ones but is certainly equivalent to them.
The position operator on a set {?} looks as
S5a S5c pc pa Sac pc
?1
Xa = U2 xa U2 = xa ?
?
+2 + ,
E (E + |p3 |) E(E + |p3 |)
E
(2.12)
S5c pc 1
?1
?
S5c = ? i?c .
X3 = U2 x3 U2 = x3 + e3 ,
E2 2
The position operator on a set of solution {?} of eq. (1.1) looks as follows
?3 S5a ?3 S5c pc pa Sac pc
?1 ?
? e3 2
?
Xa = U1 Xa U1 = xa + e3 + ,
E (E + |p3 |) E(E + |p3 |)
E
(2.13)
?3 S5c pc
?1 ?
= U1 X3 U1 = x3 ?

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