ñòð. 60 |

, 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, ¹ 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

ñòð. 60 |