стр. 25 |

??s (t, x, m)g s (m2 )dm2 ,

?s (t, x) =

(2.2)

? (t, x, m) ? R (m), x ? (x1 , x2 , x3 ),

2 s

Equations of motion in odd-dimensional spaces and T -, C-invariance 111

Pµ ?s (t, x, m) = m2 ?s (t, x, m),

2

(2.3)

µ = 0, 1, 2, 3,

?m2 ?s (t, x, m)g s (m2 )dm2 .

Pµ ?s (t, x) =

2

(2.4)

The generators of the Poincar? group on vectors (2.2) are defined in such a way

e

?Pµ ?s (t, x, m)g s (m2 )dm2 ,

Pµ ?s (t, x) = (2.5)

?Jµ? ?s (t, x, m)g s (m2 )dm2 .

Jµ? ?s (t, x) = (2.6)

The Dirac equation for the function ?s=1/2 (t, x) is:

i?0 p0 + i?k pk ? p2 ?s=1/2 (t, x) = 0. (2.7)

µ

One can easily see now that (2.7) can be reduced to the usual Dirac equation if one

formaly replaces the function g s=1/2 (m2 ) in (2.2) by ?(m2 ? m2 ). The generators of

0

P(1, 3) group defined by (2.5) and (2.6) on the manifold of solutions of eq. (2.7) are

given by (1.4), where

P0 ? H ? ?k pk + ? p2 . (2.8)

µ

We can write down the equation of motion for a particle with indefinite mass, wich

interacts with the external electromagnetic field in form

i?0 ?0 + i?k ?k ? ?µ ?s=1/2 (t, x) = 0,

2 (2.9)

where ?µ ? pµ ? eAµ . It is clear that equation (2.9) essentialy differs from the

usual Dirac equation wich discribes the motion of a particle with fixed mass in the

electromagnetic field. A detaled analysis if equation (2.9) will be performed in a

forthcoming work.

Lurcat [8] pointed out, that interpretation of function ?s (t, x) as a wave function

of particle is not correct.

More appropriate is to characterize the unstable system by the density matrix

(operator). In the Schr?dinger picture the equation of motion for the density matrix

o

looks like

??

i = [H, ?],

?t

where H is defined by (2.8).

Equation (2.7) as well as the usual Dirac equation, is P -, T -, C-invariant.

§ 3. Equation for the flat particle and T -, C-invariance

To clear up how can extend the obtained above (sec. 1) results upon any arbitrary

group P(1, 2n+1) let us consider in this section equations of motion wich are invariant

under P(1, 2) group (the group of rotations and translations in three-dimensional

Minkowski space).

112 W.I. Fushchych

The simplest equations invariant under P(1, 2) are:

?+ (t, x1 , x2 )

H + ?+ (t, x1 , x2 ) = i (3.1)

,

?t

?? (t, x1 , x2 )

H ? ?? (t, x1 , x2 ) = i (3.2)

,

?t

H ± = ?k pk ± ??, k = 1, 2,

(3.3)

?

pk = ?i

?1 = ?1 , ?2 = ?2 , ? = ?3 , ,

?xk

Here ?(t, x1 , x2 ) is a two-component spinor, and ?1 , ?2 , ?3 are Pauli manrices.

Taking into account that in this case

? p = ap · 1 + a p ?, ? w = aw · 1 + a w ?, ? c = ac · 1 + a c ? (3.4)

and arguing in a way similar to that of sec. 1, we reveal that equation (3.1) or (3.2)

is T p -, T w - and P T C-noninvariant but P - and C-invariant.

Equation

??(t, x1 , x2 )

H?(t, x1 , x2 ) = i ,

?t

(3.5)

?+ (t, x)

?(t) ? ?(t, x) = , H = ?k pk + ??, k = 1, 2,

?? (t, x)

?k 0 ?3 0

(3.6)

?k = , ?=

0 ?k 0 ?3

is T p -, T w - and C-invariant as well as equation (1.16) is, i.e. it is P T C-invariant, and

for matrices and ? p , ? w we have

0 ?3 0 ?2 ?1 0

?p = ?w = ?c = (3.7)

, , .

?3 0 ?2 0 0 ?1

Thus equations of (1.1), (3.1) type and a whole class of equations of Bargman–

Wigner type wich are derived from the equations of (1.1) type are invariant under the

limited groups:

P(1, 2) are T p -, T w -, T w C-noninvariant and C-invariant;

P(1, 4) are T p -, C-, T w C-noninvariant and T w -, T p C-invariant;

P(1, 6) are T p -, T w -, T p C-, T w C-noninvariant and C-invariant;

P(1, 8) are T p -, T w -, T w C-noninvariant and T w -, T p C-invariant.

.

.

.

To prove the assertiona given above in the case of arbitrary P(1, 2n + 1) group one

has to carry out the very similar procedure to that we employed for P(1, 4) group and

to use the fact that Dirac matrices ? (2n+1) of group P(1, 2n) are related with those

? (2n?1) of group P(1, 2n ? 2) by

(2n+1) (2n+1)

? ?2 , 1 ? ?3 , 1 ? ?1 ,

(2n+1) (2n?1)

?µ , ?2n , ?2n+1 = ?µ

µ = 0, 1, . . . , 2n ? 1.

Equations of motion in odd-dimensional spaces and T -, C-invariance 113

Puting in (3.1) and (3.2) ? = 0 one sees that equation (3.1) concides with (3.2)

and such equation is C-, T -invariant, and ? p = ?3 , ? w = ?2 .

1. Hegerfeldt G.C., Henning J., Fortschr. Phys., 1968, 16, 9.

2. Фущич В.И., Украинский физ. журн., 1968, 13, 363; 1967, 12, 741.

3. Fushchych W.I., Krivsky I.Yu., Nucl. Phys. B, 1968, 7, 79.

4. Фущич В.И., Кривский И.Ю., О волновых уравнениях в 5-мерном пространстве Минковского,

Препринт ИТФ-68-72, Киев, 1968.

5. Rosen S.P., J. Math. Phys., 1968, 9, 1593.

6. Bargman Y., Wigner E., Proc. Nat. Acad. Sci., 1948, 34, 211.

7. Фущич В.И., Сокур Л.П., Препринт ИТФ-69-33, Киев, 1969.

8. Lurcat F., Phys. Rev., 1968, 173, 1461.

9. Brauer B., Weyl H., Am. J. Math., 1935, 57, 447.

W.I. Fushchych, Scientific Works 2000, Vol. 1, 114–125.

On representations of the inhomogeneous

de Sitter group and equations in

five-dimensional Minkowski space

W.I. FUSHCHYCH, I.Yu. KRIVSKY

This paper is a continuation and elaboration of our brief notice [1] where some approach

to the variable-mass problem was proposed. Here we have found a definite realizati-

on of irreducible representations of the inhomogeneous group P (1, n), the group of

translations and rotations in (1+n)-dimensional Minkowski space, in two classes (when

2 2 2 2

P0 ? Pk > 0 and P0 ? Pk < 0). All P (1, n)-invariant equations of the Schr?dinger– o

Foldy type are written down. Some equations of physical interpretation of the quantal

scheme based on the inhomogeneous de Sitter group P (1, 4) are discussed.

The analysis of the Dirac and Kemmer–Duffin type equations in the P (1, 4) scheme

is carried out. A concrete realization of representations of the algebra P (1, 4) connected

with this equations, is obtained. The transformations of the Foldy-Wouthuysen type for

this equations are found. It is shown that in the P (1, 4) scheme of the Kemmer–Duffin

type equation describes a fermion multiplet like the nucleon-antinucleon.

1. Introduction

We recall here the initial points of our approach of the variable-mass problem

proposed in ref. [1]:

(i) The square of the variable-mass operator is defined as an independent dynamical

variable:

M 2 ? ? 2 + P4 ,

2

(1)

where ? is a fixed parameter and P4 is an operator similar to the components of the

three-momentum P , which commutes with all the generators of the algebra P (1, 3)

of the Poincar? group.

e

(ii) The relation between the energy P0 , three-momentum P and variable-mass M

of a physical system remains conventional (here = c = 1):

P0 = P 2 + M 2 ? P 2 + ?k ,

2 2

(2)

k = 1, 2, 3, 4.

(iii) The spaces p ? (p0 , p1 , . . . , p4 ) and x = (x0 , x1 , . . . , x4 ) are assumed to be

plane and reciprocally conjugated. It follows then from (i), (ii) and (iii) that the

generalized relativistic group symmetry is an inhomogeneous de Sitter group1 P (1, 4),

i.e. the group of translations and rotations in five-dimensional Minkowski space. This

group is a minimal extention of the conventional group of relativistic symmetry: the

Poincar? group P (1, 3).

e

In sect. 2 a definite realization of irreducible representations for the generators

Pµ , Jµ? of the algebra P (1, n) with arbitrary n is carried out, which made it possible

to give a proof of the P (1, n)-invariance of the Schr?dinger-Foldy type equations

o

Nuclear Physics B, 1969, 14, P. 573–585.

1 Algebras

and groups connected with them are designated here with the same symbols.

On representations of the inhomogeneous de Sitter group 115

given in ref. [1] for n = 4. Some questions of a physical interpretation of a quantal

scheme based on the group P (1, 4), are considered in sect. 3. Sects. 4 and 5 answer

the question which representations of the group P (1, 4) are realized by two types of

equations linear in ?µ ? ?/?xµ — the Dirac and Kemmer–Duffin type equations.

2. Realizations of the algebra P (1, n) representations

For the sake of generality all considerations are made here not for the de Sitter

group P (1, 4) but for the group P (1, n) of translations and rotations in (1 + n)-

dimensional Minkowski space which leaves the form

x2 ? x2 ? x2 ? · · · ? x2 ? x2 ? x2 ? x2 ,

n µ

0 1 0 k

(3)

µ = 0, 1, 2, . . . , n, k = 1, 2, . . . , n,

unchanged, where xµ are differences of point coordinates of this space.

Commutation relations for the generators Pµ , Jµ? of the algebra P (1, n) are

choosen in the form

?i [Pµ , J?? ] = gµ? P? ? gµ? P? , (4a)

[Pµ , P? ] = 0,

?i [Jµ? , J?? ] = gµ? J?? + g?? Jµ? ? gµ? J?? ? g?? Jµ? , (4b)

where g00 = 1, ?gkl = ?kl , Pµ are operators of infinitesinal displacements and Jµ? are

operators of infinitesimal rotations.

In refs. [2–5] all irreducible representations of the Poincar? group P (1, 3) are

e

studied and the concrete realization for the generators of its algebra is found. The

methods are generalized here for the case of the group P (1, n).

For representations of the class I (P 2 ? P0 ? Pk > 0) when the group O(n) of

2 2

rotations in n-dimensional Euclidean space is the little group of the group P (1, n),

the generators Pµ , Jµ? are of the form

P 0 = p0 ? ? p 2 ? ? 2 , Pk = p k ,

k

x[k pl] ? xk pl ? xl pk ,

Jkl = x[k pl] + Skl , (5)

1 Skl pl

J0k = x0 pk ? (xk p0 + p0 xk ) ? ,

p0 + ?

2

where the operators xk and pk are defined by the relations

(6)

[xk , pl ] = i?kl , [xk , xl ] = [pk , pl ] = 0,

and Skl are matrices realizing irreducible representations of the algebra O(n) which

have been studied in ref. [6].

For representations of the class III (P 2 = P0 ? Pk < 0) when the little group

2 2

of the group P (1, n) is already a non-compact group O(1, n ? 1) of rotations in

[1 + (n ? 1)]-dimensional pseudo Euclidean space, the generators Pµ , Jµ? are of the

form

P 0 = p0 ? ± p 2 ? ? 2 , Pk = p k , Jab = x[a pb] + Sab ,

k

Sab pb ? Sa0 p0 1

Jan = x[a pn] ? J0a = x0 pa ? (xa p0 + p0 xa ) + S0a ,

, (7)

pn + ? 2

1 S0a pa

= x0 pn ? (xn p0 + p0 xn ) ?

J0n ,

2 pn + ?

116 W.I. Fushchych, Yu.I. Krivsky

where a, b = 1, . . . , n ? 1, ? is a real constant, the operators xk , pk are defined by

relations (6) as before, and the operators (S0a , Sab ) are generators of the algebra

O(1, n ? 1) in corresponding irreducible representations, which have been studied by

Gelfand and Grayev [7].

Formulae (5) and (7) give the irreducible representations of the algebra P (1, n)

стр. 25 |