Academy of Sciences of the Estonian SSR

Division of Physical,

Mathematical and Technical Sciences

AS sss* Preprint F-6 (1977)• ' . • ••'•' .-•"• •, . V /

•' •... • - " • ' ' ' 0 f ?

K. Loide, R.-K. Loide

SOMEREMARKSONFIRSTORDERWAVEEQUATIONS

Tartu 1977 Щ

Academy of Sciences of the Estonian SSR

Department of Physical, Mathematical

and Technical Sciences

Preprint F-6 (1977)

K.Loide, R.-K.Loide

Institute of Physics

SOME REMARKS ON FIRST ORDER WAVE EQUATIONS

Tartu 1977•'Si

wi

C O N T E N T S

1. Introduction 3

2. Helativistic invariance 4

y. i&ore on {? matrix 8

4. Harish-Cnandra's and Umez&wa-Viscouti

conditions 13

5. New s = 1/2 equations 23

6. Space reflection 32

7. Scalar product yj

3. Derivability from lagrangian . . . . 39

9. Defifi-iteness of energy and charge . . 44

10. Multiple representations 49

Appendix , 57

References 62

Academy of Scienc&s, Estonian SSR, 1977

1. Introduction

The general principles of the theory of first order

wave equations are well known and written down by Bhabha,

Harish-Chandra, Wild 1/1-4] and others. Bhabha gave the

general form of the ft -matrices in terms of matrices

u(k) and v(k) introduced by Dirac [5]. One particle equa-

tions without subsidiary conditions in a such a formalism

were recently studied for some special representations in

[6,7]. In the case of high-spin equations the above men -

tioned formalism is not very convenient to use because

the dimensions of matrices needed are great. Recently it

has become more popular the formalism given by Gelfand

and Yaglom [8] (see for example [9]). Gelfand and Yag-

lom gave the theory in special basis, which is diagonal

by spin and spin projection (the Gelfand-Yaglom basis).

It allows to reduce the investigation of the equation to

the investigation of so-called spin blocks. The investi-

gation of spin block reduces to the investigation of

such a dimensional matrix correspondingly how many rep-

resentations with the given spin there are in equation.

The Gelfand-Yaglom formalism is recently used for examp-

le in [10-13]. It should be remarked, that the Gelfand-

Yaglom formalism can be used also in the case of infini-

te dimensional representations.

The present work is a semi-review one. The aim of

our paper is twofold: 1) to give the formalism based on

the spin projection operators and write down the condi-

tions anew we get from the Lorentz invariance and other 'f

requirements; 2) to analyse some recently given equations ||$

and to comment on some used principles. The formalism Щ

used here is based on the spin-projection operators ex- д

ploited in papers [14-16] . The relations of construe- f

of ft -matrices in the case of an arbitrary order У

equation were given in C?]» here we present only the

nessecary basic relations. We regard this formalism more

useful in pure-algebraic investigations of equations and

more surveying than the investigation of Gelfand-Yagloa

spin blocks.

To illustrate a given foraaliea wt< consider the new

s = 1/2 equations, analyse the Uaezawa-lTisconti condi -

tion, the definiteness of charge and energy, and the use

of multiple representations. As we shall see, there are

still some unsolved problems.

It should be remarked that the constxuction of high-

spin equations may represent simply a mathematical pas-

time if we remember the troubles connected with the inte-

ractions [18,19]•

In this paper we deal with the first-order relati -

vistic wave equations

( pr pf - л ) ф(р) = О , (1.1)

where it>0 and <Wp) transformes according to some

finite dimensional representation of the Lorenti group.

We use mostly the moaentura representation, but it is easy

to go over to the coordinate representation by the substi-

tution p.* -> \Ър and

2. Relatlviatic Invariance

The relativistic invariance means that the ftl*-aat- Цrices of equation (1.1) must satisfy the following со»- Щmutation relations |l

' , (2.1) I

where sl*g are the generators of the Lorents group (cf. i

Appendix). I.

In following we give the general f o m of ft* matrix '£'?

.*'«

- 5 -

in the representation where S^* is the direct sum of

irreducible representations: (a«,b4)ф (aj.,ba)<£ ... ф

(a«.,b>.). More shortly 1 ф 2 ф ...®г , denoting к =

(aK|b

K) . Then ф(р) and S**' are in form

ф(р)=

Ф,(р) 0

0(2.2)

where s£" are the generators of the representation к =

(аи,Ък)

a n^ 4i P *

n e corresponding components. ft°

is written in form

(2.3)

where ar t are numerical coefficients and t

Kg are cer-

tain fixed matrices. Due to the hoaogenity of (2.1) there

are no conditions to а ^ ( the eigenvalues of &° de -

pend on the choice of aKt )•

How we represent the prescription for the construc-

tion of matrices tKe given in [1?]. Invariance under

space rotations gives from (2.1)

where u,v = 1,2,3. The last relation allows to express

tKt with the help of spin-projection operators

Appendix) in the following general form

(2.5)

t$(cf.

where the sum is over all spins s comaon in representa-

tions к and 1. ol(s) are SOB» coefficients, spin-

projection operators t£e are expressed with the help of

Clebsh-Gordan coefficients

:;1.13

- 6 -

Cc = иГ* U,e , (2.6)

where U4 t is matrix formed from the Clebah-Gordab coef-

ficients <• € I (1)> which separated »pin s from therepresentation 1 and UtK is matrix formed correspon-dingly fron < s e |(k)> » U? K is UtK hermitiallyconjugated. In the general analyse of equations the con-crete form of t?t is not important, but the followingrelation is very useful

(no sum over 1 ) whieh allows to multiple them.As we have mentioned in introduction, the spin-pro-

jection operators t 3? axe used in papers f 14-16]. Inthese papers there are analysed certai:. high-order equa-tions (cf. [17] )•

As one can see, the invarianee under space rotationsleaves the expression of t *t ( and also (S* ) quitegeneral: t Kt i« equal to zero only for representationsк and 1 which have no common spins. Further we shall

deal with the space reflection operator П (Sec. 6) and

scalar product operator Л (Sec. 7) both satisfying

(2.4) and therefore havimg the same general form (2.3) as

Invariance under the pure Lorents transformation»res-

tricts the choice of representations к and 1, and de-

termines the coefficients o<(s) . As it is known, in the

case of the first order equations there exists a acnsero

matrix tut only la Ida* case of "linked" representations

i.e. when к = (а,о) then the representation 1 = (c,d)

must be one of representations (а+1/2,Ъ+1/ё)*(а+1/2,Ь-

1/2), (a-1/atb*1/2) or (a-1/2,b-V2). If we denote after *

Gelfand and Taglom the representation linked with к by

k* we may state that only t**./ 0, but if U k ' then

= 0-

further on we give the relations tor the calculation

- 7 - I

of c*(s). In the general ease o((e) are also expressab-

le with the help of Clebsh-Gordan coefficients [17], but

in the case of the first order equations we get the analo-

gical relations as Gelfaad and Yaglom [&]. We apply an

additional convention that the oC(s) corresponding to

mnrlmai common spin sn a z

is equal to one, i.e. o t

( *m a x

)S !

1. Then we get in the case of the representation k=(a,b)

for different 1 = k1 the following relations

1°. k* = (a+1/2,b+1/2).

2°. kf (

Г (a+b+1+a)(a»b-a)Г

- I3°.

°4°. к' = (a+1/2,b-1/2).

These formulae are connected by substitution 1°«-»2°

a+b+1 *-* a+b and 3°** 4° a** b .

The matrices tKK< are uniquely determined by (2.5),

(2.6) and (2.8). Concerning the relation (2.5) it should

be remarked that in the case of the first order equations

tKK> contains the projection operators t^. of all com-

mon spins in representations к and k1. In the case of

the higher order equations there are less restrictions on

ot(s) and we may,at request, set some «<(») equal to

«его [17].

As an example we write down some t ^ operators

used below. When the linked representations have only one

common spin s then obviously ti«> = t^t . For examp-

s "

- 8 -

le, if к = (1/2,0) and к* а (0,1/2) or vice versa, we

get tKK, = t^', ; if к = (0,0) and k' = (1/2,1/2) or

vice versa - t^» = «*& • or more generally, if к =(s,0)

and k» = (s-1/2,1/2) (or к = (0,e) and kf = (1/2,e±

1/2)) or vice versa, then tKt> - t'&t . In the case of

several common spins we oust use the relations (2.8). For

example, if к = (1,1/2) and k' =(1/2,1) then there are

common spins 3/2 and 1/2, and therefore tKKi = *VK/ +

4- 1/2 tl$ . The same is valid for к = (1/2,1) and к* =

= (1,1/2). In the same way it is possible to find tx*'

in the case of two arbitrary linked representations.

Further on we deal with the construction of fi" mat-

rix more detailly.

In the previous section we gave the general prescrip-

tion for construction of |S!" matrix. In comparison with

the Gelfand-Yaglom formalism the difference is mostly in

notations and in choice of basis: we use the so-called di-

rect-product basis, the coefficients a*e differ from

the Gelfand-Yaglom coefficients C** only by the numeri-

cal factor. Various bases and relations between them are

studied in Г20].

To get more detailed information about the equation

it is useful to dismember the ft* matrix. As in the case

of the Gelfand-Yaglom basis it is useful to arrange the

fl* matrix to spin blocks, in our case it is useful to

represent ft" in form

where ft ' includes only spin-projection operators

of spin a-,, [21]. Due to the fact that spin-projection

operators satisfy (2.7), matrices (ltt4> satisfy

r&

_ 9 -

(3.2)

The operators ft*** are in general not projection opera -tore, because in the general case ( jJts°)2 * {5°°. Matri-ces (£ttcJ determine the properties of the particles with

\ spin Sj, . $or example, if an equation describes partic-les with spin вi > then (i

1*** must have nonzero eigen-

values, but if an equation does not describe particles

with spin в i then all the eigenvalues of {&0° must

be zero and therefore jbey must be nilpotent.

;- In the following we mostly deal with the equations

0:: which described particles with one fixed rest bass m ,

3 i.e. the single-mass equations. The generalization of the

•№.. case of several masses is not principially complicated.

;•/ Now we write down minimal equations (minimal polynomials)

K; for ft0** and projection operators which separate the so-

)•: lutions corresponding to nonzero eigenvalues. In the case

л of single rest mass m the matrix ft* has only two non-

-, zero eigenvalues 4b and -b while without the lost of

' generality we may set b = 1 and then in the equation

f (1.1) Л = m . As it was shown by Harish-Chandra [3]

I ft° satisfies the minimal equation

J ( (ie )

L(( (b

e )

2 - I) = 0 . (3.3)

I As we have mentioned in the previous section ( cf.

.; (2.5)) tKk' contains the spin projection operators of

I all spins s common in the representations к and k1. • •;

I Therefore fc° contains in general all spins s4 , s».,..., [•

I 8 K common in linked representations and thus the cor- '••

% responding matrices |&№) in the representation (3.1). Щ

I The representation (3*1) contains also such spins sc '

§ which we do not want to describe with our equation. Let 2

I us suppose that our equation describes the particles with /

I spins s4, в

г, ..., s, .It means that the eigenvalues Щ

of corresponding (J>(>i> must be ±1 and О and A

w° sa - V

tisfy the minimal equations я

4II

- 10 -

where a t > 1 ( a t = 0 only in the case of the Oirae bi-spinor). The matrices ft0''corresponding to the remain-der spins Sq,.4 , . . . , s^ are nilpotent

tti»)*J = 0, (3.5)

where bj > 2,Using the expression (3.1) of A° , the relation

(3.2) and minimal equations (3.4) and (3*5) *e get that(b0 satisfies (3.3) with L = max{ai,bj} . As the t{£

satisfy

Z C = i. (з.б)

«here the sura i s over a l l spins in the representation k,then L may be even less than max { a; ,bj} . Thereforein general

L i maxjai.bjj. (3.7)

In the case when our equation describes particles

with single spin в к , only (J,

4*0 satisfies (3.4) ,the

remainder ones satisfy (3.5).

фде minimal equation» (3*4) and (3.5) restricted the

choice of coefficients aKK< in (i° matrix. Since we do

not have any general relation for the powers au and bj

(cf. the next Sec.) one must start in (3*4) and (3.5)

from the minimal possible value of a; or bj and cla-

rify whether there exists the coefficients aKK< which

satisfy the corresponding minimal equations. Due to (2.5)

the different аШ1 are not generally independent because

if we fix the coefficient а ^ before tKi0 « it also

determines the coefficients before t^i - aKKi c*(s).

To separate the solutions with given spin it is use-

ful to write down the corresponding projection operators.

In the case of the minimal equation (3.4) the projection

operators corresponding to eigenvalues ± 1 are [21,22]

t? =•':

- 11 -

at - even.

The relations (3.2), (3.4) and (3.8) give

& -*s Г

(3.8)

(3.9)

Using tbe expression (3.1) of ft* and the relation(3.9) it is possible to verify that'

and (З.Ю)

are the solutions of an equation (1.1) in the rest system

P = 0, ± is the sign of p°/ | p*|. The solution in the

system with the momentum pt* is obtainable from the solu-

tions (3.10) by boost-transformation

U = exp ( -4.SeM) , (3.1D

where \/$ = pu/lP*l and th$ =|pl/p

a. Studying the ge-

neral properties of equations we may confine oneself to

rest system.

To illustrate the before-mentioned prescription we

give some f>° matrices.

1) The representation

RL = (1/2,0) Ф (0,1/2) (3.12)

(Dirac bispinor). Denoting 1 = (1/2,0) and 2 = (0,1/2)

we get

(3.13)Ф,«k

How in (3.1) fbe « p.

00. In the case of a,

t= a

M= 1 we

get the (Ь* matrix of the Dirac equation.

2) The representation

Нг = (1,0) Ф (1/2,1/2) Ф (0,1) . (3.1*)

Denoting 1 = (1,0), 2 = (1/2,1/2) and 3 = (0,1) we get

•л

I

- 12 -

e

Now (i°«this is

3)

VDenotingwe get

f-а„

0

0

theThe

( 1 ,

a u C 00 aj,tt\

a^tVt 0ф=

ф,

Ф,Ф»

In the case of a,x= a t, - aM = (Kemmer-DuffiQ

representation

1/2) ф (0,1/2) Ф (

1=(1,1/2), 2=(Ot1/2),

0

0

*?•

0 a,

0 &г

0

0

4amatrix for spin

1/2,0) ф (1/2,1).

3=(1/2,0) and

a«(*^+ ^

0

0

(3

1

(3.

.15)

I/V2

16)

*=(1/2,1)

• (3. 17)

Now j4e- (

.«*>

0

0

0

0

0

where

0

0

0

0

0 a

0

?

0

0

0

0

0

0

0

(3-18)

00 1

П гtit»

«О

О

О

О

о

(3.19)

of a

in«ч

and (Ь are not independent because the fixing

and a4, in ft**' determines two coefficients

- 13 -

4. Harish-Chandra's and Umezawa-Visconti Conditiona

As we have mentioned that in the case of single rest

mass Harish-Chaodra has shown that fi>° satisfies the mi-

nimal equation (3.3). Proceeding from quite general as-

sumptions it was proved by Umezawa and Visconti [23]

that in the minimal equation (3*3)

Ь = 2 S j a a x

- 1 , (4.1)

where втя^. is the maximum spin contained in the repre-

sentation of an equation (1.1). Although the relation

(4.1) is valid in the case of many equations, it is not

universal. Firstly it was demonstrated by Glass [24]

who also found the equation where L > 2 sm a z

- 1 . The

' | Umezawa-Visconti condition was also analysed in papers

f [25,26], where (4.1) was enlargened to

2B - 1 £ L £ 28„„ - 1 , (4.2)

where в is the spin described by equation. As we see

this relation is also violated. From [25,26] it turns

out that the condition L £ 2s]naz-1 is acceptable, but

L > 2sma:K-'1 i s n o t quite understandable.

The proof of Umezawa and Visconti is based on the

statement that the Klein-Gordon "divisor" d(p): d(p)«

"(Pc fi.1* - m) = p1 - m 1 transforms like the direct pro-

duct ф » ф . Then the order of d(p) in the powers of

momentum p is restricted by the maximal representation

of the rotation group in ф(р) which is equal to 2s

and it in turn restricts the minimal equation of |Se .

By our opinion the assertion that d(p) transforms like

ф ® ф is not quite correct. d(p) can be expressed

as polynomial of quantities p,, jjt* . Under the Lorents

transformations Pj» -» pj. s pr pi* - » p'

r ftf therefore

d(p) -• d(p'). Altogether it is problematical, if the

matrix elements of matrices ft1* and the quantities

i9.S

1

formed from it (as the matrix elements of d(p)) can be

treated as the tensorlike quantities.

Let as once more examine the coalitions on the order

of micimal equation of jl° and demonstrate that it also

depends on the number of irreducible representations in

an equation (1.1) and therefore not only from the spin.

We do not know whether there exists a general expression

for the order of minimal equation, but it is possible to

declare that L is not greater than the number of irre-

ducible representations in an equation (1.1). We can see

below that the study of ft4*** reduces to the study of

such a dimensional matrix as much irreducible representa-

tions there are. Then in (3*?) b*. cannot be greater

than the number of irreducible representations which in

turn gives the obstruction to L.

Now we study ft matrices when there are few ir-

reducible representations and we become convinced that

in such cases the minimal equations are quite determined.

Let us suppose that there are only two linked repre-

sentations

the followingк and к*. Then the general form of [J

a> is

b twK'tC

(4.3)

Taking into consideration the relations (2.7) it is easy

to see, that the study of nonzero eigenvalues of p>L>> is

equivalent to the study of nonzero eigenvalues of the two

row matrix

Indeed

ab

A =

0

0

b

0

a

0

I

1

A2 = abI

0

- 15 -

- Л р « -+ A3= a b A .

As we see, the maximal degree of the minimal equation is

always equal to 3 and nonzero eigenvalues are ±\/ab .

Setting ab = 1 we get the minimal equation of (iu> in

form (3.4) where aL = 1. a

t= 0 is only in this case

when tj." = I and t?>*<= I * i»e. if the representations

к and k1 contain only one spin (representations (1/2,

0) and (0,1/2)). The minimal equation of A has here a

degree equal to 2, but the degree of (J>tt> is greater by

one, although A2^ I , but in general ((J

u;)

a - I is not

valied because the spin projection operators t"t had

in general also the eigenvalues equal to zero. More gene-

rally, if the r-row matrix A has Ar ~ I, then the deg-

ree of £(i) is greater by one.

Demanding the nilpotency of (ilv) we get ab = 0 :

A1" = 0 and (fto))** = 0 and therefore in minimal equa -

tion (3.5) Ъ-= 2, while the higher powers of |5<$> gave

no new conditions on coefficients a and b.

Summing up we get that in the case of two represen-

tations the maximal degree of minimal equation of (***' is

equal to 3 independently on the spin value s (only Di-

rac bispinor has degree 2). The coefficients are restric-

ted by the condition ab = 1. The maximal degree of nilpo-

tency is equal to 2 and ab = 0 that is not a good con-

dition if we want nonzero a and b.

Equations with such a fb° are already studied. In

the paper [27] Hurley has given equations which descri-

be spin 8, using representations (s,0) ф (s-1/2,1/2) or

(0,s) ф (1/2,s-1/2). Analogical equations one can get

using the representations (s,0) ф (s+1/2,1/2) or (0,s) Ф

ф (1/2,8+1/2). In both caees (3° is in form [28]

I.

- 16 -

where 1 = (s,0) and 2 = (8*1/2,1/2) or 1 = (0,8) and

2 = (1/2,sfc1/2). Here |b° is in for» (4.3) where a=b=1

and ft0 * ft

O). In the case of such equations L = 1

(L = 0 only in 8=1/2 case). The representations (s,0) ф

© (s+1/2,1/2) and (1/2,8+1/2) ф (0,s) have s ^ s

= s + 1 and L < ^Bjaax

- 1 = 2s + 1 if s > 0. The rep-

resentations (s,0) ф (s-1/2,1/2) and (1/2,8-1/2) ф

ф (0,s) have s^j = s and L =• 1 < 2 8

щяТ - 1 if s > 1.

Here the degree of minimal equation does not depend on the

spin. As one can see, the conditions (4.2) given in f25,

26] are not valid here.

In the case of the representation Bi (3.12) we ha-

ve the Dirac equation, the representations 1 = (1/2,1/2)

and 2 s (0,0) give the Kemmer-Duffin в = О equation.

Equations, where (1° satisfies the minimal equa -

tion ( jbe )* = fb° are studied in [13].

Now we suppose that there are 3 linked representa-

tions in an equation, we show that also in this case the

minimal equation is uniquely determined. The general lin-

kage is following 1 ** 2 ** 3 and fttt* takes tee form

с tti

0

The problem of finding eigenvalues reduces to the fin-

ding of eigenvalues of a 3-row matrix

I Here

A* -

A =

ab

0

od

0

b

0

0

ab+cd

0

<

ac

0

cd

a.

D

1

0

с

0

AS = (ab+cd) A .

- 17 -

As we see A3 — A and one may take ab+cd = 1. The con-

dition of nilpotency A = 0 is not a good one if we want

nonzero coefficients a, b, с and d. The next possibility

A3= 0 is satisfied by ab+cd = 0.

Therefore in the 'case of three representations the

maximal degree of minimal equation of ^>u> is equal to

3, i.e. in (3*4) at is not greater than 1 and in (3.5)

b| is not greater than 3»0nce again the minimal equa-

tion does not depend on a spin.

The example of such an equation is the Keamer-Duffin

s = 1 equation for representation Ra - (3«14). In this

case |ie« ft* and a = b = c = d = 1/^2 (or in (3.15)а„=

As we have seen, in the case of two or three linked

representations the minimal equation is uniquely determi-

ned and there are oaly two nonzero eigenvalues - equal by

modulas and opposite. When we have more than three linked

representations there are more possibilities and the mini-

mal equation is not so determined. It is possible to get

different nonzero eigenvalues and each linkage does not

permit a single mass.

We also consider the case of four linked representa-

tions, ere are three different linkages.

1) Linkage 1 «-» 2 *-• 3

4

The corresponding /1 matrix is the following

,<*)

a tмn.

**' d

**'

0

(4.7)

The study of nonzero eigenvalues reduces to the study of

matrix

- 18 -

A =

О

b

оо

а

О

вf

ос

О

о

оd

оо

It Is easy to verify that A3 = (ab+ec+df) A and the ma-

ximal degree of jbu> is equal to 3 and It Is possible

to get a single oass. In the ease of four linked repre -

santations it is the only one which gives the unique re-

sult.

2) linkage 1 *•* 2 *-*• 3 *-• *• Corresponding pf i»

f

оо

0

0

(*.8)

Here we must study the eigenvalues of matrix

0

b

0

0

a

0

d

0

0

с

0

f

0

0

e

0

i

Computing the powers of A it is possible to verify that

the the case of nonzero coefficients one cannot get sing-

le-aass conditions A* ~ A or A4 ~ A* . Therefore in

single-mass equations such a linkage cannot occur.

3) Linkage 1 «-» 3t IЦ. «-» 2

Corresponding ftW) is the following

гоо

оо

а

с**> .t»

чг

О

О

and it reduces to the matrix

0 0a

0 0c

e f 0

g h 0

b t£

о

о

b

d

0

0

(4.9)

It allows to get different minimal equations. Depending

on the choice of coefficients it is possible to get both

single- and multimass equations.

Here we have dealed separately with the minimal

equations of matrices |bw . In an equation there are

generally several fbu> corresponding to different spins

and they are not independent. Although the investigation

of |3>* reduces to the investigation of matrices (i(>) it

should be taken into consideration that the fixing of

coefficients in some |btt> determines some of them in ot-

her matrices p>w .

As an example we now consider an equation given with

the representation R, - (3.16), the (J* matrix of which

was given by the formula (3*17)• At first we deal with the

matrix (J><*t> (formula (3.18)). There are two linked rep-

resentations and it reduces to (4.3). Therefore we get

that ftcvl> satisfies the minimal equation

( ( Л * = awa

4« (i

lK). (<И10)

If we want to describe the spin 3/2 , we must take

а4 Ча

Ч 4 = 1 . But if we do not want to describe the spin

3/2 we must take а4 Ча

ч 4 = О and we get ft° x a***

(cf. the next S e c ) . Now we deal with the equations that

describe s = 3/2 particles and choose the coefficients

- 20 -

in the following way

The part corresponding to the spin 1/2 - p> (3.19)

reduces to (4.9). taking into account the choice (4.11)

it is possible to verify that the only possible minimal

equations are.г)

г = 0

orС

(4.12)

(4.13)

Concerning the choice of coefficients in equation

we have so far no other conditions as those we get from

the minimal equation. Usually the additional requirements

- invariance under space reflections, derivability from

lagrangian etc. (cf. Sec. 6, 7 and 8) are imposed which

in turn restrict the choice of coefficients. Further on

followinglj

с (4.14)

we choose the coefficients of ft041

This choice is done from the requirement of invariance un-

der space reflections and othervise under the considera -

tions of symmetry (cf. Sec. 6).

Now, if we want to describe only a e = 3/2 partic-

le, ft must be nilpotent. Demanding (4.12) we get

ab = - 1/4 and с = - 1/2. (4.-Г5)

This variant corresponds to the well known Eauli-Fierz

equation. As (****' satisfies the Binimal equation (3.4),

where a%= 1 and ^ satisfies (3.5)» where Ь^= 2,

then in Idie minimal equation of fie in (3.3) L = 2 ,

while the Umezawa-Visconti condition (4.1) is also satis-

fied. Since the Fauli-Fierz equation is a relatively simp-

le nontrivial example of a high-spin equation it is there-

fore rederived in many of papers [6,10,17,20,21,29].

then we, in addition, want to describe s = 1/2 par-

- 21 -

tides the minimal equation (4.13) gives the following

possibilities [21]

ab = 1/4 and с = 1/2 (4.16)

ab = - 3/4 and с = - 3/2 (4.17)

ab = З А and с = - 1/2 (4.18)

In first two cases we in addition to a = 3/2 particle

have one 8 = 1 / 2 particle, while in minimal equation of

fte 1 = 1 . The third ease describee one e = 3/2 and

two 8 = 1/2 particles. Due to the relation (3.8) :

t(

44 + t

c.JV = I and t'JV + t'w' = 1 |i° satisfies the

minimal equation ( p»e )* = I. All these cases do not sa-

tisfy the Umezawa-Visconti condition.

It should be mentioned that in all three cases |*>11Ь>

satisfies the same minimal equation, in first two cases

the equation describee one, in the third case two 8=1/2

particles. Therefore the minimal equation gives in gene-

ral no information about the number of particles descri-

bed by equation.

The number of particles with the given spin s is

obtained calculating the trace of projection operator

P ±4 . Due to Tr t"i. = 28+1 Tr Р^« is given in

the form

Tr Р*„ = N (28+1), (4.19)

where Я is the number of particles with spin s. Calcu-

lating Tr P ^ in the casee (4.16)-(4.18) we get in

first two cases 5 = 1 and in the third case N = 2.

The equations we now dealed with we also consider

in the following sections.

In examples not satisfying the Umeftawa-Visconti con- '

dition so far L < 2eB a x

-1. It is possible to get equa- |

tions where L > 28^^-1. Evidently it is easy to const- '

ruct such equations using multiple representations be - .<

cause the greater number of representations allows to get $'•'%

- 22 -

the higher degree in minimal equation of |J* . So it

was in the equation constructed by Glass [24] who used

the aultiple representation (1,1/2) ® 2(0,1/2) ® 2(1/2,

О) Ф (1/2,1) : £ , $ & , . (b0** is here the same (nonzeroparts) as in the Fauli-Fierz equation, ftt5tJ however hastwo more representations and possesses the nilpotency con-dition ( Jbtl4))S = 0 which in turn gets ( pe )*(( pe )*"-I)== 0. A possible equations with the representation used byGlass are studied in [29] • Equations where Ь > 28^^-1are also given in [30,31] • In [30] 8 = 1 is descri-

bed with the representation (1,0) ф (1/2,1/2)ф (0,1) ф

ф 2(0,0) (cf. Sec. 10), in papers [31] are given equa-

tions for s = 0 and s = 1 using the representation

(1,0) © 2(1/2,1/2) ф (0,1) ф (0,0). In all these equations

L = 3 and s , ^ = 1.

Here we give one more possible s = 1 equation

using the representation

Нц = (1,0) Ф 2(1/2,1/2) Ф (0,1) = Ht®(1/2,1/2).(4.20)

Denoting 1=(1,0), 2=(1/2,1/2) and 3=(0,1) (be

in form

о

о

о

0

0

0

0

оо

соi l

. ••

is given

(4.21)

It is easy to verify that ft* satis -Now (i° • (lf i e s the minimi equation

((*?)*• ( ( | » Г ) * - I ) « 0.

Since sa a x s 1 and L = 2 we get L >

2 в

и а х-

1»

given equation 1з interesting in son» cases. It describes

one s = 1 particle - calculating Tr P±4 we from

(4.19) obtain 9 = 1 . Differently from the previous equa-

tions p>w satisfies the minimal equation (3.4) where

- 23 -

a^ = 2 (previously a5 = 1). As we shall see in Sec. 8

our equation gives singular hermitianising matrix Л .

In the next section we analyse some possible я =

- 1/2 equations using the representation B- - (3.16)

and shall show that these are somewhat anomalous ones.

5. New s = 1/2 Equations

Recently the new s = 1/2 equation which differs

from the Dirac equation was given by Capri [32]. Such

equations were further treated in [25,26,33-35]. Here

we once more deal with such new s = 1/2 equations and

shall show that these are somewhat anomalous, containing

the ballast representations which are not essential in

the equation or giving the energy equal to zero.

If we want to describe only s = 1/2 particles

using the representation B, - (3*16) we have found in

the previous section that in ft one must take ara

4l =

= 0. We set a^s аЧ4 = О, we choose the other coeffi -

cients by the relation (4.14). flow fi," * fitK>and

144)

0

0

0

0

с tt t

a t4 t

с t;

0

0

Ъ t

О

О

24(5.1)

By choosing the coefficients we start from the lowest mi-

nimal equation

" :(ft*)fc - 1 ) = 0. (5.2)

I

It gives the following three possibilities:

1°. b = О, с = 1, a - arbitrary;

2°. a s 0, c2 = 1, b - arbitrary;

3°. с = 0, ab = 1. I

The next minimal equations ((&*)= ( (i* ) etc. do not

give more solutions being of interest. Thus the relation

( jie )* = (ft* )

2 gives in addition to the previously gi-

ven two solutions: a = с = 0, b - arbitrary and с = b =

= 0, a - arbitrary. These solutions however give ( fi° )2=

= 0.

Now let us deal with a given equations more detailly.

get

D b s O . с g 1. a - arbitrary. Setting

fte in form

f'

0

0

.*«?0

0

0

0

0

0

(5.3)

In the case of e / O it is Capri's new s = 1/2 equa-

tion [32].

We demonstrate that this equation contains the Dirae

equation but has ballast components ф„ and <£<t (repre-

sentations (1,1/2) and (1/2,1)) which have no independent

meaning in equation and which are fully determined by the

Dirac equation.

As the representation of 4* is a direct sum of ir-

reducible representations in the equation (1.1): РцА^ф =

= a the left hand side has the same structure as fi°

0

0

0

0

0

0

0

0 4ц 4ч

(5.*)

where = a = 1,2, 3) etc.. More detailly (5.4-) takes the fora

- 25 -

f. *Гг 4» * m +«

га

= ш

«к J • (5.5)

Now it is obvious that we have here an independent equa-

tion for representations 2 and 3 » i»e. (0,1/2) and

(1/2,0). It is the usual Dirac equation (cf. (3.13)). The

first and last equations in (5.5) determined the coapo -

nents ф, a n

^ <IN by *n e solution of the Dirac equa -

tion.

Equation (5.5) is therefore essentially equivalent

to the Dirac equation but has ballast components ф* and

j •,, фч . As it is easy to see, it is impossible to reduce

\ the given equation in some natural way to the Dirac equa-

• tion (i.e. with some nonsingular transformation). Concer-

ning the ballast representations it should be mentioned

that they have no independent meaning and we can get a

"normal" equation simply discarding them from equation

(more exactly, discarding the component equations with

these representations). These representations which we

cannot be discarded from the equation without violating

the invariance condition are the "essential" ones and

thus also the corresponding components. In given equation

the essential components are фа. and фа .In [29] such

ballast components are called "barnacles".

In an analogical way it is possible to construct new

equations starting from some given equation and adding

ballast representations. We can similarly add the repre-

sentations (1,3/2) and (3/2,1) to the Pauli-Fierz s =

= 3/2 equation.

In interactions the Capri equation cannot be nesse-

carily equivalent to the Dirac equation. In that case the

: interaction term must spoil the general structure of the

equation (5.4). But if the interaction term leaves the ge-

i)

1-

- 26 -

neral structure of equation unchanged, the properties of

a particle described by it are again determined by the

Dirac equation for фг and <V* . That takes place in

the case of a minimal electromagnetic interaction where

the interaction term has a form e A j. f£t* . Thereforethe electromagnetic properties must be the same as inthe case of the Dirac particle» The preliminary calcula-tion given by Capri [33] gives the magnetic momentwhich differs from that of a Dirac particle. The latercalculation [35] shows that there are no differencethat ie obvious from the above mentioned.

The equation considered here is invariant under theapace reflections, but has no hermitianizing matrix (cf.Sec. 6 and 8).

2) a = 0. c2 = 1, b - arbitrary. We again set с == 1 . Now &° has a form

0

О notoo о

0

0

0

(5.6)

If b 4 0 it is a new spin 1/2 equation given in [25] •

The (£ -representation is algebraically equivalent tothe hermitian conjugate of the Capri's ft-representa-tion.

It is easy to verify that this equation is fullyequivalent to the Dirac equation because here alwaysф, s <!»„ = o. Evidently for that reason this variation

was not considered by Capri in £32].The general form of the equation is now

»;••'

- 27 -

о о

о о

о о

or more detailly

О

О

г *гь Фь

Фг

фч

Ф,

(5.7)

О = m ф,

(5.8)

As we see, always ф, = 4N = 0 ( m / О ) and we may lea-

ve tbe corresponding representations out from the equa -

tion. The remaining equation is the Dirac equation for

<|>г. and ф) .

If the interaction term leaves the general structure

of the equation unchanged,we also get ф„ = 4s = 0. It

is so, for example, in the case of a minimal electromag-

netic interaction. The minimal electromagnetic interac -

tion was studied in [34-], where also the magnetic moment

of a Dirac particle was obtained.

The ft -algebras of two first equations were stu -

died in the papers [25,26,35]. Although the (b -algebra

is very important in calculations it is also nessecary to

know the properties of an equation corresponding to the

given fi> -algebra. It should be remarked that the (Ъ-

algebras studied in the above aentioned papers give the

analogical anomaluos equations, as the s = 1/2 equations

considered before. These ft -matrices do not have any

nonsingular hermitianizing matrix at all.

3) с = 0. ab = 1. We set a = b = 1. (V has a

form

- 28 -

О

о

t(5.9)

Now we get the Hurley-type of an equation [27] (cf. Sec.

4 ) which describes two independent a = 1/2 particles.

Writing out the general fora of the equation we get as

before

м 4s = •

S ^ ^ = m(5.Ю)

As we see, we get two independent 8 = 1/2 equations :

one for 4>4 and Ц>» , and the second for <4>

L and <|

4 .

The equation therefore describes two independent partic-

les.

Comparing with the previous equations, here are no

superfluous representations. The equation is invariant

under the space reflections and has also a heradtianizing

matrix (cf. Sec. 6 and 8 )- It occurs that this equation

is anomalous as well because the energy and charge densi-

ties are identically zero (cf. Sec. 9 ).

•bet us write down the linkage schemes for equations

considered here. If in |i° there is an operator ttti we

denote it on the scheme by к —» k'.

In the case of a first equation we get

(5.11)

On the given scheme the bilateral linked representations

are the essential ones, the unilateral representations

directed to them are "ballast" representations.

(0,1/2)

f(1/2,1)

«-» (1/2,0)

t(1,1/2)

••3

- 29 -

To the second equation corresponds the scheme

(0,1/2) «-• (1/2,0)

4, I (5.12)(1/2,1) (1,1/2)

The unilateral representations directed from "essential"

ones indicate to components which are equal to zero.

In the case of a third equation

(0,1/2) (1/2,0)

(1/2,1) (1,1/2)

Now we have two independent equations.

The equations without "ballast" components which

differ from Dirac equation were giren using the repre-

sentation (1,1/2) Ф 2(0,1/2) Ф 2(1/2,0) Ф (1/2,1) by

Ulehla and Petrad. These equations are studied in [36].

We also deal with the 8 = 1/2 equations given in

[37»3S]« In these papers the s = 1/2 equations are gi -

•en by doubling the components in Kemmer-Duffin s = 0

equation. The Kemmer-Duffin equation transforms accor-

ding to the representation

R5 = (1/2,1/2) Ф (0,0) . (5.1*)

The doubling of components means the direct product with

the representation (1/2) or (0,1/2). We get two possi-

bilities

R = Re® (1/2,0) = (1,1/2Ж0,1/гЖ1/2,0)(5 15)

(0,1/2) = (1/2,1)8(1/2,0)6(0,1/2)

As we see, these representations are not reflection-inva-

riant and therefore in the case when the representations

(1,1/2) and (1/2,1) are the "essential" ones, the cor-

responding equations are not invariant under the space

reflections, and are not derivable from lagrangian.Using

the representations (5*15) it is possible to get two ty-

- зо -

pes of equations: in the equations of the first type the-

re are all representations "essential", in the equations

of the second type the representations (1,1/2) and

(1/2,1) are equal to zero and the equation we get is a

Dirac equation. It becomes obvious that the equations

given in [37,38"] are the first type equations and the-

refore do not satisfy the requirements demanded by its

authors.

In the ordinary Kemmer-Duffin fl -representation

the ft° has the form (cf. Appendix)

-i

0Ф*

Ф i

(5.16)

Here the components фо , ф« , <|ч , ф» transformed

according to the representation (1/2,1/2) and ф ac-

cording to (0,0). In the rest system <|ь 4 0 and

ф 4 0 the remainder components are equal to zero. How

let us consider the direct product with the representa-

tion (1/2,0) (i.e. the doubling of components). Here

ф Ф (1/2) transforms according to (1/2,0), but фвФ

9(1/2,0) does not transform according to (0,1/2) on-

ly, but as a direct sum (1,1/2) 9 (0,1/2). We do not

perform here the reduction of the direct product (1/2,

1/2)ф(1/2,0) but only mention that according to

(0,1/2) in the direct product transformed the quanti-

ties ф«, + фм + фи. -i фм. and 4

L- ф

и + +,, +1фц

where the first index of фр». is vector index and the

second one - spinor index. Now it is obvious that the

фоф (1/2,0) components ф

о< *nd Ф«г do not trans -

form according to the representation (0,1/2). Therefo-

re the doubled фа and ф components do not give us

the Dirac bispinor. For that reason one cannot connect

them by the interaction term e^Fpv of a Dirac equa-

Ik

- 31 -

tion as in papers [37*38].

If we consider the linkage of equations with doub-

led components, we in the case of Bg get

(1,1/2) «-» (1/2,0) •-* (0,1/2) . (5.17)

The corresponding f£° matrix in our notations has a

font (4.6) where 1=(1,1/2), 2=(1/2,0) and 3=(O,1/2).

Now j&° m (ЬЛ), |Ъ° satisfies a minimal equation ( (? )=

= ft° while ab+cd = 1 . With no loss of generality we

may choose a = b = c = d = i/{2, as we have no reflec-

tion invariance and no lagrangian, there are no other

restrictions to the coefficients (they must only be non-

zero).

Setting a = b = 0 and с = d = 1 we get the Di-

rac equation while the components of (1,1/2) represen-

tation are zero and we can always leave them out from

equation. But as we have mentioned before, the equations

given in [37,383 are not reducible to such a form.

. In the case of the representation of R» the situa-

tion is the same and we do not deal with it separately

here.

In conclusion of this section some remarks on pro-

jection operators which are used in the above mentioned

papers. Since here always £°(( fi*) - I) = 0 the pro-

jection operators of f&° eigenvalues ± 1 are due to

(3.8)

The sum of operators P+4 and Р.< projects out the com-

ponents which correspond to the (Ь° nonzero eigenvalues

(such components are often called the "physical" ones).

Since

P = P+ 4 +.P-, = ( ( ^ )

2

then ( |>»° ) projects out the nonzero eigenvalues of

{$• (properly the sum ф ^ + 4*- )• И » projection ope-

1

- 32 -

rators deliver here separate solutions only in the rest

system and are not the projection operators in the moving

system. To get the solution ф(р) one must perform the

boost-transformation (3.11) to the rest system solution

and therefore the projection operator also depends on a

momentum pi* , i.e. P*< = P±t(p). If we take into consi-

deration that p = m , the projection operators take the

same form as in the case of the Dirac equation

Therefore, if in the rest system the separation

ф=((*°)2ф + (I - ((*°)

2)ф = ф' + ф"

is justified and the components ф ' we can consider as

the "physical" ones, then in the moving system the prefe-

ring of components ф' are by our meaning not justified,

because they are not connected with the moving system so-

lution ф(р). In the case of the Kenmer-Duffin equation,

for example, using the fb° representation (3*16) the com-

ponents ф' are фо and Ф .In the moving system all

the components are nonzero and therefore equally "physi -

cal"(cf. also Sec. 10).

6. Space Heflection

| In this section we write down the invariance condi-

' tions under the space reflections. Suppose that the space

? reflection p % pe'= p

e, p -» p' = -p* ( r*-» x"'= x

e ,

x -* x*' = -5? ) is represented by the operator П ,

i.e. ф(р) —> Ц»'(р' ) - ч П Ф(р), where 7 is some

coefficient with modulo equal to one. Operator П sa-

tisfies, as it is known, the relations

[П C6.D

- 33 -

With no lose of generality we may require that ПХ =

= I.

Due to the fact that П commutes with space rota-

tions SM V* we, according to the results of Sec. 2, «ay

state that the general form of П is the same as of (i?

ПАмП м

. Пах

where

(2.5)

are some coefficients, Пке a r e

Пке = L

(6.2)

form

The condition { П ,SO M } = 0 restricts both - the

possible representations and also the coefficients o((s).

Let к denote the representation we get from к by ref-

lection, then as it is known (cf. [9]) in the case of

representation k=(a,a) к = к, in the case of represen-

tation k=(a,b) - к =(b,a). Therefore the reflectional

invariant representation must contain together with each

irreducible representation (a,b) also the conjugated

representation (b,a) while in the case of multiple rep-

resentations both must hare the same multiplicity.On rep-

resentations (a,a) there are no restrictions. Conse -

quently only П*к / 0 * T h e

coefficients ot(s) must

satisfy

o((e) = - o<(s * 1). (6.4)

of

Now we again apply the convention we did in the case

(ie - we set oC(s

M,,) = 1. Then f"U

K has a form

I I <*

where in the case of к = к = (а,а)case of к = (a,b) and к = (b.a)

(6.5)

= 2a, in the: a + b. Using

'0

I

I

the relations (2.7) and (3*6) «a get

flL = x (к = к) (6.6)

and "*• .n ГЦПк** I (ki<k). (6.7)

Applying Г\Х- I we get due to the relations (6.2)

-(6.7) the following restrictions on

where £ denotes the sum over all multiple representa-tioce к ( it aeaas that if the multiplicity of к isg: k

f = ... = k» - к the sum is over all g represen-

tations). If there are no multiple representations we get

d2

K K = 1 (к = Ь (6.9)

andd

Rj. d*

K = 1 (к / к) (6И0)

With no lose of generality we may consider the u»i

to be real and in the case of conjugated representations

from the considerations of symmetry set d Kj = d {

K .

After it we have the following possibilities: d** = +1

or dK K = -1; d K£ = d£ K = +1 or d ^ = d^K = -1. It

slightly differs from the choice of Gelfand and Yaglom£в] : oC(s) = (-1) W

or ot(s) = (-1)w*'iD the case ot

к = к ( £s] is the integer part of s) and otCs)=(-1)W

in the case of к £ к. In the latter case we allow thepossibility d(s) = (-1)

м < и.

In the case of the representation B- - (3.16) con-

sidered before * = 1 and 3 = 2 ( 1 = 4 and 2 = 3 )

and matrices fl^i are the following

П - *°* +w П

Denoting dM = d<(i = d< and d t J = dj^= d t we get •

-35 -

nо

о

о <*> о

о

ал» «кt

44 -t

<4

О

О

О

,(6.11)

where in addition dj" = da = 1, i.e. we may independently

set d« and &г equal to 4.1.

In the case of representation R2 - (3»1*) 3 = 1.

2 = 2 and denoting d4, = d

j 4 = d« , d

us d.

t we get

По

о

о

о

d,

О

о

(6.12)

Now let us deal with the invariance of equation

(1.1) under the space reflections. In addition to the

restrictions on representations we have the condition

and Г Гм

Л4

Using the fact that П = I

is written in the form

the invariance condition

,П1=о. <6-1з)

It restricts both the coefficients aKH.< of ft°

and d K£ of П . Taking into consideration the gene-

ral form of (V (2.3) and that of П (6.2) and also

the restriction» due to the linkage and reflections we

get from (6.13)

where the sum on the left is over the multiple represen-

tations k' and on the right over к respectively. But

if there are no multiple representations in equation then

I!5

(6.15)

- 36 -

As we see the reflectiooal invariance connects the coeffi-

cients between к and k' , and between к and k' whi-

le they must be simultaneously nonzero or equal to zero.

Let us deal with (6.15) closely and distinguish four

possibilities in the choice of к , к*, к and к!

1°. к = к and к*= к'. When к = (а,а), к'=(а+1/2,

а+1/2) or (а-1/2,а-1/2). Now *'Пк.'|с' = - nwt

K <i and

we get

die» = - duv . (6.16)

2°. к = к . к' / к', «hen к = (а,а), к1 is (а+1/2,

а-1/2) or (a-1/2,a+1/2). How tKie«n»'fc' = П

we get

and

• к*1 (6.17)

3 . k 4 k . k' = k'. I t i s analogical to the pre -vious case and gives

4°. k 4 к . k1 af k f . Here we get

aKlti dKiii = + dni a,;.,

(6.18)

(6.19)

where the sing must be chosen in the following way: sup-

pose k = (a,b), then the + sign is in the case of rep-

resentations k* = (a+1/2,b-1/2) and k1 * (a-1/2,b+1/2),

the - sign is in the case of k' = (a+1/2,b+1/2) and

k1 = (a-1/2,b-1/2).

In a special case k = (a,a-1/2) and kf = (a-1/2,a)

we get from (6.19) aKfci= а

к.

к .

Now we consider the restrictions on equation with

representation fij - (3.16). The general form of (i° was

given by (3.17) and that of П by (6.11). Starting di-

rectly on (6.13) or using the relations (6.19) we obtain

two principially different possibilities in the choice

of dK and a

KK> (since on both cases П m y differ

even on sign): I

- 37 -

1°. d< = -d4 : а

4 ч = а

Ч 1 , a ^ = а

Ч1. = a,

агч = а

м = b and а „ = а,

г = с. (6.20)

2°. d4 = dj. :*а„ = а

ч< , а,, = -а

Ч2. = а,

аа д = - а

и = b and а

г 1 = a,

t = с. (6.21)

Our choice of coefficients in sections 4- and 5

corresponds to the first possibility where in the s=3/2

case we choose ai 4 = a

4 1 = 1. By the suitable choice of

a, b, с spin 1/2 can be eliminated or not: (4.15) -

(4.18) .

In section 5 «e choose a4 4 = a

4i = 0 and a

x i =

г агг= 1. From (6.20) it is easy to verify that the new

s = 1/2 equations are invariant under the space reflec- . I

tions.

In the case of the representation fi2 - (3.14) ji,

c

was given by (3.15) and П by (6.12). The reflectional

invariance gives two possibilities once again

1°. д.л г +dj. : а

4 г = а

э г = a, a

M = а

м = b;

о (6.22)

2 . £4 s -dz : a i. = -аи.= а, а«.4 = -а

и= Ь.

In conclusion it should be remarked that the greater

freedom allowed by us on the choice of d*< gives corres-

pondingly more possibilities for aKlt>. Gelfand and Yag-

lom [8] had in fourth case (6.19) always aKK>

but also the ацх'= -ati' is possible.

7. Scalar Product

Often it is convenient to use the lagrangian forma-

lism. It allows to derive the expressions of energy and

charge densities. It needs to introduce the invariant

bilinear hermitian form, in other words the scalar pro- Д

duct. We define the scalar product of ф., and <j>*. by £'

the help of matrix Л

(ф^ 14*) = 4v Л Ф».. (7.1)

The scalar product is real - ( ф,, фг ) = ( ф,, , ф, )*

if Л is hermitean

Л+ * Л . (7.2)

We do not impose more conditions on Л . It is also often

demanded the nonsingularity of Л which in turn guaran-

tees that for any ф 4 0 the scalar product is nonzero.

As we later on need scalar products from solutions of

wave equation, it is necessary that only such products

are nonzero. It is possible also in the case of singular

Л . The possibility of singular Л is ottered in

[29].The invariance of scalar product gives

or using the fact that in our case ( SO M )* = S

O M and

(S u>r )

+ = - S

M1(" we have

(7.3)

Comparing it with the relations (6.1) on Г) we see thatЛ has the same general form as П . Alsc the restric

tions on representations are the saae.

We write Л in the following general foria

Л?г« П

м

9»Пгг

(7.4)

where ке are sons coefficients and the nonzero Пи&are given by (6.5). Due to the heraitianity of Л on^K& there are additional restrictions

'•£-

В

- 39 -

which for k = k gives that <jK< are all real. With noloss of generality we may choose the other oK£ also tobe real and then from (.7*5) we have

„ . = ? i K . (7.6)

The invariance of scalar product under the spacereflections gives

Due to П = П * *** П*= I it is equivalent to

[л.пЬо. C7

-8)

Using (6.7) and the general form of Л and П we get

fei duk = I d

K i ? 4 l c , (7.9)

к, Ь

where the sum is over multiple representations. If there

are no multiple representations we have

? к ^ ы = <1KK ?*.* , (7.Ю)

which due to our choice dK^ = d

K K gives ^

K^ = <^

K ,

i.e. the condition (7*6). Therefore the reality of ^*£guarantees the invariance under space reflections.

In the case of the representations R-, - (3.16) andHg - (3*14) Л is correspondingly given by the relations

(6.11) and (6.12), where one must substitute dA —* f,

and dt —• o

b .

; 8. Derivability from Lagrangian

' To derive an equation (1с>гр,Г- m) ф(х) = О from

; the lagrangian

the matrix Л or in the other words the scalar product

(7*1) is needed. The reality of lagrangian is guaranteed

by

(pi4Г Л = A(b*\ (8.2)

The relation (6.2) is satisfied if it takes place iu

case of A* . Therefore we get one more additional condi-

tion

((i')*A = Л (Iе. • (8.3)

If we take into consideration that tKII> satisfy t\

Kts

= tic'* (cf. Appendix) we analogically to (6.14) get

the restrictions on a ^ and fK$

*'* ?*'*' W nk' * ' =Z?k4

aiic'n

icit

ufe« . (8.4)

The difference from (6.14) here Is that on the left hand

side in the place of а*к< there is at

1* • If there are

no multiple representations then

a** (8.5)

Similarly, as is Sec* 6, we get the relations fordifferent linkages.

= - ^к&кк> • (8.6)

= f **. aKt' • (8.7)

= <^кк BLS.K1 . (8.8)

a*.* {«•*• = ± ?*6 ajj- , (8.9)

where the + sign is in the case of к1 =(а+1/2,Ъ-1/2) or

1°.

2°.

3°.

4°.

к =

к =

к ^ к ,

к*a uj к

к»а*„

к1

atк»

='К

к' .

•к' .

к» .

4

'A

- 41 -

(а-1/2,Ь41/2) and the -sign is in the case of k1 =

= (a+1/2,b+1/2) or (а-1/2,Ъ-1/2).

In the special case к = (a,a-1/2) and k1 = (a-1/2,a)

we get fron (8.9) &%* = >К'К<

Similarly, as in the case of the reflectional inva-

riance, we have a few more possibilities to combine the

coefficients in comparison with Gelfand and Yaglom [в].

The difference was already remarked by Cox [11].

As to the relation (8.5) it should be remarked that

now the coefficients a*k'tc and a £ji are connected.

Now we also illustrate our relations using the same

representations as in the sixth section.

In the case of the representation R-, - (3*16) the

derivability from lagrangian imposes the following rest-

rictions on coefficients of (I* (3.17) and Л :

where <?, and <ffc must be real. As we see the relations

(8.10) allow quite great freedom on the choice of coeffi-

cients.

It should be remarked as we mentioned in Sec. 5 the

equations (5.4) and (5*7) dealed with there are not deri-

vable from lagrangian or in the other words they have no

hermitianizing matrix Л . It is easy to verify that for

matrices ft° (5.3) and (5.6) in the case of nonsingular

Л the relations (8.10) are not satisfying. Allowing

a singular Л as it is offered in [29], we may take

for example o,, = 0 and f». 4 0 and it satisfies the 1

relations (8.3). But such a choice needs further anali- Щ

sis because in this case it is easy to see that we opera- ":

te in scalar product only with the solution of Dirac equa- 1

tion. %

If we require in addition to the derivation from [

д

lagrangian the invariance under space reflections we get ЩЩ •

•£.:•

tooth for (6.20) and for (6.21) froa (8.10)

с, b * ^ = - ^ a . (8.11)

The invariance under the space reflections and deri-

vability from lagrangian imposed on ak K., dni and ?*£

quite great restrictions, which in turn restricts a num-

ber of possible equations because we have less free para-

meters in |3>° to vary the eigenvalues of fi° . Without

reflect!onal invariance there are less restrictions.

Now we write down some solutions for a u< in the

case of equations with the representation H-, - (3*16).

In addition to relations (4.15)-(4.18) we use the rest-

rictions (8.11) and after all we choose a44 = a4i - 1*

1°. (4.15). The Pauli-Fierz equation for s = 3/2.

Expressing b = - 1/4a from b*ft = - ?„а we obtain

that sgn q, - sgn Цх. . Choosing <?., = ?t (= 1) we get

the restriction on a : | aI = 1/4, while for each such

a b = - 1/4a. On further choice of coefficients we

may use arbitrary principles of symmetry. Setting for

example a = ± 1/2 we get accordingly b = T 1/2.

sgn P, ssgn?,.. (8.12)

2°. (4.16). Now

sgn <j, = - sgn (8.13)

We choose ^, = - ?t = 1 and a = b = 1/2 or a = -b =

= i/2.

3°. (4.17). Ноя

sgn q, = sgn ft. (8.14)

We choose p, = ?«. = 1 and a = -b = V~3/2 o r

a = b a

(8.15)

4°. (4.18). In this case

sgn ол = - agn

- 4 3 -

We choose ^ = - = 1 «ad a = b = /3/2.

In the сале of the representation Bg - (3*14) de-

rivability from lagrangian gives the restrictions а*,^=

= fi a»t * а » ?г = ?t*if taking into consideration the

reflections! inrarlance (6.22) in the first case we get

that Ъ*£г = ?^* and In the second case that Ъ*ф

г =

= - £, a . The single aass condition ab = 1/2 givesthat In the first case sgn ft = sgn fu and in the se-cond ease sgn f, = - sgn • As we see we may set Л == П . As it Is known this variant is the Каммг-Duffin

s = 1 equation.

In conclusion we write down Л and П matrices

for a =1 equation with the representation R4 - (4.20)

given in Sec. 4. The general form of Л is

Л-

о

о

о

?.<*£о

о

о

о

о

. (8.16)

П has a siailar form, only we denote the coefficients

The reflections! invariance, using the expres-

(i° (4.21) gives dt = d

M = 0 and d, = d, ,

^ = 1. Derivability from lagrangian

by d<

sion of

and in addition

gives ?»

Л

and therefore

0 tSJfc£ - tc5 0fe2 - t*t 0

О О

(8.17)

Although here all the possible coefficients are nonzero,Д is singular.

„ 44- -

9. Definiteneas of Bnai

In «his section w» deal with the definitenees ot

energy and charge in the case of single aass equations.

We suppose that there exists an invariant scalar product,

the natrices ft° and Л satisfy (8.3)» As we shall see

from the given examples here are yet soae unsolved ргоЪ -

leas and we have no reliable conditions to determine the

definiteness questions.

Here we also restrict ourselves to the rest system.

She definitenees of energy means the definiteness of sca-

lar product

( <J; , ф ) = <|,+Лф (9.D

for each solution of wave equation. In other words for

each eigenfunction ф4±. corresponding to the eigenva-

lues +1 or -1 of |*° the scalar product (9*1) oust

have the sane sign. The definitenees of charge deaands

the definiteness of

(ф ,(4«ф ) = Ф+Л(**ф. (9.2)

4 Since the solutions ф»* are given with the help

of projection operators P \< , also ( <\>i± , ф«± ) and

( ф№, fi>*4>i±. ) are expressable with the help of P ±

4 .

We get

i, Here we used the fact that due to {Ь**Л - Л (i* also

p (|*w )*Л = Л |i

tt> and therefore (P^, )*Л = Л Р 41 , and

ii also Pi* P*. :

Л (i°<k± = ф+(Р|, )*Л (4* P|, ф =

where we in addition used the relation P|< fi*= ± P±i .The definiteness of ejaergy therefore reduces to the defi-

-45 -

niteness of

(Ф*±»Ф*) = Ф*ЛР±*,<|» (9.3)

and the definite&ese of charge correspondingly to

( «K*. рФй:) = ± Ф*Л F^ ф. (9.4)

As we see we have to deal with the expression ф*Л Р*,Ф

and we are interested on the sign of it.

The defioiteness of the first order equations are

studied in soee papers. In [39] it is proved that for

definitenees it is necessary that only one particle cor-

responds to spin s . In the papers [11,12,22*1 there

are given the more general conditions of definiteness.

These conditions are derived with the help of relation

sgn( ф+Л Р £ , ф ) = sgnTr (APi, ).(9.5)

As our direct calculations have shown these definiteness

conditions are in some cases the contradictionary ones,

and the reason of it, we think, lies in the relation

(9.5).Below we rederive some definitenees conditions given

in [11,12,22]. The calculation of Тг (ЛР** ) reduces

to the calculation of Tr (Л ((***')*")t since due to (3.8)

P*±« is expressed by the powers of (ia> . As it is shown

by Cox [11] in the case of integer spin always

Tr A(|iw)

x*

+ 1= 0, (9.6)

but TrA(|J»w)

t* may be nonzero. We denote

Tr Л ( (itt>) t f = 2 <f

% . (9.7)

Analogically in the case of half-integer spin

Tr Л( (i^)1* =0, Tr Л ( ft"?*"

1 = 2 /y, . (9.8)

The ваши results with respect to ТгЛ( (b*)'' are givenin [12].

Using the relations (9.6)-(9.8) and (3.8),we get:

1°. for integer spin (bosons)

to ( Л Р £ „ ) = /у» . (9.9)

2°. for half-odd-integer spin (feraione)

Тг ( Л Р ^ ) = -/у». (9.Ю)

Now the relation (9.5) gives from (9.9) and (9.10)

the following definitenese condition,:

1°. Integer spin

the sign of energy sgn (ф4*»4ч*) = egn qe

%

(9.11)the sign of charge sgn (ф

41., |ОД)

2°. Half-odd-integer spin

the sign of energy sgn (44t-,Фи) = - sgn of»

the sign of charge egn (ф14,|Л|/

1±) = sgn/j4.

As we see from (9.11) and (9*12) the definiteness issatisfied if /y, 4 0, and if the different гц

х have the

same sign.

The definiteness conditions on ft3 are in similar

way given in [22]:1°. The definitenese of energy

(-1 f* [(te Л ( ft' )L +" )

l - (Tr Л ( ft° )

L )

Г ] > 0,

1 ' (9.13)

2°. The definitenese of charge

(-1)L[(TrA(fi°)

L+V-

where L is power in (3.3). As it is shown in [12] the-

se conditions are automatically satisfied1 in the case of

single mass equations and if these are the definiteoess

conditions, the definiteness is also automatically satis-

fied. |

Further on we show that the conditions (9.11), У

(9.12) and (9.13) give in general different results О » ] . ;

We consider the equations with representation B3-O.I6). i

Using the choice of coefficients glren in Sec. 8, the mat-

rices

P"о

о

о

and Л are the following

Р О *«

0 0 0

0 0 0(9.14)

о

о

Л

ъ1

о

о

о

a t«£

4l

о

о

) о

43

С tit

о

о

о <

о

о

1 . (*)

5*«

о

о

о

о

о

(9.15)

(9.16)

Now we compare the results we hare from (9*12) and

(9.13) (in (9.13) we must calculate Тг Л£° ).1°. The Pauli-Fierg eouation (4.15). (8.12).

Тг (Л

Тг Л (»># = 4

) (9.17)

(9.18)

Both defioiteness conditions are satisfied.

The following equations describe in addition to 8=

= 3/2 particle also one or two s s 1/2 particles.

2°. (4.16). (8.13).

/jfvt is given by (9.17).

If* • - 2?«. (9.19)

Тг Л fS* • 2 {„ . (9.20)

How we see that the definiteness conditions (9.12) are

I

n

not satisfied because sgn f^s - egn fy^ , i.e. the

charge densities of s=3/2 and 8=1/2 particles are

different. The condition*(9.13) are due to (9*20) satis

fied. Therefore the results we get are different*

3°. (4.17). (8.14).is given by (9.17).

Tfo =-•*!« (9.21)

Tr Л(Ь*= О. (9.22)

Here both conditions give the saae result that definiteness is not satisfied.

(4.1Б). (8.15).

given by (9.17).

Tr

(9.23)

The definiteness condit: one (9.12) are not satisfied, but

(9.13) is.

To clarify the problem of definiteness to some ex-

tent we have used the most direct method - using the ex-

pressions of (bt%t> , |J,

<<O and Л (9.14)-(9.16) and mat-

rices, tut given in Appendix we calculated the expres-

sions (9.4): ( фл, £ в

фл) = t Ф * Л Р*. 4» •

Now we present our results. In all cases

* < f •where the constants

projection.

depend on spin

For фэд we in the cases 2° and 3° get

where $r , |

b depend on spin projection.

As we see the charge densities of s=3/2 and s==1/2 particles have indeed opposite sign and therefore

1

I

- 4 9 -

the definiteaess of charge is not satisfied. This resultis in agreement with the conditions (9*19) and (9*21),but not with (9.20) aziid (9.22).

In the fourth case we get

The last expression gives the indefinite s=1/2 chargedensity when we have two 8=1/2 particles. This resultis not in agreement with the trace conditions (9.23) and(9.24).

Our example shows that we have no general definite-ness conditions and therefore the definiteness questionsneed further investigation. It seems that in some casesone can exploit the trace conditions for different spins.As we have seen we get a right result when the trace con-dition for a given spin is nonzero, but when the traceis equal to zero the corresponding density may be indefi-nite.

<Ve also calculated the traces in the case of s=2equation given in £10, 11,121 • A s i n the case of theprevious example we get that if the equation describesin addition to the s=2 particle particles with spin 1or spin 0 or both the traces have opposite signs andtherefore the energy is not definite. It seems that in-definiteness is the general property of first order waveequations in many particle case and in quantization onemust use indefinite metric.

10. Multiple Representations

In this section we give some remarks about the ex-ploitation of multiple representations. There arc many junsolved problems here as well. ?i[

Concerning the exploitation of multiple represents- U

i •

- 50 -

tioos it should Ъе remarked that in the case of low epics

the use of multiple representations is not necessary and

it only gives йоге possibilities to construct equations.

In the low spin case there are good equations without Mul-

tiple representations (Dirac, Kemoer-Duffin and Pauli

-Piers equations). But in the case of high spins we evi-

dently cannot manage without multiple representations if

we want to construct a single-particle equation. There

are many additional spins and therefore one must add ad-

ditional representations to eliminate them. It is shown

in [10] that in the case of s = 5/2 one cannot get a

single-particle equation without multiple rfpresentations.

The same equation (s=5/2) as in fiO] was also given

in [6]. The aforesaid naturally concerns equations with-

out subsidiary conditions which are derivable from lag-

rangian because in the case of high spins the existence

of scalar product enlarges sufficiently the number of

representations.

The exploitation of multiple representations has

been recently studied in papers [40]. The addition of

multiple representations allows to transform them between

themselves and therefore to change the form of |i* .

In [4ol the conditions when the addition of one

or two conjugated multiple representations allows to

eliminate some linkages are given.

In our notations it means that in A*-matrix (2.3) it

is possible to transform all the coefficients in some

row to zero, for example in the case of fixed 1 all

*к* = °* Tben the components ф

ь corresponding to the

representation 1 are also equal to zero and this rep-

resentation may turn out to be superfluous in equation.

In [40] the losing of linkages in the equation is na-

med falling to several independent equations. We consi-

der it generally not always so because the condition

that all aKft = 0 in the case of fixed 1 means only

the losing of unilateral linkages. If we have before the

-51 -

transformation that ак е/ О and a^jto the correspon -

ding linkage le к *-*• 1 but if after tbe traoeforaation

we bare art= О and a

u^ О the remaining linkage is

unilateral к «- 1 .

As an example we consider here the e = 1 equation

with multiple representation given in ([30] by Aaar and

Dozzio. In the representation

Bg = (1.0Ж1/2,1/2)9(0,1)92(0,0) = B^SB(O,O) (10.1)

(0,0) is doubly. The (b° matrix corresponding to the lin-

kage 1 «-» 3 «-» 2

is the following

f-#

0

0

0

0

0

0

0

0

0

0

0

0

0

0

t»

0

0

Ф,

Фч.(10.2)

where we denote 1 = (1,0), 2 = (0,1), 3 = (1/2,1/2) and

4 x (0,0).

The given equation is invariant under space reflec-

tions and has a heraiteanizing matrix Л • The general

form of П and Л according to our fte is the follo-

wing

ГЬа, 0

0

0

0

0

0

0

0

0

0

0

О

0

0

0

0

0

0

(10.3)

I

- 52 -

Л

о* J

о

о

о

о

о

о

о

о

о

о

о

О

О

О

-t с«о

О

О

О

О

(10.4)

лIn the case of Aaar-Dozzio equation p,e r (*" ^

while (^(((i'4)

2 - I ) = 0, ( j ^ V = 0 and therefore

the minimal polynomial of |i° is ( p>° )3(( |*° )

2 - I )=

= 0. The given equation also satisfies the definiteness

of energy.

Due to the results of [4O] the equation correspon-

ding to the representation fig - (10.1) must always fall

into parts since transforming the components фч and <K

it is possible to transform the coefficients is the fourth

or fifth row of (b° (10.2) equal to zero. The general

unitary transformation which leaves the generators

unchanged and nixes the components of representations

(0,0) and (0,0)' is given in fora [40]

1„ О О О О

0

0

0

0

0

0

О

О

I»

0

о -

О

О

О

О (Ю.5)

where 0(41 &od <*u are arbitrary complex numbers satisfyingfe<«lN- Ып!

2" - 1 sx»d f is an arbitrary real parameter.

After the transformation A°' = A j»>e A"* ( A" = A* )

get ft°' in formwe

I:

- 53 -

f k

о

о

*?о

о

о

о*»»о

о <

.to 0

0

о

о

о

о

.00.6)

As it can be easily seen it is always possible to set

etn -o(n = 0 or *f* + o(£ = 0. *** u s 8 e* * o r example<*u -<<* = 0 and

following (e!*« 1)

о

о

о

о

о

оо о

О О {2 tJJ

The corresponding linkage is

= 1//2 then the is the

О О

о о

tjV О . (10.7)

о о

о о

1 •-• 3 •-• 2

As we see from (10.6) it is not possible to lose all the

linkages of (0,0) or (0,0)*. Since our linkage schema

does not fall into parts it is not possible to say that

the equation falls.

Comparing the given linkage schema with schemas

(5.11) and (5.12) we can conclude that here the represen-

tation (0,0)' is the ballast one and the (0,0) represen-

tation component is identically equal to zero. It should

be remarked that the components фч and 4\ correspon-

ding to (0,0) and (0,0)' are always equal to zero in the

given equation because it describes a s = 1 particle

(one may prove it in directly wriring down the projec -

tion operators P ±4 ). It also should be remarked that

the given equation does not fall directly into parts.

Omitting, for example, the representation (0,0) we get

the s = 1 equation with singular Л matrix. Omitting

(0,0) and (00)' we get the ordinary Kemmer-Duffin equa -

tion. Proceeding from toe above mentioned the represen-

tations (0,0) and (0,0)' are in some sence superfluous in

spite of the fact that the Amar-Dozzio equation does not

violate any of the general principles demanded from the

equation.

As we see the problem of multiple representations

needs further analysis.

In conclusion we give some remarks about the inter-

pretation of the equation. Let us deal with the equation

(1.1) In the rest system p* = 0, p* = 6m

(ftm |*# - *. )<|»(p) = 0. (10.8)

We can treat it as the eigenvalue problem of the opera-

tor (**

) = ~

where as it is well known the nonzero eigenvalues of ft*

- h; are connected with the masses described by the

equation in the following way

m- = (10.10)

Since in general the only condition on и, is to be non-

zero ( Jt ± 0 ) we have here some freedom. When JC V 0

the sign of energy and the sign of eigenvalue are usually

chosen to be the same: & = sgn h; , and when *c «1 0

it is vice versa. Choosing the к to be Imaginary it is

possible to describe real masses by the help of imaginary

eigenvalues of j&° .

One of the roles of an equation is therefore to give

the mass condition and to the mass m; corresponds the

solution ф щ г «Jr . .in addition to the mass it is

possible to determine the spin and spin projection because

U

- 55 -

commutes with "22 = -(S**)

2-(S")

2-(S

S*)

2 and IS**

.,, (10.11)

IS +м«»в = б фи,д.в

The particle with mass it t

spin s and spin

projection 6 is described by the quantity <\>*:LG

which as we have seen is the eigenfunction of three ope-

rators ft* , S2 and iS

4 1 . <\>*;i< can be treated two-

fold: on one hand it transforms according to the Lorents

group representation and therefore has the index «.

фмсъб labelling the components of the given represen-

tation, on the other hand from the point of view of par-

ticle description it is a quantity with indices mt , s

and 6 . As the physical particle is characterised by

mass and spin the latter indices have primary importance.

The index ы. which labels the components of the given

representation is an auxiliary index without no direct

physical meaning. Therefore there is no reason to consi-

der the components of фк軈 - фтцв as physical and

nonphysical ones because the really physical components

are as a matter of fact the eigenfunctions >Sf^a< of

ft° , 1J and iS"2" . As we have mentioned before some-

times the classifying of irreducible representations of

the Lorentz group in an equation is useful since some rep-

resentations may be superfluous or ballast ones. But such

classification.needs further specification. Concerning the

classification of the components ф м » ; to physical and

nonphysical ones, it should be remarked that if it is

necessary, to our mind most natural way to do it is to

regard the components of all irreducible representations

containing a given spin в as the physical ones. For |

example in the case of the Dirac equation all the compo- -|

nents are then physical because both representations '

(1/2,0) and (0,1/2) contain a spin 1/2. As to the usual |

classification to regard the components which are non- §

. ' iv-r-

t

zero in the rest system as the physical ones it should be

remarked that such classification depends on a given rep-

resentation. If in the case of the Dirac equation fc° is

diagonal

I О

О -I

the nonzero or physical components are ф, and фь or <j>,

and ф«# «i.e. two components. Using the representation

where S**w is the direct sum of irreducible representa-

tions ft* has the form

0 I

FNow the nonzero or physical components are all four com-

ponents. Therefore such a classification is not satisfac-

tory. According to our classification all the four compo-

nents of the Dirac equation ere equally the physical ones.

Similarly, in the case of the Kemmer-Duffin s = 0 equa-

tion all the components are the physical ones. In our

ft* -representation (5.16) two components - ф, and ф

are nonzero in the rest system and therefore they are

usually called the physical components (cf. for example

£3в]). In the moving system all the components of ф(р)

are nonzero and therefore physical because $ (p) must

be the solution of the equation and eigenfunction of ope-

rators S2(p) and iS^Cp) where

S2(p) S

2

andIS** (p) = U (L) IS"

and U(L) is the boost-transformation (3.11)

We are grateful to Madis K6iv and Bein Saar for

several enlightening discussions.

- 57 -

A p p e n d i x

The generators of. the Lorentz group S I** satisfythe eomutation relations

where f*»v,?,€ = 0,1,2,3 and g-= -g* = -g**= -g**= 1.The finite dimensional irreducible representations

(a,b) are given in the direct product basis and fixed bythe eigenvalues a(a+1) and b(b+1) of operators tand trt where

•M - —- - — ,« о •* „емA = 1/2(1/2 6 n S - S"" )and u (A.2)

B u = 1/2(1/2 e V S • S )

and u,v,r = 1,2,3, eu<r% is totally antisymmetric ten-

sor with fc*** = 1. Au and B

M separately satisfy the

commutation relations of the generators of the rotation

group S0(3) and £AU , B**] = 0. In the basis where A3

is diagonal the operators Au with given eigenvaluea are given in form

(A4 ± iA* ) a V = [(a*a")(a*a" •1)]'4<5

ау«

* •• x.. (A

*3)

where а',а>( = a, a-1, ..., -a. Operators B

w are gi-

ven similarly. The relations (A.2) and (A.3) allow to wri-

te down the expressions of S*** for an arbitrary repre-

,, sentation (a,b).

II In the case of the vector representation (1/2,1/2)

I another representation is often used

In the representation used by us the following re-

lations are valid

(Se u)*=S~ . ( S ^ ^ - S

1" (A.5)

I - 5 8 -

Different bases for equations are treated in [20]. v

We denote the Clebsh-Gordan coefficients which sepa-

rate froa the representation (a,b) spin a in the follo-

wing way

<s 6 | (a,b) a'b«> , (A.6)

where в = a+b, a+b-1, ...,|a-bl; € s s, s-1, ..., -s;a* s a, a-1, ..., -a; b' = b, b-1, ..., -Ъ. In the caseof the representation given by (A.2) and (A.3) the Clebsh-

-Gordan coefficients are real and the coefficients of the

invarse transformation are equal to th* coefficients of

the direct transformation.

<a'b' (a,b)|e<> =<s«"|(a,b) a'b»> (A.7)

лThe following relations are also valid

зб|(а,Ь) a'b«> = (-1)aH*

J<8 6|(b,a) b'a»>,(A.8)

«> = o**»'5«' (A.9)

The general form of Clebeh-Gordan coefficients is given

for example in the book £421* In the case of low rep-resentations it is useful to exploit the tables ofClebsh-Gordan coefficients.

The spin-projection operators t"£ used by us aredefined by the relation

Ч е = и*к

и*е. СА.11)

where in the case of 1 = (a,b) 0^ is the matrix for-med from the coefficients <e«|(a,b) a*b*> with mat-rix elements (Use )«,aV and UTK is 1& the case ofк = (c,d) formed from the coefficients <£c*d'(c,d)ls6>with matrix elements (Ut< )е?а',« •

The components of the matrix t ^ are given fnform

/

-59-

Using (A.12) and (A.9) *• can get the following use-ful relstion

The latter relation is basics! in genera! calculations.From (A.11) we get

(tJS)*«tg. (A.14)

The matrices tKt are written in form

£. <*(s) tig . (A.15)

Due to the relation (2.9) are the coefficientsfor linked representations к and 1, and 1, к

sane and therefore from (A.14)

* = teic (A.16)

the

As an example we write down t *c operators for so-

me representations.

1) к = (1/2,0), 1 = (0,1/2). As in our representation

(A.2) and (A.3) S4 Z i» diagonal we have V*£l =& 14* =

s I and therefore

= tVT = I • (A.17)

2) 1 = (1,1/2), 4 = (1/2,1) (we use the notationsof formula (3*16)), At first we write down the matricesU formed from the Clebsh-Gordan coefficients.

000

00

0100

^20

0

0"00

- 10

00

0

01

0010

0

00

00

(A.18)

I

".-.:• • I.. :.-• " -•"

- 60 -

Ц' sоомм

оо

о{2О

1

О

о о0 11 Оо о

оо

о -iо -

ооо

лоо

t 14

t

Now it is easy to calculate the matrices

uo

30

0

0

0

0

0

0

0

0

0

0

0^2"

2

0

0

0

0

{2-1

С

0

0

00

0

{Г1

О

О

О

О

/2

- 2

О

О1

V2О

О

О

О

- 2

{£О

О

О

ОО

О2

Го

ооо

-1

О

ООО

О

О

3

О

О

О

О

О

О

(A.19)

and

(A.20)

ДА.21)

According to (A.14)

it« ;

ftOyt-<*«> and

41

Let us have the representation 2 = (0,1/2). Since

= I we have

*» s °J44

a n i *<№ = (*гч )*• (A.22)

Similarly in the case of 3 = (1/2,0) U ^ = I and

and t Л; = (t y, ) . (A.23)

In the «ам way it is possible to write down anot-

her operator* t ^ , for ехащр1е t^*, t ^J etc.

these aatrlcea we wrote down here we nave used in

the direct calculations of section 9.

- 61 -

3) In the cause of the vector representation к =

= (1/2,1/2) не «ay шве the S>*w representation (АЛ).

Now the Clebsb-Gordan coefficients axe not real and the

aatriz U takes the form

U0

0

0

0

1

i

0

0

i

- 1

0

For the representation 1 = (0,0)

tД = UOK =I 1 0 0i

and t S = (tS*J )

(A.24)

and

(A.25)

In the Kemmer-Duffin s = 0 equation A° is in

the standard representation the following [433

where

•*•'(A.26)

is in fore (A.25) =

I

- 62 -

B > f i r t t t d

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2. H.J. Bhabha, Hariah-Chandra, Proc.Hoy.Soc., 1185.

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3. Harish-Chandra, Eroe.Roy.Soc.. A192. 195 (19*7);

РЬуа.Нет., 23L» 793 (19*7).

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ления грушш врацена! л групш Лоренца, М., 1958.(I.M. Gel

1rand, B.A. Mlnloe, Z.Ta. Shapiro, Sepxw-

eeatatiotw of the Botation aad Lorants Group and

their Applications. Fergaaon Preae 19&3*)

10. V. Frank, Uucl.Phy*., Bgg, 429 (1973); Thesis, Uni-

versitat Earlsruh* 1972.

11. W. Cox, J.Phye.A: Math.,Duel. ,G«n.» 2» 1» 5 , 2249

(1974); Thesis, Univereity of Aston 1972.

12. Ф.И. Федоров, в.А. Пжеписов, Изв. АН БССР, сер.ф.-м., 6, 63 (1974). ji

13 . W.J. Hurley, B.C.G. Sodarehan, Апп.РЬув.(Я.Т.), 8g, [,546 (1974). i

\ 14. S. Weinberg, Phys.Bev., 133B. 1318 (1964); 134B. fj

I 882 (1964); 181, 1893 (1969). Я\ 15. D.L. Pursey, Ann.Phys.(H.I.), JJ2, 157 (1965). ЦT 16. Wu-Ki Tung, Phys.Ber.Lett., 16, 763 (1966); Phys. I

Hev., 156, 1385 (1967). |

: 17. R.-K. Loide, Sose fissarfcs on Helatiristically Inva- ]

riant equations, Preprint PAI-10, Tartu 1972. I

к

- 6 3 -

18. G. V»lo, D. Zransieer, Phye.Ber., 186, 1337 (1969);1£g, 2218 (1969).

19. А. Sbavaly, A.Z. Capri, Ann.JhyB.(M.Y.), £4, 503(1972).

20. H. Kfiir, H. SCAT, The 0 1 > 4 Тур* fielativiatically

Invariant equation for Hepreaentation О,, >4[ЙМЙ,^[,

Preprint И-33, Tartu 1974.

21. P.-K. Ловдв, Изв. АН ЭССР, Фаз.-Уатем., 23, 203(1974).

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Kroer.1'h*or.ay«., *Z. 671 (1972).26. T.S. Saatbanax, А.Й. Takoaalla, Portachr.Hjya.,

22, 431 (197*).27. W.J. Hurlaj, Phye.ReT.; D4, 3605 (1971).28. P.-K. Хоадв, Изв. АН ЭССР, Фяз.-Натем., 22, 317

(1973).29* W.J. Hurley, X.C.G. SudarabAn, J.Matb.Phya., 16,

2093 (1975).30. 7. Aaar, U. Oossio, IUOTO da., All, 87 (1972).

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32. A.Z. Capri, Иста.Вет.. 187. 1811 (1969).

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34. И.В. Menon, Frogr.Tbeor.Ehye., Q, 987 (1973).

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52. 992 (1973).

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IC/74/104, Iftnuura-Xriaat* 1974. •

;•-'

- 64 -

39. V. Amar, U. Doszio, Buovo Cim., Bg, 53 (1972).

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41. II. Kftiv, Irreducible Bepreeentatione of 0(5)

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FAI-2, Tartu 1969.

42. £• Верхе, Релятивистоная теория реакции,М.,1969.43. * Умвдзава, Квантовая теория поля, М., 1958.44. К. ДоЁде, Р.-К. Лойде, К. Сельямяэ, Труда ТЛИ,

* 408, 32 (1976).

Received Sept 3, 1976

- 6 5 -

НЕКОТОРЫЕ ЗАМЕЧАНИЯ О ВОЛНОВЫХ УРАВНЕНИЙ*

ПЕРВОГО ПОРЯДКА

К.Лойде, Р.-Я.Лойде

Р е з ю м е

Настоящая работа представляет обзор о волновыхуравнениях первого порядка. Построен формализм, основан-ный на спиновых операторах проектирования, написаны ус-ловия вытекающие из релятивистской инвариантности и дру-гих соображений. Лроанализованы некоторые недавно пред-ставленные уравнения. Затронуты следующие вопросы: реля-тивистская инвариантность, ^"-матрица, условия Хариш-Чандры и Умезава-Висконти, новые уравнения для 3 -1/2,пространственные отражения, определенность энергии изаряда. Ьибл. 44 назв.

!/

АКАДШИЯ НАУК ЭСТОНСКОЙ ССР.Отделение физико-математических и технических наук.Катрин Пауловна Л о й д е , Рейн-Карл Руудович Л о й -д е . Некоторые замечания о волновых уравнениях перво-го порядка. Препринт -6. На английском яз. Редакционно-издательский совет АН ЭССР, Таллин.

Toimetaja J. LShmue. Triikkida antud 23. 11. 77. Ofeet-

paber 30x40/8. TrOkipoognald 4.25. TingtrOkipoognaid 3*95.

Arvestuepoognaid 3*07. Trukiarr 250. MB-07640. BHSV ТА

Toimetue- ja Kirjaetuen3ukogu, Tallinn, Sakala 3. E>SV ТА

rotaprint, Tallinn, Sakala 3. Tellimuee nr. 377.

Hind 45 kop.

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