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1 1 1 1
, 0 ? D? 0, ? D? , 0 ? D+ 0, >
D+
2 2 2 2
1 1 1 1
, 0 ? D+ ? , 0 ? D? 0, ? D? 0, ?
> D+ ? (31)
2 2 2 2
1 1 1 1
?D? , 0 ? D? ? , 0 ? D+ 0, ? D+ 0, ? ,
2 2 2 2
C
the coupling scheme, brought in ref. [1], the correction D+ (s, 0) - D? (0, s) should be done.
* In
P , T , C properties of the Poincar? invariant equations for massive particles
e 269

where members 1 and ? 1 are the eigenvalues of the operators Sa pa /E and Ta pa /E.
2 2
These operators commute with the generators Pµ , J?? when m = 0. From (31) follows
that there exist 28 types of mathematical nonequivalent two-component equations for
massless particles.
Note 3. In order that Poincar?-invariant equation m = 0 was totally P , T , C invariant
e
it is necessary and sufficient that the wave function was transformed on the following
direct sum of representation of P (1, 3)

D+ (s, ? ) ? D? (s, ? ) ? D+ (?, s) ? D? (?, s), if (32)
? = s,

D+ (s, ? ) ? D? (s, ? ), if ? = s. (33)

The representation D+ (s, ? ) is in general reducible with respect to the P (1, 3) algebra,
therefore the wave function describes a multiplet of particles with variable-spin, but
fixed mass. The spin of the multiplet can take the values from (s ? ? ) to (s + ? ). The
equations of motion describing a physical system with variable-mass and variable-spin
were considered in ref. [5].

1. Fushchych W.I., Lett. Nuovo Cimento, 1972, 4, 344.
2. Fushchych W.I., Theor. Math. Phys., 1971, 7, 3; Preprint ITF-70-32, Kyiv, 1970.
3. Foldy L.L., Phys. Rev., 1956, 102, 568.
4. Shirokov Yu.M., Zurn. Eksp. Teor. Fiz., 1957, 33, 1196.
5. Fushchych W.I., Theor. Math. Phys., 1970, 4, 360; Preprint ITF-70-4, Kyiv, 1970;
Fushchych W.I., Krivsky I.Yu., Nucl. Phys. B, 1969, 14, 573.
W.I. Fushchych, Scientific Works 2000, Vol. 1, 270–273.

On the possible types of equations
for zero-mass particles
W.I. FUSHCHYCH, A.G. NIKITIN

A number of papers dedicated to the description of free particles and antiparticles
with zero mass and spin 1 has recently appeared [1–6].
2
A great many equations with different C, P , T properties have been proposed
and the impression could be formed that there are many nonequivalent theories for
zero-mass particles. The purpose or this paper is to show that it is not the case and
to describe all nonequivalent equations.
1. First we shall formulate the result [1] obtained for a particle of spin 1 in such a
2
form that all principal assertions will be valid for massless particles of arbitraly spin.
It has been shown [1] that for a particle of spin 1 three types of nonequivalent two-
2
component Poincar?-invariant equations exist. These three of equations are equivalent
e
to the Dirac equation
??(t, x)
= H?(t, x), H = ?0 ?a pa , (1)
i a = 1, 2, 3,
?t
with one out of three (actually, one out of six) subsidiary conditions imposed on a
wave function
1
? ±
P1 = (1 ± i?4 ), ?4 = ??0 ?1 ?2 ?3 , (2)
+
or
P1 ? = 0 P1 ? = 0,
2
H
1
? ±
(1 ± i?4 ?),
+
or (3)
P2 ? = 0 P2 ? = 0, P2 = ? ?=
? ,
2 E
1
? ±
(1 ± ?),
+
p2 + p2 + p2 .(4)
or
P3 ? = 0 P3 ? = 0, P3 = ? E= 1 2 2
2
±
Conditions (2)–(4) are Poincar? invariant since the projection operators Pa commute
e
with the generators of the Poincar? group P (1, 3)
e
? 1
P0 = H = ?0 ?a pa , Pa = pa = ?i J0a = tpa ? (xa P0 + P0 xa ),
,
?xa 2
(5)
i
Jab = xa pb ? xb pa + Sab , = (?a ?b ? ?b ?a ).
Sab
4
±
It should be emphasized that only the operator P1 is local in co-ordinate space. If we
introduce the four-component (as a matter of fact, two-component) modes
?± = Pa ?,
±
(6)
a

equations (1) with subsidiary conditions (2)–(4) can be written in the form
??±
= (?0 ?b pb ± ?a ?0 Pa )?± ,
?
a
(7)
i a
?t
Lettere al Nuovo Cimento, 1973, 7, 11, P. 439–442.
On the possible types of equations for zero-mass particles 271

where ?a are arbitrary constants. The wave functions ?± satisfy conditions (2)–
a
(4) automatically. One of the equations (7), namely the equation for ?+ (or ?? ),
1 1
is equivalent, as is well known, to the two-component Weyl equation. Subsidiary
conditions (2)–(4) have been generalized in [7] to massless particles of arbitrary spin
starting from the 2(2s + 1)-component equation.
These results are almost evident from the group-theoretical point of view. Indeed,
on the set {?} of solutions of the equation (1) the following direct sum of irreducible
representations of the group P (1, 3) is realized:

D+ (? = 1) ? D? (? = ?1) ? D+ (? = ?1) ? D? (? = 1), (8)

where D? (?) is the one-dimensional irreducible representation of the P (1, 3) group
characterized by the eigenvalue ? = ±1 of the sign energy operator ? and by the
?
eigenvalue ? = ±1 of the helicity operator
J12 P3 + J23 P1 + J01 P2
? (9)
?=2 = i?4 ?.
?
E
Two-dimensional subspaces of representations

D+ (? = 1) ? D? (? = ?1) D+ (? = ?1) ? D? (? = 1),
or (10)

D+ (? = 1) ? D? (? = 1) D+ (? = ?1) ? D? (? = ?1),
or (11)

D? (? = 1) ? D? (? = ?1),
D+ (? = 1) ? D+ (? = ?1) or (12)

are selected by subsidiary conditions (2)–(4) from {?} in a Poincar?-invariant manner.
e
??
The operators P , T , C (their definitions see e.g. in [8]) and ?, ? satisfy the
relations
? ? ?
[P (1) , ?]+ = [P (1) , ?]? = [T (2) , ?]? = [T (2) , ?]? = [C, ?]? = [C, ?]+ = 0. (13)
? ? ?

Taking into account (13) one obtains the relations

P (1) Pj± = Pj? P (1) , ± ± ± ±
P (1) P3 = P3 P (1) , T (2) Pa = Pa T (2) , (14)
j = 1, 2,

± ? ± ± ± ±
(15)
CP1 = P1 C, CP2 = P2 C, CP3 = P3 C.

From (14), (16) it follows that
1) the system of equations (1), (2) is T (2) , P (1) , C-invariant but P (1) , C-noninva-
riant,
2) the system of equations (1), (3) is T (2) , C-invariant but P (1) -noninvariant,
3) the system of equations (1), (4) is T (2) , P (1) -invariant but C-noninvariant.
To obtain these result we have used only the relations (13) which are valid for
massless particles of arbitrary spin. The above discussion is followed by tins conclusi-
on: if the particle (and antiparticle) of zero mass is characterized by helicity and
by the sign of energy only (without additional quantum numbers) three and only
three types of two-component Poincar?-invariant essentially different (in respect to
e
C, P, T propertis) equations exist. It is interesting to note that the hypothesis of
272 W.I. Fushchych, A.G. Nikitin

Lee and Yang and Landau on CP -parity conservation is not valid for the equati-
ons (1), (3); (1), (4). Moreover the system of equations (1), (3) is CP (1) T (2) - and
CP (1) T (1) -noninvariant.
Note 1. Equation (1) with subsidiary conditions
?, ? = ±1,
??
(16)
P2 P3 ? = 0,
??
(17)
P2 P3 ? = ?,

is equivalent to three- and one-component equations
1
(?µ pµ + ?0 P2 P3 )??? = 0, (1 ? P2 P3 )?,
? ?? ??? = ??
(18)
µ = 0, 1, 2, 3,
2
? ? ?? ? ?? ? ?? ??
(?µ Pµ + ?1 P2 P3 + ?2 P2 P3 + ?3 P2 P3 )??? = 0, ??? = P2 P3 ?,(19)
??
? ?

respectively, where ?µ are arbitary constants. It is not difficult to calculate that
?
there are fifteen equations (2)–(4), (16), (17) exhausting all possible nonequivalent
Poincar?-invariant subsidiary conditions which can be imposed on {?}.
e
Note 2. If a zero-mass particle is characterized by two (but not by one) quantum num-
bers, there exist more than three types of nonequivalent two-component equations.
Theoretically such a possibility exists due to commutativity of Dirac’s Hamiltonian
for a particle of spin 1 with SO4 ? SU2 ? SU2 algebra. It means that besides the
2
mass two conserved quantum numbers s and ? exist. For the zero-mass case the
eigenvalues of helicity-type operators
S a pa ?a pa
? ?
?1 = , ?2 = ,
p p
(20)
1 1 1 1
?abc Sbc ? S4a
Sa = ?abc Sbc + S4a , ?a =
2 2 2 2
are conserved. If the massless particle is characterized by cigenvalues of operators
(20), the number of theoretically possible equations increases. This follows from the
fact that the two-dimensional irreducible representation of the group P (1, 3) for m = 0
is reduced in the case m = 0 to the following direct sum of one-dimensional irreducible
representations:
1 1 1
D± 0, > D± 0, + ? D± 0, ? ,
2 2 2
(21)
1 1 1
D± > D± + , 0 ? D± ? , 0 .
,0
2 2 2
We shall not analyze all possible equations in this case (it is difficult to do this using
the results of paper [8]) because it is not clear from the physical point of view how
one can distinguish, say, the representations D± 0, ? 1 and D± ? 2 , 0 . 1
2
2. Let us now show that four- and two-component equations obtained in [4, 5] are
isometrically equivalent to the Dirac equation (1) and to the Weyl equation.
Consider the four-component equation of the type [4]
??(t, x)
= H? ?(t, x) = (?a pa + ?)?(t, x), (22)
i ?a = ?a ?a ,
?t
On the possible types of equations for zero-mass particles 273

where ? is an operator satisfying the condition

?a pa ? = ???a pa , ?2 = 0. (23)

Equation (22) can be obtained from (1) with the help of the isometric transformation

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( 122 .)



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