ñòð. 59 |

has the form Q = ?. Equation (1) with subsidiary conditions (4)–(6) can be written

?

in the form

?

(?µ pµ ? ?4 m + ?a Pa )Pa ?(t, x) = 0,

+

(11)

where ?a are the arbitrary constant numbers. For eq. (11) the conditions (4)–(6) are

automatically satisfied.

Equation (1) with the subsidiary conditions (4), (5), (6) (or three eqs. (11)) has

different P -, T -, C-properties. These properties can be read easily from the following

coupling scheme or irreducible representations of the Poincar? group

e

P

<>

D+ (s, 0) D+ (0, s)

Tp Tp

C C

P

D? (s, 0) <> D? (0, s)

T p is the Pauli - time-reversal operator. These questions will be considered in more

detail in another paper.

1. Fushchych W.I., Nucl. Phys. B, 1970, 21, 321; Theor. Math. Phys., 1971, 9, 91 (in Russian).

2. Fushchych W.I., Grishchenko A.L., Lett. Nuovo Cimento, 1970, 4, 927; Preprint ITF-70-88E, Kiev,

1970.

3. Fushchych W.I., Theor. Math. Phys., 1971, 7, 3; Preprint ITF-70-32, Kiev, 1970.

W.I. Fushchych, Scientific Works 2000, Vol. 1, 265–269.

?

P , T , C properties of the Poincare invariant

equations for massive particles

W.I. FUSHCHYCH

Recently [1] we have shown that for free particles and antiparticles with mass

m > 0 and arbitrary spin s > 0, in the framework of the Poincar? group P (1, 3), there

e

exist three types of nonequivalent equations. In the present paper we study the P , T ,

C properties of these equations.

It will be convinient to investigate these properties in the canonical representation

where the Hamiltonian is diagonal (as matrix) and other operators (position operator

and spin operator) have adequate physical interpretation. For the transformation to

this representation let us make unitary transformation [2]

? ?0 H(8) ?0 H(8)

1 1

=v

U p, s = = exp 1+ ,

2 4E E

2 (1)

H(8) ? ?0 ?k pk , p4 ? m, k = 1, 2, 3, 4,

over the eight-component equation of the Dirac type

??(8) (t, x)

= H(8) ?(8) (t, x). (2)

i

?t

Equation (2) after the transformation (1) transfers into

??(8) (t, x)

= Hc ?(8) (t, x), Hc = ?0 E, ?(8) = U ?(8) . (3)

i

?t

In the canonical representation the generators of the P (1, 3) group have the form [2]

?

P0 = Hc = ?0 E, Pa = pa = ?i , a = 1, 2, 3,

?xa

Mab = xa pb ? xb pa , (4)

Jab = Mab + Sab ,

1 Sab pb + S04 m

J0a = x0 pa ? [xa , Hc ]+ ? ?0 x0 ? t,

,

2 E

where Sab , S04 matrices are generators of the SO4 ? SU2 ? SU2 group. On the

solutions {?(8) } of eq.(2) thhese matrices have form

i

(8)

(?k ?l ? ?l ?k ),

Skl = Skl = k, l = 1, 2, 3, 4.

4

The representation for the generators P (1, 3) in the form (4) differs from the Foldy–

Shirokov [3, 4] representation. In the form (4) it is explicity distinguished the fact that

in the space where a representation of the P (1, 3) group is given, also a representation

Lettere al Nuovo Cimento, 1973, 6, ¹ 4, P. 133–137.

266 W.I. Fushchych

of SO4 ? SU2 ? SU2 is realized. This follows, in particular, from the fact [Hc , Skl ]? =

0, i.e. it means that the matrices

1 1 1 1

?abc Sbc ? S4a ,

Sa = ?abc Sbc + S4a , Ta =

2 2 2 2

commute with the Hamiltonian* . In other words this means that the space, where the

representation of P (1, 3) group is realized, must be characterized (besides the mass

m and the sign of the energy) by pair of indices s and ?

1 3

2 2

Sa ? = s(s + 1)?, Ta ? = ? (? + 1)?, s, ? = , 1, , . . . .

2 2

we shall denote by D± (s, 0) and D± (0, ? ) the irreducible representation of P (1, 3)

group. For futher understanding it should be noted that the irreducible representations

D(s, 0) and D(s, 0) of SO4 group are indistinguishable with respect to the matrices

Sab from the SO3 algebra.

From the canonical eight-component equation (3) we can obtain the following

three types of nonequivalent four-component equations

??a (t, x)

= Ha ?a (t, x), (5)

i a = 1, 2, 3,

?t

H1 = H2 = ??0 E, H3 = ?E, ? = ±1, (6)

where ?0 is the hermitian and diagonal 4 ? 4 matrix** . Under a transformation of

the P (1, 3) group the four-component wave functions ?1 , ?2 , ?3 transform on the

representations (for the sake of brevity we consider only case ? = +1)

1

D+ (s, 0) ? D? (0, ? ), (7)

s=? = ,

2

1

D+ (s, 0) ? D? (s, 0), (8)

s= , ? = 0,

2

1

D+ (s, 0) ? D+ (0, ? ), (9)

s=? = .

2

On the manifolds {?1 }, {?2 }, {?3 } the generators Pµ , J?? have the forms

(1) (1)

= H1 , (1)

P0 Pa = p a , Jab = Mab + Sab ,

(10)

1 Sab pb + Sa4 m

(1)

J0a = x0 pa ? [xa , H1 ]+ ? ?0 ;

2 E

(2) (2)

= H2 , (2)

P0 Pa = p a , Jab = Mab + Sab ,

(11)

Sab pb + 1 ?abc Sbc m

1

(2)

= x0 pa ? [xa , H2 ]+ ? ?0 2

J0a ;

2 E

fact, eq.(2) or (3) is invariant with respect to SO6 ? SO4 group [2]. A relativistic equation of

* In

motion for particle with spin 3 is invariant also with respect to the SO6 group.

2

** The fact that the H and H have identical forms in two eqs.(5) must not lead into confusion since the

1 2

equation of motion is defined completly if only we determine both the Hamiltonian and the representation

of P (1, 3) group.

P , T , C properties of the Poincar? invariant equations for massive particles

e 267

(3) (3)

= H3 = E, (3)

P0 P a = pa , Jab = Mab + Sab ,

(12)

H

1 Sab pb + Sa4 m

(3)

= x0 pa ? [xa , E]+ ? ? x0 pa ? xa E + S0a ,

J0a

2 E E

where

i

H = ?0 ?k pk , (?µ ?? ? ?? ?µ ),

Sµ? = µ = 0, 1, 2, 3, 4.

4

It should be noted that only in the last representation (12) the Hamiltonian H3 = E

is the positive-definite operator. If we add to the algebra (12) an operator of the

change Q = ?0 , then such algebra (in the quantum mechanics framework) has the

same properties as the corresponding Poincar? algebra, obtained by the procedure of

e

the Dirac equation quantization.

It is well known [3] that there exist two nonequivalent definitions of the space-

reflection operator P :

2

P (1) ?(t, x, m) = r1 ?(t, ?x, m), ? 1,

P (1) (13)

2

P (2) ?(t, x, m) = r2 ?? (t, ?x, m), ? 1,

P (2) (14)

[P (1) , P0 ]? = 0 = [P (1) , Jab ]? , [P (1) , Pa ]+ = 0 = [P (1) , J0a ]+ , (15)

[P (2) , P0 ]+ = 0 = [P (2) , Jab ]+ , [P (2) , Pa ]? = 0 = [P (2) , J0a ]? . (16)

Also there exist two nonequivalent definitions of the time-reflection T :

2

? 1,

(1) (1)

(17)

T ?(t, x, m) = t1 ?(?t, x, m), T

2

T (2) ?(t, x, m) = t2 ?? (?t, x, m), ? 1,

T (2) (18)

[T (1) , P0 ]+ = 0 = [T (1) , J0a ]+ , [T (1) , Pa ]? = 0 = [T (1) , Jab ]? , (19)

[T (2) , P0 ]? = 0 = [T (2) , J0a ]? , [T (2) , Pa ]+ = 0 = [T (2) , Jab ]+ . (20)

Besides these conditions usually imposed on the discrete operators P and T we

shall require also the subsidiary conditions

? ?

[Xa , P (1) ]+ = 0 = [P (2) , Xa ]+ , (21)

? ?

[T (1) , Xa ]? = 0 = [T (2) , Xa ]? (22)

?

to be satisfied where Xa is a position operator. The conditions (21) and (22) guarantee

that quantities r1 , r2 , t1 , t2 are the matrices which do not depend on the momentum.

If the conditions (21), (22) are not imposed, then the operators P and T may be

nonlocal (in this case the quantities depend on the momentum).

In addition to the discrete operators P and T we shall introduce some more discrete

operators:

M ?(t, x, m) = rm ?(t, x, ?m), M 2 ? 1, (23)

268 W.I. Fushchych

Mt ?(t, x, m) = mt ?(?t, x, ?m), Mt2 ? 1, (24)

Mx ?(t, x, m) = mx ?(t, ?x, ?m), Mx ? 1,

2

(25)

(26)

[M, Pµ ]? = 0 = [M, Jµ? ]? , µ, ? = 0, 1, 2, 3,

(27)

[Mt , P0 ]+ = 0 = [Mt , J0a ]+ , [Mt , Pa ]? = 0 = [Mt , Jab ]? ,

(28)

[Mx , P0 ]? = 0 = [Mx , Jab ]? , [Mx , Pa ]+ = 0 = [Mx , J0a ]+ ,

where rm , mt , mx are the 4 ? 4 matrices.

There is no need to define specially the operator of the charge conjugation C since

it is equal to the operator T (1) · T (2) (or P (1) · P (2) ).

If we use the explicit forms (10)–(28) for the generators Pµ and J?? and carrying

out the analysis of the conditions (13)–(28) we come to the following results:

1) Equation (5) for the function ?1 (taking into consideration the representa-

tion (10)) is C, Mx , Mt , P (1) T (2) invariant, but P (1) , P (2) , T (2) , M noni-

nvariant;

2) Equation (5) for the function ?2 (taking into consideration the representa-

tion (11)) is P (2) , T (1) , Mx , P (1) T (2) invariant, but P (1) , T (2) , C, M , Mt

noninvariant* ;

3) Equation (5) for the function ?3 (taking into consideration the representa-

tion (12)) is P (1) , T (2) , M , Mx , P (1) T (2) invariant, but T (1) , C, P (2) , Mt

noninvariant.

These assertions may be proved also starting from eight-component equation (2)

(or (3)) in which constraints have been imposed on the wave function [1]. To establish

this it is necessary to analyse the commutation relations between the discrete opera-

± ± ±

tors and the projections P1 , P2 , P3 .

Note 1. It can be easily checked that

P (1) Sa = Ta P (1) , T (1) Sa = Sa T (1) . (29)

M Sa = Ta M,

The transformation connecting the cannonical representations (10)–(12) and the Fol-

dy–Shirokov representation has the form

m + E + ?4 ?a pa

(30)

U1 = .

{2E(E + m)}1/2

Note 2. If we put m = 0 in the reducible representation (4), then it reduces into the

following direct sum of the irreducible representation of the P (1, 3) algebra

ñòð. 59 |